STOCHASTIC TRANSPORT IN COMPLEX SYSTEMS
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STOCHASTIC TRANSPORT IN COMPLEX SYSTEMS FROM MOLECULES TO VEHICLES
ANDREAS SCHADSCHNEIDER DEBASHISH CHOWDHURY KATSUHIRO NISHINARI
AMSTERDAM • BOSTON • HEIDELBERG • LONDON •NEW YORK • OXFORD PARIS • SAN DIEGO • SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO
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Dedicated to
Martina, Indrani, and Yumiko
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CONTENTS
Preface Acknowledgments
xv xix
Part One Methods and Concepts
1
1
3
Introduction to Nonequilibrium Systems and Transport Phenomena 1.1. 1.2. 1.3. 1.4.
1.5.
1.6.
1.7. 1.8.
2
Introduction Classification of Nonequilibrium Phenomena Hierarchy of Description at Different Levels Individual-Based Models 1.4.1. Newton’s Equations, Hamilton’s Equations, and Individual Trajectories 1.4.2. Langevin Equation Population-Based Models 1.5.1. Master Equation and Irreversibility 1.5.2. Fokker–Planck Equation Fluid Flow: Theoretical Descriptions at Different Levels 1.6.1. Liouville Equation and Flow in Phase Space 1.6.2. From Liouville Equation to Boltzmann Equation: BBGKY Hierarchy 1.6.3. From Kinetic Theory to Navier–Stokes Equation Back to Discrete Models: Mimicking Hydrodynamics with Fictitious Particles 1.7.1. Driven-Diffusive Lattice Gas Models Phase Transitions, Critical Dynamics, and Kinetics of Phase Ordering 1.8.1. Critical Dynamics: Role of Symmetry and Conservation Laws 1.8.2. Kinetics of Phase Ordering: Metastable and Unstable Initial States 1.8.3. Phase Transitions in Driven Systems
3 4 6 7 7 9 10 10 13 14 15 16 17 19 20 22 22 23 24
Methods for the Description of Stochastic Models
27
2.1. Quantum Formalism 2.1.1. Master Equation and Stochastic Hamiltonian 2.1.2. Spectrum and Expectation Values 2.1.3. Discrete Time Dynamics 2.2. Mean-Field and Cluster Methods 2.2.1. Mean-Field Approximations 2.3. Bethe Ansatz 2.4. Matrix-Product Ansatz 2.4.1. MPA in Quantum Formalism 2.4.2. MPA for Discrete Time Updates 2.4.3. Dynamical MPA 2.4.4. Relation with Bethe Ansatz
28 28 31 33 37 37 41 43 45 48 51 52
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3
2.5. Other Methods 2.5.1. Hydrodynamic Limit 2.5.2. Field-Theoretic Methods and Renormalization Groups 2.5.3. Similarity Transformations 2.5.4. Ultradiscrete Method 2.6. Numerical Methods 2.6.1. Computer Simulation (MC Methods) 2.6.2. Exact Diagonalization 2.6.3. Density-Matrix Renormalization Group 2.6.4. Transfer-Matrix DMRG 2.7. Appendices 2.7.1. Some Mathematics 2.7.2. MPA and Optimum Ground States for Quantum Spin Chains 2.7.3. Krebs–Sandow Theorem and Extensions
52 52 53 54 54 57 58 60 61 63 66 66 66 68
Particle-Hopping Models of Transport Far from Equilibrium
71
3.1. Elements of Random Walk Theory 3.2. Asymmetric Simple Exclusion Process 3.3. Zero-Range Process and Exact Results 3.3.1. Exact Solution 3.3.2. Bethe Ansatz Solution 3.4. Extensions and Generalizations 3.4.1. Parallel Dynamics 3.4.2. Other Lattice Structures 3.4.3. ZRP with Disorder 3.4.4. ZRP with Fluctuating Particle Number 3.4.5. Generalizations 3.4.6. Dynamical Urn Models 3.4.7. Misanthrope Process 3.4.8. Relation of ZRP to Other Models and Some Applications 3.5. Physics of the ZRP 3.5.1. Condensation Transition 3.5.2. Dynamics and Coarsening 3.5.3. Criterion for Phase Separation 3.6. Particle-Hopping Models with Factorized Stationary States 3.6.1. Models with Pair-Factorized Steady States 3.7. Generalized Mass Transport Models 3.7.1. Models with Continuous States 3.7.2. Asymmetric Random Average Process 3.7.3. Chipping Model 3.8. Appendix 3.8.1. Derivation of the Factorization Criterion
72 74 75 76 79 81 81 82 83 83 86 86 87 88 90 90 94 95 97 100 102 102 103 105 106 106
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4
Asymmetric Simple Exclusion Process – Exact Results
109
4.1. ASEP with Periodic Boundary Conditions 4.1.1. Random-Sequential Dynamics 4.1.2. Bethe Ansatz for Translationally Invariant Systems 4.1.3. Mean-Field Theories for Parallel Dynamics 4.1.4. Mapping to ZRP 4.1.5. Paradisical Mean-Field Theory 4.1.6. Combinatorial Solution for Parallel Dynamics 4.1.7. Ordered-Sequential and Sublattice-Parallel Updates 4.1.8. Shuffled Dynamics 4.2. ASEP with Open Boundary Conditions 4.2.1. Mean-Field Theory 4.2.2. Recursion Relations 4.2.3. Matrix-Product Ansatz 4.2.4. Exact Phase Diagram 4.2.5. Phase Transitions 4.2.6. Relation with Combinatorics 4.2.7. Bethe Ansatz 4.2.8. Dynamical MPA 4.2.9. Hydrodynamic Limit 4.3. Partially Asymmetric Version 4.3.1. MPA Solution 4.3.2. Bethe-Ansatz Solution 4.3.3. Phase Diagram of the PASEP 4.4. Extension of the ASEP to Other Update Types 4.4.1. Ordered-Sequential Updates 4.4.2. Sublattice-Parallel Update 4.4.3. Parallel Update 4.5. Boundary-Induced Phase Transitions 4.5.1. Domain Wall Picture 4.5.2. Extremal Principle and Steady-State Selection 4.5.3. More on Shock Dynamics 4.5.4. Fluctuations and Large Deviation Functions 4.6. Extensions of ASEP 4.6.1. Quenched Disorder 4.6.2. Disorder in Open Systems 4.6.3. Langmuir Kinetics 4.6.4. Extended Particles 4.6.5. Other Boundary Conditions 4.6.6. Long-Range Hopping 4.6.7. ASEP Beyond One Dimension 4.7. Multispecies Models 4.7.1. Models with Second-Class Particles
111 111 113 116 123 123 124 126 128 131 132 134 135 136 141 142 142 143 143 145 146 148 148 151 151 154 154 158 158 161 161 162 163 164 173 174 177 177 179 180 181 181
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4.7.2. ABC Model 4.7.3. AHR Model 4.8. Other Related Models 4.8.1. Staggered Hopping Rates 4.8.2. Two-Parameter Model 4.8.3. Restricted ASEP 4.8.4. KLS Model 4.8.5. Asymmetric Avalanche Process 4.8.6. Higher Velocities 4.8.7. Reconstituting Dimers 4.9. Appendices 4.9.1. Mapping of ASEP to Surface Growth Model 4.9.2. Mapping of the ASEP to an Ising Model 4.9.3. Solution of the Mean-Field Recursion Relations for the ASEP 4.9.4. Results Obtained from Normal-Ordering of Matrices 4.9.5. Dimension of Matrices in the MPA for the ASEP 4.9.6. Representations of the Matrix Algebra of the ASEP 4.9.7. Mean-Field Approximation of the DTASEP
Part Two 5
6
Applications
183 184 185 185 186 188 190 191 193 194 195 195 196 197 199 200 201 205
207
Modeling of Traffic and Transport Processes
209
5.1. Introduction 5.1.1. Some Practical Questions 5.1.2. Some Fundamental Questions 5.2. Classification of Models 5.2.1. Model Characteristics 5.2.2. Model Classes
209 210 211 212 212 214
Vehicular Traffic I: Empirical Facts
215
6.1. Measurement Techniques and Detectors 6.2. Observables and Data Analysis 6.3. Formation and Characterization of Traffic Jams 6.3.1. Jams Induced by Bottlenecks 6.3.2. Spontaneous Traffic Jams 6.3.3. Experiment on Spontaneous Jam Formation 6.4. Fundamental Diagram 6.5. Metastability and Hysteresis 6.6. Phases of Traffic Flow 6.6.1. Level of Service Classification 6.6.2. Traffic Phases and Phase Transitions 6.6.3. Gas–Liquid Analogy 6.7. Ramps, Intersections, and Other Inhomogeneities 6.8. Headway Distributions
215 216 221 222 223 224 226 228 230 230 231 234 235 236
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7
8
6.9. Optimal-Velocity Function 6.10. Correlation Functions 6.11. Psychological Effects
238 239 240
Vehicular Traffic II: The Nagel–Schreckenberg Model
243
7.1. Definition of the Model 7.1.1. Update Rules 7.1.2. Relation with Other Models 7.2. Fundamental Diagram and Limiting Cases of the NaSch Model 7.2.1. Fundamental Diagram 7.2.2. NaSch Model in the Deterministic Limit p = 0 7.2.3. NaSch Model in the Deterministic Limit p = 1 7.2.4. NaSch Model with vmax = 1 7.2.5. NaSch Model in the Limit vmax = ∞ 7.3. Analytical Theories for NaSch Model with vmax > 1 7.3.1. SOMF Theory for the NaSch Model 7.3.2. Cluster-Approximations for the NaSch Model 7.3.3. pMF Theory of the NaSch Model 7.3.4. Car-Oriented Mean-Field Theory of the NaSch Model 7.4. Spatio-Temporal Organization of Vehicles 7.4.1. Microscopic Structure of the Stationary State 7.4.2. Spatial Correlations 7.4.3. Headway Distributions 7.4.4. Distributions of Jam Sizes and Gaps between Jams 7.4.5. Distribution of Lifetimes of Jams 7.4.6. Temporal Correlations and Relaxation Time 7.4.7. Structure Factor 7.4.8. Phase Transition 7.4.9. Boundary-Induced Phase Transitions 7.5. Appendices 7.5.1. Details of SOMF for NaSch 7.5.2. Details of PMF for NaSch 7.5.3. Details of COMF for NaSch
244 244 247 248 248 250 251 251 253 255 255 256 257 259 260 260 261 262 263 265 266 267 268 270 272 272 276 277
Vehicular Traffic III: Other CA Models
281
8.1. Slow-to-Start Rules, Metastability, and Hysteresis 8.1.1. General Remarks 8.1.2. The Velocity-Dependent-Randomization Model 8.1.3. Takayasu–Takayasu Slow-to-Start Rule 8.1.4. The BJH Model of Slow-to-Start Rule 8.1.5. Other Slow-to-Start Rules 8.1.6. Flow Optimization and Metastable States 8.2. Cruise-Control Limit 8.3. CA Models of Synchronized Traffic 8.3.1. Brake-Light or Comfortable Driving Model
282 282 283 288 289 289 290 291 294 295
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8.4.
8.5.
8.6.
8.7.
8.8. 8.9.
9
8.3.2. Kerner–Klenov–Wolf Model 8.3.3. Mechanical Restrictions Model of Lee et al. Other CA Models 8.4.1. Fukui–Ishibashi Model 8.4.2. Velocity-Dependent Braking Model 8.4.3. Time-Oriented CA Model 8.4.4. Models with Anticipation 8.4.5. Galilei-Invariant Model 8.4.6. Car-Following CA CA from Ultradiscrete Method 8.5.1. Generalizations of BCA 8.5.2. Euler–Lagrange Transformation 8.5.3. Traffic Models in Lagrange Form CA Models of Multilane Traffic 8.6.1. Classification of Lane Changing Rules 8.6.2. CA Models of Bidirectional Traffic Effects of Quenched Disorder 8.7.1. Randomness in the Braking Probability 8.7.2. Random vmax 8.7.3. Randomly Placed Bottlenecks 8.7.4. Ramps Bus-Route Model Accidents
Vehicular Traffic IV: Non-CA Approaches 9.1. Fluid Dynamical Theories 9.1.1. Lighthill–Whitham–Richards Theory and Kinematic Waves 9.1.2. Diffusion Term in LWR Theory and Its Effects 9.1.3. Greenshields Model and Burgers Equation 9.2. Second-Order Fluid Dynamical Theories 9.2.1. Special Models 9.2.2. Instabilities and Jam Formation 9.2.3. Problems with Second-Order Models 9.2.4. Aw–Rascle Model 9.2.5. Fluid-Dynamical Models and Synchronized Traffic 9.2.6. Fluid-Dynamical Theories of Traffic on Multilane Highways and in Cities 9.3. Gas-Kinetic Models 9.3.1. Prigogine Model 9.3.2. Paveri-Fontana Model 9.3.3. Derivation of Fluid-Dynamical Equations from Gas-Kinetic Equations 9.4. Car-Following Models 9.4.1. Follow-the-Leader Model 9.4.2. Optimal Velocity Model and Its Extensions
299 301 304 304 305 306 307 309 311 313 314 315 316 319 319 322 324 324 326 326 328 329 332
335 336 337 340 341 342 344 345 348 348 349 350 351 351 353 356 357 358 360
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9.4.3. Generalized Force Models 9.4.4. Intelligent Driver Model 9.4.5. Kerner–Klenov Model 9.4.6. Inertial Car-Following Model 9.5. Coupled-Map Models 9.5.1. Gipps Model 9.5.2. Krauss Model (SK Model) 9.5.3. Yukawa-Kikuchi Model 9.5.4. Nagel-Herrmann Model 9.6. Other Approaches 9.6.1. Probabilistic Traffic Flow Theory 9.6.2. Cell Transmission Model 9.6.3. Queueing Models
10 Transport on Networks 10.1. Networks and Transport 10.2. BML Model of City Traffic 10.2.1. Phase Transition 10.2.2. Generalizations and Extensions of the BML Model 10.2.3. More Realistic Description of Streets and Junctions 10.3. Chowdhury–Schadschneider Model 10.3.1. Crossroads with Signals 10.3.2. ChSch Model 10.3.3. Traffic Signal Optimization 10.4. Highway and City Networks 10.4.1. Online Simulation of Traffic Networks 10.4.2. Network Analysis 10.4.3. Braess Paradox 10.5. Computer Networks and Internet Traffic
11 Pedestrian Dynamics 11.1. Introduction 11.2. Empirical Observations and Collective Phenomena 11.2.1. Individual Properties 11.2.2. Observables 11.2.3. Fundamental Diagram 11.2.4. Flows at Bottlenecks 11.2.5. Collective Phenomena 11.3. Cellular Automata Models 11.3.1. Fukui–Ishibashi Model 11.3.2. Blue–Adler Model 11.3.3. Gipps–Marksjös Model 11.4. Floor Field CA 11.4.1. General Principle 11.4.2. Update Rules
364 366 368 369 371 372 373 375 376 377 377 379 381
383 383 384 385 386 388 390 390 391 395 398 398 400 401 402
407 408 409 409 410 413 415 418 423 424 428 428 430 430 432
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11.4.3. Construction of the Static Floor Field 11.4.4. Conflicts and Friction 11.4.5. Other Generalizations and Interactions 11.4.6. Moving Beyond Nearest Neighbors: vmax > 1 11.4.7. Collective Effects 11.4.8. Evacuation Simulations 11.5. Other Models 11.5.1. Fluid-Dynamic and Gas-Kinetic Models 11.5.2. Social-Force Models 11.5.3. Lattice Gas Models 11.5.4. Optimal Velocity Model 11.5.5. Active Walker Models
12 Traffic Phenomena In Biology 12.1. Introduction 12.1.1. Different Types of Traffic in Biology 12.2. TASEP for Hard Rods: Minimal Model of Transcription and Translation 12.2.1. TASEP for Hard Rods: Minimal Models of Traffic of Ribosomes and RNAPs 12.2.2. TASEP for Hard Rods with Internal States: Effects of Individual Mechano-Chemistry 12.3. TASEP for Particles with Langmuir Kinetics: Minimal Model of Kinesin Traffic 12.3.1. TASEP-Like Generic Models of Molecular Motor Traffic 12.3.2. Traffic of Interacting Particles with “Internal States” and Langmuir Kinetics: Effects of Individual Mechano-Chemistry of KIF1A 12.4. Traffic in Social Insect Colonies: Ant-Trails 12.4.1. Model of Single-Lane Unidirectional Ant-Traffic 12.4.2. Model of Single-Lane Bidirectional Ant-Traffic 12.4.3. Model of Two-Lane Bidirectional Ant-Traffic 12.4.4. Experimental Investigations of Ant-Traffic 12.4.5. Empirical Results for Fundamental Diagrams of Ant-Trails Guide to the Literature Bibliography Index
435 436 437 440 441 444 447 447 449 454 456 458
461 461 462 462 464 466 466 467 468 472 473 478 483 485 486 489 491 549
PREFACE
Historically, statistical mechanics was developed as the molecular kinetic theory of matter. Its formalism for systems in thermodynamic equilibrium has been established on a solid foundation. Twentieth century has witnessed many triumphs of this formalism in solving many mysteries of bulk matter in thermodynamic equilibrium. Encouraged by the overwhelming success of the concepts and techniques of statistical mechanics in understanding the physical properties of inanimate matter, some statistical physicists have taken bold steps to explore virgin territories beyond the traditional boundaries of physics using the same tool box. Some of these systems were subjects of earlier investigation in other traditional disciplines. Such unconventional applications of statistical mechanics have opened up new horizons of interdisciplinary research; one of the most successful among such joint ventures is the area of soft matter. But, despite relentless efforts of statistical physicists over the last century, an equally strong foundation of nonequilibrium statistical mechanics remains elusive. One class of nonequilibrium systems that has received lot of attention in recent years consists of mutually interacting particles, which are driven by an external field. Such intrinsically nonequilibrium systems can settle down to a nonequilibrium steady state which, in principle, can be far from the equilibrium state that it could attain in the absence of the external field. Many physical systems have been successfully studied in the last two decades, using the techniques developed for models of such field-driven interacting particles. In more recent times, nonequilibrium statistical mechanics has found even more unusual applications in research on traffic flow of various types of objects. A list of such systems would include not only vehicular traffic, but also traffic-like collective phenomena in living systems, as well as in colonies of organisms. The common modeling strategy is to represent the motile elements (i.e., vehicles or an organism) by self-propelled particles, which convert chemical energy (derived from fuel or food) into mechanical energy required for the forward movement in a dissipative environment. In such generic models, the mutual influences of the motile elements on the movement of each other are captured by postulating appropriate interparticle interactions. Just as nonequilibrium statistical mechanics has helped in getting deep insight into the general physical principles governing the spatio-temporal organization and traffic flow in a wide variety of complex systems, which were earlier subjects of investigation in other branches of science and engineering, these disciplines have also enriched physics by proving wider testing grounds for the tool box of nonequilibrium statistical mechanics. The problems that we are studying here are classic examples of complex systems. Their collective behavior is not just a simple superposition of the individual behaviors, but the xv
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interactions among the particles/vehicles/agents. . . lead to new features of the dynamics with, sometimes, surprising consequences. In Part I of this book, we give a pedagogical introduction to the conceptual framework and technical tools of nonequilibrium statistical mechanics. Although our main focus is on driven systems of interacting particles, we present the subject from a much broader perspective. In Part II, we illustrate the use of the formalisms developed in Part I by applying these to vehicular and pedestrian traffic, as well as to traffic-like collective phenomena in biology. Modern life is crucially dependent on vehicular traffic. Our travel to our work places, to supermarkets, to hospitals, to schools and universities all depend on vehicular traffic. The supply of our daily needs also mostly come by trucks. We are all too familiar with the irritating traffic jams. But, traffic jam is not a modern phenomenon; it was familiar to inhabitants of cities even before the invention of motorized vehicles. Even in ancient cities like Rome, carts and chariots were often stuck in traffic jam and elaborate plans for traffic control were made even by Leonardo da Vinci. However, study of traffic as a branch of science and engineering started growing only in the second half of the twentieth century. Interestingly, new approaches of investigations in traffic science have been opened up mostly by physicists, perhaps, because of the close analogy between vehicular traffic and systems of interacting particles driven far from equilibrium. The notable among the early contributors include, for example, Montroll, Potts, Prigogine, and others. The conceptual framework and technical tool box of statistical physics has turned out to be extremely useful in modeling traffic phenomena and analyzing the properties of the models. Nature has dealt with traffic-like phenomena for billions of years. In the recent years, it has become clear that the environment inside a cell is, in many respects, similar to urban traffic system where molecular motors carry cargo over long distances by moving along filamentary tracks. Molecular traffic jam can lead to diseases. Therefore, fundamental understanding of these traffic phenomena will not only help in diagnosis but also in the control of those diseases that arise from malfunctioning of the molecular motor traffic in living systems. Traffic-like phenomena occur in biological systems at almost all levels of organization- from individual molecules and molecular-self assemblies to cells and organisms. The collective movements of ants toward the food source and their return to the nest appears very similar to vehicular traffic. Surprisingly, quantitative studies of this traffic phenomenon started only a couple of years ago despite the possible applications of the results in various ant-based routing algorithms in communication networks, in swarm intelligence, and even in decentralized management. In the past, capturing large number of empirical facts and phenomena with a few mathematical equations have always led to great progress in scientific theories. Maxwell unified all the empirical phenomena in electricity and magnetism in terms
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of the four equations named after him. Similarly, our attempt has been to highlight the “conceptual unity” among the apparent diversity of systems covering a long range of length scales and time scales. A common conceptual thread runs through all these traffic-like phenomena. We hope this monograph will stimulate further exchange of ideas across disciplinary boundaries of physics, chemistry, biology, as well as technology and engineering enriching all.
HOW TO USE THIS BOOK In Part I, we present a systematic pedagogical treatment of the theoretical formalisms which we, then, use in the Part II to develop and analyze models of transport and traffic phenomena. This book is self-contained in the sense that all the formal theoretical methods required in Part II are available in Part I. The formalisms discussed in Part I are essentially based on the concepts and techniques of nonequilibrium statistical mechanics. For the convenience of nonexperts, we minimize subtle technical details in the main chapters of Part I. Instead, for the benefit of readers interested in such details, lengthy mathematical calculations have been given in the appendices. In Chapter 1, while familiarizing the reader with the inventory of the toolbox of nonequilibrium statistical mechanics, we also mention their potential use in the subsequent chapters in the context of traffic science. In Chapter 2, we train the beginner in using some of the most powerful tools. In particular, we introduce several analytical, numerical, as well as phenomenological approaches for studying (quasi-) one-dimensional driven diffusive systems. In Chapter 3, we focus more specifically on a technique that, in recent years, has been used successfully in the mathematical treatment of many particle-hopping models. In Chapter 4, we give a comprehensive overview of our current understanding of the asymmetric simple exclusion process (ASEP), which is the most important particle-hopping model relevant for theoretical modeling of traffic. In fact, most of the particle-hopping models of vehicular traffic, which are described in Part II, are extensions of ASEP appropriate for capturing some specific traffic phenomena. We classify the traffic models in Chapter 5 so that the reader does not lose sight of the forest for the trees. We review the well-known empirical facts about traffic phenomena in Chapter 6. In Chapter 7, we discuss all the theoretical treatments of the Nagel–Schreckenberg model, the minimal and most basic model of vehicular traffic on highways, in detail. All the other CA models of vehicular traffic, many of which are related to the Nagel–Schreckenberg model, are considered in Chapter 8. For the sake of completeness of the overview of theoretical approaches to vehicular traffic, we also present the non-CA models in Chapter 9. The models of highway traffic have been modified and extended appropriately to capture some of the key features of vehicular
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traffic on networks of roads and highways in cities and greater urban areas. We present these models in Chapter 10 along with similar models of internet traffic. Theories of pedestrian traffic and traffic-like collective phenomena in biological systems are now making rapid progress. We provide a glimpse of this fast emerging frontier area in Chapters 11 and 12. There are at least two different routes how to use this book.
ROUTE 1: INTRODUCTION TO STOCHASTIC SYSTEMS Part I of this book can be used as a textbook for a specialized course on stochastic transport in systems driven far from equilibrium. In such a course, depending on the academic background and research interest of the students, a few topics from Part II can be selected to show interesting applications of the methodology.
ROUTE 2: INTRODUCTION TO THE MODELING OF TRAFFIC PHENOMENA Part II of this book can serve as a comprehensive introduction to the models of traffic and traffic-like collective phenomena. A reader, who is mainly interested in applied research in traffic, may need to refer to Part I if and only if (s)he wants to delve into any particular formalism. A long list of references is provided as a guide to the literature. Therefore, this compendium will also serve as a valuable reference for experts actively engaged in research on traffic models and related phenomena in complex systems.
ACKNOWLEDGMENTS
We have benefitted from our interactions with large number of students and colleagues over the last decade. First of all, we thank Ludger Santen who started this endeavor with us. But due to other commitments, he graciously resigned from this project. Several parts of this book are based on our joint review published almost 10 years ago in Physics Reports. We would like to thank all our collaborators with whom we have carried out our original work over the last two decades in this area of research. We thank Robert Barlovic, Aakash Basu, Maik Boltes, Jordan G. Brankov, Elmar Brockfeld, Carsten Burstedde, Debanjan Chowdhury, Mohcine Chraibi, Rashmi Desai, Georg Diedrich, Christian Eilhardt, Bernd Eisenblätter, Nils Eissfeldt, Minoru Fukui, Ashok Garai, Andrej Gendiar, Kingshuk Ghosh, Sascha Grabolus, Philip Greulich, Vishwesha Guttal, Torsten Huisinga, Nobuyasu Ito, Rui Jiang, Alexander John, Jens Kähler, Andreas Kemper, Janos Kertész, Macoto Kikuchi, Ansgar Kirchner, Kai O. Klauck, Hubert Klüpfel, Wolfgang Knospe, Tobias Kretz, Dima Ktitarev, Ambarish Kunwar, Maria Elena Larraga, Jack Liddle, Wolfgang Mackens, Arnab Majumdar, Kai Nagel, Akihiro Nakayama, Alireza Namazi, Lutz Neubert, Tomotoshi Nishino, Stefan Nowak, Yasushi Okada, Abhay Pasupathy, T.V. Ramakrishnan, Vladislav Popkov, Andrea Portz, Andreas Pottmeier, Vyatcheslav B. Priezzhev, Nikolaus Rajewsky, Antonio del Rio, Christian Rogsch, Tobias Rupprecht, Ludger Santen, Michael Schreckenberg, Gunter M. Schütz, Armin Seyfried, Ajeet Sharma, Shishir Sinha, Bernhard Steffen, Robin Stinchcombe, Yuki Sugiyama, Shin-ichi Tadaki, Christian Thiemann, Akiyasu Tomoeda, Tripti Tripathi, Satoshi Yukawa, Peter Wagner, Jian-Sheng Wang, Andreas Winkens, Robinson Will, Dietrich Wolf, Marko Wölki, Frank Zielen, and J. Zittartz for enjoyable collaborations. We apologize to everyone whose name might have been left out unintentionally. The collaboration of the three authors has been supported, at various stages, by the Deutsche Forschungsgemeinschaft (DFG, Germany), Federal Ministry of Education and Research (BMBF, Germany), Alexander von Humboldt Foundation (Germany), Council of Scientific and Industrial Research (CSIR, India). Over the last two decades, our ideas on traffic and related phenomena have been strongly influenced by our interactions with our collaborators, as well as with Cecile Appert-Rolland, Mustansir Barma, Richard Blythe, Martin Burd, Bikas Chakrabarti, Bernard Derrida, Deepak Dhar, Audrey Dussutour, Martin Evans, Erwin Frey, Raghavendra Gadagkar, Stephan Grill, Stefan Grosskinsky, Tom Hanney, Dirk Helbing, Joe Howard, Frank Jülicher, Boris Kerner, Stefan Klumpp, Anatoly Kolomeisky, Stefan Krauß, Joachim Krug, Reinhard Lipowsky, Reinhard Mahnke, Satya Majumdar, Gautam Menon, David Mukamel, and Dietrich Stauffer. xix
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However, we, the authors, are solely responsible for any conceptual error that might have crept in. We are indebted to many students and friends for their constructive criticism of our style of presentation. Special thanks go to Richard Blythe, Mohcine Chraibi, Christian Eilhardt, Philip Greulich, Tomohiro Sasamoto, Gunter Schütz, Armin Seyfried, and Daichi Yanagisawa who devoted lot of their valuable time to read earlier versions of the manuscript and for their comments and suggestions. However, we are responsible for the error(s), if any, which could not be detected before sending the manuscript to the printer. We could not have completed this book without the full cooperation and support from our family members who had to sacrifice many holidays so that we could devote the time to meet the deadline. Last, but not the least, our most sincere thanks to Kristi Green and Anita Koch at Elsevier for their patience and for extending the deadline many times in the last 4 years.
PART ONE
Methods and Concepts
1
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CHAPTER ONE
Introduction to Nonequilibrium Systems and Transport Phenomena Contents 1.1. Introduction
3
1.2. Classification of Nonequilibrium Phenomena
4
1.3. Hierarchy of Description at Different Levels
6
1.4. Individual-Based Models 1.4.1. Newton’s Equations, Hamilton’s Equations, and Individual Trajectories 1.4.2. Langevin Equation
7 7 9
1.5. Population-Based Models 1.5.1. Master Equation and Irreversibility 1.5.2. Fokker–Planck Equation
10 10 13
1.6. Fluid Flow: Theoretical Descriptions at Different Levels 1.6.1. Liouville Equation and Flow in Phase Space 1.6.2. From Liouville Equation to Boltzmann Equation: BBGKY Hierarchy 1.6.3. From Kinetic Theory to Navier–Stokes Equation
14 15 16 17
1.7. Back to Discrete Models: Mimicking Hydrodynamics with Fictitious Particles 1.7.1. Driven-Diffusive Lattice Gas Models
19 20
1.8. Phase Transitions, Critical Dynamics, and Kinetics of Phase Ordering 1.8.1. Critical Dynamics: Role of Symmetry and Conservation Laws 1.8.2. Kinetics of Phase Ordering: Metastable and Unstable Initial States 1.8.3. Phase Transitions in Driven Systems
22 22 23 24
1.1. INTRODUCTION This book is mainly concerned with traffic and traffic-like collective phenomena. We shall first present an overview of the techniques of modeling from a broader perspective within the framework of the formalisms of nonequilibrium statistical mechanics and nonlinear dynamics of interacting particles and fluids. The traffic-like systems that we will consider here involve purely classical transport phenomena, and quantum mechanics plays no role in these processes. However, there are subtle differences between vehicular traffic and the collective transport of interacting classical particles. Furthermore, the mechanisms of transport by particles of inanimate Stochastic Transport in Complex Systems DOI: 10.1016/B978-0-444-52853-7.00001-4
Copyright © 2011, Elsevier BV. All rights reserved.
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matter and those driving transport in the living world also differ in details in spite of all the apparent similarities [219]. In this chapter, we provide a pedagogical introduction to wide varieties of formalisms of nonequilibrium statistical mechanics [62, 1530]. After introducing each formalism from a broad general perspective, we indicate how the formalism can be adapted to treat vehicular traffic and other traffic-like collective phenomena.
1.2. CLASSIFICATION OF NONEQUILIBRIUM PHENOMENA We shall use the symbol C to describe the configurations of the system. Suppose, PC is the probability distribution of the configurations. A characteristic feature of the thermodynamic equilibrium of a system at the absolute temperature T is that PC is given by the Gibbsian form PCeq ∝ exp (−HC /kB T ),
(1.1)
where H is the Hamiltonian of the system such that HC is the energy corresponding to the configuration C and kB is the Boltzmann constant. Broadly speaking, we can classify nonequilibrium systems into two groups: (1) systems at or near equilibrium, and (2) systems far from equilibrium (see Fig. 1.1). For systems near thermodynamic equilibrium, Onsager’s regression hypothesis [186] states that the relaxation of macroscopic nonequilibrium disturbances follows the same laws, which govern the regression of spontaneous microscopic fluctuations in a system that is in equilibrium. For such systems, linear response theory holds, and the
Systems in statistical mechanics
Systems in, or near, equilibrium
Systems for from equilibrium
Systems evolving toward equilibrium steady-state
Metastable initial state
Unstable initial state
Systems evolving toward nonequilibrium steady-state
Metastable initial state
Figure 1.1 Equilibrium and nonequilibrium systems in statistical mechanics.
Unstable initial state
Introduction to Nonequilibrium Systems and Transport Phenomena
fluctuation-dissipation theorem connects response and dissipation with correlations in the fluctuations. However, for systems that are not in stable thermodynamic equilibrium, the distribution PC (t), in general, depends on time t. However, given enough time for spontaneous time evolution, such systems may eventually reach a stationary state (also called steady state), which corresponds to a time-independent distribution PCst . But, for an arbitrary stationary state, PCst is not necessarily identical to the Gibbsian formula (1.1). It is worth emphasizing that not all stationary states are thermodynamic equilibrium states; the state of thermodynamic equilibrium is a very special stationary state, as we will show later in this chapter. In fact, for a large class of truly nonequilibrium systems, no Hamiltonian can be defined; such models are defined in terms of the rates of the transitions between different configurations. Such a system may attain a nonequilibrium stationary state or exhibit nonstationary behavior for ever. Therefore, systems far from equilibrium can be divided further into two subclasses: (1) systems for which Hamiltonian can be defined and the stable thermodynamic equilibrium exists, so that the system evolves spontaneously toward this equilibrium; and (2) systems for which neither the Hamiltonian nor the Gibbsian equilibrium state exists. Systems of type (1) can attain thermodynamic equilibrium state after sufficiently long time. But, a system of type (2) may attain only a nonequilibrium steady-state (see Fig. 1.1). Consider, e.g., a window glass which is a supercooled liquid and is in a metastable state. It can, in principle, attain the corresponding crystalline equilibrium structure after sufficiently long time. But, in contrast, a conducting wire carrying an electrical current can, at best, attain a nonequilibrium steady state after the transient currents die down. Systems in the latter category will be referred to as driven nonequilibrium system where, loosely speaking, an external “drive” maintains the corresponding current. A difference of chemical potential between the two ends of a sample can drive a current of material particles. Similarly, an electric potential gradient can drive an electrical current. How do systems evolve from initial states that are far from equilibrium? If an equilibrium exists, such a system would tend to equilibrate. However, such systems follow different mechanisms depending on whether the initial state is metastable or unstable [475]. From a metastable initial state, the equilibration of the system occurs through nucleation of droplets (nuclei) of the stable phase and their growth, as well as coalescence [712]. A system in an unstable state evolves in time by the coarsening of a random interconnected structure leading to the phase separation of the components (spinodal decomposition or Ostwald ripening). We will discuss these issues in more detail in Section 1.8.2. In a thermodynamic system, i.e., a system for which Gibbsian equilibrium exists, the difference between the metastable and unstable states can be explained easily by considering an appropriate thermodynamic potential with more than one extremum. The maxima of the potential correspond to the unstable states, whereas all the minima, except the lowest one, are identified as metastable. The global minimum corresponds to the stable equilibrium state. The concepts of unstable and metastable steady states of
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a driven nonequilibrium system cannot be distinguished in a similar manner because analogs of the thermodynamic potentials, which can be defined for all driven systems, are not known. Nevertheless, fluctuation-driven spontaneous transitions from metastable steady states to stable steady states have been observed, for example, in vehicular traffic.
1.3. HIERARCHY OF DESCRIPTION AT DIFFERENT LEVELS The theoretical approaches of nonequilibrium statistical mechanics can be broadly divided into two categories: (1) “individual-based” and (2) “population-based” (Fig. 1.2). The individual-based models describe the dynamics of the individual elements explicitly. Just as “microscopic” models of matter are formulated in terms of molecular constituents, the individual-based models of transport are also developed in terms of the constituent elements. Therefore, the individual-based models are often referred to as microscopic models. When watched closely, traffic flow appears very similar to particles that are not only driven forward but also influence each other (although direct collisions take place only when they accidentally crash against each other). By exploiting these similarities with systems of interacting driven particles, we can treat traffic problems as those of nonequilibrium statistical mechanics. However, if vehicular traffic is watched from a long distance (say, from a hill top or a helicopter), traffic flow at peak hours resembles fluid flow where we do not see each individual vehicle; the fluid is characterized by a local density at different locations along the highway. In the population-based models individual elements do not appear explicitly and, instead, one considers only the population densities (i.e., number of individual elements per unit area or per unit volume). The spatio-temporal organization of the elements are emergent collective properties that are determined by the responses of the individuals to their local environments and the local interactions among the individual elements. Therefore, in order to gain a deep understanding of the collective phenomena, it is essential to investigate the linkages between these two levels of organization. Usually, but not necessarily, space and time are treated as continua in the populationbased models and partial differential equations (PDEs) or integro-differential equations are written down for the time-dependent local collective densities of the elements. The individual-based models have been formulated following both continuum and Models Individual-based models
Population-based models
Figure 1.2 Individual-based models versus population-based models.
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discrete approaches. In the continuum formulation of the Lagrangian models, differential equations describe the individual trajectories of the elements. The theoretical models of traffic and traffic-like collective dynamics have been formulated using several different mathematical formalisms. The choice of the formalism depends on the level of description. Descriptions at different levels are certainly not equivalent to each other; however, in principle, the dynamical equations at a higher level can be derived from one at a lower level. Moreover, even at a given level, alternative but equivalent formulations of the dynamics are possible.
1.4. INDIVIDUAL-BASED MODELS 1.4.1. Newton’s Equations, Hamilton’s Equations, and Individual Trajectories The natural starting point of Newtonian dynamics is to write down the equations of motion for all the individual particles. Suppose, mi is the mass of the i-th particle. Then, for a system of N particles, the Newton’s equations for the individual particles are given by mi
d2 xi = F iext + F iint, 2 dt
i = 1, . . ., N ,
(1.2)
where Fiext is the external force on the i-th particle and the internal force Fiint = N int j=1(j= i) fij leads to coupling of the N differential equations (1.2). Conservative forces are derivable from a potential U (x). The state of the entire N -particle system is uniquely given by (x1 , . . ., xN ; p1 , . . ., pN ; t) at any arbitrary instant of time t. Thus, each such state corresponds to a single point in the 2Nd-dimensional phase space of the N particles in a d-dimensional space. The 2Nd coordinates of the phase space are spanned by Nd positions of the N particles and the corresponding Nd momenta. Therefore, the time evolution of a single N -particle system is described by a trajectory in this 2Nd-dimensional phase space. Hamilton’s canonical equations of motion for the generalized coordinate q and generalized momentum p of the i-th particle are given by q˙ i =
∂H , ∂p
p˙ i = −
∂H , ∂q
(1.3)
where the Hamiltonian H is a function of q and p. These equations provide an alternative formulation of the dynamics of the N -particle system that is completely equivalent to the description in terms of the Newton’s equations.
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In principle, depending on the available computational resources, the Newton’s equations can be solved numerically for a finite system consisting of N particles provided N is not very large. In those situations where each particle corresponds to a molecule, this method is usually referred to as molecular dynamics (MD). However, precise initial conditions for all the N particles are difficult, if not impossible, to extract from any independent measurements. Furthermore, even a slight error in the initial conditions, which can never be ruled out because of the finite resolution of any measurement, can have enormous effects on the accuracy of the results of the simulations. This is a consequence of the well-known phenomenon that the separation between two trajectories, which initially differ by an insignificant amount, can grow exponentially with time if the equations of motion are nonlinear. Newtonian Approach to Modeling Traffic: A real piece of matter consists of molecules.
Therefore, atomic or molecular constituents of matter can be represented by a particle. What would be the counterpart of a particle in models of vehicular traffic? Although a vehicle consist of an enormously large number of molecules, we can still treat each of these as a particle for appropriate traffic phenomena, depending on the length scales involved in the problem. We should not forget that, while developing his laws of mechanics, Newton treated sun and the planets as point particles which was appropriate for the celestial phenomena he studied with that formalism. Therefore, from now onwards, we will use the more general term “particle” to denote a motile object (a molecular motor or a living cell or a cell or an organism or a vehicle). What would be the counterpart of force in the analog of Newton’s equation that models vehicular movement in traffic? In order to get an intuitive clue, note that the Newton equation can be regarded as a linear response relation, i.e., a linear relation between a cause and the corresponding effect. In this case, force acting on a particle is the cause and the corresponding acceleration is the effect, 1/m being the analog of a response coefficient, i.e., a measure of the extent of response to unit force. This is analogous to the linear response relation for ohmic conductors where electric current is caused by an electric field (potential gradient); electrical conductivity being the corresponding linear response coefficient. Thus, Response ∝ Stimulus .
(1.4)
Based on this philosophy, many years ago, statistical physicists formulated the so-called car-following models of vehicular traffic which will be described in detail in Section 9.4. The interaction between different vehicles cannot be captured by interparticle physical interactions alone because the drivers respond to the surrounding stimuli. Therefore, for modeling traffic or pedestrian dynamics with Newton-like equations, one sometimes includes “social forces” in the stimulus while acceleration of the vehicle or pedestrian corresponds to the response. Similar approaches have been used also in formulating the
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equations of motion of organisms (e.g., ants) for modeling their collective movements. It is worth pointing out here that Newton’s third law does not hold for social forces.
1.4.2. Langevin Equation The true microscopic description in terms of Newton’s equations (or, equivalently, Hamiltonian’s equations) cannot account for the irreversible phenomena in the macroscopic world because these equations are invariant under time reversal. For a more concrete discussion, let us consider a steel ball falling under gravity in a viscous medium. As we learn in elementary Newtonian mechanics, the particle attains a terminal (steady) velocity because of the viscous drag force. But, if instead of focussing our attention only on the steel ball, we also wrote down the Newton’s equations for all the N particles of the surrounding medium, the system of N + 1 equations would have perfect time reversal symmetry. In other words, the dissipative viscous drag appears only when we study the dynamics in a subspace of the entire phase space of the system by projecting out the other degrees of freedom. Now, instead of the steel ball, consider a pollen grain which is small by macroscopic standards but is still several orders of magnitude larger than the molecules of water. If the pollen grain is immersed in water, it executes an apparently erratic movement, which is known as Brownian motion. Guided by deep intuitive insight, Paul Langevin wrote down an equation of motion for the Brownian particles which was later named after him [837]. The Langevin equation for a Brownian particle of mass m can be written as
m
d2x = F ext + Fdiss + Fr , dt 2
(1.5)
where the second term on the right-hand side captures the dissipative forces while the third term on the right-hand side is a random force. Since, by definition, the random force is unpredictable, only its statistical properties can be postulated. The simplest thing one can assume is that the average of this random force vanishes so that the average motion is still similar to that of the steel ball we discussed earlier. In his original work, Langevin assumed the form Fdiss = −γ
dx dt
(1.6)
for the dissipative viscous force, where γ is a phenomenological coefficient. Fr (t) is the random force (noise) whose time dependence is not known. However, in order that the average velocity satisfies the Newton equation for a particle in a viscous medium, we further assume for the average ξ(t) = 0
(1.7)
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and the autocorrelation function ξ(t)ξ(t ) = 2δ(t − t ),
(1.8)
where ξ = Fr /m and, at this level of description, is a phenomenological parameter. The prefactor 2 on the right-hand side of Eqn. (1.8) has been chosen for convenience. At this level of description, one effectively separates the entire composite system into two subsystems one of which is the motile element, i.e., the system of interest; the other subsystem, which includes the rest of the composite system, is treated as a reservoir or bath. The time evolution of an individual motile element is now given by a differential equation that, unlike Newton’s equation, is both stochastic and irreversible. Langevin Approach to Modeling Traffic : The formalism we have described so far in this
section describes “passive” Brownian particles in external fields. In the last decade, this formalism has been extended to model “active” Brownian particles [1278] where one assumes that the dissipative force is such that it is positive at small velocities (and, hence, accelerates rather than decelerating the particle), whereas it is negative for all velocities beyond a critical value. In the context of vehicular traffic phenomena, the Langevin equation has not been used widely (for an exception, see [899]). However, it has been used for few trafficlike collective phenomena in biology where the external force acting on the particle (which represents the motile object) is derived from a potential U (x) using the relation (x). In this scenario, the potential U (x) is a landscape in which the Fext (x) = −∇U “particle” feels a force along the local gradient. However, in many systems the potential U (x) is time-dependent. In such cases, one needs an additional equation which governs the slow time evolution of the landscape itself. For example, as we will show later, in the case of molecular motors, the landscape changes periodically and completes one cycle with the consumption of each molecule of the chemical fuel. In case of collective traffic-like movements of ants on trails, the dynamics of the landscape and the forward movement of the ants are coupled in a very interesting manner; we will consider this in detail in Section 12.4.
1.5. POPULATION-BASED MODELS 1.5.1. Master Equation and Irreversibility Let us again use the abstract symbol C for labeling the various possible configurations of a system and denote the probability of finding the system in configuration C at time t by PC (t). In the context of traffic we will often face situations where time can still be considered continuous, but the configurations of the system form a discrete set.
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The dynamics of the system corresponds to transitions between the configurations. The master equation governing the time evolution of the PC (t) is given by [1409] dPC (t) WCC PC − WC C PC , = dt C
(1.9)
C
where WCC is the transition probability,1 per unit time, from configuration C to configuration C. The first and the second terms on the right-hand side are called the gain and loss terms, respectively. The summations over the configurations have to be replaced by integrations when the configurations form a continuum; for a continuous random variable, z, the master equation has the form ∂P(z, t) (1.10) = Wzz P(z , t) − Wzz P(z, t) dz ∂t In the commonly used forms (1.9) and (1.10), time t is treated as a continuous variable. In many applications, particularly in modeling traffic, time is discretized in short intervals t. In such situations, given the configuration of the system at time t, the master equation yields the corresponding configuration at time t + t: ⎛ PC (t + t) = ⎝1 −
⎞ WC C t ⎠ PC (t) +
C =C
WCC t PC (t)
By introducing WCC = WCC t for C= C and WCC = 1 − written in the more compact form PC (t + t) =
(1.11)
C =C
WCC PC (t) .
C =C WC C t,
this can be
(1.12)
C
WCC is the probability for a transition from state C to state C during the timestep t. The transition probabilities satisfy the normalization C WCC = 1 due to probability conservation. The form of the discrete master equation (1.12) is quite intuitive since the state C at time t + t must have evolved from one of the configurations C at time t. The number of such transitions is both proportional to the transition probability WCC and the probability PC (t) to find the state C . 1 The terms with C = C cancel so that W does not need to be specified. CC
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Note that if
WCC PC =
C
WC C PC ,
(1.13)
C
PC is independent of time (see Eqn. 1.9), i.e., the system is in a stationary state (also called steady state). But, not all steady states are states of stable thermodynamic equilibrium. Existence of equilibrium requires the stronger condition WCC PC = WC C PC for all configurations C and C which, in turn, imposes the condition WCC /WC C = PC /PC
(1.14)
on the transition rates WCC ; this is the condition of detailed balance. Situations may arise in systems where the strong condition (1.14) of detailed balance does not hold, but a steady-state exists because of the pairwise-balance condition WCC PC = WC C PC .
(1.15)
It implies that for every configuration C which can be reached directly from C a unique configuration C exists so that the condition (1.15) is satisfied. The differences between the conditions (1.13), (1.14), and (1.15) are explained graphically in Fig. 1.3. Consider the 10 configurations labelled by the integers 1, . . . , 10. If the only transitions into and out of the configuration 1are those shown in Fig. 1.3a, then dP1/dt = 0 provided j=2,..,6 W (j → 1)Pj = j=7,..,10 W (1 → j)P1 . In other words, for any arbitrary configuration i, stationarity of Pi , i.e., the condition dPi /dt = 0, is satisfied if merely the total probability current into the configuration i is exactly balanced by the total probability current out of i. In contrast, the equilibrium, which is a special stationary state, requires detailed balance (see Fig. 1.3b): the net forward and reverse probability current in between the members of each pair of configurations must balance exactly. Clearly, this is a much stronger condition than what was needed to guarantee a nonequilibrium stationary state. Finally, for many models, a pairwise-balance is also satisfied by the nonequilibrium stationary state (see Fig. 1.3c): The probability current from configuration 1 to 2 is exactly balanced by that from 2 to 3 so that dP2 /dt = 0. Similarly, 2
3
4
5
6
2
3
4
1 7
8
6
1
1 9
(a)
5
10
7
8
9 (b)
Figure 1.3 Illustration of detailed and pairwise balance.
10
3
2 (c)
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the probability current 2 to 3 is equal to that from 3 to 1, and so on. Such cyclic processes are important for many phenomena, particularly in enzymatic catalysis, which occur in molecular motor transport. For a more general discussion of the master equation and its relation with mathematical graph theory, we refer to the classic review article by Schnakenberg [1252]. This has been extended, e.g., to classify nonequilibrium steady states in terms of their stationary probability distribution and the associated currents [1525, 1526] and to derive fluctuation theorems [31, 498]. In a more formal way, the master equation (1.9) can be recast as dP ˆ = MP(t), dt
(1.16)
dPC (t) ˆ MCC PC (t), = dt
(1.17)
i.e.,
C
ˆ is given by where the master operator M ˆ CC = WCC − δCC M
WC C
(1.18)
C
and the components of the vector P(t) are the configurations PC (t). We will come back to this formulation in the next chapter where it is the starting point of the so-called quantum formalism. Master Equation for Traffic Models: The master equation approach has been used exten-
sively for formulating models of vehicular traffic. While time is discretized in most of these models, master equation in continuous time has also been used. This will be discussed in detail in Part II of this book.
1.5.2. Fokker–Planck Equation The Fokker–Planck equation is a valid description at the same level at which the Langevin description holds. The Fokker–Planck equation can be derived from the master equation under an appropriate truncation of an expansion known as Kramers–Moyal expansion [1409]. The truncated equation is essentially a master equation ∂P(z, t) = MFP P(z, t), ∂t
(1.19)
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where the operator M is a second-order differential operator given by MFP = −
∂ 1 ∂2 A2 (z) A1 (z) + ∂z 2 ∂z2
(1.20)
with the jump moments ∞ Aμ(z) =
ξ μ W(z; ξ)dξ.
(1.21)
−∞
The Fokker–Planck equation has also been used for the description of traffic, see e.g., [899].
1.6. FLUID FLOW: THEORETICAL DESCRIPTIONS AT DIFFERENT LEVELS Fluid flow has been studied at several different levels of description (Fig. 1.4). It is worth pointing out that not all the intermediate steps of the hierarchy are necessarily essential in a comprehensive treatment. For example, the hydrodynamic counterparts of many lattice gas models can be derived directly without going through the intermediate steps of deriving any lattice Liouville or Lattice Boltzmann version. Therefore, in this section, we provide a brief overview of these theoretical approaches at different levels. A real fluid consists of interacting molecules. Therefore, in principle, one may naively attempt formulating the mathematical theory of a fluid in terms of the Newton equations of its constituent particles. Even if any computer, in forseable future, could handle such enormous number of coupled ordinary differential equations, it is Navier–Stokes equation
Boltzmann
Lattice Boltzmann
Liouville
Lattice Liouville
Newton
Lattice gas
Figure 1.4 Relation between the different levels of description.
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practically impossible to get the states of all the particles at a given instant of time, which can be treated as the initial state of the system. Moreover, even if it were possible, integration of these equations (or, equivalently, MD simulation) would yield enormous amount of information most of which is redundant for describing fluid flow at the macroscopic level. The large number of molecules in the fluid is actually not a bane but a boon for an alternative approach based on the principles of statistics. Any statistical prediction improves with increase in the sample size! In the following subsection, we outline the general formal approach for deriving the equations for such a statistical description of a fluid.
1.6.1. Liouville Equation and Flow in Phase Space Newton’s equation describes the time evolution of each individual particle. A given initial condition leads to a unique trajectory in the phase space. But, since the initial conditions are usually not available, let us assume a probability distribution for the individual conditions. Thus, instead of a one single point, now we have a “swarm” of points corresponding to several possible initial states of the system. This swarm “flows” in the phase space, each point of the swarm tracing its own trajectory. This alternative, but equivalent, statistical approach is based on the N -body distribution function fN (x1 , . . ., xN ; p1 , . . ., pN ; t), which is the joint probability of finding the molecule 1 at position x1 with momentum p1 , 2 at position x2 with momentum p2 , and so on up to the molecule N at location xN with momentum pN , all at the same time t. The dynamical equation governing the time evolution of the distribution fN is given by
∂ x )i + ai · (∇ v )i vi · (∇ + ∂t N
fN (x1 , . . ., xN ; p1 , . . ., pN ; t) = 0,
(1.22)
i=1
where vi = pi /m, ai = Fi /m and x = ∂ + ∂ + ∂ , ∇ ∂x ∂y ∂z
v = ∂ + ∂ + ∂ . ∇ ∂vx ∂vy ∂vz
(1.23)
The equation (1.22) is known as the Liouville equation. Note that this equation can be recast as ; t) ∂fN (X ˙ ; t) = 0, · ∇X fN (X + X ∂t
(1.24)
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is a 2Nd-dimensional vector in phase space and its components are the positions where X and momenta of the N particles. The form (1.24) of the Liouville equation reveals its close formal similarity with the equation of continuity for incompressible fluids, the density of fluid corresponds to the probability density fN . Using Hamilton’s canonical equations, the equation (1.22) can also be written as ∂fN ˆ N, = Lf ∂t
(1.25)
where the Liouville operator Lˆ is given by Lˆ =
N ∂H i=1
∂ ∂H ∂ · − · . ∂qi ∂pi ∂pi ∂qi
(1.26)
Newtonian dynamics is invariant under time reversal. The Liouville equation also does not account for irreversible processes. Both contain equally large amount of information about the system lot of which is, for most practical purposes, redundant. Besides, to our knowledge, no Liouville-like equation has ever been written down for traffic flow. Nevertheless, the Liouville equation is a good starting point for systematically reducing the amount of irrelevant information and for introducing elements that account for the irreversibility observed in the macroscopic world. Moreover, it also allows a transparent link between the reversible dynamics, which it describes, and irreversible processes which are described by the master equation. The form (1.16) of the master equation appears formally similar to the form (1.25) of the Liouville equation. But, there are crucial fundamental differences between the two equations; the Liouville equation captures the reversible Newtonian (or, equivalently, Hamiltonian) dynamics, whereas the master equation describes irreversible processes. To our knowledge, the relation is best elucidated with the help of the Kac ring model [320, 321].
1.6.2. From Liouville Equation to Boltzmann Equation: BBGKY Hierarchy The Liouville equation provides a kinetic description of a many-body system. It contains as much information as contained collectively in all the Newton’s equations of the N particles. Nevertheless, the Liouville equation forms a convenient starting point for a formal way of systematically eliminating irrelevant information and cast the theory in terms of reduced distribution functions fM (x1 , . . ., xM ; p1 , . . ., pM ; t) = fN (x1 , . . ., xN ; p1 , . . ., pN ; t)dzM +1dzM +2 . . .dzN , (1.27)
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where dzk = dxk dpk . The chain of time evolution equations for fM with decreasing M is known as the BBGKY hierarchy (named after Bogoliubov–Born–Green–Kirkwood– Yvon [134, 142, 757, 1511]). This hierarchy, finally, leads to the equation for the onebody distribution f (x, p; t) which can be formally written as
∂ + v · (∇x) + a · (∇v ) f = C[ f , f ], ∂t
(1.28)
where f (x, p; t) is the probability density of finding a single particle at position x with momentum p at time t. The right-hand side symbolically represents the effects of binary collisions; this is usually a complicated integral whose actual form depends on the detailed nature of the intermolecular interactions. This is known as the Boltzmann equation; it provides a kinetic description of one body in a many-body system. Derivation of this equation is based on the assumption of molecular chaos (what Boltzmann called Stosszahlansatz) according to which f12 (x1 , x2 ; p1 , p2 ; t) = f (x1 , p1 ; t) f (x2 , p2 ; t),
(1.29)
i.e., absence of correlation between molecules encountering a binary collision. This assumption, however, has profound implications as it breaks the time-reversal symmetry inherent in the Newton’s equations and, equivalently, in the Liouville equation. Boltzmann-Like Equation in Kinetic Theory of Vehicular Traffic: Extending the basic concepts underlying the Boltzmann equation, Prigogine and coworkers developed a kinetic theory of vehicular traffic. However, several subtle questions had to be addressed in formulating the strategy. First, for a dilute gas, an arbitrary initial distribution relaxes to the corresponding equilibrium distribution (Maxwell’s distribution) after a sufficiently long time. What would be the analog of equilibrium distribution when the same method is applied to vehicular traffic? Second, what would be the counterpart of the Stosszahlansatz for the kinetic theory of vehicular traffic? These questions will be addressed in Section 9.3 while we present the kinetic theory of vehicular traffic.
1.6.3. From Kinetic Theory to Navier–Stokes Equation Let us first define the averaged variables, namely local mass density, momentum density, and energy density by the equations ρ(x, t) = m f (x, p; t)dp , (1.30)
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ρu(x, t) = m
f (x, p; t)vdp ,
ρe(x, t) = m
f (x, p; t)
v2 dp , 2
(1.31) (1.32)
where m is the mass of a molecule. In the hydrodynamic regime (where the deviations from thermodynamic equilibrium is small), one gets the standard equation of fluid dynamics at the macroscopic level (for simplicity, we drop the equation (1.32) from our consideration and this is valid for isothermal fluids). The Navier–Stokes equation is given by ∂u + f, u = ν∇ 2 u − ∇P + (u · ∇) ∂t
(1.33)
where ν is the viscosity, P is the pressure, and f is the external force. Moreover, for · u = 0 must be satisfied. incompressible fluids, the condition ∇ The Navier–Stokes equation is a force-balance equation for elementary volumes of fluid. Although small by macroscopic standard, these elementary volumes contain a sufficiently large number of molecules to make the density-based description a sensible approach. Thus, at the level of Navier–Stokes equation, the dynamics of individual constituent particles do not appear explicitly. Because of the conservation of the fluid, in addition to the Navier–Stokes equation, the fluid density also satisfies the equation of continuity ∂ρ · J = 0, +∇ ∂t
(1.34)
∂u ∂u ∂ 2u +u =ν 2 ∂t ∂x ∂x
(1.35)
where J is the current density. The Burgers equation
is a nonlinear diffusion equation in one-dimensional space. Although the setting up of this equation can be motivated by hydrodynamics, the equation itself is too simple to capture several realistic features of real fluids. For example, it neither accounts for shear nor for vortices in a fluid and its solutions correspond to an infinitely compressible medium!
Introduction to Nonequilibrium Systems and Transport Phenomena
Fluid-Dynamical Approach to Vehicular Traffic: When viewed from a long distance, the flow
of vehicles in a crowded traffic on a highway resembles fluid flow. Naturally, mathematical methods that have been successfully used for fluids have been also tried for the study of traffic flow. Because of the conservation of the vehicles, an equation of continuity can be easily written down for vehicular traffic. Pioneering work in this direction was done by Lighthill and Whitham [874, 875] more than half a century ago. However, establishing an appropriate analog of the Navier–Stokes equation for traffic flow has been much more difficult. Since vehicles in a traffic are subjected to “social forces,” in addition to real physical forces, the force balance becomes more subtle than in the case of a real fluid. We will study the fluid-dynamical models of vehicular traffic in Section 9.1. Some of the most successful particle-hopping models of vehicular traffic reduce to the Burgers equation [87, 167] in the continuum limit. One of the key features of this equation is that two solutions of it can coexist in such a way that u(x) varies from a high value to a low value over a narrow region whose width vanishes in the limit ν → 0. Such a shock wave solution, in the limit of vanishing viscosity in the Burgers equation, has a physical interpretation as the front of a traffic jam.
1.7. BACK TO DISCRETE MODELS: MIMICKING HYDRODYNAMICS WITH FICTITIOUS PARTICLES The Navier–Stokes equation is a PDE for a fluid in the continuum space-time. Over the last few decades, a few alternative approaches have been developed in terms of “fictitous particles.” The rules of dynamical evolution of these fictitious particles capture only the most essential features of the corresponding real particles. These dynamical rules are prescribed in such a way that an appropriate coarse-graining of the model would yield the corresponding correct PDEs for the “density” variables that characterize the real macroscopic system in the continuum space-time limit. Cellular Automata Hydrodynamics: Numerical solution of Navier–Stokes equation (or
similar nontrivial equations of hydrodynamics) requires discretization of both space and time. Therefore, the alternative discrete formulations may be used from the beginning. Besides, the theory can be formulated in terms of fictituous particles each of which can, in principle, capture many degrees of freedom of the real fluid, i.e., each particle can be viewed as an object obtained after coarse-graining over many real particles around a point in the fluid. In these models, fictituous particles are constrained to move along the bonds of a regular lattice following well-defined prescriptions or update rules. These rules mimic
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A. Schadschneider, D. Chowdhury, and K. Nishinari
both collisions of real particles, as well as their free streaming in between collisions. The update rules govern the evolution of the discrete positions and discrete velocities of the fictituous particles in time which is also discretized. Lattice Boltzmann Equations and Hydrodynamics: The lattice Boltzmann equation (LBE) is based on the discretization of the phase space to allow a numerical solution of the Boltzmann equation. It may be regarded as an extension of the cellular automata (CA) hydrodynamics, where the dynamical variable associated with each discrete space-time point is a real variable [1317].
1.7.1. Driven-Diffusive Lattice Gas Models The terms “lattice” and “gas” may appear contradictory at first sight because lattice is a characteristic property of a crystalline solid; there is no real lattice in the gas phase of matter. The basic idea of lattice gas is to discretize continuous space into a regular array of cells. Let us imagine that the three-dimensional space is divided into a regular array of cubical cells such that each cell shares common surface with six neighbouring cells on six sides. The size of each cell is such that it cannot simultaneously accommodate more than one particle. The center of the cells form a discrete lattice. At any given instant of time, a fraction of the cells (or, equivalently, a fraction of the lattice sites) is occupied by the particles. For such a lattice gas, the set of configurations C is discrete. However, there are more than one ways of labeling the configurations. To illustrate this point, let us consider a lattice gas in one dimension. Suppose, the lattice sites are labelled by the integers j such that j = 1 and j = L correspond to the leftmost and the rightmost sites, respectively, of the finite system. Let the dynamical variable σj denote the instantaneous status of occupation of the j-th site, i.e., whether or not site j is occupied and, if so, what is the discrete speed of the particle. Using σj the configuration of the system can be unambiguously specified by the set {σj }. This convention of labeling the states of the system is used in what is commonly called the site-oriented approach or Euler representation. The alternative particle-oriented approach, which is sometimes called Lagrange representation, is closer to the Newtonian description where the state of the system is specified by listing the positions of the particles in the lattice and the corresponding discrete velocities. All the particle-hopping models of vehicular traffic are essentially lattice gas models in which each particle represents a vehicle. In other words, the size of each cell is chosen to be such that it can accommodate just one vehicle at a time. There are some practical advantages in modeling traffic systems with CA-type rules. It is quite realistic to think in terms of the way each individual motile element responds to its local environment and the series of actions they perform. The lack of detailed knowledge of these behavioral responses is compensated by the rules of CA. Usually, it is much easier to devise a reasonable set of logic-based rules, instead of cooking up some effective force
21
Introduction to Nonequilibrium Systems and Transport Phenomena
for dynamical equations, to capture the behavior of the elements. Moreover, because of the high speed of simulations of CA, a wide range of possibilities can be explored which would be impossible with more traditional methods based on differential equations. The forward motion of the vehicles is captured by imposing a drive, which biases the motion of the vehicle. In some models, the bias is introduced by a driving external field, whereas in others, the particles are self-driven. The difference between these two types of models are explained the following subsections. Field-Driven Particles: To our knowledge, the Katz–Lebowitz–Spohn (KLS) model [715,
716, 1523] is the simplest among the models for systems of interacting field-driven particles. Suppose the variable ni describes the state of occupation of the site i (i = 1, 2, . . . , N ) on a discrete lattice; ni is allowed to take one of the only two values, namely, ni = 1 if the site i is occupied by a particle and ni = 0 if it is empty (or, equivalently, occupied by a “hole”). The Hamiltonian for the system, in the absence of any external driving field, is given by ni nj , (1.36) H = −4J
where the summation on the right-hand side is to be carried out over all the nearestneighbor pairs and J takes into account the corresponding interparticle interactions. The instantaneous state (configuration) of the system at time t is completely described by ({n}; t). For example, in case of a system of a one-dimension system of length L one has ({n}; t) ≡ (n1 , n2 , . . ., nL ; t). Similarly, for the Lx × Ly square lattice ({n}; t) ≡ (n11 , n12 , . . ., nij , . . ., nLx Ly ; t). The average density ρ of the particles is given by s N 1 = lim ni , N ,Ns →∞ Ns N ,Ns →∞ Ns
N
ρ=
lim
(1.37)
i=1
where Ns is the total number of available sites, e.g., Ns = L for a linear chain or Ns = LxLy for a square lattice of size Lx × Ly. Note that, because of the conservation of the particles, the density ρ is conserved by the dynamics. The dynamics of the system is governed by the well-known Kawasaki dynamics: at any nonzero temperature T , a randomly chosen nearest-neighbor particle-hole pair is exchanged with the probability min[1, e−β(H+ E) ] where β = (kB T )−1 (kB being the Boltzmann constant) and H = H({n}new ) − H({n}old ) is the difference in the energy of the new and old configurations while = (−1, 0, +1) for jumps, respectively, along, transverse to, against the direction of the driving field E. For the KLS model with attractive interparticle interactions ( J > 0) on a square lattice, there is not only an ordered state at all T < Tc (E), but also the critical temperature Tc (E)
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A. Schadschneider, D. Chowdhury, and K. Nishinari
increases with E, saturating at a value Tc (E → ∞) 1.4Tc (E = 0), where Tc (E = 0) is the critical temperature of the corresponding Ising model in thermodynamic equilibrium [715, 716, 1250, 1251]. On the other hand, Tc (E) decreases with E when the interparticle interactions are repulsive (i.e., J < 0); the ordering is altogether destroyed by sufficiently large E. However, there is no ordered structure at any nonzero temperature in the one-dimensional KLS model, irrespective of the sign of the interaction J . Self-Driven Particles: The totally asymmetric simple exclusion process (TASEP) is probably the
simplest model for a system of interacting self-driven particles. In this model, the configurations of the system are updated according to the following update rule: a particle moves forward with probability q if and only if the target site is empty. The particles are picked up either in parallel or in a random sequential manner for updating. The dynamics of a finite sample depends on the boundary conditions; most common boundary conditions are the periodic boundary conditions and open boundary conditions typically realized by coupling the system to particle reservoirs. Mathematically, TASEP is a special case of the KLS model. However, almost all the models of traffic can be regarded as extensions of the TASEP. Therefore, we present detailed discussion on the TASEP and briefly treat the KLS model as an extension of the TASEP in the following chapters.
1.8. PHASE TRANSITIONS, CRITICAL DYNAMICS, AND KINETICS OF PHASE ORDERING 1.8.1. Critical Dynamics: Role of Symmetry and Conservation Laws Landau theory for equilibrium phase transitions begins by identifying the most appropriate order parameter(s) in terms of which a phenomenological free energy (more appropriately, a free energy functional in the Landau–Ginzburg theory) is written. The order parameter is related to the symmetry breaking at the phase transition and distinguishes the symmetry-broken states in the ordered phase. It vanishes continuously at a critical point and the corresponding phase transition is also called a second-order transition. In contrast, for a first-order phase transition, the order parameter vanishes discontinuously at the transition point. It is the singularities of the free energy that expose the nature of the phase transition in a thermodynamic system. At a critical temperature, for example, the vanishing of the order parameter ψ is characterized by a power law of the type ψ ∼ (Tc − T )β where the value of the exponent β does not depend on many details of the system except the dimension of space, symmetry of the order parameter ψ, the range of the interactions among the constituents of the system, and the presence (or, absence) of quenched disorder. Similarly, one defines the critical exponents α, γ , δ, etc., which characterize
23
Introduction to Nonequilibrium Systems and Transport Phenomena
the singularities associated with different thermodynamic quantities for the system. All physical systems can be grouped into different universality classes where each class is labelled by a distinct set of numerical values of the exponents. In a spatially inhomogeneous system, the order parameter ψ depends on the spatial location x, and the Landau free energy is replaced by the more general Landau–Ginzburg free energy functional F[{ψ(x)}]. Even for systems which are far from equilibrium, but which evolve toward the thermodynamic equilibrium state, such a Landau–Ginzburg free energy functional can be written down based on symmetry considerations. For such models, the corresponding sets of time-dependent Ginzburg–Landau (TDGL) equations, which must satisfy conservation laws, describe the kinetics of the system. For example, for a system with nonconserved scalar order parameter, the phenomenological TDGL equation reads δF[{ψ(x)}] ∂ψ(x) = − , ∂t δψ(x)
(1.38)
where is a phenomenological parameter, and the symbol δF/δψ denotes the functional derivative of F with respect to ψ(x). Usually, sufficiently close to a critical point, the relaxation time τ for a spontaneous fluctuation in an equilibrium system is related to the corresponding correlation length ξ by the power law τ ∼ ξz,
(1.39)
where z is called the dynamical critical exponent. Since ξ itself diverges at the critical point, τ also diverges, and this phenomenon is called critical slowing down. The TDGLtype approach has successfully provided deep insight into the generic features of critical dynamics, including estimation of dynamic critical exponents [573]. It has been established, over the last half century, that two systems which belong to the same static universality class may have different exponents z if the dynamics of the two systems are qualitatively different (e.g., if one conserves the order parameter while the other does not).
1.8.2. Kinetics of Phase Ordering: Metastable and Unstable Initial States In this subsection, we summarize the key concepts in the studies of the kinetics of ordering in systems that are initially in metastable or unstable equilibrium [475]. Nucleation of droplets (nuclei) of the stable phase and their growth, as well as coalescence, leads to the equilibration of a system from a metastable initial state [712]. For example, in a supercooled vapor, droplets of the liquid nucleate and grow. Similarly, as the crystallites of the solute nucleate from a supersaturated solution and grow spontaneously, the
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A. Schadschneider, D. Chowdhury, and K. Nishinari
system approaches its thermodynamic equilibrium. Homogeneous nucleation occurs by the spontaneous thermal fluctuations in the metastable system. But, if nucleation starts on a “seed” of the stable phase inserted into the metastable system, the process is called heterogeneous nucleation. As another concrete example, consider a binary mixture of two immiscible components. An off-critical quench of this system from the single phase regime to the two-phase regime would create a metastable situation; then the morphology of the evolving pattern in the system is dominated by droplets of the minority phase dispersed in a continuous background of the majority phase. A critical quench of the binary mixture, however, creates an unstable state from which the system evolves in time by the coarsening of a random interconnected structure leading to the phase separation of the two components. This process is referred to as spinodal decomposition or Ostwald ripening. The ordered regions grow with time following some universal laws [155]. For example, in a pure system, the linear size of a typical ordered domain R(t) grows with time t following a power law of the form R(t) ∼ t x where the exponent x is determined by only a few characteristics of the system and not on the other details. The properties of the dynamics that affect x are the presence (or absence) of symmetries of order parameters and the associated conservation laws. Statistical field theories of nucleation and spinodal decomposition were also developed a few decades ago using TDGL or its appropriate extensions.
1.8.3. Phase Transitions in Driven Systems In the standard theoretical treatment of microscopic models of thermodynamic systems, one first calculates a partition function from the microscopic Hamiltonian. The microscopic description of statistical mechanics and the macroscopic description of thermodynamics are linked together by the relation between the partition function and the corresponding free energy. However, such an elaborate standard procedure is still lacking for studying the nonequilibrium phase transitions in driven systems. It has been proved rigorously in equilibrium statistical mechanics that even infinitesimal thermal fluctuations destroy long-range order in a classical one-dimensional system with only short-range interactions. However, one-dimensional open system of interacting driven particles, which are minimal models of traffic systems, are known to exhibit boundary-induced phase transitions even if the interactions between the particles have a short range. Under open boundary conditions, the TASEP exhibits three different dynamical phases of the system in a plane spanned by α and β, which are the rates of entry and exit of the particles, respectively. Yang and Lee [849] developed a general theory of phase transitions in systems that are in thermodynamic equilibrium in terms of the zeroes of the partition functions. An analogous treatment of phase transitions in driven nonequilibrium systems has been
Introduction to Nonequilibrium Systems and Transport Phenomena
developed for a class of models [47, 48, 99, 100, 124, 125, 260, 460, 638, 639]. But, to our knowledge, it is not clear whether the approach can be followed generally for all models of driven nonequilibrium systems. In principle, the general concepts of order parameter, criticality, critical slowing down, universality classes, etc., which were originally developed in equilibrium statistical mechanics, should be applicable also to all nonequilibrium phase transitions, provided the role of equilibrium state is played by a nonequilibrium steady state [557, 567, 568, 1074, 1302].
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CHAPTER TWO
Methods for the Description of Stochastic Models Contents 2.1. Quantum Formalism 2.1.1. Master Equation and Stochastic Hamiltonian 2.1.2. Spectrum and Expectation Values 2.1.3. Discrete Time Dynamics
28 28 31 33
2.2. Mean-Field and Cluster Methods 2.2.1. Mean-Field Approximations
37 37
2.3. Bethe Ansatz
41
2.4. Matrix-Product Ansatz 2.4.1. MPA in Quantum Formalism 2.4.2. MPA for Discrete Time Updates 2.4.3. Dynamical MPA 2.4.4. Relation with Bethe Ansatz
43 45 48 51 52
2.5. Other Methods 2.5.1. Hydrodynamic Limit 2.5.2. Field-Theoretic Methods and Renormalization Groups 2.5.3. Similarity Transformations 2.5.4. Ultradiscrete Method
52 52 53 54 54
2.6. Numerical Methods 2.6.1. Computer Simulation (MC Methods) 2.6.2. Exact Diagonalization 2.6.3. Density-Matrix Renormalization Group 2.6.4. Transfer-Matrix DMRG
57 58 60 61 63
2.7. Appendices 2.7.1. Some Mathematics 2.7.2. MPA and Optimum Ground States for Quantum Spin Chains 2.7.3. Krebs–Sandow Theorem and Extensions
66 66 66 68
In this chapter, we will adapt certain general ideas outlined in Chapter 1 to the needs of describing traffic-like problems. We will also introduce several analytical and numerical methods that are successfully used for the investigation of stochastic systems. Many of them are in fact modifications of well-known approaches introduced originally for Stochastic Transport in Complex Systems DOI: 10.1016/B978-0-444-52853-7.00002-6
Copyright © 2011, Elsevier BV. All rights reserved.
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A. Schadschneider, D. Chowdhury, and K. Nishinari
quantum systems. Therefore, we begin this chapter by introducing the so-called quantum formalism, which emphasizes certain formal analogies of stochastic and quantum systems. It is a starting point of several analytical and numerical methods and systematic approximations.
2.1. QUANTUM FORMALISM The quantum formalism [15, 313, 314, 555, 1273] is a reformulation of the master equation that emphasizes similarities with quantum mechanics. It facilitates the adoption of well-established quantum methods for the investigation of stochastic processes and goes back to early work by Glauber [438]. We will demonstrate the strengths of this formalism explicitly for the asymmetric simple exclusion process (ASEP) and its relatives in the following chapters.
2.1.1. Master Equation and Stochastic Hamiltonian We consider a discrete stochastic system where the state at each site j ( j = 1, . . ., L) is specified by a state variable sj . In the following, typically sj will be discrete, e.g., an occupation number. The time evolution of the probability P(s, t) to find the system in the configuration s = (s1 , . . ., sL ) is determined by the master equation. For continuous time dynamics, it has the form ∂P(s, t) w(˜s → s)P(˜s, t) − w(s → s˜ )P(s, t) = ∂t ˜s
(2.1)
˜s
with transition rates w(˜s → s) from state s˜ to state s. Equation (2.1) can be rewritten in the form of a Schrödinger equation in an imaginary time [15, 555], ∂ |P(t) = −H|P(t), ∂t
(2.2)
with the state vector |P(t) =
P(s, t)|s.
(2.3)
s
The vectors |s = |s1 , . . ., sL corresponding to the configurations s form an orthonormal basis of the configuration space, and we have P(s, t) = s|P(t).
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Methods for the Description of Stochastic Models
The stochastic Hamiltonian H, also known as generator of the stochastic process, Liouville operator, or Markov matrix, is defined through its matrix elements −w(˜s → s) for s= s˜ s|H|˜s = . s =s w(s → s ) for s = ˜s
(2.4)
Thus, all nondiagonal elements of H are nonpositive, and the sum of all elements of each column is zero. Such matrices are usually called stochastic matrices [1409]. Note that these are not identical with random matrices which have elements that are stochastic variables! Many stochastic processes are defined by homogeneous, time-independent transition rates where only transitions between nearest neighbors are allowed. In such a case, the stochastic Hamiltonian can be written in the form [15] H=
L
hj, j+1 ,
(2.5)
j=1
i.e., as a sum of local Hamiltonians hj, j+1 with nearest-neighbor interactions that act nontrivially only on sites j and j + 1. Here, we have assumed periodic boundary conditions. Generalization to other cases is straightforward. Later we will often consider systems coupled to boundary reservoirs. In this case, the stochastic Hamiltonian has the form H = h1 +
L−1
hj, j+1 + hL ,
(2.6)
j=1
where h1 and hL act only on the first and last site, respectively, and describe the coupling to the corresponding particle reservoirs. In the following, we consider a two-state model1 where each site j can be either in state “0,” interpreted as an empty site, or in state “1,” which represents a site occupied by a particle. However, this is no restriction, and a generalization to processes with more than two states per site is straightforward. 1 The terminology “two-state model” is used in two different ways in the literature. It can refer to either the number of
states of a site or the number of states of the particle (if it has an internal degree of freedom). In the latter case, the number of states of a site is 3. Sometimes, one speaks of an n-species model, which then corresponds to n + 1 different states of a site. In general, these two formulations are equivalent, but the natural symmetries in the two classes of models are different.
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A. Schadschneider, D. Chowdhury, and K. Nishinari
Table 2.1 Definition and interpretation of rates allowed in a two-state stochastic process Interpretation
Process
Rate
Diffusion to left
01 → 10
01 10
Diffusion to right
10 → 01
10 01
Pair annihilation
11 → 00
11 00
Pair creation
00 → 11
00 11
Death on left
10 → 00
10 00
Death on right
01 → 00
01 00
Birth on left
00 → 10
00 10
Birth on right
00 → 01
00 01
Fusion on left
11 → 10
11 10
Fusion on right
11 → 01
11 01
Branching to left
01 → 11
01 11
Branching to right
10 → 11
10 11
Branching and fusion are sometimes called coagulation (or coalesττ are detercence) and decoagulation. The diagonal elements ττ ττ mined by the condition σ ,σ σ σ = 0. sj sj+1
In terms of the local transition rates sj sj+1 for a local transition of sites j and j + 1 ) to (s s from (sj sj+1 j j+1 ), the local Hamiltonian in the basis (00, 01, 10, 11) is given by ⎞ 00 + 00 + 00 01 10 11 −00 −00 −00 01 10 11 ⎟ ⎜ 00 01 + 01 + 01 10 11 −01 00 −01 −01 ⎟ ⎜ 10 11 hj, j+1 = ⎜ ⎟. 00 01 10 + 10 + 10 11 ⎠ ⎝ −10 −10 00 − 01 11 10 00 01 10 11 11 11 −11 −11 −11 00 + 01 + 10 ⎛
(2.7) The interpretation of these rates is given in Table 2.1. A particle-hopping model conserves sj sj+1
the number of particles, at least in the bulk.2 Then, only those sj sj+1 with sj + sj+1 = 01 , 10 . Our main example for a two-state particle hopping are nonzero, i.e., 10 sj + sj+1 01 01 = 0, 10 = p. will be the ASEP, which has 10 01 2 Sometimes, an exception from this terminology is made in the context of models of molecular motors, which are based
on the ASEP with additional attachment and detachment processes in the bulk.
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Methods for the Description of Stochastic Models
In the case of coupling to boundary reservoirs, particles can be injected or extracted at sites j = 1 and j = L. The corresponding rates are denoted by α, γ at site 1 and δ, β at site L. The matrix representations of the boundary Hamiltonians are then given by
α −γ , h1 = −α γ
δ −β hL = . −δ β
(2.8)
Often insertion is only allowed at the left end and extraction at the right end. In this case, γ = δ = 0.
2.1.2. Spectrum and Expectation Values The stochastic Hamiltonian is generically nonhermitian. Therefore, it has complex eigenvalues that have nothing to do with energy levels of a classical system. In the following, we will investigate the spectrum in a rather general way and see how the eigenvalues are related to the dynamical properties of the underlying stochastic system. The stationary state of the stochastic process corresponds to the eigenvector |P0 of H with eigenvalue 0, H|P0 = 0,
(2.9)
i.e., it becomes the analog of the ground state of a quantum mechanical system. Note that in contrast to the latter, the “ground state energy” E0 of the stochastic Hamiltonian is known a priori (E0 = 0)! The existence of the eigenvalue E0 = 0 can be proven explicitly by considering the vector 0| =
s|.
(2.10)
s
In the basis of (2.7), one has 0| = (1, . . ., 1). Since the sum of all column elements is zero, 0| is a left eigenvector of H with eigenvalue 0. The corresponding right eigenvector is then the stationary state vector |P0 . The special properties of a stochastic matrix also guarantee that the real parts of all eigenvalues Eα of H are nonnegative [1409]. This can be shown using Gershgorin’s theorem [455] (see Appendix 2.7.1). For ergodic systems,3 |P0 is unique, i.e., independent of the initial conditions. The configuration space does not consist of independent parts. There are sufficient criteria for ergodicity (see, e.g., 3 One common (but restrictive) definition of ergodicity in a finite state space is the existence of a unique stationary
distribution for every initial condition where every configuration occurs with nonzero probability [721].
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A. Schadschneider, D. Chowdhury, and K. Nishinari
[113, 721, 1176, 1203, 1409]), which, e.g., involve the irreducibility of the transition matrix. The formal solution of the master equation (2.2) is given by |P(t) = e−Ht |P(0),
(2.11)
where |P(0) is the initial state of the system at time t = 0. Conservation of probability implies 0| exp(−Ht) = 0|. Using the left eigenvector 0|, we can calculate the average of observable A(s) at time t through A(s)P(s, t) = 0|A|P(t). (2.12) A(t) = s
Note that this is different from quantum mechanics where averages are given by ψ|A|ψ. In the stochastic quantum formalism, the elements of the vector |P(t) are already probabilities, whereas in quantum mechanics, the square of the modulus is interpreted as probability density. Expanding the initial state |P(0) = λ aλ|Pλ in terms of the eigenvectors |Pλ with eigenvalues Eλ of H, the expectation values can be rewritten as aλe−Eλ t 0|A|Pλ . (2.13) A(t) = 0|Ae−Ht |P(0) = λ
This shows that the behavior for large times t is governed by the low-lying excitations, i.e., the eigenvalues with smallest real parts Re(Eλ ). These determine the leading relaxation times τλ through τλ−1 = Re(Eλ )
(2.14)
provided that the corresponding matrix element 0|A|Pλ is nonvanishing. Using the terminology of field theory, the stochastic system is referred to as massless or critical if the spectrum is continuous in the thermodynamic limit. Otherwise it is called massive or noncritical. Two different kinds of behavior for the relaxational properties of an observable can occur,4 i.e., e−t/τλ (2.15) A(t) − A0 ∝ −α , t where A0 = a0 is the stationary value and α is some constant. 4 For a discussion of the subtleties that can occur in the thermodynamic limit, we refer to [1273].
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Methods for the Description of Stochastic Models
Table 2.2 Dictionary for the similarities and differences of stochastic dynamics and quantum mechanics
Local state Global state Model definition Fundamental equation State representation Hamiltonian Lowest eigenstate Lowest eigenvalue Spectrum Eigenvectors (EV) Averages Excitations
Stochastic dynamics
Quantum mechanics
sj = 0, 1, . . ., M Probability P(s1 , . . ., sL ; , t) Transition rates w(˜s → s) Master equation |P(t) = s P(s, t)|s Nonhermitian, stochastic matrix Stationary state 0 Eα complex, Re(Eα ) ≥ 0 Right EV = left EV A(t) = 0|A|P(t) Relaxation τλ −1 ∝ Re(Eλ )
Sjz = −S, . . . , S Wavefunction ψ(S1z , . . ., SLz ; t) Energies Eσ (+ matrix elements) Schrödinger equation Vector |ψ in Hilbert space Hermitian Ground state E0 Eα real Right EV = left EV A = ψ|A|ψ Energy gap
As example for a quantum mechanical system, a quantum spin chain with spin S has been used. S is related to the number of (local) states M of the stochastic system by S = M /2.
Table 2.2 summarizes the main aspects in the formal description of stochastic and quantum systems and provides a dictionary for the relevant quantities In the following sections, we will use these analogies to adopt methods and concepts from quantum mechanics for the investigation of stochastic systems.
2.1.3. Discrete Time Dynamics The above scheme applies to stochastic processes in continuous time. In computer simulation, this is approximated by the random-sequential update: sites (or particles) are updated in random order where at each step 1 is chosen randomly. Note that the same object could be chosen in the next step again,5 so there is no well-defined time step in which all sites or particles are exactly updated once. Therefore, the update has no intrinsic timescale, and rescaling t → αt of time can be absorbed in the definition of the transition rates (w → w/α). This is different for discrete time updates. Here, a well-defined time step t exists in which all sites or particles are updated exactly once. However, this does not specify the update completely. In fact, the updating can be done in different order, and so various discrete time update schemes have to be distinguished. We will later see explicitly that 5 The probability that two particles are updated at the same time in the random-sequential dynamics is of the order O((dt)2 ) and thus be neglected.
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A. Schadschneider, D. Chowdhury, and K. Nishinari
the properties of a model can indeed depend on the specific type of update used, which therefore has to be considered a relevant part of the model definition. Before introducing the most important variants of discrete time updates, we discuss the general aspects of the quantum formalism in discrete time. The master equation takes the form P(s, t + t) =
W (˜s → s)P(˜s, t),
(2.16)
s˜
where W (˜s → s) are transition probabilities between the configurations ˜s and s. Introducing a transfer matrix T with elements Ts,˜s := W (˜s → s)
(2.17)
instead of a stochastic Hamiltonian H, this can be rewritten as |P(t + t) = T |P(t).
(2.18)
Thus, the stationary state |P0 = limt→∞ |P(t) corresponds to the eigenvector |P0 of the transfer matrix T with eigenvalue 1. Similar arguments for the existence and uniqueness of the stationary state as in the case of continuous time apply. Now 0| as defined in (2.10) is a left eigenvector of T with eigenvalue 1. The corresponding right eigenvector P0 | is the stationary state. All other eigenvalues of T have real parts ≥ 1. Note that the stochastic process with discrete time dynamics is defined by probabilities, not rates as in the continuous time case. Therefore, rescaling of time can no longer be absorbed in the transition probabilities. The formal solution of the master equation (2.18) can be obtained by iteration |P(t) = T n |P(0)
for t = n t.
(2.19)
For t → 0, the random-sequential case is recovered. This can be seen by expanding ∂|P(t) (2.20) t + O(( t)2). ∂t After inserting into (2.16) and taking into account that ˜s W (˜s → s) = 1, one obtains the continuous time master equation (2.1) with rates |P(t + t) = |P(t) +
w(s → s˜) =
W (s → s˜ ) . t
(2.21)
35
Methods for the Description of Stochastic Models
The rest of the formalism is identical to the continuous case, e.g., for the calculation of expectation values . If the stochastic process has nearest-neighbor interactions only, the transfer matrix T can be written as a product of local transfer matrices Tj, j+1 . This is the analog of (2.5) in the continuous time case. The Tj, j+1 are completely determined by the (local) rates. However, this still leaves some freedom for T , which has a structure that depends on the precise order in which the individual sites or particles are updated.6 The structure of the transfer matrix is slightly different in these cases. The most common variants of discrete time updates are as follows: •
Ordered-sequential update: In ordered-sequential updates, sites (or particles) are updated sequentially in a fixed order. There are different forms: the most common being forward- and backward-sequential updates. This terminology comes from models with a unidirectional motion of particles where one can distinguish between an update in the direction of motion (forward) from one in the opposite direction (backward). More precisely, for a system with open boundaries, the backward-ordered (site-) sequential update starts with updating the last site L and then proceeds backwards by updating site L − 1, L − 2, and so forth, until the left end of the chain is reached. The corresponding transfer matrix has the structure T← = L · T12 · . . . · TL−1,L · R
(2.22)
with7 . L = Lr ⊗ 1 ⊗ . . .⊗ 1, R = 1 ⊗ . . . ⊗ 1 ⊗ Rr ,
(2.23)
Tj, j+1 = 1 ⊗ 1 . . . ⊗ Tj, j+1 ⊗ 1 . . . ⊗ 1, where 1 is the unit matrix and Tj, j+1 is a local transfer matrix defined by the transitions between sites j and j + 1 analogous to the global transfer matrix (2.17). Lr and Rr describe the effects of the left and right boundary (e.g., the coupling to a reservoir), respectively. In principle, one has to distinguish two different types of ordered-sequential updates, which one could name site-ordered-sequential and particle-ordered-sequential, respectively. In contrast to the site-ordered-sequential update described above, in the particle-ordered-sequential update, the rules are only applied to occupied sites, i.e., to particles. This might have a strong effect, as can be seen most easily for the case 6 “Updating” means the application of the (local) transition rules. 7 For a definition of the tensor product, see Appendix 2.7.1.
36
A. Schadschneider, D. Chowdhury, and K. Nishinari
the deterministic ASEP ( p = 1) with a small number of particles. In the case of sitesequential update, a single particle injected at the left end moves through the lattice in one time step (i.e., one sweep through the lattice). For the particle-sequential update, a time step means updating occupied sites only, and so the particle moves only one site. The two updates might also introduce rather different correlations. In a system with two particles separated initially by d empty sites, in the site-ordered-sequential case, the left particle will move to the right until it reaches the right particle, which then starts to move. On the other hand, in the case of particle-ordered-sequential update, the particles will always stay d or d − 1 sites apart. • Sublattice-parallel update: In this update scheme, odd and even bonds ( j, j + 1) are updated alternatingly (for nearest-neighbor interaction). The dynamics can be divided into two half-steps. In the first step, the bonds between sites (0, 1), (2, 3), (4, 5), etc. are updated, where sites 0 and L + 1 correspond to the reservoirs. In the case of nearest-neighbor interactions, all these local updates are independent of one another. In the second step, the remaining bonds (1, 2), (3, 4), etc. are updated. Assuming L to be even, the transfer matrix can be written as a product of local terms Tsub = Teven · Todd
(2.24)
with Teven = Lr ⊗ T23 ⊗ T45 . . . ⊗ TL−2,L−1 ⊗ Rr , Todd = T12 ⊗ T34 . . . ⊗ TL−3,L−2 ⊗ TL−1,L .
(2.25)
• Parallel or synchronous update: In this update, all sites (or particles) are updated at the same time. Because of its natural timescale that corresponds to the time step, the parallel update is the most important one for applications. In contrast to the orderedsequential update, where one has to choose a starting point for the update sequence, all sites are treated on equal footing. The parallel update has a strong similarity with the forward-ordered (particle-) sequential update. The main difference for periodic boundary conditions is in the treatment of last particle which, in the sequential case, sees a particle in front which has already been updated, whereas in the parallel case, it does not. However, the parallel update leads to problems if the dynamics is not given by a strictly one-dimensional (1D) directed motion of particles. This can lead to ambiguities where two or more particles try to move to the same cell in one time step. Such conflicts play an important role, e.g., in models of pedestrian dynamics. This will be discussed in some detail in Chapter 11.
37
Methods for the Description of Stochastic Models
Note that all the discrete updates described above do not contain any randomness through the ordering of the particle updates, in contrast to the random-sequential case. Therefore, in these cases, the models usually have a deterministic limit if the transition probabilities are chosen appropriately. On the other hand, it is not possible to rescale the transition probabilities that have to satisfy a normalization condition.
2.2. MEAN-FIELD AND CLUSTER METHODS Various analytical approaches exist that can be applied systematically to any kind of stochastic process. Usually, these methods provide only approximate results, but they are “general” in the sense that their applicability does not require any restrictions of the model parameters. In contrast, exact methods like the Bethe Ansatz usually work only for special parameter values.
2.2.1. Mean-Field Approximations 2.2.1.1. Site- and Particle-Oriented Mean Field Mean-field theories (MFTs) neglect all correlations between state variables. Formally, the mean-field approximation (MFA) corresponds to factorizing the probability P(s) into single-site contributions P1 (sj ): P(s1, . . ., sL ) ≈ P1(s1 )P1 (s2 ) . . .P1 (sL )
(2.26)
Inserting this Ansatz into the master equation8 simplifies the problem considerably and leads to an equation that only involves the function P1 . For translationally invariant systems, P1 is independent of the site j. It amounts in approximately expressing the transition rates w(s → ˜s) through the single-site contributions P1 (sj ). Alternatively one could write down the exact master equation for P1 , which would require knowledge of P2 and P3 in the case of nearest-neighbor interactions. Then, the higher Pn are expressed as products of P1 similar to (2.26). Note that often a configuration can be characterized by different variables. The main example in this book is particle-hopping models. Instead of occupation numbers nj = 0, 1 or particle positions xj , any (translational-invariant) state can equivalently be characterized by the interparticle distances (gaps) dj , also called headway in traffic engineering. The MFTs for nj and dj will usually lead to different results. In order to distinguish these different approximations, one sometimes speaks of site-oriented and particle-oriented mean-field 8 In most applications, MFA is used for the steady-state properties. But in principle also, the full dynamics could be studied
with time-dependent P1 .
38
A. Schadschneider, D. Chowdhury, and K. Nishinari
approach. Since the latter is used quite frequently in the context of traffic modelling, the terminology car-oriented MFT (COMF) is more standard and will be used in the following. There is a large class of models for which MFT even gives the exact result. These models are said to have a product-measure or factorized steady state. It should be stressed that such systems usually are not completely trivial. In Chapter 3, we will discuss the important example of the zero-range process for which MFT is exact. Nevertheless its physics is far from boring. 2.2.1.2. Cluster Approximation MFT can be systematically extended to take into account short-range correlations. This leads to the cluster approximation [93, 1258], which is also known under the name local structure theory [131, 479], n-step Markovian approximation [244], and goes back to the path probability method [748]. The n-cluster approximation treats a cluster of n neighboring sites exactly. The 1-cluster approximation is identical to the (site-oriented) mean-field approach [see (2.26)]. The 2-cluster approximation corresponds to the factorization P(s1, . . ., sL ) ∝ P2 (s1 , s2 )P2 (s2 , s3) . . .P2 (sL−1 , sL ).
(2.27)
For periodic boundary conditions, an additional factor P2 (sL , s1 ) appears. For larger clusters, one has the freedom to choose different overlaps of neighboring cluster [93]. For instance, there are two different 3-cluster approximations P(s1, . . ., sL ) ∝ P3 (s1 , s2 , s3)P3 (s2 , s3 , s4 ) . . .
(2.28)
called (3, 2)-cluster approximation, and P(s1, . . ., sL ) ∝ P3 (s1 , s2 , s3)P3 (s3 , s4 , s5 ) . . .
(2.29)
called (3, 1)-cluster approximation. In general, there are n − 1 different n-cluster approximations (n, m), where m is the overlap between neighboring clusters. For fixed size n, the quality of the approximation typically increases with the overlap m [93]. Figure 2.1 shows a graphical representation of the cluster decomposition. In order to ensure a proper normalization, conditional probabilities have to be used [479]. As an example, we give the (2, 1)-approximation for L = 4, P(s1 , s2 , s3, s4 ) = P(s1 |s2 )P(s2, s3 )P(s3|s4 ),
(2.30)
39
Methods for the Description of Stochastic Models
sj − 2
sj − 1
sj
sj + 1
sj + 2
(a)
sj − 2
sj − 1
sj
sj + 1
sj + 2
sj + 3
(b)
sj − 2
sj − 1
sj
sj + 1
sj + 2
sj + 3
sj + 4
(c)
Figure 2.1 Graphical representation of the n-cluster approximation for (a) n = 1 (i.e., mean-field theory), (b) n = 2, and (c) n = 3 (more specifically the (3, 2)-cluster approximation). Each circle corresponds to one factor in the factorization.
where the conditional probabilities P(s1, s2 ) P(s1|s2 ) = , τ P(τ , s2 )
P(s1, s2 ) P(s1 |s2 ) = τ P(s1, τ )
(2.31)
take care of the overlap. The 2-cluster approximation represented geometrically in Fig. 2.1 can be expressed mathematically as [1258] P(sj−2 , sj−1 , sj , sj+1 , sj+2 , sj+3) = P2 (sj−2 |sj−1 )P2 (sj−1 |sj )P2 (sj , sj+1 ) · P2(sj+1 |sj+2 )P2 (sj+2 |sj+3 )
(2.32)
where P2 (sj−1 |sj ) =
P2 (sj−1 , sj ) . sj−1 P2 (sj−1 , sj )
(2.33)
Similarly, the (3,2)-cluster approximation in Fig. 2.1 corresponds to the factorization P(sj−2, sj−1 , sj , sj+1 , sj+2 , sj+3 , sj+4 ) = P3 (sj−2 |sj−1 , sj )P3 (sj−1 |sj , sj+1) · P3(sj , sj+1 , sj+2 )P3 (sj+1 , sj+2 |sj+3 )P3 (sj+2 , sj+3 |sj+4 ).
(2.34)
40
A. Schadschneider, D. Chowdhury, and K. Nishinari
Analogous factorizations hold for an arbitrary number of sites on the eft-hand side of (2.32) and (2.34). The probabilities corresponding to clusters of different sizes are not independent, but related through the Kolmogorov consistency conditions: Pn−1 (s1 , . . ., sn−1 ) =
Pn (τ , s1 , . . ., sn−1 )
τ
=
Pn (s1 , . . ., sn−1 , τ ),
(2.35)
τ
e.g., τ P2 (s1 , τ ) = P1 (s1 ). These relations give strong conditions especially for the 2-cluster approximation of particle-hopping models. There it also provides a relation with physical parameters, e.g., the density ρ in translational invariant systems through P1 (1) = ρ and P1 (0) = 1 − ρ. 2.2.1.3. Interparticle Distribution Functions The method of interparticle distribution functions (IPDF) [92, 232, 312, 1307], also known as method of empty intervals, is closely related to the particle-oriented mean-field approach. Its basic quantity is the probability En (t) that a randomly chosen segment of n consecutive sites does not contain any particles. Once this is known for all n, one can derive various quantities of interest from it, e.g., the average density since 1 − E1(t) is the probability that a site is occupied. It is related to the particle-oriented MFT because En − En+1 is the probability that a site is occupied after n consecutive empty sites. Then, the probability pn that the nearest neighbor is n sites away (i.e., that there a exactly n − 1 empty sites in between) is given by [92] ρpn = En−1 − 2En + En+1 (n > 1)
and p1 = 1 − 2E1 + E2.
(2.36)
Therefore, the probabilities En (t) are usually sufficient to write down closed kinetic equations that describe the dynamic rules of the stochastic process. In some cases, this is possible even when the number of particles is not conserved by the dynamics. In order to simplify the analysis, especially in cases where En (t) is not translationally invariant, a spatial continuum limit is performed. Assuming that site j is located at position xj = j x, a continuous spatial coordinate x is introduced by performing the limit x → 0. The probabilities En (x, t) are then replaced by E(x, t) so that the density is given by
∂E(x, t)
. (2.37) ρ(t) = − ∂x x=0
41
Methods for the Description of Stochastic Models
In this limit, pn becomes the IPDF c(t)p(x, t) =
∂ 2 E(x, t) . ∂x2
(2.38)
The empty-interval method can be applied to a large class of reaction-diffusion models where additional interactions or reactions can be added [7, 8, 556, 746, 917, 940, 1103]. 2.2.1.4. Hierarchical Equations of Motion A common feature of interacting many-particle systems is the fact that the time evolution of an observable A is generically determined by its correlation functions. In other words, in the equation of motion for A(t), higher correlation functions like AA(t) occur. The same is true for general n-point functions of the variable which typically require the knowledge of n + 1-, n + 2-,… functions. Therefore, in general, one has to deal with an infinite hierarchy of equations of motion if one wants to determine the expectation values exactly. This is indeed one main source of difficulty when dealing with systems of interacting particles. One option to simplify this problem is to truncate the hierarchy at some level. This involves some kind of approximation and leads, e.g., to the cluster approximation (see Section 2.2). However, one can also try to look systematically for models where this truncation occurs automatically, typically only for a special choice of parameters. For these autonomous models, the equations for the correlation functions decouple at some level so that in principle they can be solved exactly. This program has been carried out for a large class of reaction-diffusion models. Naturally, the chances of finding such a subspace of decoupled correlation functions become larger if the process is rather general and involves a large number of parameters. For the general 12-parameter process (2.7), Schütz found a 10-parameter family of autonomous models [1265]. Further results and various generalizations can be found, e.g., in [5, 711, 941, 942, 1285]. A systematic way to identify observables (including empty-intervals) for which closed systems of equations can be obtained has been proposed by Peschel et al. [1103] using the quantum formalism.
2.3. BETHE ANSATZ The Bethe Ansatz is probably the most successful method for the exact solution of interacting many particle systems [58]. It has been developed by Hans Bethe [110] in his celebrated solution of the (1D) isotropic Heisenberg model of interacting quantum spins. The method is closely related to the concept of integrability, i.e., the existence of an infinite number of (mutually commuting) conservation laws, and the occurrence of
42
A. Schadschneider, D. Chowdhury, and K. Nishinari
nondiffractive scattering [1326]. With hindsight to the quantum formalism described in detail in Section 2.1, it is natural to try application of the method also to stochastic systems. As the name already indicates, a special form of the wavefunction is assumed as an “Ansatz.” Typically, its structure is ⎡ ⎤ N A(P) exp⎣i kP(j)xj ⎦ (2.39) (x1 , . . ., xN ) = P
j=1
if particles are located at x1 < x2 < . . . < xN . The sum is over all permutations P of the numbers (1, . . . , N ), and the kj are (quasi-)momenta that have still to be determined. “Particles” is here to be understood in a general sense, e.g., it could also refer to downspins in a sea of up-spins in the case of the Heisenberg model. The form is motivated by the fact that first regions in configuration space are considered where particles do not interact. For N particles, there are N ! such regions. The next step is to try whether the wavefunctions can be matched at the interfaces between these regions. The physics behind this Ansatz is given by restrictions on multiparticle scattering processes. The Bethe Ansatz requires that all scattering processes are equivalent to a sequence of two-particle scattering processes. Thus, amplitudes corresponding to two “neighboring” permutations P and P˜ (which are identical except for P( j) = P ( j+1) =: l and P( j + 1) = P ( j) =: l ) describe a scattering process of two (quasi-)momenta k = kl and k = kl and are related through9 A(P ) = −e−iθ(k,k ) , A(P)
(2.40)
where θ (k, k ) is the two-body phase shift. Finally, (periodic) boundary conditions impose additional constraints on the allowed quasimomenta kj since the total phase factor when a particle passes once around the chain of length L (and thus interacting with all other particles, collecting a phase factor − l= j θ (kj , kl )) must be unity: eiLkj =
−eiθ(kj ,kl ) .
(2.41)
l= j
This set of coupled nonlinear equations is called Bethe Ansatz equations. Its solutions {kj } allow to determine energy and momentum, which are usually simple functions of the kj . 9 Note that different conventions for signs are used, e.g., depending on the statistics of the particles.
Methods for the Description of Stochastic Models
A considerable simplification of the Bethe Ansatz equations occurs in the thermodynamic limit where the macroscopic number of coupled equations can be transformed into a single (linear) integral equation for the density ρ(k) of the quasimomenta. The Ansatz as described above applies to systems of particles without any internal degree of freedom or quantum spins 1/2. In other cases, a generalized approach known as nested Bethe Ansatz has to be applied. Basically, it means that another Bethe Ansatz is required to determine the amplitudes A(P). The nested Bethe Ansatz has to be used for models of itinerant electrons like the Hubbard model [353]. The Leningrad school has developed this approach further into the so-called quantuminverse scattering method (QISM) or algebraic Bethe Ansatz [787]. It allows a better understanding of the relation with two-dimensional (2D) classical models and the concept of integrability. A central role is played here by the transfer matrix of the underlying classical system. In contrast to the so-called coordinate Bethe Ansatz (2.39), the QISM is an algebraic approach that constructs a spectrum-generating operator algebra. The Bethe Ansatz equations then appear in the form of consistency conditions. As we have seen, the applicability of the Bethe Ansatz imposes serious restrictions on the form of the interactions, and therefore, only few models can be treated by this method. Amazingly two of the most important models of interacting many-particle quantum systems, the Heisenberg and the Hubbard model, belong to this class! However, the restrictions on the allowed scattering processes imply that nontrivial results can only be expected for (quasi-)1D quantum systems (or their 2D classical counterparts). For the Bethe Ansatz solvable models, usually energies in the thermodynamic limit can be calculated. This gives valuable information about the excitations, e.g., energy gaps. The explicit form of corresponding eigenvectors is usually impossible to determine in a useful form. Therefore, it is also extremely difficult to calculate correlation functions, and even the norm of the Bethe-eigenvectors can only be determined explicitly in a few special cases. However, for critical systems, characterized by a vanishing excitation gap, one can use conformal invariance [177] which gives a direct relation between the finite-size corrections to the energies and the asymptotics of corresponding correlation functions. The Bethe Ansatz as outlined above has also been successfully applied to various stochastic particle systems (see, e.g., the review [1273]). In the following chapters, we will present several explicit examples.
2.4. MATRIX-PRODUCT ANSATZ A very powerful method for the determination of stationary solutions of the master equation is the so-called matrix-product Ansatz (MPA). A rather extensive review can be found in [123]. For a system with open boundaries, the weights P(s) in the stationary
43
44
A. Schadschneider, D. Chowdhury, and K. Nishinari
state can be written in the form L 1 sj D + (1 − sj )E |V . P(s1, . . ., sL ) = W | ZL j=1
(2.42)
For periodic boundary conditions, the MPA takes the translational invariant form ⎛ P(s1 , . . ., sL ) =
1 ⎝ Tr ZL
L
⎞
sj D + (1 − sj )E ⎠ .
(2.43)
j=1
For simplicity, we have assumed a two-state system where, e.g., sj = 0 corresponds to an empty cell j and sj = 1 to an occupied cell so that sj can be identified with the occupation number of site j. ZL is a normalization constant that can be calculated as ZL = W |C L |V ,
with C = D + E.
(2.44)
In (2.42), (2.43), E and D are matrices, and W | and |V are vectors characterizing the boundary conditions. The explicit form of these quantities has to be determined from the condition that (2.42) or (2.43) solves the master equation (2.1). This leads in general to algebraic relations between the matrices E and D and the boundary vectors W | and |V [758, 800]. Once these have been determined, one has a simple recipe for determining P(s1, . . ., sL ) for any configuration in the stationary state. First, translate the configuration (s1 , . . ., sL ) into a product of matrices by identifying each empty cell (sj = 0) with a factor E and each occupied cell (sj = 1) with D. In that way, the configuration 011001 . . . corresponds to the product EDDEED . . . = ED 2 E 2D . . .. The weight P of the configuration is then just the matrix element with the vectors W | and |V . So far we are just describing a specific configuration, and the matrices are not yet determined. This is achieved by involving the dynamics of the stochastic process. In the case of the ASEP (see Section 4.2.3), it is rather easy to determine these restrictions by inserting the MPA into the master equation. Another approach that is often used starts from the exact digitalization of small systems and tries to “guess” the relations between the matrices. Below we describe a systematic method based on the quantum formalism. The price to pay is the introduction of additional auxiliary matrices that do not explicitly appear in the weights (2.42), but determine the algebraic relations between D and E. The MPA can be extended in a straightforward way to multispecies models or particles with internal degrees of freedom. In these cases, one defines for each species or
45
Methods for the Description of Stochastic Models
internal state α a separate matrix Dα instead of just one single D. The MPA is then analogous to (2.42) with each factor sj D + (1 − sj )E replaced by (1 − sj )E +
α
(α)
sj D α .
(2.45)
(α)
Here, sj is 1, if site j is occupied by a particle of species α and 0 otherwise, and (α) sj = α sj . Expectation values and correlation functions can also easily be expressed in terms of the MPA. For the one-point function sj , we obtain sj =
sj P(s)
(2.46)
s
=
W |C j−1 DC L−j |V W |C L |V
(2.47)
since sj2 = 1 and sj (1 − sj ) = 0 after inserting (2.42). Similarly, two-point functions are calculated from sj sl =
W |C j−1 DC l−j−1 DC L−j |V W |C L |V
(2.48)
with obvious generalization to higher correlation functions. Expectation values like the current can usually be expressed by multipoint correlations and thus be calculated within the framework of the MPA.
2.4.1. MPA in Quantum Formalism The MPA can be written in a more formal and compact way in the quantum formalism. This also emphasizes its relation with the MPA as developed for quantum spin systems [384, 761–763] and the optimum ground state concept (see Appendix 2.7.2). In the quantum formalism, we have introduced a state vector |P(t) [see equation (2.3)] whose components are the steady-state probabilities P(s1, . . ., sL ) for all configurations. This allows to rewrite (2.43) in more compact form. First, we introduce the vector E A= D
(2.49)
46
A. Schadschneider, D. Chowdhury, and K. Nishinari
with the matrices E and D as elements. Then, the elements of the L-fold tensor product (see Appendix 2.7.1) ⎛ ⎞ E . . .EE ⎜ E . . .ED ⎟ ⎜ ⎟ . A ⊗ ...⊗ A = ⎜ (2.50) ⎟ . ⎝ ⎠ . D . . .DD are in one-to-one correspondence with the configurations s = (0, . . . , 0, 0), (0, . . . , 0, 1), . . ., (1, . . . , 1, 1) of the system. Thus, (2.43) becomes |P0 =
1 Tr(A ⊗ A ⊗ .. .⊗ A), ZL
(2.51)
1 W |A ⊗ . . .⊗ A|V . ZL
(2.52)
and (2.42) can be written as |P0 =
Now that we have derived the form of the matrix-product states in the quantum formalism, we can apply the stochastic Hamiltonian to it to see under which conditions |P0 becomes the stationary state. For the periodic case, it turns out that a single local condition (only involving hj, j+1 ) is sufficient ¯ hj, j+1 (A ⊗ A) = A¯ ⊗ A − A ⊗ A,
(2.53)
E¯ ¯ where the auxiliary vector A = ¯ can be different from A. The divergence-like D term [570, 1270, 1315] on the right-hand side can be viewed as a generalization of the optimum ground state condition (2.98). Equation (2.53) is sometimes called cancellation mechanism. In fact, it is easy to see that (for periodic boundary conditions) the divergence-like terms cancel after summing over j. Applying hj, j+1 to A ⊗ . . . ⊗ A (with L factors) creates two terms where A¯ is in position j and j + 1, respectively. Since these terms have different signs, they are canceled upon summation over j by the terms generated by hj−1, j and hj+1, j+2 . Hence, the stationary state of the stochastic process described by (2.4) is of the form (2.51). For open boundary conditions, the cancellation mechanism has to be supplemented by boundary terms (see below). In this case, it can be shown that in a certain sense, the
47
Methods for the Description of Stochastic Models
stationary state of 1D stochastic systems is generically of matrix-product form [758, 800]. However, for periodic boundary conditions, certain problems in its general applicability have been pointed out in [797]. ¯ D, ¯ the problem has become Due to the introduction of the auxiliary matrices E, more complex. Usually, the auxiliary matrices are not determined uniquely, and there is a certain freedom in their choice. In fact, in some cases, it has turned out that they can be chosen proportional to the unit matrix 1 (see, e.g., [14]). For hermitian Hamiltonians, one always has A¯ = A, and (2.53) reduces to (2.107). Therefore, one can regard (2.53) as the generalization of the optimum ground state concept to nonhermitian Hamiltonians. It is possible to generalize these ideas to treat open boundary conditions with singlesite boundary interactions. These are described by the local Hamiltonians h1 and hL acting only on the first and last site, respectively. The stochastic Hamiltonian in the case of nearest-neighbor bulk interactions is then of the form H = h1 +
L−1
hj, j+1 + hL .
(2.54)
j=1
Executing the sum (2.5) in (2.53) leads to a cancellation of all terms in the bulk of the chain in the same way as for the periodic case. However, now terms with A¯ as first and last factor survive. These remaining boundary terms have to be annihilated by h1 and hL , which can be achieved through ¯ W |h1 A = −W |A,
¯ . hL A|V = A|V
(2.55)
Thus, in the case of open boundary conditions, the cancellation mechanism (2.53) has to be supplemented by the conditions (2.55) on the boundary terms to guarantee that |P0 is the stationary state. Inserting the explicit matrix representations (2.7) of the local Hamiltonians hj, j+1 (a 4 × 4-matrix), h1 and hL (2 × 2-matrices) into (2.53), (2.55), using, e.g., A ⊗ A = ¯ and D, ¯ (E 2, ED, DE, D 2)T , one obtains a system of quadratic equations in E, D, E, which is called the algebra of Ansatz (2.53). Due to the occurrence of the auxiliary matri¯ this algebra can look at first sight different from the one derived by the ces E¯ and D, method in the previous section. However, the latter usually corresponds to a special ¯ which often can be taken to be proportional to the unit matrix 1. choice of E¯ and D, This leads to a serious simplification of the problem since basically now it has been reduced to a local problem no longer involving the system size L (explicitly). If one is able to find explicit representations for the algebra, one can determine in principle all expectation values in the stationary state exactly. The dimension of the matrices is not determined by the Ansatz itself.
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Sometimes, results can also be obtained directly from the algebra, e.g., by a clever iteration scheme, without the use of a representation. An example, which we will discuss in more detail later, is the TASEP (see Section 4.2.3). Up to this point, the MPA appears to be just an Ansatz for the stationary state. However, it turns out that it is in some sense the generic form for the stationary state of 1D stochastic processes [758, 800, 1163]. Krebs and Sandow have shown [800] that the MPA (2.52) with the cancellation mechanism (2.53) is an equivalent reformulation of the master equation in the sense that for a given 1D stochastic process with random-sequential update, nearest-neighbor interaction, and fixed system size L, the stationary state can be written in the form (2.52), where the matrices (and vectors) obey essentially the cancellation mechanism (2.53). Details of the argument can be found in Appendix 2.7.3. Therefore, it is of general interest to study the quadratic algebras generated by (2.53) and to try to find explicit representations [14, 50, 354, 908]. Generically, the matrices are expected to be infinite dimensional.10 Even though finding general representations of the resulting algebras is difficult in general, it is possible to search systematically for finite-dimensional representations in subspaces of the parameter space of the model, e.g., by inspection. In [564], a necessary condition for the existence of m-dimensional representations in an n-state model has been derived together with an explicit construction method for the case n ≤ m based on the exact stationary states of small systems.11 The theorem of Krebs and Sandow assumes open boundary conditions. Its extension to the periodic case is not possible. In fact, several problems might arise in the periodic case [123, 797]. For multispecies models that rotational invariance of the trace operation can lead to global constraints on particle numbers and the MPA becomes inconsistent if these constraints are not satisfied. Furthermore, zero-energy eigenstates can exist, which are not of matrix-product form [797]. Since the mathematical structure of the stationary state is known through the Krebs– Sandow theorem, it is sometimes possible to derive rather general results. As an example in [1163], relations between expectation values for ordered-sequential and sublatticeparallel dynamics have been derived based on the MPA structure of the stationary states.
2.4.2. MPA for Discrete Time Updates As we have seen in Section 2.1, the structure of the master equation is slightly different in the case of discrete time. Instead of the quantum Hamiltonian H, the time evolution is now controlled by a transfer matrix T , which has a structure that depends on the 10 Or they depend on the system size. In contrast, the matrix algebra derived for the ASEP (see Section 4.79) has
representations that are valid for arbitrary system sizes. 11 Note that these results can also be used to exclude the existence of finite-dimensional representations by studying small
systems exactly.
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Methods for the Description of Stochastic Models
precise nature of the update. Here, we extend the MPA to models of type (2.6) with (site-)ordered-sequential and sublattice-parallel dynamics [356, 566, 1160, 1162, 1264]. In the following, we consider a general stochastic process of the type defined by the rates in Table 2.1 and with open boundaries. Later we will be more specific and apply the results to the TASEP in Chapter 4. Due to the nearest-neighbor interactions, the transfer matrix T is a product of local terms, as in (2.22) or (2.25). 2.4.2.1. Ordered-Sequential Updates The MPA for the stationary state with backward ordered-sequential update has the same form |P0 ← =
1 W |A ⊗ . . . ⊗ A|V ZL
(2.56)
as in the random-sequential case [see (2.52)]. However, the mechanism for stationarity is different [1160, 1162]: Tj, j+1 A ⊗ A¯ = A¯ ⊗ A, W |LA¯ = W |A,
(2.57)
¯ , RA|V = A|V where Tj, j+1 is the local transfer matrix [see (2.22) and (2.24)]. It is easy to see that these algebraic conditions on the matrices and vectors indeed guarantee the stationarity of |P0 ← : Applying the transfer matrix to |P0 first creates a “defect” A¯ at the right end j = L which is the successively transported to the left end j = 1 through the action of Tj, j+1 and finally becomes annihilated at the left end j = 1. The ordered-sequential update in the opposite direction (left to right) can be treated in the same way. It is described by the transfer matrix T→ which has the opposite order of factors than T← [see Eqn. (2.22)] T→ = R · TL−1,L · . . . · T12 · L
(2.58)
The stationary state is given by |P0 → = with the same mechanism (2.57).
1 ¯ W |A¯ ⊗ . . . ⊗ A|V ZL
(2.59)
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A. Schadschneider, D. Chowdhury, and K. Nishinari
2.4.2.2. Sublattice-Parallel Update For the sublattice-parallel update, the MPA for the stationary state is of the form [566, 576, 1160] |P0 sub =
1 W |A¯ ⊗ A ⊗ A¯ ⊗ A . . . ⊗ A¯ ⊗ A|V . ZL
(2.60)
One can now use exactly the same mechanism (2.57) as in the ordered-sequential case and thus obtains the same algebra. The fact that the ordered-sequential and the sublattice-parallel update lead to the same algebra implies the existence of an intimate relationship between the averages of observables. Although the stationary states themselves are different, they are connected via transformations, and it can be shown that the density profile of the ordered-sequential update from the left (right) to the right (left) corresponds to the density of the even (odd) sites produced by the sublattice-parallel update [1163]. This result holds for arbitrary stochastic models with nearest-neighbor interactions. As an example, the densities ρ(j) = ni are related by ρ→ (j) ρsub (j) = ρ← (j)
for j odd, for j even.
(2.61)
Analogous results hold for higher correlation functions. 2.4.2.3. Parallel Update As mention earlier, a fully general treatment of the parallel case has not been achieved so far. Only very few models with synchronous update have been solved by MPA, the most notable example being the ASEP. For the case of periodic boundary conditions, an operator-based solution was given by Evans [356]. It is also possible to rewrite the 2-cluster approximation, which becomes exact for the ASEP, in matrix-product form (see Section 4.1.3). For the ASEP with open boundaries, two different MPA solutions have been found, a site-oriented Ansatz [375] and a bond-oriented Ansatz [272] which can be shown to be equivalent [1481]. This will be discussed in detail in Section 4.4.3. A rather general strategy to derive a MPA for systems with parallel dynamics uses the close relation of the parallel and the forward-(site)-ordered-sequential update [1160]. The latter can always be written in local form as in (2.58) and thus allows for a MPA as described above. The main difference between the two updates occurs for particles moving forward. In the sequential case, these particles are updated again by the next local term Tj+1, j+2 , in contrast to the parallel case. This can be fixed by introducing an intermediate state, which indicates that the corresponding particle has already been
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Methods for the Description of Stochastic Models
updated [1160]. Effectively this means that the parallel update can be mapped onto an equivalent ordered-sequential update with additional states. These states are “virtual” in the sense that they do not appear explicitly in the steady state. The price to pay is an algebra, which becomes more complex due to the additional states. This program has been carried out for the TASEP in [1160] where it was necessary to introduce just one ¯ set of auxiliary matrices F, F. In another approach, one can also generalize COMF (see Section 2.2.1) to MPAform [1229]. In this case, the matrices do not represent occupation numbers, but interparticle-distances, i.e., the number dj of empty sites in front of a particle j. This implies that one has to introduce an infinite number of matrices D(d), one for each interparticle distance d = 0, 1, 2, . . .. It turns out that the resulting infinite algebra can be simplified to a finite one by an appropriate Ansatz.
2.4.3. Dynamical MPA The MPA has also been extended to treat the full dynamics, not only the stationary state [17, 1222, 1315, 1316]. A general outline of the approach can be found in [1270, 1312]. It starts by allowing time-dependent matrices D(t) and E(t) in the Ansatz (2.52). Assuming that the vectors W | and |V are time-independent, one inserts the Ansatz into the full master equation (2.2). As in the stationary case, this leads to a matrix algebra, ˙ ˙ which now also contains the time-derivatives D(t) and E(t) of D(t) and E(t). In [1271], the dynamical MPA (DMPA) for the most general two-state process with boundary reservoirs at both ends has been studied. It can be described by a local stochastic Hamiltonian of the form (2.7). The algebra becomes 1d 2 E − [S, E] = A(1) , 2 dt 1d 2 D − [T , D] = A(2) , 2 dt
1d (ED) − SD + ET = B (1) , 2 dt 1d (DE) − TE + DS = B (2). 2 dt
(2.62)
The inhomogeneities A( j) and B ( j) are quadratic forms of the matrices E and D with coefficients that are linear combinations of the matrix elements of the local Hamiltonian (2.7). S(t) and T (t) are auxiliary matrices that have to satisfy certain conditions involving the boundary vectors W | and |V (see [1271]). In addition to the bulk algebra, the boundary terms lead to four relations where linear combinations of the matrices E and D and their time derivatives act on the vectors W | and |V . This algebra can be simplified to a smaller algebra with less generators and relations under certain conditions [1116, 1271], e.g., by assuming that C is also time independent and has a representation which is invertible. However, a general solution of these equations has so far not been obtained, but several special cases have been investigated in more
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detail, e.g., the symmetric exclusion process with general open and periodic boundary conditions [1315, 1316] or the TASEP with periodic boundaries [1222], as well as other special cases [1116]. The key step of this analysis is the introduction of the auxiliary the equations to linquantities Dn = C n−1 DC −n , etc., which allow to reduce some ofink ear difference equations. The Fourier transforms Dk = k Dn (t)e turn out to have a simple time-dependence Dk (t) = e−(k)t Dk (0), and the algebra is reduced to the form [1222, 1315] Dk1 Dk2 = S(k2 , k1 )Dk2 Dk1 ,
(2.63)
which is closely related to the so-called Zamolodchikov algebra [1512, 1513]. Consistency of the algebra then requires that the Bethe Ansatz equations S(kl , kj ) (2.64) eiLkj = l= j
are satisfied. The formalism has also been applied to spin relaxation by interpreting the Glauber dynamics as reaction-diffusion system [1271]. Extensions to the calculation of timedependent correlations functions [1215] and two-species system [1116] are also possible.
2.4.4. Relation with Bethe Ansatz The DMPA has allowed deeper insights into the relation between stochastic many-body dynamics and quantum systems, especially quantum spin systems [1223]. As demonstrated in [1116, 1123], it is possible to derive a sufficient criterion for integrability directly from the quantum Hamiltonian. It is not necessary to refer explicitly to an infinite set of conservation laws or construct transfer matrices that satisfy the Yang-Baxter equation. Later these ideas have been extended by Alcaraz and Lazo [17, 18]. They reformulated the Bethe Ansatz as a MPA and used it to obtain a unified description of a large class of integrable quantum systems, including the XXZ chain, the t − J and Hubbard models, and the Sutherland model. In [452], the inverse problem was investigated. It was shown that the algebraic Bethe Ansatz for the ASEP can be used to derive representations for the MPA algebra.
2.5. OTHER METHODS 2.5.1. Hydrodynamic Limit One guiding idea often used in nonequilibrium physics is that of local equilibrium. Here, locally on a macroscopic scale, one can define macroscopic variables (e.g., density,
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Methods for the Description of Stochastic Models
temperature), which vary smoothly. The system reaches local equilibrium on a timescale, which is much shorter than the timescale of the macroscopic evolution (described by hydrodynamic equations). On large scales, the noise in stochastic interacting particle systems scales out, and the system can effectively be described by deterministic evolution equations. This is similar to the derivation of hydrodynamic equations from the Newtonian dynamics of molecules in a gas or a fluid. The procedure involved in the derivation of macroscopic evolution equations for systems with conserved quantities is therefore referred to as the hydrodynamic limit. It can be made mathematically rigorous for several models; see, e.g., the books by Kipnis and Landim [751] and Spohn [1305]. In the cases we are interested in the generic procedure for the derivation of the hydrodynamic limit is the Eulerian scaling a → 0, t → t/a,
at fixed system length L,
(2.65)
where a is the lattice constant. For the ASEP, one main ingredient for the derivation of the hydrodynamic equations is the lattice continuity equation and the fundamental diagram, i.e., the relation between particle density ρ and the (stationary) particle current J . This will be discussed further in Section 4.2.9.
2.5.2. Field-Theoretic Methods and Renormalization Groups Field-theoretic methods have also been applied to stochastic reaction-diffusion models on a lattice (for reviews, see, e.g., [921, 1355, 1356]). A standard field theoretic formulation of stochastic interacting particle systems is the Doi–Peliti formalism [313, 314, 1099]. It starts from the master equation in a second quantization representation. Using a coherent state representation, a field theory is derived in terms of two fields. There are, however, certain problems when dealing with hard-core particles [1410]. An application to the zero-range process (ZRP) and ASEP can be found, e.g., in [850]. The field theoretic formulation then allows a perturbative treatment of the problem. It is also starting point for renormalization group approaches [176, 573]. Other real-space renormalization group methods that are not explicitly based on a field theoretic approach have also been used to study reaction-diffusion processes [495, 588]. Variants of the real-space renormalization group that allow to predict the steady-state and dynamic properties of driven diffusive systems have been developed by Hanney and Stinchcombe [494, 495] based on the mapping between nonequilibrium and quantum systems. It allows to take into account conservation laws and symmetries and yields exact fixed points for some exclusion models.
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2.5.3. Similarity Transformations Sometimes, a deeper relation between different stochastic systems exists on the level of their quantum Hamiltonians. In [558, 1273], similarity transformations H˜ = BHB−1
with
B=
L
Bi
(2.66)
i=1
between two Hamiltonians H and H˜ have been investigated. Here, Bi is a change of basis on site i. Although the stochastic processes described by these Hamiltonians can be quite different, their time evolution operators have identical spectral properties and equivalent correlations functions. In [558], two cases have been distinguished. Two stochastic systems are called similar if a nonsingular transformation B of the form (2.66) exist such that all rates are positive in both systems. A weaker statement is the stochastic similarity, which only requires that for all probability distributions |P also B|P is a probability distribution of the transformed system. This might relate a stochastic system with a nonstochastic one, which could provide valuable information if the nonstochastic system is exactly solvable. Another form of equivalence is called enantiodromy [1273]. Here, the stochastic Hamiltonians are related by H˜ = BHT B−1 ,
(2.67)
which also implies certain relations between the expectation values of the two systems [1273]. Examples for the application of similarity transformations to reaction-diffusion processes can be found, e.g., in [6, 15, 558, 799, 939, 1103, 1214].
2.5.4. Ultradiscrete Method A direct connection between difference equations and (fully) discrete systems like cellular automata (CA) is known as the so-called ultradiscrete method (UDM) [1368]. The problem in the derivation of CA from difference equations lies in the discretization of the “state” (or dependent) variable. CA is fully discrete in its dependent and independent variables, whereas the difference equations have only discrete independent variables. The key formula for overcoming this difficulty is A B + exp + · · · = max(A, B, . . .). (2.68) lim ε log exp ε→+0 ε ε Let us illustrate how this formula works for obtaining CA by using the following simple difference equation: 1 t t + ujt + uj+1 ). (2.69) ujt+1 = (uj−1 3
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Methods for the Description of Stochastic Models
Here, ujt represents the value of u at time t and position j on the 1D lattice Z. Note that a continuous limit of (2.69) gives the diffusion equation ut = Duxx ,
(2.70)
by taking x, t → 0 with keeping D = x2 /(3 t) to be a constant, where we put t = u(t, x ± x), ujt+1 = u(t + t, x). uj±1 The first step is to transform the dependent variable by introducing an exponential function with the small positive parameter ε as t Uj t . (2.71) uj = exp ε Next, substituting (2.71) into (2.69), we have t t t Uj Uj+1 Uj−1 1 t+1 + exp + exp + ε log . Uj = ε log exp ε ε ε 3
(2.72)
Then, taking the limit ε → +0 and using the formula (2.68), we have t t , Ujt , Uj+1 ). Ujt+1 = max(Uj−1
(2.73)
We immediately know that if we take Ujt ∈ Z for all j at a time t = T , then Ujt ∈ Z holds for all j and t > T . Thus, (2.73) can be regarded as a CA associated with (2.69) or (2.70). This example shows how CA is obtained from a partial differential equation (PDE): 1. First, we discretize all independent variables in the PDE but keep its mathematical structures, e.g., the stability of the solution, by using standard methods for computation [1139] or soliton theory [261]. Then, we have a difference equation associated with the original PDE. In the above example, we have discretized (2.70) into (2.69). 2. Next, we discretize the dependent variables in the difference equation, which is called the UDM. For this purpose, we change the dependent variables by using the exponential-type transformation (2.71), introducing the small parameter ε. 3. Finally, taking the limit ε → +0, and using the formula (2.68), we may have a corresponding CA, as obtained above in (2.73). Another example is the Burgers CA (BCA) [1065], which is a basic CA model for traffic flow as discussed later. BCA is associated with the Burgers equation ut = 2uux + uxx .
(2.74)
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A. Schadschneider, D. Chowdhury, and K. Nishinari
This can be linearized through the Cole-Hopf (CH) transformation fx , f
(2.75)
ft = fxx .
(2.76)
u= into the diffusion equation
To discretize (2.74), we utilize discrete analogs to (2.75) and (2.76). Discretizing both time and space variables in (2.76), we obtain a discrete diffusion equation fj t+1 − fj t t
=
t − 2f t + f t fj+1 j j−1
( x)2
.
(2.77)
Next, we define a discrete analog to the CH transformation ujt ≡ c
t fj+1
fj t
,
(2.78)
where c is a constant (c ∼ 1/ x). Rewriting (2.77) with ujt in place of fj t , we obtain a difference Burgers equation 1 − 2δ 1 1 t + t + 2 uj+1 cδ uj c , ujt+1 = ujt 1 − 2δ 1 1 + t + 2 ujt cδ uj−1 c
δ = t/( x)2 .
(2.79)
ut
1 Assuming u(j x, t t) = x log cj and taking limits x → 0 and t → 0, we obtain (2.74) from (2.79). Next, let us introduce a transformation of variables and parameters as t Uj L 1 − 2δ −M t , = exp , c 2 = exp , (2.80) uj = exp ε cδ ε ε
where L and M are parameters. Then, taking the limit ε → +0 and using the formula max(A, B) = − min(−A, −B),
(2.81)
t t , L − Ujt ) − min(M , Ujt , L − Uj+1 ). Ujt+1 = Ujt + min(M , Uj−1
(2.82)
we obtain the BCA
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Methods for the Description of Stochastic Models
We can also derive a ultradiscrete diffusion equation. Introducing a transformation t Fj , (2.83) fj t = exp ε we obtain an ultradiscrete CH transformation t Ujt = Fj+1 − Fjt +
L 2
(2.84)
from (2.78) under the limit ε → +0. Then, from (2.82), we obtain t , Fjt + Fjt+1 = max(Fj−1
L t ). − M , Fj+1 2
(2.85)
BCA can be “linearized” by substituting (2.84) into (2.82). If we take L/2 = M , then (2.85) coincides with (2.73). It is noted that BCA is related to the Burgers equation through the transformation Ujt = L/2 + ε xu(j x, t t). There are two parameters L and M in BCA. It can be shown that assuming that M > 0, L > 0, and 0 ≤ Ujt ≤ L for any site j at a certain time t, then 0 ≤ Ujt+1 ≤ L holds for any j. Thus, BCA is equivalent to a CA with a value set {0, 1, . . ., L} under the above conditions. In contrast to most other CA, multiple occupations of cells are allowed. Moreover, if we put a restriction L ≤ M and L = 1 on BCA, then BCA is equivalent to the rule–184 CA [1476], i.e., the NaSch model with Vmax = 1 and p = 0. Further related work on UDM can be found in [920, 1062, 1344, 1367]. In these works, various soliton equations have been transformed into CA by this method, and the solutions of the CA have also been obtained exactly without losing the mathematical properties of corresponding difference equations.
2.6. NUMERICAL METHODS Generally, interacting many-particle systems cannot be solved exactly. Therefore, one has to rely on approximations, preferably such that the error can be controlled. Apart from analytical approaches also numerical methods can supply useful information. The most natural numerical approach for the investigation of stochastic systems is Monte Carlo (MC) simulations, which allow to implement the stochastic dynamics directly. Besides, computer simulations also numerical methods can be used to gain insights. Here, usually a well-defined mathematical problem, which is derived from an analytical description of the dynamics, is solved using methods from numerical mathematics. As in the case of analytical methods described in the previous sections, these are
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A. Schadschneider, D. Chowdhury, and K. Nishinari
often variants of approaches that have been successfully applied for equilibrium systems, especially quantum spin chains.
2.6.1. Computer Simulation (MC Methods) The most natural approach to stochastic systems is in some sense MC simulation [714, 963, 1051]. Here, the stochastic process is realized on the computer by generating transitions with appropriate probabilities. Therefore, each MC simulation generates one specific realization of the stochastic process. Since normally one is interested in averages of certain quantities, one has to repeat such a simulation run sufficiently often and then average over the different runs, which correspond to different realizations of the process. The necessity for averaging can be rather time consuming, although in principle such a process is perfectly parallelizable. Fortunately, if one is interested in the stationary state only, this can be simplified considerably. For ergodic systems, the ensemble average is identical to the time average. Therefore, only one MC run is needed to determine the averages of interest. Steady-state expectation values can be obtained by letting the system relax into the stationary state and then taking the corresponding time average. For a model defined by local interactions, e.g., between neighboring sites only, a MC step consists of local updating all sites in the order determined by the dynamics used. If such a local update consists of a process happening with some probability p (e.g., a particle moving to the neighbor cell), one first generates a random number r from a uniform distribution in [0, 1]. If r ≤ p, the process will happen, if r > p not. This can be generalized if there are more than two options (“process will happen” versus “process will not happen”). Take, e.g., a process where with probability p1 process 1 will happen, with probability p2 process 2, and with probability 1 − p1 − p2 nothing happens. Again only one random number r is needed. If r ≤ p1 , process 1 will be performed. If p1 < r ≤ p1 + p2 , process 2 will be performed, and nothing is done for r > p1 + p2 . In this way, all process are generated with the correct probabilities. Stochastic processes with parallel dynamics can be implemented directly by identifying an update step with the time step t. Practically parallel dynamics is usually realized in a sequential way by updating all sites in a certain order. However, the new configurations are first stored in an auxiliary lattice until all sites are updated. Only then, the configuration is replaced by that of the auxiliary lattice. In this way, one makes sure that all sites are really updated synchronously, i.e., the state changes of neighboring sites in the same time step are not already taken into account. The MC simulation of other discrete time updates proceeds in a similar way, but the order in which sites are updated has to be taken into account. Then, it is not necessary to use an intermediate lattice. Discrete-time dynamics have the advantage that the stochastic process is given in terms of probabilities which can be used directly in the simulations. For randomsequential dynamics, on the other hand, the process is given in terms of transition
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Methods for the Description of Stochastic Models
rates ijkl . In order to make a MC simulation possible, one then rescales time in the process in such a way that the largest rate of the process becomes equal to 1. More forkl mally, one first determines = max ij and then t → t and ijkl → ijkl / . After this, all rescaled rates are in the interval [0, 1] and can be interpreted as probabilities. The actual MC simulation for random-sequential dynamics proceeds by first choosing a site (rather a bond connecting two sites), which is then updated using the probabilities defining the rescaled process. 2.6.1.1. Random Numbers The above description shows that random numbers play an important role in MC simulations. These are obtained from random number generators (RNG), which generate a sequence of random numbers. In fact, this sequence is not truly random, but usually even deterministic. Therefore, one speaks of pseudo-random number generators (PRNG), which create a sequence of numbers that appear to be random in the sense that they pass certain statistical tests and do not exhibit any easily discernible patterns. Another important requirement on RNG is efficiency in the sense that the PRNG is based on a simple algorithm that does not require much resources (computer time, memory) [501, 929]. Indeed, very often, the generation of random numbers is one of the most time consuming factors in the simulations. The choice of the PRNG is therefore crucial not only for the quality of the results but also for the resources needed. Most programming languages or compilers provide built-in PRNG, but it is often more convenient to implement one of the well-tested PRNG. Typically, a PNRG generates a sequence of integer numbers rj in a certain interval using a recursion of the general form rj = f (rj−1 , rj−2 , . . ., rj−n ).
(2.86)
To start the iteration, one has to provide the first n numbers r1 , . . ., rn , the seed. The iteration is deterministic so that the randomness enters through the choice of the seed. Since the random numbers lie in a finite interval, the sequence (rj ) is periodic. For applications, the period of the PRNG should be as large as possible. The quality of the randomness can be tested by measuring correlations of the sequence which should be as small as possible. One of the most frequently used PRNG is the so-called linear congruential generator. It is based on the recurrence rj+1 = (arj + b) mod m,
(2.87)
which generates a sequence of integer pseudo-random numbers rj with 0 ≤ rj < m. a and b are parameters chosen such that the correlations are small and the period N is long. N cannot be larger than m, but for some choices of a and b, it can be considerably smaller. Some popular and tested choices for the parameters are given in Table 2.3.
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A. Schadschneider, D. Chowdhury, and K. Nishinari
Table 2.3 Some standard choices for the parameters of the linear congruential generator and their period lengths N [963]
a
b
m
N
75 = 16807 69069 1313
0 1 0
231 − 1 232 259
231 − 1 = 2.15 · 109 4.29 · 109 5.76 · 1017
Another class of frequently used PRNG is the shift register generators, also known as Tausworthe generators. They generate a random sequence of binary numbers bj through the recursion bj = bj−p .XOR. bj−q ,
(2.88)
where .XOR. is the exclusive “OR” operation which can be carried out quickly. For p > q, the period cannot be larger than 2p − 1. One needs to provide p seeds, which is usually done using another PRNG, e.g., the linear congruential generator. A standard parameter choice known as Kirkpatrick-Stoll generator or R250 generator uses p = 250 and q = 103. In this case, the period is 2250 − 1 = 1.81 · 1075 .
2.6.2. Exact Diagonalization A rather different approach does not try to simulate the stochastic system directly, but to start from the mathematical formulation of the master equation, e.g., in the form (2.2), which is the starting point of the quantum formalism. As we have seen in Section 2.1, all relevant information about expectation values of observables is contained in the eigenvalues Eλ and eigenvectors |Pλ of H; see, e.g., (2.13). If one is interested in the stationary state and the final stages of the relaxation process, only the leading eigenvalues and eigenvectors are needed. Therefore, one could try to calculate these directly using suitable methods. This has the big advantage that no averaging like in the simulations has to be done. However, the matrix H is sparse but grows quickly with the system size L, and so one is restricted to rather small systems (see, e.g., [116, 326]). Except for special cases, where one can use certain symmetries of the matrix to reduce its effective dimensionality, it is currently not possible to go much beyond L = 20. The most common method used for the diagonalization of large matrices resulting from many-particle quantum systems is the Lanczos algorithm [836]. Since the matrices derived from stochastic systems are typically nonsymmetric, one needs a generalization of the Lanczos algorithm to nonsymmetric matrices. Here, the Arnoldi algorithm is considered most appropriate, especially since a highly sophisticated software package (ARPACK) [852] is freely available.
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2.6.3. Density-Matrix Renormalization Group The density-matrix renormalization group (DMRG) (for reviews, see, e.g., [201, 491, 1104, 1254, 1462]) has become the standard numerical approach for the investigation of 1D quantum systems. In contrast to textbook renormalization group methods, the DMRG is not strictly an approach that relies on integrating out degrees of freedom (decimation). Instead, it starts from small system size and tries to built an effective Hilbert space for larger systems by “adding” local Hilbert spaces. In order to keep the problem tractable in each step, the Hilbert space is reduced to a smaller one, usually of fixed size, by keeping the most important states. The finesse of the DMRG is now the clever choice of these most important states. Naively one would take the states of lowest energy, especially since usually one is interested in the low-energy physics. But this often leads to unsatisfactory results, which can be traced back incompatibilities arising due to the boundary conditions. Therefore, the DMRG algorithm tries to anticipate which states at the current system size are important for an accurate description of larger systems. As shown by White [1461], these states are not those with lowest energy but rather those corresponding to the largest eigenvalues of the density matrix (which describes the coupling to the outside world, usually called environment in the context of DMRG). So the basic principle of the DMRG can be described by the following steps (Fig. 2.2): 0. Start with a small system that can be treated exactly. 1. Add degrees of freedom (e.g., an additional quantum spin). 2. Couple this larger system to an environment that is usually just a (reflected) copy of the system. This large system is called superblock. 3. Reduce the Hilbert space of the larger system by projection with the density matrix to an effective one of small size (m states) which still is tractable. 4. Iterate steps 2 and 3. As mentioned before, the key to the success of the DMRG approach is the use of the reduced density matrix for the determination of the most important states. The states of
1 Increase size System
Environment
2 Superblock 3 DM projection 4 Increase size
m states m states
Iteration
Figure 2.2 General scheme of the DMRG algorithm.
m states
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the superblock have the form |ψ =
ψij |i| j,
(2.89)
i, j
where |i and | j are the states of the system and environment, respectively. The reduced density matrix of the system is then defined by ρii =
ψij∗ ψi j ,
(2.90)
j
which satisfies Tr ρ = 1. It contains all the information necessary to calculate any property of the system block from the wavefunction |ψ. For example, for an operator A that acts only on the system block on has A = Tr ρA. There are two different classes of DMRG algorithms. In the infinite size algorithm, basically the steps described above are iterated as long as possible thus building up an “infinite” system. However, in most cases, the finite size algorithm gives better results. It starts with the infinite size algorithm until a certain system size L is reached. After that, one applies the steps of infinite size algorithm in such a way that only one block is enlarged, while the other is shrinked in such a way that the system size remains constant. When the latter block has reached some minimal size, the roles of the blocks are interchanged. A complete growth-shrinkage sequence for both blocks is called a sweep. Already a relatively small number of sweeps can improve the quality of the results significantly. One interesting aspect of the DMRG approach is its relation with the MPA. Indeed, DMRG can be reinterpreted as a variational calculation using MPA states [29, 327, 923, 1086, 1193, 1345]. Recently, this has lead to some progress in improving the DMRG algorithm and extending it to higher dimensions [1416]. It is now not surprising that due to the quantum formalism of Section 2.1, the DMRG can also be adopted for the investigation of 1D stochastic systems. The natural approach is to apply the DMRG method to the stochastic Hamiltonian. Here, similar complications as in the case of exact diagonalization (Section 2.6.2) occur. The DMRG approach has first been extended to stochastic systems in [178] and [563]. Hieida [563] applied the DMRG to the transfer matrix of the ASEP with sublattice-parallel dynamics. He found an excellent agreement with the exact solution already for a small number of retained states. In the approach of Carlon et al. [178], the quantum Hamiltonian in the DMRG is replaced by the stochastic Hamiltonian (2.4). Since H is nonhermitian, the exact diagonalization step of the algorithm has to be modified, e.g., by using the Arnoldi algorithm. The influence of the nonhermitian operators on the numerics has been discussed in
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detail in [178]. In general, the Arnoldi algorithm is more time-consuming and less stable compared to its hermitian counterparts. A second problem is the fact that left and right eigenvectors of H are no longer related by transposition as in quantum problems. This can be solved by an appropriate choice of the density matrix, which allows to determine both eigenvectors simultaneously [178]. Working with the stochastic Hamiltonian and deriving the physics from its leading eigenvalues means that one automatically works in the limit of very large times t → ∞. In contrast, the systems treated are finite, e.g., of the order of L = 50 − 100 [178]. Therefore, an extrapolation to large lattice sizes is necessary, and finite-size corrections become relevant.
2.6.4. Transfer-Matrix DMRG The transfer matrix DMRG (TMRG) approach is a variant of the DMRG that allows to treat infinite systems, i.e., to perform the thermodynamic limit exactly. On the other hand, it cannot treat the stationary limit t → ∞, but only the dynamics at finite times. The TMRG for stochastic systems was originally introduced by Kemper et al. [723]. In analogy to the Trotter-Suzuki mapping known for quantum systems, the dynamics of 1D stochastic processes can be mapped to a 2D classical model. This natural leads to the idea of applying the TMRG to the corresponding “stochastic” transfer matrix. Later it was pointed out in [351] that one cannot expect a high accuracy for generic systems. This is related to the special structure of the (row-to-row) transfer matrix in the case of stochastic systems, which describes real-time dynamics, not an imaginary time dynamics for quantum systems. This leads to a new variant of the stochastic TMRG, called stochastic light-cone corner transfer matrix renormalization group (LCTMRG) [722]. Its basic idea is not to use rowto-row transfer matrices, but corner transfer matrices (CTM), which have the form of a light-cone (see Fig. 2.3) and only consist of causally connected points. As in the conventional quantum TMRG algorithm [168, 1450], the first step to determine the dynamic evolution of the local density n(t) = 1|n · e−t·H |P(0)
(2.91)
of a stochastic process governed by a stochastic Hamiltonian H = i hi,i+1 in the thermodynamic limit L → ∞. It is mapped to a 2D statistical model by using aTrotter-Suzuki decomposition [1328, 1399]. The resulting classical 2D lattice [351, 723] is shown in Fig. 2.3a. The local plaquette interactions are given by (τ )lr11 lr22
− t·h
= l2 r2 |e
|l1 r1 =
l2
r2
l1
r1
with
li , ri ∈ {0, 1} .
(2.92)
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Σ
Σ
Sum out
n
Σ
Σ
Σ n
Σ
Σ
Σ Time
Σ
Σ
Σ
Initial state P(0) Space (a)
(b)
Figure 2.3 (a) Graphical representation of the Trotter-Suzuki decomposition in the LCTMRG. The resulting 2D lattice consists of local plaquette interactions τ is infinite in space direction. (b) Reduction of the 2D lattice to a triangular structure.
The spatial dimension L of the stochastic process is expanded by a virtual Trotter dimension M = t/ t, which corresponds to the time direction and is split into (discrete) steps of size t. t has to be chosen sufficiently small to obtain a good approximation of n(t). Formally, the Trotter decomposition becomes exact for t → 0. The space dimension of the 2D lattice is infinite if the thermodynamic limit L → ∞ of the stochastic model is performed. Since the local density n(t) is determined at finite time t, the Trotter dimension is finite. In contrast to the quantum TMRG, the boundary conditions are fixed in Trotter direction and given by the vectors 1| and |P(0) (Fig. 2.3a). The TMRG algorithm proposed in [722] is based on the CTM. Such a corner-transfermatrix DMRG algorithm (CTMRG) is known to be numerical, more stable, and faster than TMRG [1069]. Another reason for the special choice of the CTM is the probability conservation, which implies the identity 1|e− t·h |s = 1
(2.93)
for any state |s. Thus, the plaquettes τ [see (2.92)] trivializes by summing out the “future” indices, i.e., ∀l1 , r1 :
l2 r2
(τ )lr11lr22 = 1,
Σl 2 Σr 2 l1
= 1.
(2.94)
r1
Therefore, for the computation of n(t), a huge number of plaquette interactions can be omitted because they “trivialize.” The remaining nontrivial plaquettes form a 2D lattice of finite dimension, which is shown in Fig. 2.3b. The trivialization process can easily be
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Cu
Corner transfer matrices
Cr
Cl Cd (a)
(b)
Figure 2.4 Construction of corner transfer matrices. The lattice is split into four parts (a), which defines the corner transfer matrices (b).
Time
Σ
Σ
nj
Σ
Σ
1
Local density: nj (t) = 1 nj e−tH P0 Trotter-Suzuki decomposition H = hj, j + 1 j
Σ
r1 r2
Δt
= r1r2 e−Δt·h l1l2 l1 l2 P0 Space
Causal structure: “Light cone” Σr1 Σr2 =
Trivialisation: l1 l2
Σrr
r1r2
1 2
e−Δt·h l1l2 = 1
Figure 2.5 General scheme of the LCTMRG.
understood by a causality argument: only a “light-cone” of plaquette interactions can influence the site where the local density is measured. The next step is the construction of a CTMRG algorithm, which genuinely fits to the triangle structure of the 2D lattice. The lattice is divided into four parts as shown in Fig. 2.4. The cuts are natural because they form the boundaries of the “future” and “past light-cone” of the center point of the triangle (Fig. 2.5). In the DMRG algorithm, the CTM corresponding to the four parts is enlarged sequentially. After each extension step, the CTMs have to be renormalized by a density-matrix projection. Here, a careful choice of the reduced density matrix is important. Details can be found in [722].
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2.7. APPENDICES 2.7.1. Some Mathematics 2.7.1.1. Matrix Theorems Gershgorin’s theorem [455] states that the eigenvalues λj of a complex n × n-matrix A = (aij ) lie in the union of the disks ⎧ ⎫ ⎨ ⎬ |ajl | . Dj = z ∈ C : |z − ajj | ≤ ⎩ ⎭
(2.95)
l= j
# " For a stochastic matrix, we therefore have Dj = z ∈ C : |z − ajj | ≤ ajj . Another useful result is the Perron–Frobenius theorem. If A = (aij ) is a real N × N matrix with strictly positive entries aij > 0, then there is a real number λ0 > 0, which is an eigenvalue of A, and any other (possibly complex) eigenvalue λ is strictly smaller than, i.e., |λ| < λ0 . As a consequence, the spectral radius of A is equal to λ0 . In compact form, the Perron–Frobenius theorem states that a real square matrix with positive entries has a unique largest real eigenvalue with a corresponding eigenvector which has strictly positive components. The theorem can be extended to the case aij ≥ 0, but then, additional conditions have to be satisfied, i.e., the matrix has to be irreducible. 2.7.1.2. Definition of Tensor Product The tensor product of a n1 × n2 matrix A = (aij ) and a m1 × m2 matrix B is defined as the following (n1 m1 ) × (n2 m2 ) matrix: ⎛
a11 B a12 B ⎜ a21 B a22 B ⎜ . A ⊗ B = ⎜ .. . ⎝ . . an1 1 B an1 2 B
⎞ · · · a1n2 B · · · a2n2 B ⎟ ⎟ . ⎟. .. . . . ⎠ · · · an1 n2 B
(2.96)
Here, aij B stands for the submatrix obtained by multiplying all elements of B by aij .
2.7.2. MPA and Optimum Ground States for Quantum Spin Chains In the following, we give an overview over the application of the MPA to quantum spin chains and its relation with the concept of optimum ground states. Let us consider a Hamiltonian for a quantum spin chain (with periodic boundary conditions) of the form H = Lj=1 hj, j+1 , where hj, j+1 is the local hermitian Hamiltonian and independent of j, acting only on spin j and j + 1.
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It is always possible to set the lowest eigenvalue of hj, j+1 equal to zero by adding a suitable constant. Then, hj, j+1 is positive-semidefinite and since H is the sum of positivesemidefinite operators, it follows that zero is a lower bound for the ground state energy E0 of H , i.e., E0 ≥ 0. Usually, E0 is greater than zero (E0 > 0), and the global ground state involves also excited states of hj, j+1 . Therefore, a construction of the global ground state is usually very difficult. However, there are special cases where E0 is equal to zero, H |ψ0 = 0
(2.97)
hj, j+1 |ψ0 = 0.
(2.98)
and therefore, for all j,
A state |ψ0 is called optimum ground state of H if and only if condition (2.98) holds. This implies that the ground state energy is independent of the system size, i.e., there are no finite-size corrections. The construction of optimum ground states for quantum spin chains via matrix products was introduced in [761–763] (see also [384, 487] and [1056] for further references). The idea is to describe ground states by means of a product of matrices, |ψ0 = Tr(m1 m2 . . . mL ),
(2.99)
where the entries of matrix mj are single-site states and the symbol denotes the usual matrix multiplication of matrices with a tensor product of the matrix elements. Note that m1 m2 . . . mL is still of the same size as the matrices mj , but its elements are large linear combinations of tensor product states. The trace assures the translation invariance of the ground state. For nonperiodic boundary conditions, it has to be replaced by a suitable linear combination of the elements of m1 m2 . . . mL . As an example [761–763], the ground state of a large class of antiferomagnetic spin-1 chains can be constructed using the S z eigenstates |0j and |±j by a|0j b|+j mj = , (2.100) c|−j d|0j where the a, b, c, d are real numbers. Condition (2.98) requires hj, j+1 (mj mj+1) = 0,
(2.101)
i.e., all four elements of mj mj+1 are local ground states of hj, j+1 . Let us now write mj = A0 |0 + A−|− + A+|+
(2.102)
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with suitable 2 × 2 matrices A0 and A± . It is obvious that (2.99) can be written equivalently as |ψ0 = Tr Aα1 . . .AαL |α1 , . . ., αL , (2.103) α1 ,...,αL
with αj = 0, +, −, or, in an appropriate basis, ⎡⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎤ A0 A0 A0 |ψ0 = Tr ⎣⎝ A− ⎠ ⊗ ⎝ A− ⎠ ⊗ . . . ⊗ ⎝ A− ⎠⎦. A+ A+ A+ Here, now “⊗” denotes the tensor product, or defining a column vector ⎛ ⎞ A0 A = ⎝ A− ⎠ , A+
(2.104)
(2.105)
as |ψ0 = Tr(A ⊗ A ⊗ . . .⊗ A).
(2.106)
The condition (2.101) can then be rewritten as hj, j+1 (A ⊗ A) = 0.
(2.107)
This means that there are two equivalent ways to represent |ψ0 . While (2.99) uses a product of matrices with vectors as entries, (2.106) expresses |ψ0 as a product of vectors with matrices as entries. The original idea of Derrida et al. [293] was to construct the stationary state |P0 of a stochastic process defined by (2.5) as a suitable linear form of a product of matrices where each matrix corresponds to a single-site state precisely as in (2.106). Matrix-product states have recently attracted a lot of interest in investigations of quantum entanglement; see, e.g., [1102]. Different generalization to higher dimensions have been proposed, e.g., vertex-state models [9, 1054, 1055] and projected entangled pair states (PEPS) [1101, 1416–1418]. However, up to now, these ideas have not been implemented within the framework of nonequilibrium physics.
2.7.3. Krebs–Sandow Theorem and Extensions Krebs and Sandow [800] have shown that the stationary state of a rather general class of 1D stochastic processes can be written in matrix-product form. Their proof is nonconstructive in the sense that the stationary state is known and that no explicit results for the matrices, like representations, can be given. The construction relies on the cancellation mechanism (2.53) supplemented by the boundary conditions (2.55).
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Although the original result was restricted to models with nearest-neighbor interactions and random-sequential dynamics, it can be extended to interactions of arbitrary finite range [758] and, using the general results of [1163], most discrete updates. For the case of nearest-neighbor interaction and random-sequential dynamics described by a stochastic n-state Hamiltonian of the form (2.5) for periodic boundary conditions or (2.6) in the presence of boundary interactions, the first part of the Krebs–Sandow theorem can be stated as follows (see [758]): •
¯ 0, . . ., D ¯ n−1 such that the vectors A = If one can find matrices12 D0 , . . ., Dn−1 , D ¯ 0, . . ., D ¯ n−1 )t satisfy (D0 , . . ., Dn−1 )t and A¯ = (D ¯ hj, j+1 (A ⊗ A) = A¯ ⊗ A − A ⊗ A,
•
(2.108)
(p) then |PL 0 = Tr[A⊗L ] is a zero-energy eigenvector of HL = Lj=1 hj, j+1 , i.e., a stationary state of the underlying stochastic process with periodic boundary conditions. If in addition to (2.108), one can find vectors |V and W | such that ¯ W |h1 A = −W |A,
¯ , hL A|V = A|V
then |PL 0 = W |A⊗L |V is a zero-energy eigenvector of HL = h1 + hL with boundary interactions h1 and hL .
(2.109) L−1 j=1
hj, j+1 +
The proof is rather straightforward by noticing that after applying hj, j+1 to A⊗L = A ⊗ A ⊗ . .. ⊗ A, two similar terms A ⊗ . . . A¯ ⊗ A . . . are generated with “defects” A¯ at sites j and j + 1, respectively. Since these two terms have different signs, the contributions of consecutive terms cancel partially, similar to a telescopic sum. For the case of open boundaries, the boundary terms have just the form to guarantee that the remaining terms are annihilated by the boundary interactions. In the presence of boundary interactions, one can show more. Here, the matrixproduct state is not an Ansatz, but merely a reformulation of the fact that the stationary state is a zero-energy eigenvector of HL for all system lengths: Given L−1 a stochastic process described by a stochastic Hamiltonian of the form HL = h1 + j=1 hj, j+1 + hL which has a unique stationary state for any system length L. Then, the eigenstate |PL 0 with eigenvalue 0 corresponding to this stationary state can be written as a matrix-product state W |A⊗L |V with A = (D0 , D1 , . . ., Dn−1)t and vectors W |, |V . ¯ j such that the cancelling-mechanism (2.108)–(2.109) Moreover, one finds n matrices D is fulfilled. 12 For a particle-hopping model, D = E is usually identified with an empty site as in the ASEP. 0
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ˆ s which extend the lattice by one The basic idea of the proof is to define operators D site in state s: ˆ s |s1 , . . ., sL := |s, s1, . . ., sL D
(2.110)
The vector |V is defined as the vacuum vector |vac and W | :=
∞
PL (s1 , s2 , . . ., sL ) s1 , s2, . . ., sL |.
(2.111)
L=1 s1 ,s2 ,...,sL
Therefore, one has P(s1, . . ., sL ) = W |Ds1 Ds2 . . .DsL |V , i.e., |PL 0 = W |A⊗L |V . ¯ j are also defined by their action on basis vectors: The auxiliary matrices D ⎡ ⎤ L−1 A¯ ⊗ A⊗(L−1)|V := ⎣ (2.112) hˆ j, j+1 + hˆ L ⎦ A⊗L−1 |V . j=1
¯ one can show that the theorem is indeed correct With these definitions of A and A, [800]. The construction in the proof only works for open boundary conditions. So far no extension to the periodic case is known. The problems arising there are discussed in detail in [797] (see also Section 2.4.1). The theorem can be extended to models with an arbitrary finite interaction range r [758]. In this case, the cancellation mechanism takes the form r times
$ %& ' hˆ k,k+1,...,k+r−1 (A ⊗ A ⊗ . . .⊗ A) = X ⊗ A − A ⊗ X,
(2.113)
r−1 times
$ %& ' W |hˆ l (r)(A ⊗ A ⊗ . . .⊗ A) = −W |X, hˆ r (r)(A '$. . .⊗ A%)|V = X |V , & ⊗A⊗
(2.114) (2.115)
r−1 times
where the boundary interactions hˆ l (r) and hˆ r (r) have a range of r − 1 sites. X is a column vector with N r components, which operates on r − 1 neighboring sites and usually does not factorize into single site contributions. In [354], the special case X = (x + y) ⊗ A + A ⊗ (y + z) + t with vectors x, y, z and a number matrix t has been considered. Using (2.113)–(2.115), the proof of the theorem is analogous to the case r = 2. Note that the proof implicitly assumes that the stationary state is known. It then shows that it can be rewritten in matrix-product form using the cancellation methods without encountering internal inconsistencies. This offers new ways to determine the stationary way directly by studying the cancelling mechanism directly instead of the “full” master equation.
CHAPTER THREE
Particle-Hopping Models of Transport Far from Equilibrium Contents 3.1. Elements of Random Walk Theory
72
3.2. Asymmetric Simple Exclusion Process
74
3.3. Zero-Range Process and Exact Results 3.3.1. Exact Solution 3.3.2. Bethe Ansatz Solution
75 76 79
3.4. Extensions and Generalizations 3.4.1. Parallel Dynamics 3.4.2. Other Lattice Structures 3.4.3. ZRP with Disorder 3.4.4. ZRP with Fluctuating Particle Number 3.4.5. Generalizations 3.4.6. Dynamical Urn Models 3.4.7. Misanthrope Process 3.4.8. Relation of ZRP to Other Models and Some Applications
81 81 82 83 83 86 86 87 88
3.5. Physics of the ZRP 3.5.1. Condensation Transition 3.5.2. Dynamics and Coarsening 3.5.3. Criterion for Phase Separation
90 90 94 95
3.6. Particle-Hopping Models with Factorized Stationary States 3.6.1. Models with Pair-Factorized Steady States
97 100
3.7. Generalized Mass Transport Models 3.7.1. Models with Continuous States 3.7.2. Asymmetric Random Average Process 3.7.3. Chipping Model
102 102 103 105
3.8. Appendix 3.8.1. Derivation of the Factorization Criterion
106 106
Particle-hopping models conserve the particle number in the bulk, i.e., in the case of nearest-neighbor interactions, only those processes in Table 2.1 that satisfy the condition sj sj+1
sj sj+1 = 0 for sj + sj+1 = sj + sj+1
Stochastic Transport in Complex Systems DOI: 10.1016/B978-0-444-52853-7.00003-8
(3.1) Copyright © 2011, Elsevier BV. All rights reserved.
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are allowed. This conservation law implies that particles are transported in the system through hopping to neighbor sites. In general, this transport will be asymmetric in the sense that rates for motion to the left and right are different. If the particles have an internal degree of freedom, corresponding to a multistate model, only the total number of particles needs to be conserved, not that of each species separately. In the following, we will consider not only models obeying an exclusion principle such that each site can be occupied by at most one particle, but also models without exclusion (often called mass transport models), where each site can be occupied by an arbitrary number of particles, but with particle conservation (in the bulk). From discrete mass transport models, it is only a small step to continuous mass transport models, often referred to as mass transfer models. Here, one no longer distinguishes discrete particles, but the dynamical variable at each lattice site is interpreted as a continuous mass. The elementary dynamical step then consists of moving a certain fraction of this mass (which is determined by some prescribed probability density) to a neighboring site. Although we focus on discrete models, we will briefly discuss the case of continuous variables in Section 3.7.
3.1. ELEMENTS OF RANDOM WALK THEORY First, we consider a single particle performing a random walk in continuous time on a one-dimensional lattice of L sites with periodic boundary conditions. In the infinitesimal time interval dt, the particle hops with probability pdt to the right and with probability qdt to the left. In terms of the transition rates p and q, the master equation is given by ∂P( j, t) = pP( j − 1, t) + qP( j + 1, t) − (p + q)P( j, t), ∂t
(3.2)
where P( j, t) is the probability to find the particle time t at site j. The first two terms correspond to processes where the particle jumps from the left (right) to site j, and the last term represents the rate at which the particle leaves site j. The master equation (3.2) is a diffusion equation in discrete space and can be solved exactly, e.g., through Fourier transformation (in space):
˜ t) = P(k,
1 P( j, t)e2πijk/L . L j=1 L
(3.3)
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Particle-Hopping Models of Transport Far from Equilibrium
This yields the Fourier-transformed master equation ˜ t) dP(k, ˜ t) = λk P(k, dt
(3.4)
λk = p(zk − 1) + q(zk−1 − 1),
(3.5)
with
where we have introduced the abbreviation zk = e2πik/L . Therefore, ˜ t) = P(0, ˜ t)eiλk t P(k,
(3.6)
and after inverting the Fourier transform, one obtains the general solution P( j, t) =
L
˜ 0)eiλk t zk−j . P(k,
(3.7)
k=1
The stationary state corresponds to λ = 0, i.e., the mode k = L. The other eigenvalues λk determine the decay times Tk =
1 | Re λk |
(3.8)
of the corresponding mode. For large L, the eigenvalues are approximated by λk ≈ (p − q)
2π ik 4π 2k2 − (p + q) 2 L L
(3.9)
which implies Tk ∼ L z
with z = 2,
(3.10)
i.e., the dynamic exponent is z = 2 as expected for a diffusion process. If one considers an infinite lattice instead of a periodic one, the master equation (3.2) is solved using a generating function approach. In this case, the probability P( j, t) can be expressed by modified Bessel functions of the first kind In (x): −(p+q)t
P( j, t) = e
∞ ( j−l)/2 p √ P(l, 0)Ij−l (2 pqt). q
l=−∞
(3.11)
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For the special case of a symmetric random walker (p = q = 1/2) starting at the origin j = 0, one obtains P( j, t) = Ij (t)e−t .
(3.12)
The biased random walker (p = q) is a simple example for a system that does not satisfy the detailed balance condition (1.14). For a periodic lattice of L sites, the stationary probability is P( j) = 1/L so that Eqn (1.14) cannot be satisfied for p = q. In discrete time, the probability PN ( j) to find the walker at site j after N steps is given by the binomial distribution N j N −j pq (3.13) PN ( j) = j since in each step an independent decision about the direction is√ made. Therefore, the 2 2 2 average position is j = 0, and the variance σ = j − j = N . A natural generalization of the random walk is to consider a collection of walkers moving on the same lattice. Then, interactions can be introduced if walkers come close to one another. In the simplest case, this is just volume exclusion where two particles cannot occupy the same space. In the one-dimensional case, this implies that overtaking is not allowed and the order remains unchanged during the diffusion process. This process is usually called single-file diffusion (for a review, see [1275]). In the following, we will introduce a discrete variant in more detail.
3.2. ASYMMETRIC SIMPLE EXCLUSION PROCESS The asymmetric simple exclusion process (ASEP) can be viewed as a model of interacting random walks. Instead of one random walker now a large number N of them is considered. On a one-dimensional lattice, these walkers move with rate p to the right and with rate q to the left (Fig. 3.1). In addition, an exclusion principle allows this motion only if the target site is empty. The ASEP can also be viewed as a special case of the general two-state stochastic process or reaction-diffusion models introduced in Section 2.1. It is a particle-hopping model characterized by the nonvanishing transition rates 01 = q, 10 p
p
and p
10 01 = p.
(3.14)
p
Figure 3.1 Dynamics of the (totally) asymmetric simple exclusion process: particles move to empty right neighbor sites with rate p.
Particle-Hopping Models of Transport Far from Equilibrium
The most important special case, theoretically as well as in the applications to traffic problems, is the totally asymmetric simple exclusion process (TASEP), which has q = 0 so that motion is allowed only in one direction. The TASEP has obvious interpretation as traffic model. In fact, it is the paradigmatic model of all transport processes discussed in Part II of this book! Therefore, it can be called the “mother of all traffic models.” As we will see later, most CA-based traffic models can be considered variants of the ASEP. Depending on the kind of transport process, different extensions are required, e.g., a range of allowed velocities or particlecreation and -annihilation. The ASEP can also be considered the “Ising model” of nonequilibrium physics because it has a similar importance as the latter for equilibrium systems. The two models share several important features. First, they are rather simple, and only the very basic interactions are kept, e.g., the Ising model is a model of classical spins. Then, both models are exactly solvable. The famous exact solution of the two-dimensional Ising model [1084] has certainly played an enormous role for statistical physics, especially the understanding of phase transitions. It allows to test many general ideas at a specific model without the need for any approximations that could obscure the results. Last, but not least, both models have many applications. The Ising model has not only been used for the investigation of magnetism but also been interpreted as a model for a (lattice) gas, opinion formation, etc. Because of its importance, Chapter 4 will be devoted to the ASEP, its exact solution, and the results derived from it.
3.3. ZERO-RANGE PROCESS AND EXACT RESULTS Before we discuss the ASEP in some detail in Chapter 4, we study another simple process which, at first glance, looks completely unrelated to the ASEP. However, in the following Section 3.4.8, we will see that the ASEP, after an appropriate exact mapping, can indeed be considered a special case of this zero-range process (ZRP). On the other hand, the ZRP can be considered a particle-hopping model in its own right, or, more precisely, a mass transport model. The ZRP is characterized by its ultralocal transition rules. Originally, it was introduced by Spitzer [1304] in 1970. Meanwhile many general and rigorous results are known for it coming from mathematics [751]. In physics, the importance and relevance of the ZRP was only realized widely almost 30 years later [357, 363, 440] As in the ASEP, the ZRP describes the dynamics of N (classical and indistinguishable) particles on a lattice of L sites. In the following, we will not make any specific assumption about the geometry of the underlying lattice, but for simplicity, we will usually have the one-dimensional case of a periodic lattice in mind. In contrast to the ASEP, particles in the ZRP are not subject to any exclusion rule. Each lattice site j can be occupied by an
75
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A. Schadschneider, D. Chowdhury, and K. Nishinari
u(2) u(3)
u(1)
Figure 3.2 Dynamics of the zero-range process: each site of the lattice can be occupied by an arbitrary number of particles. The departure rate u(n) of a particle depends only on the number n of particles at the departure site, not on that at the target site.
arbitrary number nj of particles (Fig. 3.2). Using random-sequential dynamics, in each update step, a particle is chosen at random, say on site j, which is allowed to move to any other site l. The essential point now is that the corresponding hopping rate wjl is independent of the state of the target site l, i.e., the number of particles present at that site. It might only depend on the number of particles nj on the starting site j. Therefore, the rate can conveniently be denoted by u(nj ). This property motivates the name ZRP since the range of interaction can be interpreted to be zero. Obviously, this only works if the number of particles allowed at each site is not restricted. Any restriction would imply that the hopping rate cannot be independent of the state of the target site since one has to check whether there are less particles than the maximally allowed number. If this number is reached, the hopping rates from any other site to this site vanish.
3.3.1. Exact Solution The ZRP has the important property of being exactly solvable. In fact, its steady state is given by a simple factorized form, where the probability P(n) of finding the system in a configuration n = (n1 , n2 , . . ., nL ) is given by a product of (scalar) factors f (nl ):
P(n) =
1
L
ZL,N
l=1
f (nl ).
(3.15)
The normalization ZL,N , which is sometimes called (canonical) partition function, ensures that the probabilities for all configurations containing N particles sum up to one:
ZL,N =
L n l=1
f (nl ) δ
L
nl − N ,
(3.16)
l=1
where the Kronecker delta guarantees that only configurations containing N particles are included.
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Particle-Hopping Models of Transport Far from Equilibrium
In order to prove the factorization property (3.15) and determine the factors f (nl ), we insert this as an Ansatz into the stationary master equation: L
u(nl−1 +1)P(. . ., nl−1 +1, nl −1, . . .) =
l=1
L
u(nl )P(n).
(3.17)
l=1
In the steady state, the probability current due to hops into a particular configuration n [left-hand side of (3.17)] balances the probability current due to hops out of the same configuration [right-hand side of (3.17)]. The choice u(0) = 0 takes into account that a site has to be occupied before a particle can move away from it.1 Now we substitute the factorized probability (3.15) into the master equation (3.17) and equate each term in the sum separately. After cancelling common factors, this leads to the conditions u(nl−1 + 1) f (nl−1 +1) f (nl −1) = u(nl ) f (nl−1 ) f (nl ),
(3.18)
which can be rewritten in the form u(nl−1 +1)
f (nl−1 +1) f (nl ) ! = constant, = u(nl ) f (nl−1 ) f (nl −1)
(3.19)
for all values of l. The equality of the first two terms implies that they are constant since the first term depends only on nl−1 , whereas the second one depends only on nl . The constant can be set equal to unity without loss of generality (by rescaling time and thus the rates) so that we obtain the iteration f (nl ) =
f (nl −1) . u(nl )
(3.20)
Setting f (0) = 1, again without loss of generality, this iteration is solved by n 1 f (n) = u( j)
for n > 0.
(3.21)
j=1
Thus, we have shown that the factorization (3.15) indeed holds if we choose the factors according to (3.21). This is also true for other types of update. The case of parallel dynamics will be considered explicitly in Section 3.6. A derivation of the stationary state based on the quantum formalism has been given in [161]. 1 It is also assumed that P(n , . . ., n ) = 0, if at least one of the arguments is negative. 1 L
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Alternatively one could also start with prescribing the stationary state by defining the function f (n) and then determine the corresponding hopping rates through u(n) =
f (n−1) . f (n)
(3.22)
This shows that the ZRP gives a complete account of all the possible steady-state behavior in particle-conserving models with factorized steady state. Knowing the explicit form (3.15), (3.21) of the steady-state probabilities, one can calculate all stationary properties of the ZRP, at least in principle. One important quantity that helps to understand the structure of the stationary state is the probability p(n) that a given site contains n particles. Fixing the occupation of site j = 1 to be n, one has
p(n) =
P(n, n2 . . .nL ) δ
n2 ,...,nL
L
nl − (N − n) ,
(3.23)
l=2
where the Kronecker delta takes into account that the remaining sites contain only N − n particles. Using the factorization (3.15), one finally obtains p(n) = f (n)
ZL−1,N −n . ZL,N
(3.24)
Note that this probability p(n) to find n particles at a given site is different from the single-site weight f (n). This is a consequence of the fact that the constraint of fixed particle number induces correlations between sites, although the steady state factorizes. To determine the partition function ZL,N on a computer, the following iteration is very useful: ZL,N =
N
f (n)ZL−1,N −n,
(3.25)
n=0
with Z1,N = f (N ). It can be easily derived by summing (3.24) over n. An explicit expression for the partition function in terms of the Lauricella hypergeometric function has been derived in [704]. Another interesting quantity that can be obtained straightforwardly is the mean hopping rate u(n) , which coincides with the particle current or flow only in the case of
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Particle-Hopping Models of Transport Far from Equilibrium
totally asymmetric dynamics. Using (3.20), we find u(n) =
1
ZL,N
n1 ,...,nL
u(n)
L
f (nl ) δ
l=1
L
nl − N
l=1
ZL,N −1 . = ZL,N
(3.26)
Thus, the mean hop rate is independent of the location in the system. Note that in the case of symmetric dynamics, u(n) remains finite although the current vanishes. The average velocity of the particles in the steady state is defined by vL,N =
N
u(n)p(n).
(3.27)
n=0
Using (3.20), (3.24), and (3.25), one finds that it is identical to u(n) . In [704], the average velocity has been calculated for various choices for the transition rates. In the generic case for a transport model, where u(0) = 0 and u(n) = 1 for n > K where K is some integer parameter, the velocity could be expressed by the Lauricella hypergeometric function. Further traffic-related results are discussed in [719] and in Section 9.6.1.
3.3.2. Bethe Ansatz Solution The solution given in the previous subsection does only provide information about the steady state. In order to determine dynamical properties, e.g., relaxation times, the full master equation ∂Pt (n) = ∂t
L
u(nk−1 + 1)Pt (. . ., nk−1 + 1, nk − 1, . . .) − u(nk )Pt (n)
(3.28)
k=1,nk =0
for the time-dependent probabilities Pt (n) has to be solved. For a solution by Bethe Ansatz (see Section 2.3), it is more convenient to specify a configuration of the system not by the occupation numbers n = (n1 , . . ., nL ) (Euler representation), but by the coordinates x = (x1 , . . ., xN ) of the N particles (Lagrange representation). Obviously these two descriptions are completely equivalent. The master equation in the coordinate description becomes more complicated. It allows for a Bethe Ansatz solution, but only under certain conditions [1130]. Technically, it is easier to determine the function Pt(0)(x1 , . . ., xN ) which is defined by Pt (x1 , . . ., xN ) =
L l=1
(0)
f (nl ) Pt (x1 , . . ., xN ),
(3.29)
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A. Schadschneider, D. Chowdhury, and K. Nishinari
where the first factor is, up to a normalization, the stationary measure (3.15). It is of the typical Bethe Ansatz form
(0)
Pt (x1 , . . ., xN ) = eλt
A()
∈SN
N
−xj
z( j) ,
(3.30)
j=1
where z1 , . . ., zN are complex numbers that need to be determined, the summation is taken over all N ! permutations = ((1), . . ., (N )) of the numbers (1, . . ., N ), and the coordinates of particles are in increasing order x1 ≤ x2 ≤ · · · ≤ xN . By inserting the Ansatz into the master equation (3.28), one finds that it only works for rates satisfying the recurrence relation [1130] u(n) = 1 − (1 − u)u(n − 1)
(3.31)
where, without restriction, u(1) = 1 and u(2) = u have been chosen. The explicit solution of this recurrence has the form u(n) = [n]q :=
1 − qn 1−q
with q = u − 1.
(3.32)
The parameters in the Ansatz (3.30) are given by λ=
N
zi − N ,
(3.33)
i=1
and the zj have to satisfy the Bethe Ansatz equations [1130] zj−L
= (−1)
j−1
j (2 − u) − (1 − u)zl − zj l=1
(2 − u) − (1 − u)zj − zl
,
(3.34)
which follow from the condition of compatibility of periodic boundary conditions. In order to obtain information about the long-time behavior of the dynamics, one has to solve the Bethe Ansatz equations and look for the solutions with the smallest λ [see Eqn (3.33)]. Solutions of (3.34) are obtained by assuming that in the thermodynamic limit (L → ∞, N → ∞, N /L = ρ), the roots of the Bethe Ansatz equations are distributed along a continuous contour in the complex plane. Then, these equations can be transformed into a single integral equation, which, at least in some special cases, can be solved analytically. The Bethe Ansatz solution as discussed here only works for hopping rates of the form (3.32). In the limit q → 1, the q-numbers become [n]q=1 = n. This corresponds to the
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Particle-Hopping Models of Transport Far from Equilibrium
diffusion of noninteracting particles, and the Bethe Ansatz equations decouple in this limit: zj−L = 1. For q > 1, the rates grow exponentially with n, and the effective interaction between the particles accelerates their diffusion. For q → ∞, the model becomes equivalent to the drop-push model [19, 1268] (see Section 4.8.2). For 0 < q < 1, the rates u(n) grow monotonously from u(2) = q + 1 to u(∞) = 1/(1 − q). Here, the interaction slows down the diffusion. The case q = 0 can be mapped on the TASEP by inserting an additional bond in front of every particle.
3.4. EXTENSIONS AND GENERALIZATIONS 3.4.1. Parallel Dynamics For parallel dynamics, where all sites are updated simultaneously, the transition probabilities from a configuration n to a configuration n are given by ⎡ ⎤ 1 1 L
⎣ u(nj )δ nj , nj − νj + νj−1 ⎦ . ··· W (n → n) = (3.35) ν1 =0
νL =0
j=1
Here, νj (= 0, 1) particles hop to the next site with probability u(nj ). The argument of the Kronecker delta reflects that after the transition, the number of particles at site j has changed from nj to nj = nj − νj + νj−1. The balance of probability currents for each configuration n in the steady state implies [W (n → n)P(n ) − W (n → n )P(n)] = 0, (3.36) n
where the first term is the current flowing into a configuration n from other configurations n , and the second one is the current flowing out of n. The steady-state probability P(n) is again given as a product (3.15) of single-site weights (marginals) f (n). As in the random-sequential case, this can be confirmed directly, and the recursion u(n + 1)f (n + 1) = f (n) − u(n)f (n)
(3.37)
leads to the single-site weights ⎧ ⎪ ⎨1 − u(1) n f (n) = 1 − u(1) 1 − u( j) ⎪ ⎩1 − u(n) u( j) j=1
(for n = 0) (for n ≥ 1).
(3.38)
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The average velocity vL,N now satisfies the recursion vL,N +1 ZL,N +1 = ZL,N − vL,N ZL,N , which is solved by vL,N = −
(3.39)
N −1
(−1)n ZL,n . (−1)N ZL,N
n=0
(3.40)
3.4.2. Other Lattice Structures Many of the results of the previous sections can be generalized to arbitrary lattice structures (in any dimension) if certain restrictions on the hopping rates are satisfied. The steady state of a ZRP on an arbitrary lattice [357] factorizes if the hopping rate ukl (nl ) from site l to k is of the form [363] ukl (nl ) = ul (nl )Wkl .
(3.41)
Here, ul (nl ) is a site-dependent function that can be interpreted as the total rate at which a particle leaves site l if l is occupied by nl particles. Then, Wkl is the probability that the particle hops to site k. The conservation of probability implies that these Wkl define a stochastic matrix, i.e., Wkl = 1. (3.42) k
Up to now, no lattice structure has been introduced explicitly. This can be done by defining a connectivity through the matrix Wkl : only sites k, l with Wkl = 0 are connected by a bond such that particles may hop from site l to k. The hopping does not need to be symmetric if Wkl = Wlk , and the rates need not to be heterogeneous. Under these conditions, the steady state is given by a generalization of (3.15) with site-dependent factors fl given by fl (nl ) =
nl sl ul ( j)
(3.43)
j=1
for nl > 0 and fl (0) = 1. The weights sl are given by sk Wlk , sl =
(3.44)
k
i.e., the steady-state weights of a single random walker moving on a lattice with rates Wkl .
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Particle-Hopping Models of Transport Far from Equilibrium
This general result includes the partially asymmetric one-dimensional ZRP where particles move to the right with rate ul+1,l (n) = pu(n) and to the left with rate ul−1,l (n) = qu(n). This asymmetry leads to the probabilities Wl+1 l = p
and
Wl−1 l = q.
(3.45)
and, via (3.44), the weights sl = 1. Thus, the steady-state weights (3.43) are identical to those of the totally asymmetric case [see (3.21)]. These general results can also be applied to one-dimensional systems with interactions beyond nearest neighbors and ZRPs on higher-dimensional lattices [363]. For studies of the ZRP on various types of networks, see, e.g., [914, 1071, 1072, 1354, 1421, 1422].
3.4.3. ZRP with Disorder The solution of the ZRP can be extended to include quenched disorder, i.e., sitedependent rates ul (n) [see (3.41)]. In one dimension, the totally asymmetric case has been studied in [648]. In this case, a dynamical phase transition takes place from a low-density phase, where one observes the condensation of holes, to a homogeneous high-density state. At low densities, the average speed of the particles is determined by the lower cutoff of the effective hopping rates. The properties of the phase transition depend on the asymptotic behavior of the hopping rate distribution near the lower cutoff. The dynamics in the symmetric case has been studied in [81]. A generalization to the partially asymmetric case can be found in [697, 698]. A ZRP with open boundaries and disorder has been investigated in [1151]. Grosskinsky et al. [468] have studied the influence of quenched disorder on the condensation transition. They could show rigorously that the disorder changes the critical exponent in the interaction strength below which condensation may occur (see Section 3.5.1) from σ = 1 to σ = 1/2. The ZRP can be mapped onto an ASEP where the disorder in the ZRP is transformed into particle dependent hopping rates. The physics of the ASEP with inhomogeneous hopping rates will be discussed in more detail in Section 4.6.1.
3.4.4. ZRP with Fluctuating Particle Number 3.4.4.1. Open Boundary Conditions So far, we have consider ZRP with periodic boundary conditions. In this case, the particle number is conserved. The steady-state properties of the ZRP with open boundary conditions have been studied in full generality in [859] for the case of partial asymmetry. The model is defined in a one-dimensional lattice of L sites (Fig. 3.3). Considering a site j in the bulk which contains n particles, one of these moves with rate pu(n) to the right and with rate qu(n) to the left. At site 1, a particle is injected with rate α and removed
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A. Schadschneider, D. Chowdhury, and K. Nishinari
α
δ qu(n)
γ u(n) pu(n)
1
pu(n)
β u(n)
qu(n)
2
L
Figure 3.3 Definition of the partially asymmetric ZRP with open boundary conditions.
with rate γ u(n). Furthermore, particles on site 1 can move to the right with rate pu(n). The dynamics at the last site L is analogous: particles are inserted with rate δ, removed with rate βu(n), and move to the left with rate qu(n). The stationary state is given by a product measure P(n1 , . . ., nL ) = P(n1 ) . . .P(nL ) [859] where P(nj = n) =
n zjn 1 Zj u(n)
(3.46)
l=1
with site-dependent fugacities zj . Zj is the local analog of the grand-canonical partition function Zj = Z(zj ) =
∞ n=0
zjn
n 1 . u(n)
(3.47)
l=1
In terms of the rates, the fugacities are given by p L−1 p j−1 − γ δ + αβ q [(α + δ)(p − q) − αβ + γ δ] q . zj = p L−1 γ (p − q − β) + β(p − q + γ ) q
(3.48)
In the bulk, it is given, up to exponentially small corrections in L, by zeff = α/(p − q + γ ). For the totally asymmetric case q = 0, the fugacities simplify to zj = α/(p + γ ) for j = L and zL = ((α + δ)p + γ δ)/(β(p + γ )). The current is given by p L−1 −γ δ + αβ q (3.49) J = (p − q) p L−1 . γ (p − q − β) + β(p − q + γ ) q In the totally asymmetric case q = 0, this reduces to J = pα/(p + γ ). The existence of a steady state for given u(n) is determined by the radius of convergence of Z. If some zj are outside of this radius, condensation occurs in the system.
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Particle-Hopping Models of Transport Far from Equilibrium
In [859], the ZRP with rates of the form u(n) = 1 +
b n
(3.50)
was investigated. Later in Section 3.5.3, we will see that the corresponding periodic system exhibits a condensation phenomenon at large densities for b > 2 where a macroscopic proportion of particles accumulates on a single site. For the open system, it was shown in [859] that for a weak boundary drive, the system reaches a steady state given by the product measure. If the boundary drive is sufficiently strong, a condensate may develop on one or both boundary sites, even for the case b < 2 where no bulk condensation occurs in the periodic system. The number of particles in the condensate increases linearly in time. Further investigations of the open ZRP can be found in [496], where the current fluctuations were studied, and in [1151], where the influence of disorder was considered. 3.4.4.2. Langmuir Kinetics Another extension of the ZRP that violates the conservation of the total number of particles allows for particle creation with rate c(nj ) and annihilation with rate a(nj ) at bulk sites j, sometimes called Langmuir kinetics. Choosing the rates such that the condition a(n)z = u(n)c(n − 1) is satisfied [363] implies that the nonconserving dynamics obeys detailed balance with respect to the steady state P(n) =
L 1 nl z f (nl ) , Z
(3.51)
l=1
where f (n) satisfies the factorization condition f (n) = f (n − 1)/u(n). Then, the special choice a(n) = u(n) and c(n) = z generates in the steady state the grand-canonical ensemble. In [32, 33], different choices for the creation and annihilation rates were used, namely c(n) =
1 , Ls
and a(n) =
n k l
(n),
(3.52)
where denotes the Heaviside step function and the positive indices k and s. The annihilation rate increases with the number of particles at the site and thus provides a mechanism to suppress condensation. For the rates u(n) = 1 + b/n, a rich phase diagram with five distinct phases is found [33]. In [914], a ZRP-type model with creation-annihilation processes was studied on a network. There are other variations of the ZRP with nonconserved particle number,
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A. Schadschneider, D. Chowdhury, and K. Nishinari
e.g., by allowing both the particle number N and the number of sites L to fluctuate, with N + L kept constant [363]. The case in which only the number of sites is allowed to fluctuate by creating (annihilating) vacant sites with rate w (rate 1) in addition to the usual ZRP dynamics is related to lattice gravity and was studied in [114].
3.4.5. Generalizations A natural generalization of the ZRP is the inclusion of different particle species. We briefly review some important results for two-species models on a one-dimensional lattice of L sites with periodic boundary conditions. The number of A and B particles at site j is denoted by nj and mj , respectively. The total number of particles is N + M with N = Lj=1 nj and M = Lj=1 mj. Particles move to site j + 1 with rates u(nj , mj ) and v(nj , mj ). Like the one-species ZRP, the two-species models also have a factorized steady state [362, 472] of the form P(n, m) =
L 1 f (nj , mj ) Z j=1
(3.53)
provided that the hopping rates satisfy the constraint u(nj , mj ) v(nj , mj ) = u(nj , mj − 1) v(nj − 1, mj )
(3.54)
for nj , mj = 0. There are no constraints on u(nj , 0) and v(0, mj ). The single-site weights are given by 1 1 l = 1m f (n, m) = j = 1n (3.55) u(i, m) v(0, l) with f (0, 0) = 1. Although not obvious, the rates u and v play symmetric roles in (3.55). These results can be extended, as in the one-species case, to arbitrary lattices and inhomogeneous hopping rates. In the general case of an arbitrary number of species, constraints of the form (3.54) have to be satisfied by each pair of species [472].
3.4.6. Dynamical Urn Models The ZRP can be considered a special case of dynamical urn models [440], which are sometimes also called migration processes. After picking a departure site j at random, one of its neighboring sites l is chosen randomly as target site. A bias can be introduced if not all sites are chosen with the same probability. Then, a particle moves from j to l at a rate Wk,l = W (nj , nl ) that only depends on the occupations nj and nl of departure and target site. Obviously the rates W (0, nl ) = 0 have to vanish since no particle is present at the starting site. The ZRP is recovered if Wk,l = uk with u0 = 0.
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Particle-Hopping Models of Transport Far from Equilibrium
Another special case of the dynamical urn model leads to a model which is in some sense dual to the ZRP [890]. This process, which is characterized by Wk,l = (1 − δk,0)vl ,
(3.56)
is called target process. Here, the transition rates depend only on the occupation of the target site. Therefore, the roles of target and departure site are interchanged compared to the ZRP. However, this is not entirely true. The factor 1 − δk,0 in (3.56) implies a dependence on the departure site to guarantee the vanishing of the rates W0,l . For the one-dimensional case, where the right neighbor is chosen with probability p as a target site and the left neighbor with probability 1 − p, the stationary state can have a product measure form. For symmetric dynamics, i.e., p = 1/2, the rates have to satisfy the pairwise balance condition (1.15) pk+1 pl Wk+1,l = pk pl+1 Wl+1,k .
(3.57)
The stationary state is an equilibrium state. The asymmetric case p = 1/2 leads to a genuine nonequilibrium steady state which is of product form if Wk,l − Wk,0 = Wl,k − Wl,0.
(3.58)
The symmetric target process always has a stationary state of product measure form, irrespective of the choice of the rates vl . For the asymmetric case, the condition (3.58) implies that a product measure is only realized if v0 for k = 0, (3.59) vk = v for k ≥ 1.
3.4.7. Misanthrope Process The misanthrope process is characterized by hopping rates u(nl , nl+1 ), which depend on the occupancies of departure and target site. It has a factorized stationary state [231, 363] of the form (3.15) with weights n f (1) n u(1, j − 1) (3.60) f (n) = f (0) f (0) u( j, 0) j=1
if the hopping rates (for all n and m > 0) satisfy the condition u(n, m) − u(n + 1, m − 1)
u(1, n)u(m, 0) = u(n, 0) − u(m, 0). u(n + 1, 0)u(1, m − 1)
(3.61)
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A. Schadschneider, D. Chowdhury, and K. Nishinari
Alternatively two separate conditions u(1, n)u(m, 0) u(n + 1, 0)u(1, m − 1) u(n, m) − u(m, n) = u(n, 0) − u(m, 0) u(m, n) = u(n + 1, m − 1)
(3.62) (3.63)
can be imposed. The misanthrope process can be mapped to an exclusion process, similar to the relation of the ZRP and the ASEP, which we will discuss in detail in Section 4.1.4. It turns out to be equivalent to a special case of the Katz–Lebowitz–Spohn (KLS) model [716]. The KLS model is an exclusion process where the hopping probability depends on four sites and will be discussed in more detail in Section 4.8.4.
3.4.8. Relation of ZRP to Other Models and Some Applications The ZRP has been applied to a variety of different phenomena and problems. In Part II, we will see that the condensation phenomena observed in the bus-route model (Section 8.8) and the ant-trail model (Section 12.4) are well approximated by a ZRP. There are other condensation phenomena that can be described by a ZRP. Particles suspended in a fluid flowing through a narrow pipe can become weakly attached to the pipe which reduces their mobility and can finally lead to clogging. This can be described by a model that is related to the bus-route model by a particle-hole transformation [1082]. The ZRP has also been used to study growing and rewiring networks (for reviews on the theory of networks, see, e.g., [10, 319]). In growing networks, nodes and links are added, whereas in rewiring networks (or equilibrium networks), links are deleted and reattached stochastically such that the number of nodes and links is fixed. In such dynamical networks, condensation, also called gelation in this context, occurs when a node captures a finite fraction of the total number of links. For the case of rewiring networks, there is a close relation with the condensation phenomena found in the ZRP [32, 34, 363, 1153]. A relation of certain growing networks with the ZRP has been established in [943]. Another area of application is shaken granular matter, which shows interesting phenomena like pattern formation [43, 267, 561, 637]. A simple but fascinating example is a vertically shaken box that consists of L connected compartments (for a review, see [1406]). Particles can move through slits at some height h to one of the neighboring compartments (Fig. 3.4). For strong driving, particles are evenly distributed over the compartments, but below a critical driving strength, they will cluster in one of the compartments. The dynamics of the system can be described approximately by a ZRP [1374] where each compartment corresponds to a site. Different forms for the transition rates u(n) have been proposed [339, 877], which depend on the total particle number N .
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h
Figure 3.4 A container with L = 7 compartments and periodic boundary conditions connecting the leftmost and rightmost compartments. If it is shaken vertically, particles can move from one compartment to a neighboring one through holes at height h.
Upon variation of the system parameters (e.g., the effective temperature) that determine the transition rates, the ZRP model shows a phase transition from a phase where the particles are homogeneously distributed among the compartments to a phase in which one compartment contains (almost) all particles. For the case L = 2, this transition is of second order [339, 877, 1290], whereas for L > 2, it becomes first order [239, 1291, 1404, 1405, 1408]. For example, generalizations to the case of a bidisperse granular gas [933] or holes at different heights [1403] have also been investigated. Other applications of the ZRP are macroeconomics [166] and traffic flow [719, 1373]. In [1407], it has been proposed to consider highways as made up of compartments bounded by inductive loops. Shaken compartmentalized systems are not the only application of the ZRP to granular matter. It has also been used to study sandpile models [179, 647] in the context of self-organized criticality. The model proposed in [647] is a fixed energy sandpile model where one particle moves out of a site if the number of particles at that site exceeds a threshold. It shows a active-absorbing phase transition. The active high-density phase is equivalent to a ZRP, which allows to obtain exact results. The backgammon model has been developed as a simple model for glassy dynamics [441, 1189]. It is a generalization of the Ehrenfest urn model [442] to finite temperatures and many urns such that in the late time regime and at low temperatures entropic barriers and slow dynamics arise. In the backgammon model, a unit mass hops under dynamics, which satisfies detailed balance with respect to the energy function E(n) = −δn,0 , i.e., the total energy is given by minus the number of occupied sites. This corresponds to a misanthrope process with hopping rates u(n, m) = 1 + e−β − 1 δm,0
for n > 0,
(3.64)
where β is the inverse temperature. Then, condition (3.62) is satisfied, and the system has a factorized steady state. In the low temperature limit T → 0, the steady state is dominated by a condensate where all masses are at the same site.
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3.5. PHYSICS OF THE ZRP After we have discussed the basics of the ZRP process and derived the product structure of its stationary state, we now discuss its physics in more detail. Although the steady state factorizes, i.e., is of mean-field type, its physics is far from trivial.
3.5.1. Condensation Transition The simple structure of the steady state allows to analyze the physical properties of the ZRP exactly thus providing valuable general information relevant for more complex models and applications. The main property that has attracted the attention of many researchers is the possibility of condensation. Here, condensation means a transition to a phase where one site (or a finite number of sites) contains a finite fraction of particles in the system. It occurs in very different contexts (see Section 3.4.8) ranging from gelation, coalescence in granular systems, to wealth condensation in macroeconomics. Later, we will discuss applications to jamming in traffic in more detail. A more general result is a criterion for phase separation [701], which is extremely useful if a given system can be described effectively by a ZRP. This will be discussed in more detail in Section 3.5.3. We now sketch the derivation of the conditions for condensation in a homogeneous ZRP with site-independent hopping rates u(n). Usually one would try to determine the distribution of the number of particles p(n) to demonstrate the occurrence of condensation. Such an analysis involves the (canonical) partition function ZL,N defined in (3.16). However, technically it is simpler to work in a grand-canonical ensemble by allowing fluctuations of the particle number N . In analogy to equilibrium statistical physics, one defines the grand-canonical partition function as ∞
ZL (z) =
zN ZL,N .
(3.65)
N =0
The fugacity z fixes the particle density ρ through ρ=
z ∂ ln ZL (z) . L ∂z
(3.66)
Using the definition (3.16), one obtains ZL (z) =
∞ {nl =0}
z
l nl
L l=1
f (nl ) = [F(z)]L ,
(3.67)
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where the function F(z) is defined by F(z) =
∞
zm f (m).
(3.68)
m=0
This yields an explicit relation between density and fugacity: ρ =z
F (z) . F(z)
(3.69)
The distribution of the number of particles at a given site becomes in the grand canonical ensemble p(n) = zn f (n)
ZL−1 zn f (n) = . ZL F(z)
(3.70)
Since the right-hand side of (3.69) is a monotonically increasing function of z, the maximum density ρc consistent with (3.69) is determined by the radius of convergence β of F(z) as defined in (3.68): ρc = β
F (β) . F(β)
(3.71)
Condensation then occurs if ρc is finite and ρ > ρc because (3.69) can no longer be satisfied. The excess density ρ − ρc will form the condensate. In fact, it can be proven rigorously that the condensate corresponds to a single site [470]. This can also be seen by an intuitive argument based on the scaling of the canonical weights with the system size [363]. The occurrence of condensation is connected to the asymptotic behavior of f (n). To be specific, we assume the asymptotic form f (n) ∼
A β n nb
(3.72)
so that indeed β is the radius of convergence of (3.68). Since F (β) converges for b > 2, ρc will be finite in this case and condensation occurs. This can also be seen from the asymptotic behavior of the single-site occupation p(n) [see (3.24)], which is given by p(n)
(z/β)n A . F(z) nb
(3.73)
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For b > 2, the power law 1/nb has a finite mean, and even the limit z → β corresponds to a finite density. Thus, a condensate must form. For b ≤ 2, on the other hand, one can achieve any density ρ = n np(n) by choosing z appropriately. Usually it is more convenient to specify the condition for condensation in terms of the hopping rates which define the stochastic process. This leads to the following condensation criterion for the ZRP: Condensation occurs if the hopping rates decay more slowly than 2 u(n) β 1 + for n 1. n
(3.74)
The corresponding asymptotic form of f (n) is given by (3.72). If u(n) → 0 as n → ∞, condensation occurs at all densities.2 If u(n) decays to a nonzero constant, a single condensate is observed above a nonzero critical density coexisting with a background fluid at the critical density.
For increasing u(n), the radius of convergence is infinite, and condensation never occurs. The generic rates for which the condensation criterion (3.74) is satisfied are of the form b (3.75) u(n) = a 1 + σ n with either σ <1 and b > 0 or σ = 1 and b > 2. The corresponding stationary weights decay as bn1−σ for 0 < σ < 1 , (3.76) f (n) ∼ exp − 1−σ f (n) ∼ n−b
for σ = 1 .
(3.77)
In Section 3.5.3, we will discuss how the condensation criterion can be used in a larger context to understand the occurrence of phase separation in systems which can effectively be described by a ZRP. The condensation transition can also be discussed in the canonical ensemble. Using Cauchy’s theorem, the partition functions are related via ds −(N +1) ds −(N +1) ZL (s) = [F(s)]L , (3.78) s s ZL,N = 2π i 2π i where the integral is around a closed contour about the origin in the complex s plane. For large L and N , the integral is dominated by a saddle point z, which exactly corresponds to the condition (3.69). 2 In this case, the radius of convergence is zero (β = 0).
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One can analyze the condensate in more detail in both the canonical and grandcanonical ensemble. In [470], it has been shown that for ρ > ρc , the fluid part of the canonical distribution coincides with the critical grand-canonical distribution. However, the condensate itself is only correctly described in the canonical ensemble [903]. An analysis of the part of p(n) that corresponds to the condensate shows that it can be described by distinct universal forms for 2 < b < 3 and b > 3 [903]. Finally, for illustration, we discuss the generic choice b (3.79) u(n) ∝ 1 + n in more detail [902]. For large n, it leads to a weight function f (n), which shows powerlaw behavior, f (n) ∼ n−b [see (3.72)]. For b > 2, the system undergoes a condensation transition if the density is increased beyond the critical value ρc =
1 . b−2
(3.80)
The phase diagram is shown in Fig. 3.5. The single-site probabilities show qualitatively different behavior in the two phases: ⎧ ∗ ⎪ for ρ < ρc ⎨exp −n/m −b (3.81) p(n) ∼ n for ρ = ρc . ⎪ ⎩ −b n + “condensate” for ρ > ρc It decays exponentially with a characteristic mass m∗ for ρ → ρc . At ρc , the characteristic mass diverges leading to power-law behavior, which remains for ρ > ρc , but all
6
ρc(b)
Condensed phase ρbulk = ρc
ρ
Fluid phase ρbulk = ρ
ρbulk
4
Fluid
Condensed
ρc
2
0
0
1
2
3 b (a)
4
5
ρc ρ (b)
Figure 3.5 Phase diagram of the ZRP with hopping rates u(n) ∝ 1 + nb . (a) Density in the bulk as function of b; (b) bulk density as function of the total density ρ for fixed b.
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the additional mass (ρ − ρc )L condenses onto a single site. This condensate shows up as a bump at the tail of the power-law form. Thus, a single condensate coexists with a background critical fluid for ρ > ρc . Due to the absence of interactions between neighboring sites, condensates in the ZRP always occupy a single site. Spatially extended condensates can occur in interacting models like models with pair-factorized steady states (PFSS) [365, 1424] (see Section 3.6.1). However, recently it was shown that a different kind of condensation phenomenon can occur in the ZRP [1361]. In this saturated condensation, an extensive number of finitesize condensates are formed in the steady state if certain conditions are satisfied by the hopping rates u(n). In the case where the rates depend not only on the particle number, but also on the system size, the phase transition becomes discontinuous, and the system exhibits metastable phases and breaking of ergodicity [469]. A surprising observation is the nonequivalence of ensembles in some regions of the phase diagram. These results are of relevance for the granular clustering described in Section 3.4.8. Another factor that can have a considerable influence on the condensation transition is correlated dynamics [571]. In the one-dimensional case, it can lead to extended condensates that occupy two adjacent sites and drifts with a finite velocity. In [1277], the transition rates u(n) were chosen in a nonmonotonic way. This can lead to the destabilization of a single condensate. Instead either a finite number of extensive condensates or a subextensive number of subextensive mesocondensates are formed. In both cases, a true condensation transition is observed where a finite fraction of the mass occupies a vanishingly small fraction of sites in the thermodynamic limit. More on condensation can be found, e.g., in the reviews [358, 902]. In Section 3.6, we will discuss condensation transitions in a slightly more general class of models.
3.5.2. Dynamics and Coarsening The dynamics of condensation, which is described by a nontrivial coarsening process, has also been investigated in some detail. Most of these studies are not rigorous, but rely mostly on heuristic arguments and Monte Carlo simulations. An extensive review can be found in [440]. Of special interest is the late-time coarsening of the excess mass into a decreasing number of growing clusters [439, 443, 470]. Then, a process of particle exchanges between the remaining clusters starts. In the steady state of finite systems, the condensate occasionally melts and then reforms at a different site [443, 470, 903]. Quantitative results have been derived, e.g., by a random walk argument [470] that estimates the time until a particle escaping from a condensate site reaches the next
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condensate site. It is found that the growth of the mean condensate size is determined only by the time needed until a condensate site looses all its particles. Thus, the mean condensate size m(t) , defined as the total number of particles at condensate size divided by the number of condensate sites at time t, scales as m(t) ∼ t δ
with
1 δ= . 2
(3.82)
The same behavior is found in the mean-field analysis of [439]. For asymmetric dynamics, it also agrees with the results from Monte Carlo simulations. For symmetric dynamics, the exponent changes to δ = 13 [470].
3.5.3. Criterion for Phase Separation As mentioned before, the ZRP can be used as an effective description that provides new insight into the question of phase separation in one-dimensional driven systems [701, 1276]. A quantitative criterion for the existence of phase separation in driven onedimensional systems that conserve density has been conjectured. The central idea is a correspondence between such systems and a ZRP, which is similar to mapping of the ASEP on the ZRP. In this mapping, clusters of particles or domains (i.e., a sequence of consecutive occupied sites) are represented by sites of a ZRP. The dynamics of these clusters is then described by effective hopping rates u(n) that depend on the cluster size n. Effectively this means that the clusters become uncorrelated and u(n) corresponds to a conserved current flowing out of the cluster. Thus, one effectively has to consider a single domain only, and the relevant information is encoded in the effective current u(n). Qualitatively one can expect coarsening if smaller domains exchange particles with the environment at a faster rate than larger domains. This can be quantified by the approach described above, which implies that the criterion for phase separation corresponds to the criterion for condensation within the ZRP [370, 701, 1276]. For condensation to occur, either the hop rates u(n) should decrease to 0 as n increases or else u(n) should decay more slowly to an asymptotic value β than β(1 + 2/n) [see (3.74)]. Returning to the original system, the criterion for phase separation is related to the size-dependence of the steady-state currents of domains, which is identified with the hopping rate u of the ZRP. Then, denoting the steady-state current that flows through a domain of length n by Jn , phase separation should occur if either Jn → 0 for n → ∞,
(3.83)
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or, in the case limn→∞ Jn = J∞ = 0, if b Jn J∞ 1 + σ n
for σ < 1, b > 0,
(3.84)
or Jn J∞
b 1+ n
for b > 2.
(3.85)
Practically, it is usually assumed that Jn is given by the current through an (open) system of length n in the steady state (although n fluctuates!). The last condition is especially relevant since in many models the current of a finite b domain of size n is to leading order of the form Jn J∞ 1 + n . For b > 0, longer domains grow at the expense of shorter ones since their current is smaller. Obviously this is a necessary condition for phase separation. The case Jn → 0 corresponds to strong phase separation [702]. The phase separated state consists of coexisting domains where density fluctuations are limited to finite regions around the domain boundaries. Phase separation is expected at any density (ρc = 0) in this case. The phase separated states in the case of nonvanishing J∞ are rather different. Here, phase separation exists only at high densities (ρc > 0), whereas for low densities, the system will be homogeneous. Density fluctuations will also occur in the bulk of macroscopic domains. Because of the similarities with Bose–Einstein condensation, this phase is often called condensed and one speaks of soft phase separation [1276]. The criterion for phase separation is extremely useful and has helped to clarify several controversial issues regarding the occurrence of phase separation in certain systems. In [701], the criterion has been used to argue that in certain models, no phase separation will occur in the thermodynamic limit. One prominent example is the AHR model, which will be discussed in Section 4.7.3, and a related two-lane model introduced in [788, 930]. The currents are of the form (3.85) with b = 3/2 in the AHR model and b ≈ 0.8 in the two-lane model. On the other hand, the KLS model [716] (see Section 4.8.4) has a parameter range where σ = 1 and b > 2 [702, 1276]. In [703], the origin of sharp crossovers and anomalously large correlation lengths has been investigated. It has been found that here higher order corrections to the current become relevant. It should be mentioned that phase separation is closely related to hysteresis and ergodicity breaking. These phenomena have been explored further in the present context in [1166, 1167].
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3.6. PARTICLE-HOPPING MODELS WITH FACTORIZED STATIONARY STATES We have seen that the ZRP has a factorized steady state. This implies that meanfield theory becomes exact. A natural question now is whether this can be extended to more general models of mass transport. This problem has been first studied for models with a continuous state space [372, 1524, 1528, 1529], i.e., the asymmetric random average process (ARAP) and its extensions (see Section 3.7). Here, we follow the argumentation of Evans, Majumdar, and Zia [372, 1524], but specialize to models with discrete state space. The goal is the derivation of the conditions under which a particlehopping model has a factorized steady state, i.e., described exactly by a mean-field theory. The class of models that will be considered is usually called mass transfer models. We will first study the case of a one-dimensional system of L sites with periodic boundary conditions and parallel (synchronous) dynamics. As in the ZRP, each site j can be occupied by an arbitrary number nj of particles, which in this context is usually referred to as “mass on site j.” The total number of particles N = Lj=1 nj is conserved. The dynamics is a generalization of the ZRP dynamics since now an arbitrary number μj of particles is allowed to move from site j to site j + 1 simultaneously. At each update step and each site j, the number μj of particles that are transferred to the right neighbor are drawn from a distribution φ(μj |nj ). This distribution will be called chipping function in the following. In [372, 1524], conditions on the chipping function φ(μj |nj ) that lead to the factorization of the steady state
F(n1 , . . ., nL ) =
L
f (nj )
(3.86)
j=1
into single-site weights f (nj ) just as in the ZRP case have been derived. The details of the calculation for a model with discrete states are given in Appendix 3.8.1. One finds that every (one-dimensional) mass transfer model defined by a chipping function φ(μ|n) that can be written in the form v(μ)w(n − μ) φ(μ|n) = n m=0 v(m)w(n − m)
(3.87)
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with arbitrary functions v and w has a factorized steady state of the form (3.86). This condition is necessary and sufficient [372]. The weight function is explicitly given by
f (n) =
n
v(m)w(n − m).
(3.88)
m=0
In the case of continuous time, which can be obtained in the limit dt → 0 from the discrete case, the corresponding hopping rates γ (μ|n) defining the process have to be of the form [372] γ (μ|n) =
v˜ (μ)w(n − μ) , w(n)
(3.89)
and the single-site weights in the steady state are given by f (n) = w(n).
(3.90)
A more practical form of the factorization condition (3.87) can be derived, which tells directly whether a given chipping function φj (μ|n) yields a factorized steady state. It can be seen in [1524] that the chipping function must satisfy the factorization test φ(μ + 1|n + 2)φ(μ|n) ! = R(n), φ(μ + 1|n + 1)φ(μ|n + 1)
(3.91)
i.e., the cross-ratios on the left-hand side (defined when all the φ’s are positive) are functions of n alone.3 Then, from (3.87), one has R(n) =
f (n + 1)2 , f (n) f (n + 2)
(3.92)
which implies the recursion f (n + 2) 1 f (n + 1) = . f (n + 1) R(n) f (n)
(3.93)
3 For the corresponding test for the case of random-sequential dynamics, φ(μ|n) has to be replaced by γ (μ|n) [1524].
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This recursion is easily solved by iterating it twice, which yields the explicit form of the single-site weights ⎡ ⎤ n n−2 j f (1) 1 ⎣ ⎦ f (n) = f (0) f (0) R(k)
j=0
for n ≥ 2.
(3.94)
k=0
As an example, we consider the ZRP with parallel dynamics for which φ(0|n) = 1 − u(n),
φ(1|n) = u(n),
φ(k|n) = 0 (for k > 1).
(3.95)
The crossratio (3.91) is only defined for the case μ = 0, and therefore, R(n) =
u(n + 2)(1 − u(n)) u(n + 1)(1 − u(n + 1))
(3.96)
is automatically a function of n alone. Thus, the steady state factorizes and the single-site weights for the ZRP with parallel dynamics are obtained from (3.94) as
f (1) f (n) = f (0) f (0)
n
(u(1))n 1 − u( j) , 1 − u(n) j=1 u( j) n
(3.97)
which is slightly more complex than the corresponding result (3.21) for randomsequential dynamics. An interesting case is the generalized ZRP where one or two particles are allowed to move simultaneously to the neighboring sites [461, 1524]. The probabilities for these processes are given by u1 (n) and u2 (n), respectively. Then, the factorization test (3.91) requires u2 (n + 1)(1 − u1(n) − u2 (n)) = A for n ≥ 1, u1 (n + 1)u1 (n)
(3.98)
where the constant A > 0 is independent of n [1524]. This recursion fixes u2 (n) in terms of A, u2 (1) and u1 (n). The single-site weights [1524] are then given by
f (1) f (n) = f (0) f (0)
n
n (u1 (1))n 1 − u1( j) − u2( j) , 1 − u1 (n) − u2 (n) j=1 u1 ( j)
which can be derived using (3.94).
(3.99)
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Another possible generalization is the inhomogeneous case where the chipping function φj (μ|n) depends on the site j. The (3.87) takes the form v(μ)wj (n − μ) φj (μ|n) = n m=0 v(m)wj (n − m)
(3.100)
with weights4 f (n) =
n
v(m)wj (n − m) .
(3.101)
m=0
An analogous generalization applies for the continuous time case. In [461], results for a class of generalized ZRP on periodic hypercubic lattices have been derived where ν ≤ νmax particles are allowed to move at same time. In this case, (3.89) is replaced by γq (μ|n) =
v˜q (μ)w(n − μ) w(n)
(3.102)
with some nonnegative v˜q (ν). The index q specifies the direction of hopping on the hypercubic lattice. An extension to arbitrary graphs has been derived in [374]. One possible application of these results is one-dimensional lattices where hopping processes beyond nearest neighbors are allowed. The possibility of a condensation transition has been studied in detail for the onedimensional case. In [903], a condition for the occurrence of condensation was derived. Two distinct condensate regimes exist where the condensate is either Gaussian or nonGaussian distributed. Here, the use of the canonical ensemble allows to analyze the condensation transition and the nature of the condensate in more detail [373]. In [371], the condensation transition was investigated from the point of view of extreme value statistics.
3.6.1. Models with Pair-Factorized Steady States Going beyond a simple factorization the next step is a steady state of two-cluster structure ( pair-factorized steady state (PFSS)). A generalization of the ZRP that leads to a PFSS has first been proposed in [365]. The hopping rate u of a particle from site j to site j + 1 not only depends on the number of particles at those two sites, but also depends on that of site j − 1. It therefore has the general form u(nj |nj−1 , nj+1 ). 4 Note the j dependence of w in comparison to (3.88).
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The stationary state probabilities for a configuration (n1 , . . ., nL ) of a one-dimensional periodic system with N particles are of the form
P(n1 , . . ., nL ) =
1
L
ZL,N
j=1
⎛ ⎞ L g(nj , nj+1 )δ⎝ nj − N ⎠
(3.103)
j=1
if the hopping rates factorize according to u(nj |nj−1, nj+1 ) = f1 (nj−1, nj )f2 (nj , nj+1 ) .
(3.104)
The functions f1 and f2 determine the pair-functions g through ⎛ ⎞−1 n m g(m, n) = ⎝ f1 (m, j) f2 (n, j)⎠ j=1
for m, n > 0 and g(0, 0) = 1
(3.105)
j=1
and have to satisfy the constraints f1 (m − 1, n) f2 (m, n − 1) = . f1 (m, n) f2 (m, n)
(3.106)
For any given form of the pair-function g(m, n), the corresponding f1 and f2 can be determined by a recursion [365]. This construction can be generalized to arbitrary graphs [1423]. The pair-function g(m, n) is the product of a local interaction factor K (|m − n|) and ultralocal factors p(n), which can be considered on-site potentials: g(m, n) = K (|m − n|) p(n)p(m) .
(3.107)
Here also long-range interactions are allowed, which correspond to power-law decay of K and p. In the ZRP, if condensation occurs, the ultralocal interactions lead to a condensate, which is strictly localized at a single site. In the PFSS models, interaction-driven condensation occurs, but with √ a spatially extended condensate [365]. In the simplest case, the extension scales as L, but it turns out that the shape of the condensate and its extension are nonuniversal [1424]. They can be tuned by the choice of ultralocal and local contributions to the weight factors.
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3.7. GENERALIZED MASS TRANSPORT MODELS So far, we have focussed on transport models that have discrete state variables. This is more suitable for the applications to traffic, which we will discuss in Part II. In the following, we will generalize the mass transport models in several directions. One possible extension is to allow for continuous state variables.
3.7.1. Models with Continuous States We will consider a rather general mass transport model, but restrict ourselves mostly to a one-dimensional lattice of L sites and periodic boundary conditions. Each site j contains a continuous mass mj . In each time step (or infinitesimal time interval), a certain part 5 mj of this mass is transported to the right neighbor. This implies that the total mass M = j mj is conserved. The dynamics is more precisely defined as follows (Fig. 3.6). At each time t, a mass 0 ≤ μi ≤ mi is chosen independently from a probability distribution φ(μi |mi ). This function is also called chipping kernel, and for simplicity, we will assume that it does not depend on the site j. It has to satisfy the obvious normalization condition m φ(μ|m) dμ = 1.
(3.108)
0
For the parallel update version of the model, the dynamics of the process is quantitatively described by mi (t + t) = mi (t) − μi (t) + μi−1 (t),
(3.109)
μi mi
i−2
i−1
i
i+1
Figure 3.6 Definition of the generalized transport model. 5 This corresponds to the totally asymmetric case with nearest-neighbor interactions. Generalizations to the partially
asymmetric case, larger interaction range, different lattice structures, and so on are obvious.
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where μi (t) is the loss of mass at site j due to transport to site j + 1 and μi−1 (t) the gain from site j − 1. The case of random-sequential dynamics can be obtained from the limit t → 0. Here, φ(μ|m) must generically be of the form ⎤ ⎡ m (3.110) φ(μ|m) = α(μ|m)dt + ⎣1 − dt α(μ |m)dμ ⎦ δ(μ), 0
which automatically satisfies the normalization condition (3.108). Here, α(μ|m) denotes the rate at which a mass μ leaves a site with mass m. The second term in (3.110) takes into account the probability that no mass leaves the site. The model class just defined contains several interesting models as special cases. For the choice α(μ|m) = u(m)δ(μ − 1)
(3.111)
with the convention u(m) = 0 for m < 1, we recover the ZRP. In the following, we briefly discuss two other special cases in more detail.
3.7.2. Asymmetric Random Average Process In the ARAP [816, 1158] at each time step, a random fraction rj mj of the mass mj on site j is transported to the next site j + 1. Here, rj ∈ [0, 1] is a random number chosen, independently for each site j, from a uniform distribution φ0 (r) = 1. For continuoustime dynamics (random-sequential update), this leads to the chipping rate [see (3.110)] α(μ|m) =
1 m
(for 0 ≤ μ ≤ m),
(3.112)
which is independent of μ and conserves the total mass M = ρL = j mj . The ARAP with discrete-time dynamics where all sites are updated simultaneously is closely related to the q-model, which describes force fluctuations in bead packs [229, 238, 881]. In [1158], it has been shown that for continuous-time dynamics, the joint distribution of masses P(m1 , m2, . . ., mL ) in the steady state is not of factorized product measure form. Surprisingly, for the case of parallel dynamics, it has been proved [816, 1158] that the joint distribution factorizes. In the large-L limit, the single-site mass distribution is of the exact scaling form m 1 P(m, ρ) = F , (3.113) ρ ρ
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which is preserved by the dynamics.6 The scaling function F(x) has been calculated explicitly. For random-sequential dynamics, it is given by 1 (3.114) e−x/2 , F(x) = √ 2π x which has been derived using mean-field theory [816, 1158]. For parallel dynamics, where mean-field theory is exact, one obtains the exact result F(x) = 4xe−2x .
(3.115)
These results for parallel dynamics have been generalized in [1528, 1529]. Here, instead of a uniform distribution φ0 (r) = 1 for random mass fractions arbitrary distributions, φ(r) were studied. It was found that exact product measure solutions P(m1 , . . ., mL ) = j P(mj ) exist if the distributions are so-called Beta densities [385] φa,b (r) =
1 r a−1 (1 − r)b−1 , B(a, b)
(3.116)
where a, b are positive real parameters, and the normalization constant B(a, b) is given by the Beta function B(a, b) ≡
(a)(b) . (a + b)
(3.117)
For these densities, the single-site mass distribution is given by the Gamma densities λλ 1 m λ−1 −λm/ρ e , (3.118) Pλ (m) = (λ) ρ ρ where the parameter λ is given by λ = a + b. Thus, the ARAPs with factorized steady state have the nice property of transforming Beta distributions into Gamma distributions. The ARAP does not show a condensation transition. However, by limiting the amount of mass that can be transferred from one site to its neighbor such a transition can be induced [1527]. In the truncated ARAP (tARAP), a phase transition between a homogeneous high flow phase and a mixed phase is observed. The latter consists of a homogeneous high flow and a condensed low flow substate without translation invariance. The finite system alternates between these substates, which both have diverging lifetimes in the thermodynamic limit, so ergodicity is broken in the infinite system. However, the scaling behavior of the lifetimes in dependence of the system size is different due to different underlying flipping mechanisms. 6 In contrast to the ZRP where the dynamics introduces an additional mass scale (unit mass) besides the density ρ.
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Particle-Hopping Models of Transport Far from Equilibrium
3.7.3. Chipping Model The dynamical variables mj of the chipping model, sometimes called (conserved) mass aggregation model, are integer parameters with mj ≥ 0. The dynamics consists of two separate processes. For the symmetric case and continuous-time dynamics, these processes can be characterized as follows: • Diffusion and aggregation: In an infinitesimal time interval dt with probability dt/2, the entire mass mj moves to its left neighbor site j + 1, and with probability dt/2, it will move to site j − 1. • Chipping: With probability w dt, only one unit of mass is chipped off from site j with mass mj and transported with probability w dt/2 to site j + 1 and with the same probability w dt/2 to site j − 1. Generalizations to asymmetric dynamics and discrete time are obvious. For asymmetric continuous-time dynamics, the chipping kernel is given by (3.110) with α(μ|m) = wδ(μ − 1) + δ(μ − m) .
(3.119)
The first term corresponds to the chipping process, and the second term corresponds to the transfer of the full mass. Unlike for the ZRP, the steady-state distribution P(m1, . . ., mL ) is not known exactly. For the symmetric case, it is believed [1159] that it is not of the factorized form (3.15). In the steady state, the symmetric chipping model undergoes a condensation phase transition at a critical density [904] ρc =
√
w + 1 − 1.
(3.120)
In fact, this remains true in any dimension [1159]. Below the transition density (ρ < ρc (w)), the mass is distributed homogeneously, and the mass distribution decays exponentially for large masses. At the critical density ρ = ρc (w), the mass distribution decays as a power law. For ρ > ρc , a condensate forms on a single site, which coexists with a critical background fluid [904]. The condensation contains the macroscopic mass (ρ − ρc )L. The single-site mass distribution P(m) is of the form (3.81). Mean-field theory gives the value b = 5/2 for the exponent [904], which is conjectured to be correct even for the one-dimensional case [1159]. In contrast to the ZRP, the condensation transition disappears in the asymmetric case although remnants can be observed in finite systems [1157]. However, in a generalization that includes both the chipping model and the ZRP, the condensation transition appears to survive even in the asymmetric case [860].
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Generalized mass transport model
Generalized ZRP
ARAP
Chipping model
Zero-range process NaSch model
ASEP SSEP
TASEP
Figure 3.7 Relation between the transport models discussed here. Also included is the generic cellular automaton model for modeling of vehicular traffic, the Nagel-Schreckenberg (NaSch) model, which will be discussed in Part II.
The chipping model has found various applications, e.g., in the context of traffic [862, 863], networks [595, 830], finance [1329, 1488], and the growth of nanoparticles [3]. Figure 3.7 illustrates the relation between various models discussed here.
3.8. APPENDIX 3.8.1. Derivation of the Factorization Criterion In the following, we determine all those distributions φ that lead to a factorized steady state in generalized particle-hopping models with discrete states. Denoting the weight of a configuration n = (n1 , . . ., nL ) at time t by F(n; t), the master equation for this process reads n
F(n; t + 1) =
j ∞ L
j=1 nj =0 μj =0
L
φ(μj |nj ) δ nj , nj − μj + μj−1 F(n ; t) ,
(3.121)
l=1
where the Kronecker delta takes into account that the net increase of the particle number at site j is given by μj−1 − μj . In the stationary state, one has F(n) = limt→∞ F(n; t). Now, we assume that the steady state factorizes F(n) =
L j=1
f (nj )
(3.122)
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into single-site weights f (nj ) just as in the ZRP case. The goal is to derive the conditions on the chipping function φ(μj |nj ) for which (3.122) becomes exact. Inserting the Ansatz (3.122) into the stationary limit of the master equation (3.121), one obtains after applying the z-transform (discrete Laplace transform) g(zj ) =
∞ nj =0
−nj
f (nj )zj
(3.123)
the following equation ⎡ L
g(zj ) =
j=1
L
⎢ ⎣
j=1
∞
nj =0
⎤
nj
f (nj )
μj =0
−(nj −μj +μj+1 ) ⎥
φ(μj |nj )zj
⎦.
(3.124)
Introducing the remaining mass σj = nj − μj ∈ {0, 1, . . .} at site j and the function P(μ, σ ) = f (n)φ(μ|n), this can be rewritten as L
g(zj ) =
∞ L ∞ j=1 μj =0 σj =0
j=1
−σj −μj zj+1 ,
(3.125)
= v(zj )w(zj+1 )
(3.126)
P(μj , σj )zj
which is satisfied if and only if ∞ ∞ μj =0 σj =0
−σj −μj zj+1
P(μj , σj )zj
with some arbitrary functions v and w that satisfy g(z) = v(z)w(z) .
(3.127)
This then gives the necessary and sufficient condition7 [372] v(μ)w(n − μ) φ(μ|n) = n m=0 v(m)w(n − m)
7 Take the logarithm of (3.125) and then derivatives with respect to z and z j j+1 .
(3.128)
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for factorization of the steady-state probabilities. This means that every (onedimensional) mass transfer model defined by a chipping function φ(μ|n) that can be written in the form (3.128) has a factorized steady state. In this case, the weight function is given by f (n) =
n m=0
v(m)w(n − m) .
(3.129)
CHAPTER FOUR
Asymmetric Simple Exclusion Process – Exact Results Contents 4.1. ASEP with Periodic Boundary Conditions 4.1.1. Random-Sequential Dynamics 4.1.2. Bethe Ansatz for Translationally Invariant Systems 4.1.3. Mean-Field Theories for Parallel Dynamics 4.1.4. Mapping to ZRP 4.1.5. Paradisical Mean-Field Theory 4.1.6. Combinatorial Solution for Parallel Dynamics 4.1.7. Ordered-Sequential and Sublattice-Parallel Updates 4.1.8. Shuffled Dynamics
111 111 113 116 123 123 124 126 128
4.2. ASEP with Open Boundary Conditions 4.2.1. Mean-Field Theory 4.2.2. Recursion Relations 4.2.3. Matrix-Product Ansatz 4.2.4. Exact Phase Diagram 4.2.5. Phase Transitions 4.2.6. Relation with Combinatorics 4.2.7. Bethe Ansatz 4.2.8. Dynamical MPA 4.2.9. Hydrodynamic Limit
131 132 134 135 136 141 142 142 143 143
4.3. Partially Asymmetric Exclusion Process 4.3.1. MPA Solution 4.3.2. Bethe-Ansatz Solution 4.3.3. Phase Diagram of the PASEP
145 146 148 148
4.4. Extension of the ASEP to Other Update Types 4.4.1. Ordered-Sequential Updates 4.4.2. Sublattice-Parallel Update 4.4.3. Parallel Update
151 151 154 154
4.5. Boundary-Induced Phase Transitions 4.5.1. Domain Wall Picture 4.5.2. Extremal Principle and Steady-State Selection 4.5.3. More on Shock Dynamics 4.5.4. Fluctuations and Large Deviation Functions
158 158 161 161 162
Stochastic Transport in Complex Systems DOI: 10.1016/B978-0-444-52853-7.00004-X
Copyright © 2011, Elsevier BV. All rights reserved.
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4.6. Extensions of ASEP 4.6.1. Quenched Disorder 4.6.2. Disorder in Open Systems 4.6.3. Langmuir Kinetics 4.6.4. Extended Particles 4.6.5. Other Boundary Conditions 4.6.6. Long-Range Hopping 4.6.7. ASEP Beyond One Dimension
163 164 173 174 177 177 179 180
4.7. Multispecies Models 4.7.1. Models with Second-Class Particles 4.7.2. ABC Model 4.7.3. AHR Model
181 181 183 184
4.8. Other Related Models 4.8.1. Staggered Hopping Rates 4.8.2. Two-Parameter Model 4.8.3. Restricted ASEP 4.8.4. KLS Model 4.8.5. Asymmetric Avalanche Process 4.8.6. Higher Velocities 4.8.7. Reconstituting Dimers
185 185 186 188 190 191 193 194
4.9. Appendices 4.9.1. Mapping of ASEP to Surface Growth Model 4.9.2. Mapping of the ASEP to an Ising Model 4.9.3. Solution of the Mean-Field Recursion Relations for the ASEP 4.9.4. Results Obtained from Normal-Ordering of Matrices 4.9.5. Dimension of Matrices in the MPA for the ASEP 4.9.6. Representations of the Matrix Algebra of the ASEP 4.9.7. Mean-Field Approximation of the DTASEP
195 195 196 197 199 200 201 205
As pointed out earlier, the asymmetric simple exclusion process (ASEP) can be considered to be the simplest possible stochastic transport model or the “mother of all traffic models.” It is a genuine nonequilibrium model, which belongs to the class of driven diffusive systems [1250, 1251, 1273]. One of its attractive properties is the fact that many exact results can be obtained. In the following sections, we will sketch how these results can be derived. At the same time, this allows us to demonstrate the application of several of the theoretical methods introduced in Chapter 2 to the specific case of the ASEP. The focus will be on the totally asymmetric simple exclusion process (TASEP), in which motion is allowed only in one direction. For the applications in Part II, this is the most relevant special case. The behavior of the partially asymmetric version (PASEP) is usually not very different and we will indicate the main results without going into the details of their derivation. Unfortunately, two different terminologies are commonly used in the literature. One calls the unidirectional variant of the ASEP as “TASEP,” whereas in the other, it is just “ASEP.” The variant where hopping in both directions is allowed is then either
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called just “ASEP” or “PASEP.” Here we will mainly use the second option although, sometimes, “TASEP” is also used as emphasis. The ASEP has also been studied intensively by mathematicians. Many rigorous results have been derived. A good overview can be found in the books [751, 872, 873, 1305]. It has also been studied in the general context of cellular automata theory [1476, 1478]. According to the Wolfram classification, it is known as rule-184 CA.
4.1. ASEP WITH PERIODIC BOUNDARY CONDITIONS First, we consider the simpler case of periodic boundary conditions that leads to a translational-invariant stationary state. We will focus on random-sequential dynamics, where the steady state is quite simple, and the parallel update, which is relevant for traffic models. Here the steady state is slightly more complicated.
4.1.1. Random-Sequential Dynamics Let us first consider the TASEP with periodic boundary conditions and randomsequential dynamics. In the following, L and N refer to the total number of sites and particles, respectively. The state of each site i is characterized by the occupation number ni such that ni = 0 if the site is empty and ni = 1 if it is occupied by a particle. For any given initial configuration n(0) = (n1 (0), . . ., nL (0)), we can write the equations governing the time evolution of ni (t) (and all the correlation functions) by taking into account all the processes, i.e., bond updates, during the elementary time interval dt. It is not difficult to establish that (for p = 1) [285, 289, 291] ⎧ ⎪ with probability 1 − 2dt ⎨ni (t) (4.1) ni (t + dt) = ni−1 (t) + ni (t) − ni−1 (t)ni (t) with probability dt ⎪ ⎩ with probability dt, ni (t)ni+1 (t) 1 and we have assumed that p = 1. The first equation reflects that during where dt = L+1 dt neither bond (i − 1, i) nor bond (i, i + 1) is updated (probability 1 − 2dt). With probability dt, bond (i − 1, i) is updated (second line) such that either nothing is changed or a particle moves from i − 1 to i. With probability dt, bond (i, i + 1) is updated (third line), and if site i is occupied, the particle moves to site i. Combining the terms in (4.1) gives
ni (t + dt) = ni (t)(1 − 2dt) + (ni−1(t) + ni (t) − ni−1(t)ni (t))dt + ni (t)ni+1 (t)dt, (4.2) which can be recast as dni = ni−1 (1 − ni ) − ni (1 − ni+1 ). dt
(4.3)
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After averaging, this leads to the equation dni = ni−1 − ni − ni−1 ni + ni ni+1 dt
(4.4)
for ni , the average occupation of the i-th site. Note that the equation for ni involves two-site correlations. Similarly, it is straightforward to see that the equations for the two-site correlations involve three-site correlations, and so on. Thus, the dynamics corresponds to a genuine N -body problem! In fact, one can write down the analogous equations for the pair correlation function [285, 291] dni ni+1 = ni−1 ni+1 − ni ni+1 − ni−1ni ni+1 + ni ni+1 ni+2 , dt
(4.5)
which involves three-points functions, and so on. It is easy to see that all states for fixed particle number appear with the same probability in the stationary state [285, 291, 924]. In a finite system, the probability distribution is then given by P(n) =
N ! (L − N )! . L!
(4.6)
This can easily be checked by interpreting the ASEP in terms of the dynamics of particle clusters. Due to exclusion, only the first particle of a cluster can move. The rate at which a given configuration is left is thus given by the number of clusters. If all configurations n = (n1 , . . ., nL ) are equally likely this is equal to the rate at which the system enters this configuration that corresponds to the motion of the last particles in the clusters [285, 291, 924]. The simple form of the stationary distribution function allows to calculate equal-time correlation functions in a straightforward way, even for finite systems. The probability to find a particle at a site is N /L. The probability to find another particle somewhere else is (N − 1)/(L − 1) and so on. Thus ni =
N , L
ni nj =
N (N − 1) , L(L − 1)
ni nj nk =
N (N − 1)(N − 2) . L(L − 1)(L − 2)
(4.7)
In the thermodynamic limit, this implies ni = ρ, ni ni+1 = ρ 2, and so on, i.e., meanfield theory is exact. Another quantity that can be used to characterize the stationary state is the cluster size distribution. Its main properties have been studied in [1152] where it was, e.g., found that the length of the longest cluster diverges logarithmically with the system size.
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Some information about nonequal-time correlations can be obtained from the diffusion constant D of a tagged particle1 [273, 829], which can be derived from Y (t), the number of jumps the particle has made up to time t. The velocity and diffusion constant are obtained from the large-time limit v = lim
t→∞
Y (t) , t
(4.8)
Y 2 (t) − Y (t)2 . t→∞ t
D = lim
(4.9)
Explicitly one finds [291, 296] v=
L−N , L−1
(4.10)
(2L − 3)! D= (2N − 1)! (2L − 2N − 1)!
(N − 1)! (L − N )! (L − 1)!
2 ,
(4.11)
which for large system size becomes v = 1 − ρ, √ π (1 − ρ)3/2 1 . D 2 ρ 1/2 L 1/2
(4.12) (4.13)
Thus D vanishes in the thermodynamic limit, i.e., the fluctuations in the travel distance of the tagged particle are subdiffusive. The result for the diffusion constant D cannot simply be obtained from the probability distribution function. Its derivation requires the matrix-product Ansatz (MPA) or Bethe Ansatz technique. The calculation of the diffusion constant has been extended to the case of open boundary conditions in [295]. In recent year, considerable progress has been made in the calculation of fluctuations and nonequal-time correlations like ni (t)nj (t ), e.g., by considering scaling limits or connections with random matrix theory. Due to space-limitations, we will not discuss these in detail but refer to [287, 389, 687, 1136, 1137, 1220, 1306] and references therein.
4.1.2. Bethe Ansatz for Translationally Invariant Systems Although the stationary state of the ASEP with periodic boundary conditions is extremely simple and shows no correlations, its dynamical properties are not so easy to determine.2 Here one has to make full use of the quantum formalism, which allows 1 Tagging does not change the dynamics of the particles. 2 In principle, it is possible to determine the short-time dynamics of correlation functions, see, e.g., [291].
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the application of the Bethe Ansatz approach to gain more insight into the spectrum of the quantum Hamiltonian and thus the relaxation toward the stationary state. 4.1.2.1. Bethe Ansatz for Periodic Boundary Conditions Gwa and Spohn [480, 481] (see also [305]) have established a deep connection of the PASEP with several other models studied before, namely the six-vertex model, the single-step model of a growing surface, the anisotropic Heisenberg quantum spin chain (XXZ model) and the noisy Burgers equation (see also [818] and references therein). This connection then can be used to determine dynamical properties of the PASEP. More precisely, identifying an occupied site with the quantum state |↑ and an empty site with |↓ the local generator of the PASEP can be written as p−q 1 1 + − z z σjz − σj+1 + + , + qσj+σj+1 + σjz σj+1 hj, j+1 = − pσj− σj+1 4 4 4
(4.14)
where σj± = (σjx ± iσj )/2 and σjα are the standard Pauli matrices. After a similarity transformation, the full quantum Hamiltonian can be brought into the form of a nonHermitian XXZ model y
L Q + Q −1 z z 1 x x y y σj σj+1 + σj σj+1 + σj σj+1 H=− 2 2 j=1
+ Q L σL− σ1+ + Q −L σL+ σ1− +
Q + Q −1 z z σL σ1 4
(4.15)
with twisted boundary conditions [451, 555, 1211, 1273, 1312]. The anisotropy = (Q + Q −1)/2 of the spin chain is parametrized by the hopping rates to the right and left, √ Q = p/q. Gwa and Spohn [480, 481] have derived a slightly different form of the quantum Hamiltonian through a mapping on the single-step model and then the six-vertex model. The transfer matrix of the latter commutes with the Hamiltonian (cp. (4.14)) L 1 y y x σ j · σ j+1 − 1 + i σjx σj+1 − σj σj+1 , H=− 4 j=1
(4.16)
y
where σ j = (σjx , σj , σjz ). The parameter parametrizes the anisotropy of the hopping rates, which are given by p=
1 (1 + ), 2
q=
1 (1 − ). 2
(4.17)
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Asymmetric Simple Exclusion Process – Exact Results
In the symmetric case = 0, the non-Hermitian term in (4.16), which can be interpreted as a driving field, vanishes and H becomes the isotropic ferromagnetic Heisenberg model. Since the XXZ Hamiltonian is solvable by Bethe Ansatz [58, 1326, 1494], one can expect that it will provide important insights into the dynamics of the ASEP. Indeed, Gwa and Spohn [480] were able to compute the dynamical scaling exponent by applying the Bethe Ansatz to the TASEP in the case of periodic boundary conditions. They considered the asymptotic behavior of the dynamical correlation function CN ( j, t) = σ1z (t = 0)σjz (t)N , and the corresponding dynamical structure factor SN (k, t) for large times t and j → ∞. Here |0N denotes the ground state for fixed particle number N . To avoid boundary effects, first the limit L → ∞ with fixed magnetization density m = (2N − L)/L has to be taken. The asymptotic behavior of the structure factor is characterized by the dynamic exponent z. However, z is also related to the asymptotic behavior of the gap of the quantum Hamiltonian H, i.e., the real part E1 of the first excited state, through E1 ∝ L −z .
(4.18)
Gwa and Spohn were able to determine E1 and thus z from the Bethe Ansatz solution. They found that z = 3/2 in the asymmetric case = 0 (see also [305]) and z = 2 in the symmetric case = 0. Further results, e.g., for scaling functions, have been obtained in [448–450, 749, 750] and the references given at the end of Section 4.1.1. The connection of the Bethe Ansatz with the MPA for the ASEP has been discussed in [452]. There has been shown that the components of the eigenvectors constructed by the algebraic Bethe Ansatz can be expressed as a matrix product. 4.1.2.2. ASEP on a Line: Determinant Representation Schütz [1266] has shown that for a finite number N of particles in an infinite onedimensional system, the probability distribution can be expressed as a N × N determinant. The probability P(x; t | y; 0) to explicitly find the N particles at time t at sites x = (x1 , . . ., xN ) given the initial configuration y = (y1 , . . ., yN ) at time t = 0, usually called Green function, can be written in the form P(x; t|y; 0) = det F(x, y; t),
(4.19)
where the matrix elements of F are given by Fij (x, y; t) = e
−t
∞
k+i−j−1 k=0
i−j−1
t k+xi −yj . (k + xi − yj )!
(4.20)
Binomial coefficient and factorial are defined by the -function, i.e., k! = (k + 1).
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In the derivation of this result, the coordinate Bethe Ansatz was used. It is however not necessary and one can check the validity of the determinant representation directly without referring to the Bethe Ansatz solution [1266]. The probability P(x; t | y; 0) also has an integral representation [1266, 1377, 1378], which can be derived using the Bethe Ansatz. Both representations are very useful because they allow to establish the relation of the ASEP with random matrix theory [687, 1168, 1169] or the analysis of the regime of universal fluctuations [287, 1306, 1379, 1380]. Determinant representations have also been derived for the case of discrete-time dynamics, see, e.g., [153, 1112, 1113, 1131, 1132].
4.1.3. Mean-Field Theories for Parallel Dynamics As we have seen the stationary state of TASEP with random-sequential update has a rather simple structure in the case of periodic boundary conditions. It shows no correlations and is described by a simple product state. This means that the simple mean-field approach is exact in this case. The case of parallel dynamics turns out to be slightly more complex. In the following, we will apply the different mean-field theories that were introduced in a rather general form in Section 2.2 to gain some insights into the nature of the correlations. This will also help to understand the strong influence of the global dynamics (update scheme) on the structure of the stationary state. Except for the simple site-oriented mean-field approach, all these theories will turn out to be exact in the case of the TASEP. Because the approaches are based on different representations of the process, their results complement each other. 4.1.3.1. Site-Oriented Mean-Field Theory A simple mean-field-type argument gives an expression for the current, which then turns out to be exact for the case of random-sequential dynamics. Assuming that there are no correlations between the occupations of neighboring sites, a cell will be occupied with probability ρ, where ρ is the particle density (which is conserved in the case of periodic boundaries). Similarly, the cell in front will be empty, with probability 1 − ρ. This means that within the time interval δt, a particle will move from one site to the next with probability pρ(1 − ρ)δt, where p is the hopping rate. Thus, one obtains the current JMF (ρ) = pρ(1 − ρ)
(4.21)
which, as mentioned before, is exact for random-sequential dynamics. This already indicates that in the random-sequential case, there are no correlations, which we will later prove by mapping the ASEP to the zero-range process (ZRP).
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Therefore, when we discuss other mean-field schemes in the following, we will concentrate on the ASEP with parallel dynamics. In addition to being the most relevant update for the applications that we study in Part II of this book, it will also lead to results which are different from the mean-field expression we have just derived. The mean-field expression (4.21) shows what is called particle-hole symmetry, borrowing terminology from solid-state physics. More explicitly this symmetry implies J (ρ) = J (1 − ρ),
(4.22)
in other words, the current does not change if all particles are replaced by empty cells (holes) and vice versa (particle-hole transformation). This is of course expected from the definition of the dynamics because every time a particle moves to the right, a hole moves to the left. This symmetry is also satisfied for other updates, e.g., the parallel one, and is thus not connected to the fact that the result (4.21) is exact for random-sequential dynamics. It is, however, destroyed by certain updating sequences like the ordered-sequential ones. Applying the particle-hole transformation to the latter case changes the direction of the update. Therefore, the currents J→ and J← of the forward- and backward-ordered updates are related by J→ (ρ) = J← (1 − ρ).
(4.23)
Exact expressions for these currents will be given in Section 4.1.7. 4.1.3.2. Cluster Approximation The two-cluster approximation for the ASEP has first been derived in [1244]. A more detailed discussion can be found in [1258] and [1230]. Actually in these works, it was recovered as the limit vmax = 1 of the Nagel–Schreckenberg model for traffic flow, which can be considered as a generalization of the ASEP with parallel dynamics to a larger interaction range. This model will be introduced and discussed in detail in Chapter 7. Let us denote the probability to find two neighboring sites j and j + 1 in the state (nj , nj+1 ), where n = 0 denotes an empty cell and n = 1 denotes an occupied cell, by P(nj , nj+1 ). The two-cluster approximation then assumes a factorized form P(n1 , . . ., nL ) ∝ P(n1 , n2 )P(n2 , n3 ) . . .P(nL−1 , nL )P(nL , n1 ).
(4.24)
Using the Kolmogorov consistency conditions (2.35) with P(1) = ρ and P(0) = 1 − ρ, one obtains the following symmetry relations: P(0, 0) = 1 − ρ − P(1, 0), P(1, 1) = ρ − P(1, 0), P(1, 0) = P(0, 1).
(4.25)
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P(1, 0) = P(0, 1) is a direct consequence of the particle-hole symmetry. This leaves only one independent two-cluster probability, which has to be determined explicitly from the master equation. This can be achieved following the procedure outlined in Section 2.2 (for details, see also [1230, 1233, 1258, 1443]). Using (2.30), one factorizes the four-cluster probabilities occurring in the exact master equation for P(1, 0) in terms of two-cluster conditional probabilities. In the first column of Table 4.1, we list all those configurations (ni−1, ni , ni+1, ni+2 ; t) which can lead to the configurations, shown in the second column, which is the exhaustive list of the four-cluster configurations each having ni = 1, ni+1 = 0; the corresponding transition probabilities W (ni−1, ni , ni+1, ni+2 |1, 0) are given in the third column. Using the configurations at t and t + 1, as well as the corresponding transition probabilities given in table in Fig. 4.1, the master equation for P(1, 0) reduces to the quadratic algebraic equation [1258] py2 − y + ρ(1 − ρ) = 0,
(4.26)
where we have used the shorthand notation y = P(1, 0). Solving this quadratic equation, we get [1244, 1258]
1 1 − 1 − 4pρ(1 − ρ) (4.27) P(1, 0) = 2p and, hence, P(1, 1), P(0, 0), P(0, 1) from the equations (4.25). Table 4.1 All four-cluster configurations, which lead to the configuration (1, 0) of the central sites are shown in the left column. Open and filled circles correspond, respectively, to empty and occupied sites, a question mark to sites which can be in either state. The transition probabilities are shown in the right column t+1
t
• ◦ • • • ◦ • ◦ •
• • ◦ ◦ ◦ • • • •
•◦ •◦ ◦◦ ◦• •◦ ◦◦ ◦◦ ◦• ◦•
• ? ◦ ◦ ◦ ? • ? •
• • • • • • • • •
◦• ◦• ◦◦ ◦? ◦• ◦◦ ◦◦ ◦? ◦?
W
p p p p p2 1−p 1−p 1−p 1−p
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Asymmetric Simple Exclusion Process – Exact Results
0.3
0.2 0.15 J (ρ)
J (ρ)
0.2 0.1
0.1 0.05
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
ρ
ρ
(a)
(b)
0.8
1
Figure 4.1 Flow–density relation for the ASEP with parallel dynamics. (a) Comparison of the exact result and mean-field theory (broken line) for p = 0.5. The mean-field result is exact for the case of random-sequential dynamics. (b) Comparison of exact results for p = 0.25, p = 0.5, and p = 0.75 (from bottom to top).
Higher order cluster approximations with n > 2 yield equivalent results. This indicates that (4.27) is exact. Indeed, this has been proven in [1258] by a combinatorial argument (see also [1487]), which is discussed in Section 4.1.6. The flow is given by J (ρ) = pP(1, 0). For all 0 ≤ p ≤ 1, it exhibits particle-hole symmetry, as in the random-sequential case. Obviously, it vanishes for p = 0. In the deterministic limit p = 1, however, one obtains Jdet (ρ) =
ρ 1−ρ
for ρ ≤ 1/2 for ρ ≥ 1/2
= min (ρ, 1 − ρ),
(4.28)
i.e., the fundamental diagram is made up of two symmetric linear branches. The stationary state consists of configurations, which either do have no pairs of occupied (for ρ ≤ 1/2) or empty sites (for ρ > 1/2). Which configurations contribute to the stationary state is decided by the initial condition. Comparing the exact flow J (ρ, p) =
1 1 − 1 − 4pρ(1 − ρ) 2
(4.29)
with that predicted by the mean-field approximation, one notices that the mean-field result (4.21), which is also valid for the case of parallel dynamics, strongly underestimates the correct flow (Fig. 4.1). This shows the importance of correlations and indicates that the choice of the update is an important factor in the model definition.
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4.1.3.3. Matrix-Product Ansatz The two-cluster approximation (4.24) can be rewritten in the form of a matrix-product state [758]. For the ASEP, it takes the form PL (n1 , . . ., nL ) = Tr
L
[nl D + (1 − nl )E]
(4.30)
l=1
with the 2 × 2-matrices
t(00) 0 E= , t(10) 0
0 t(01) D= . 0 t(11)
(4.31)
This form of the matrices does not generate terms . . .t(τ σ )t(τ σ ) with σ = τ in the product and is equivalent to (4.24). The components have to satisfy the condition3 [1481] t(11) t(10) = (1 − p) , t(01) t(00)
(4.32)
which can be guaranteed by the choice t(11) = (1 − p)t(01) and t(10) = t(00). Evans [356] has used a related approach to solve the general case where individual hopping rates pj are associated with each particle j. Based on a ZRP representation where particles in the ASEP become sites in the ZRP and the empty sites behind a particle the corresponding particles, he developed an approach similar to that of [487]. The transfer matrix is expressed as a product of operator-valued 2 × 2-matrices with operators acting on the particles in the ZRP representation. Then it can be shown that the steady state in terms of the gap variables has a simple product structure. 4.1.3.4. Car-Oriented Mean-Field Theory The car-oriented (or particle-oriented) mean-field theory (COMF)4 described in Section 2.2 takes into account certain correlations between sites. Its starting point is the observation that a configuration of the ASEP can not only be specified by the occupation numbers {nj }, but also by the gaps {dj }, i.e., the empty sites in between two consecutive particles j and j + 1. Strictly speaking, one needs also to keep track of the position of one of the particles, e.g., particle 1. But because the stationary state is translational invariant, this is not necessary here. The central quantity in COMF is the probability Pn to find exactly n empty cells (i.e., a gap of size n) in front of a particle. The essence of COMF is now to neglect 3 Using the symmetries (4.25), this condition is seen to be equivalent to (4.26). 4 We use in the following the terminology COMF which is more common, even outside of traffic modeling.
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correlations between gaps. This gives rise to the following system of equations, which result from the master equation P0 (t + 1) = g¯ (t) P0 (t) + pP1(t) , P1 (t + 1) = g(t)P0(t) + pg(t) + (1 − p)¯g(t) P1 (t) + p¯g(t)P2(t), Pn (t + 1) = (1 − p)g(t)Pn−1(t) + pg(t) + (1 − p)¯g(t) Pn (t) + p¯g(t)Pn+1(t),
(4.33) (n ≥ 2)
where g(t) = p
Pn (t) = p[1 − P0(t)]
(4.34)
n≥1
is the probability that a particle moves in the next timestep. g¯ (t) = 1 − g(t) then is the probability that a particle does not move. As an example for the derivation of these equations, we consider the case n ≥ 2. Because the particles can move at most one site per timestep, a gap of n cells at time t + 1 must have evolved from a gap of length n − 1, n, or n + 1 in the previous timestep. A gap of n − 1 cells evolves into a gap of n cells only if the first particle moves (with probability g) and the second particle does not move (probability 1 − p), i.e., the total probability for this process is pgPn−1 . Similarly, the gap remains constant only if either both particles move (probability pg) or both particles do not move (probability (1 − p)¯g). Finally, the gap is reduced by one if only the second particle moves (probability p¯g ). The equations for P0 and P1 can be justified by similar arguments. In the following, we focus on the stationary state where the Pn become timeindependent. The probabilities Pn have to satisfy the normalization conditions 1=
∞
Pn ,
n=0 ∞
1 (n + 1)Pn . = ρ n=0
(4.35)
(4.36)
Equation (4.36) is a consequence of the conservation of the number of particles because each particle with gap n occupies effectively n + 1 cells. A solution of this infinite system (4.33) of nonlinear equations can be obtained by introducing the generating function [1245] P(z) =
∞
n=0
Pn zn+1 .
(4.37)
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After multiplying the corresponding equation in (4.33) by zn+1 and summing over all equations, one finds the explicit expression P(z) =
p( g¯ + gz)zP0 . p¯g − (1 − p)gz
(4.38)
The normalization condition (4.35) and the density relation (4.36) imply that the generating function has to satisfy P(1) = 1 and P (1) = ρ1 , where P (z) denotes the derivative of P(z). From the explicit form of P(z), one can then easily obtain the probabilities:
1 2pρ − 1 + 1 − 4pρ(1 − ρ) , 2pρ n (1 − p)(1 − P0) P0 Pn = (n ≥ 1), 1 − p P0 + (1 − p)(1 − P0)
P0 =
(4.39) (4.40)
and for the flow J (ρ, p) = ρg
(4.41)
the exact solution (4.29) is reproduced. Obviously, the COMF-solution and the two-cluster result are related. The exact cluster probabilities can be expressed in terms of the probabilites Pn as [1245] P(1, 1) = ρP0 , P 2(1, 0) = ρ(1 − ρ)P1, P(0, 0) = (1 − ρ)
Pn+1 Pn
(4.42) (n ≥ 1).
The factors ρ and 1 − ρ appear due to the different normalization of the Pn and P(nj , nj+1 ). The Pn are normalized by the number of particles, whereas the P(nj , nj+1 ) are normalized by the number of sites. The fact that the COMF yields the exact result comes not unexpected because it takes into account all relevant correlations for the case vmax = 1, where only nearest-neighbor correlations are nontrivial [1244, 1258]. This changes if hopping beyond nearest neighbors is allowed as in the NaSch model with vmax > 1 (see Chapter 7). Here the nature of the correlations is different from the ASEP [1230, 1258] and COMF is no longer exact. Nevertheless, it yields good results because it takes into account the longer-ranged correlations at least partially.
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1
2
3
4
5
A
B
C
6 D
(a)
1
2
A
C B
3
4
D 5
6
(b)
Figure 4.2 Mapping of the ASEP onto the ZRP for L = 10 and N = 6. Particle in the ASEP correspond to sites in the ZRP (indicated by numbers). Empty sites in the ASEP become particles in the ZRP (indicated by letters). The arrows show a possible local transition with probability u(1).
4.1.4. Mapping to ZRP The ASEP can be mapped onto the ZRP discussed in Section 3.3. In this mapping, the particles of the ASEP become the sites in the ZRP. The particles in the ZRP picture are then given by empty sites (holes) in the ASEP picture: the empty sites between particles j and j + 1 in the ASEP become the particles on site j in the ZRP (see Fig. 4.2). Therefore, the ASEP with L sites and N particles is equivalent to a ZRP with N sites and L − N particles. This mapping holds for time-continuous dynamics and for parallel and orderedsequential updating [356] as well. The hopping rate in the ZRP is given by u(n) = p
for n ≥ 1.
(4.43)
Note that the particles in the ZRP hop to the left, like the holes in the ASEP. Using the exact results for the ZRP derived in Section 3.3, the fundamental diagram [704] and other quantities can be calculated explicitly. However, the calculation of spatial correlations is more difficult since in the ZRP representation particles belonging to the same site are indistinguishable [85]. The mapping from the ASEP to the ZRP is not one-to-one. For each configuration of the ZRP, there are L/(L − N ) configurations of the ASEP [1152].
4.1.5. Paradisical Mean-Field Theory In the following, we describe a method, which again reproduces the exact results for the ASEP with parallel dynamics. However, it provides interesting new insights, especially into the nature of the correlations that appear in the steady state although correlations are completely absent in the case of random-sequential updating. The first step in this approach is to slightly generalize the problem by assigning an additional internal state variable vj , called velocity in the following, to the particles. This internal state has no influence on the dynamics itself, but should be mainly considered
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as a way of book-keeping. Later it will become clear that we consider here the vmax = 1-limit of the Nagel–Schreckenberg model of traffic flow (see Chapter 7). The internal state can take the values vj = 1 or vj = 0 depending on whether particle j has moved in the previous timestep or not5 . The important observation [1247] is now that for parallel dynamics, so-called Garden of Eden (GoE) states [945] exist. These are states which are dynamically forbidden, i.e., can not be reached by the dynamics. Because these configurations have no predecessor, they are also called paradisical states. An example is any state that has a particle with velocity vj = 0 immediately behind a particle with velocity vj+1 = 1, e.g., ..01.. where “.” denotes an empty site. This would mean that the right particle has moved one site in the last timestep, whereas the left particle has not moved. Therefore, they would have been on the same site, which is not allowed due to the exclusion principle. The basic idea of the paradisical mean-field theory is now to identify all GoE states and define a reduced configuration space where all GoE have been eliminated. For the ASEP, all GoE states can be identified by a simple local criterion [1247]: they contain either at least one pair “01” (as explained in the example above) or “11.” Note that for random-sequential update, no GoE states exist because the particles are not updated in a fixed order and any particle might even be updated in consecutive timesteps. After constructing the reduced configuration space, a standard (site-oriented) meanfield theory is performed in it. The mathematical details for the ASEP-case will be described in the context of NaSch model in the Part II of the book. The surprising result is that this paradisical mean-field theory reproduces the exact result for the ASEP. This shows that no “true” correlations exist. All correlations have their origin in the existence of GoE states. The absence of true correlations and the fact that for random-sequential dynamics no GoE states exist explains why MFT is exact in that case.
4.1.6. Combinatorial Solution for Parallel Dynamics In [1258], it has been shown that the stationary state of the TASEP with parallel dynamics and periodic boundary conditions is fully determined by the cluster probabilities P2 (n1 , n2 ) derived in Section 4.1.3 with an appropriate normalization constant Z. This was proven using combinatorial methods (see also [1487]). The complete set of evolution equations for parallel update reads: P(n; t + 1) =
n
5 That is why the variable is called “velocity”!
W (n → n) · P(n ; t),
(4.44)
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Asymmetric Simple Exclusion Process – Exact Results
where P(n; t) denotes the probability for state n = {n1 , . . ., nL } at time t. The transition probability W (n , → n) to move in one timestep from state n to state n factorizes into local terms:
W (n → n) =
L
W (ni , ni+1 → ni , ni+1 ).
(4.45)
i=1
The only nonvanishing transition probabilities are given by the following: W (0, 1 | 1, 0) = 1 − p, W (1, n |n, 1) = 1 W (1, 0 | 1, 0) = p,
W (n, 0|0, 0) = 1
(4.46)
W (0, n|0, 1) = 1, for all n, n = 0, 1, whereas W (n , m → n, m) = 0 in all other cases. We now make the Ansatz that the probability in the stationary state P(n) factorizes into local two-site terms Pni ni+1 : P(n) =
L
Pni ,ni+1
(4.47)
i=1
with periodic boundary condition nL+1 = n1 . We will denote the number of pairs of next neighbors (n, n ) in a particular state n by Mn,n (n). Due to the particle-hole symmetry of the system, one has M01 = M10 and the following simple relations hold: L = 2M01 + M11 + M00 ,
(4.48)
N = M01 + M11.
(4.49)
In the stationary state, equation (4.44) becomes time-independent and the states n in the summation on the right-hand side can be classified according to the number of particles l, which have to be moved in order to obtain the new state n: M11 L−2M01 −M11 P00 = (P01 P10 )M01 P11
M01
g(M01 , l, )(1 − p)l pM01 −l− ·
l=0 M11 + L−2M01 −M11 + P00 . · (P01P10 )M01 − P11
(4.50)
The summation index is defined as = M11 (n ) − M11(n). Therefore, the range of the -summation depends on the particular state n. The function g(M01 , l, ) counts
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the number of possible states n leading to state n given fixed values of M01 , l, and and reflects a kind of degeneracy. Summing g over yields simply
M01 g(M01, l, ) = . l
(4.51)
Dividing Eqn (4.50) by the left-hand side, one gets
1=
M01
g(M01, l, )(1 − p)l pM01 −l
l=0
P00P11 pP01P10
.
(4.52)
This is satisfied for P00 P11 = pP01 P10,
(4.53)
which can be seen with the help of (4.51). Note that Pnn is not normalized for a finite system, and therefore, a normalization constant Z has to be taken into account.
4.1.7. Ordered-Sequential and Sublattice-Parallel Updates For the case of periodic boundary conditions, we have already seen that the choice of the updating procedure has an important influence on the properties of the steady state. The TASEP with random-sequential update reaches a trivial stationary state without correlations (Section 4.1.1). In contrast, the parallel update produces a particle-hole attraction (Section 4.4.3). In the following, we will discuss various discrete time update schemes. These updates have a proper deterministic limit, in contrast to the random-sequential update that is intrinsically stochastic due to the random order of the particle updates. Therefore, a rescaling of the rates does not correspond to the rescaling of time. 4.1.7.1. Ordered-Sequential Updates In the following, we will consider the site-ordered-sequential updates6. As we have seen in Section 4.1.3 [see Eqn (4.23)], the forward- and backward-ordered updates are related by a particle-hole transformation. Therefore, we can restrict ourselves to a discussion of the backward-sequential case. 6 To which we will refer as ordered-sequential update for brevity.
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As in the random-sequential case, the steady state of the backward-sequential update is given by a product measure, i.e., mean-field theory is exact. This can also be seen from the MPA
1 Tr A⊗(L−1) ⊗ Aˆ (4.54) |P0 = ZL because the corresponding bulk equation of the matrix algebra has a one-dimensional solution. In terms of the MPA, the current through the bond ( j, j + 1) can be calculated as J← ( j) =
p ˆ L−j−1 Tr C j−1 D EC ZL
(4.55)
and is given by J← (ρ, p) = pρ
1−ρ . 1 − pρ
(4.56)
This result can be obtained directly using a mean-field argument. The site to the right of an occupied site is empty with probability 1 − (ρ − J← ). Here one has to take into account that the density is reduced by J← after the update of that site. Therefore, the current satisfies J← = pρ(1 − (ρ − J←)) which leads to (4.56). max ( p) = 1 (1 − √1 − p) ≥ 1/2 The maximal flow is reached for a density ρ← p (Fig. 4.3). Thus the flow–density relation is asymmetric and shifted toward higher densities compared with the random-sequential update. The maximum value of the flow √ J← (ρ max , p) = 2p (1 − 1 − p) − 1 is larger than the maximal flow for the parallel update. The particle-hole symmetry can be used to determine these quantities for forwardsequential case simply by replacing ρ with 1 − ρ. In this case, the fundamental diagram is shifted toward lower densities. The particle-ordered-sequential update has been considered by Evans [356]. It also can be mapped onto a ZRP and thus has a factorized steady state. This corresponds to a COMF-type factorization as in the parallel case which is not surprising since the updates are closely related. The only difference is the treatment of the last particle: In the forward-ordered update the preceding particle might have moved away since it was updated before in the same timestep. 4.1.7.2. Sublattice-Parallel Update As shown in [1163] there is a close relation between the sublattice-parallel and the ordered-sequential updates. The TASEP with sublattice-parallel dynamics has first been solved by Schütz [1264] for deterministic hopping. Later a solution for the stationary state was given in terms of the MPA [566]. This MPA solution can be generalized to the case of
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0.4 p = 0.75 p = 0.50
J (ρ)
0.3
0.2
0.1
0
0.2
0.4
0.6
0.8
1
ρ
Figure 4.3 Flow–density relation of the ASEP with backward-ordered sequential update and periodic max ( p). boundary conditions for p = 0.5 and p = 0.75. The vertical lines indicate the location of ρ← 4
3
2
1
4
3
2
1
Figure 4.4 Shuffled update of a cluster consisting of four particles numbered from right to left. The update sequence is 3, 4, 1, 2 and particles move according to the ASEP rules with p = 1. Particle 3, chosen first, cannot move because the cell in front is occupied by 2, and similarly for particle 4. Particle 1 then moves to the right. Then 2 also moves because it was drawn after 1. Although particle 4 is drawn after 3, it cannot move because both were drawn before 1 and 2.
stochastic hopping [576, 1160] which leads to the same algebra as the ordered-sequential updates. Therefore in the case of periodic boundary conditions mean-field theory gives the exact solution. However, the densities on the even and odd sublattices are different. The current in the case of deterministic hopping is given by J (ρ) = 2ρ for ρ ≤ 1/2. For ρ > 1/2 it can be obtained using the particle-hole symmetry J (ρ) = J (1 − ρ).
4.1.8. Shuffled Dynamics The update schemes discussed so far either correspond to discrete or continuous time dynamics. The so-called shuffled update has elements of both. In this scheme at each timestep the order in which particles are allowed to move is determined by a random permutation of the particle numbers [743, 1479]. An example for the ASEP with shuffled update and p = 1 is shown in Fig. 4.4. The difference between this update scheme and the random-sequential update is analogous to an urn problem with and without replacement: If one imagines an urn with N balls, numbered 1, 2, . . . , N , the
Asymmetric Simple Exclusion Process – Exact Results
random-sequential update can be realized by choosing a ball at random, updating the particle with the ball-number and replacing the ball into the urn afterwards. In the shuffled update, one chooses a ball and updates the corresponding particle without replacing the ball into the urn. Then, one chooses the next ball, and so on, until the last particle is updated. Finally the urn is refilled and the procedure is repeated in the next timestep. The shuffled update is discrete in the sense that it has a well-defined timestep during which each particle is updated exactly once. On the other hand, the order of updating the particles is not fixed. It may happen that a specific particle is updated last during a timestep and first during the next one! This makes it difficult to identify the timestep with a kind of reaction time as it is natural for the case of parallel dynamics [1258]. One of the advantages of the shuffled update is that no conflicts can occur in cases where the particles can move in different directions [743]. As mentioned before this is a problem of the parallel update where, e.g., in the PASEP particles might attempt to move to the same site. These conflicts have then to be resolved in the sense that one needs an algorithm to decide which of the particles is allowed to move. The same problem occurs in two-dimensional variants and is of some relevance for the modeling of pedestrian dynamics (see Chapter 11). The TASEP with shuffled dynamics shows surprising properties [1479]. For all other updates schemes discussed so far, the stationary state of the TASEP with periodic boundary conditions is given by a factorized steady state. This is different for the shuffled update. In fact, it was shown in [1479] that the master equation cannot be solved by a product Ansatz. Nevertheless, COMF [1479] and cluster approximation [1298] give excellent results, e.g., for the fundamental diagram. Using the procedure described in Section 4.1.4, the TASEP with shuffled dynamics can be mapped on a ZRP with shuffled dynamics (Fig. 4.5). As shown in [1479], the ZRP with shuffled update does not have a factorized steady state. This can be seen by considering the dynamics of the configuration where all particles are located at the same site. As a consequence, COMF cannot be exact for the TASEP with shuffled update. There is another mapping of the shuffled ASEP to a ZRP-type model, where the empty sites of the TASEP are identified as the sites of a generalized ZRP (GZRP) with parallel dynamics. The particles in the GZRP are then the number of particles in front of the corresponding empty site, i.e., the GZRP has N − L sites and N particles (Fig. 4.5). Unfortunately, this GZRP has also not been solved so far. The shuffled ASEP can also be interpreted as a model of cluster dynamics where all clusters are updated synchronously. The specific random sequence in which the particles are updated determines how many particles in a cluster are allowed to move forward. The l-th particle of a cluster can move at time t + 1, if and only if at least the first l particles are chosen in the order from the right to the left (with probability 1/l!) and if they all move (with probability pl ). The probability that exactly l particles of a cluster of length m
129
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A. Schadschneider, D. Chowdhury, and K. Nishinari
(a)
(b)
(c)
Figure 4.5 Mapping of the (a) TASEP onto (b) GZRP and (c) ZRP for L = 10, N = 6 and periodic boundary conditions. The arrows indicate a possible local transition with probability u1 (3).
(with m > l) move is given as the probability for l particles to move, as calculated earlier, minus pk+1 /(k + 1)!, the probability for the (l + 1)-th particle to move. Thus the probability ul (m) that l particles leave a cluster of length m (where 0 ≤ l ≤ m) is given by ul (m) =
pl pl+1 − θ (m − l) l! (l + 1)!
(4.57)
and 0 otherwise. The Heaviside step function ensures that in the case l = m, the right term vanishes. As mentioned before, the shuffled update combines elements of discrete and continuous time schemes. This is also reflected in the properties of the stationary state. In the case p = 1, the stationary state for densities ρ ≤ 1/2 is identical to the deterministic limit of parallel dynamics, i.e., a free-flow phase where all particles can move unhindered. For ρ > 1/2, the randomness of the update scheme becomes important, and the system is in a jammed phase different from that in the parallel update case. The fundamental diagram is strongly asymmetric because the dynamics is no longer particle-hole symmetric (Fig. 4.6). The probability P0 (ρ) can be considered as a kind of order parameter. In the case p = 1, it can be exactly calculated [1479]. It vanishes for ρ ≤ 1/2 and increases continuously for ρ > 1/2 indicating a second-order transition. This is similar to the deterministic limit of parallel dynamics [341]. With decreasing hopping probability, the asymmetry of the fundamental diagram becomes smaller. In the limit p → 0, the result for random-sequential update, J = pρ (1 − ρ), is recovered. For p < 1, the free-flow and jammed regimes are not separated by a phase transition because P0 (ρ) no longer shows nonanalytic behavior.
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0.2
0.5
0.15
0.3
J
J
0.4
0.1
0.2 0.05
0.1 0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
ρ
ρ
(a)
(b)
0.8
1
Figure 4.6 Fundamental diagram of the TASEP with shuffled dynamics for p = 1 (a) and p = 0.5 (b). Shown are the results from simulations (squares) in comparison with the COMF results (lines). The system size is L = 500.
Using COMF, an approximate analytical form for the fundamental diagram can be obtained: ρ for ρ ≤ 1/2,
(4.58) J (ρ, p = 1) = ρ(1−ρ) 2ρ−1 −1 for ρ > 1/2. 2ρ−1 exp ρ As can be seen in Fig. 4.6, this approximation is in excellent agreement with simulation results. As we have mentioned before, no exact solution of the TASEP with shuffled update is known. Even the seemingly simpler case where only one fixed random sequence is chosen initially has not been solved up to now. The only exceptions are the special cases of forward and backward sequential update.
4.2. ASEP WITH OPEN BOUNDARY CONDITIONS The ASEP with open boundary conditions is much richer than the periodic case. We will see that it is the paradigmatic model for so-called boundary-induced phase transitions [812]. Usually, the open boundaries are realized by (infinite) particle reservoirs (Fig. 4.7). At the left end, particles are inserted with a rate α if the first site is empty. At the right end of the chain, the particles are removed from the last site ( j = L) with rate β. Various methods have been applied for its exact solution, which all have its merits. In the following, we will sketch the main results. Unless stated otherwise, we will consider a system of L sites with random-sequential update and boundary rates α and β. The input and output reservoirs will be identified
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α
p
p
p
β
Figure 4.7 Asymmetric simple exclusion process with open boundary conditions realized by particle reservoirs at the chain ends. At the left end, particles are inserted with rate α if the first site is empty. Particles are removed from the last site with rate β.
with site j = 0 and j = L + 1, respectively. nj = 0, 1 denotes the occupation number of site j. If one is mainly interested in the properties of the stationary state, often time t is rescaled so that the hopping rate p in the bulk becomes p = 1. This implies a rescaling of the boundary rates α→
α , p
and β →
β . p
(4.59)
For discrete-time, updates such a rescaling is not possible.
4.2.1. Mean-Field Theory The simplest approach for the description of the steady state is a mean-field approach that neglects correlations. As we have seen for the periodic TASEP with random-sequential dynamics, this even yields the exact result. An important complication in the case of open boundary conditions arises from the broken translational invariance. Therefore, we expect a nontrivial density profile nj . First, we will modify the exact dynamical equations for the occupation numbers nj (t) of site j at time t derived in Section 4.1.1 to account for the open boundaries. For sites in the bulk (2 ≤ j ≤ L − 1), these are identical to (4.1) and thus the average occupations ni are determined by (4.4). By a similar reasoning, the dynamical equations for the boundary sites (rather: bonds) j = 1 and j = L can be obtained. The dynamics of the single-site occupations of the open system is then governed by the dni = ni−1 − ni − ni−1 ni + ni ni+1 , dt dn1 = α(1 − n1) − n1(1 − n2 ), dt dn2 = nL−1 (1 − nL ) − βnL . dt
(4.60) (4.61) (4.62)
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In the stationary limit t → ∞, the averages become time-independent, and the density profile is determined by nj − nj nj+1 = nj−1 − nj−1nj , n1 − n1n2 = α(1 − n1), βnL = nL−1 − nL−1 nL .
(4.63) (4.64) (4.65)
These equations exhibit the typical hierarchical structure: In order to determine onepoint correlations nj , higher correlation functions like nj nj+1 have to be known. As explained in Section 2.2.1, in mean-field approximations, these correlations are neglected. This amounts in replacing nj nj+1 by nj nj+1 , and so on, and leads to a closed set of equations. Introducing the abbreviation tj = nj , the mean-field approximation corresponds to the following L iterative equations for the L unknowns tj [289]: tj − tj tj+1 = tj−1 − tj−1tj
(for 2 ≤ j ≤ L − 1),
t1 − t1t2 = α(1 − t1), βtL = tL−1 − tL−1 tL .
(4.66) (4.67) (4.68)
An important consequence from these iterations is J tj+1 = 1 − , tj
(4.69)
because (4.66) implies that tj − tj tj+1 = J with a constant J that is independent of j. This result can be derived even simpler from the continuity equation, which implies that the current Jj = nj (1 − nj+1 )
(4.70)
in the stationary state is site-independent, which in mean-field approximation gives (4.69). The recursion (4.69) is analyzed further in Appendix 4.9.3. In [289], it was shown graphically that three different types of solution exist. These correspond to three different phases that can be realised in the system. A detailed discussion of the phases based on the exact solution will be presented in Section 4.2.4. It will show that mean-field theory indeed produces the correct phase diagram! However, other quantities like the density profiles are not exact.
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4.2.2. Recursion Relations The first step toward an exact solution of the ASEP with arbitrary boundary rates has been made by Derrida, Domany, and Mukamel [289]. They have derived exact recursion relations in the system length L for the stationary state. This recursion expresses the weights fL (n1 , . . ., nL ) of a system of length L, which are related to the probabilities by 1 fL (n1 , . . ., nL ), ZL
(4.71)
fL (n1 , . . ., nL )
(4.72)
PL (n1 , . . ., nL ) = where
ZL =
n1 ,...,nL
to those of smaller systems. In [289], it has been shown that fL (n1 , . . ., nL ) = αnL fL−1 (n1 , . . ., nL−1 ) + αβ(1 − nL )[fL−1 (n1 , . . ., nL−2 , 1) + fL−1 (n1 , . . ., nL−2 , 0)] + ··· + αβ(1 − nL )(1 − nL−1 ) · · · (1 − n2 )n1 [ fL−1 (1, n2 , . . ., nL−1 ) + fL−1 (0, n2 , . . ., nL−1 )] + β(1 − nL )(1 − nL−1 ) · · ·(1 − n1 )fL−1 (n1 , n2 , . . ., nL−1 ).
(4.73)
For small system size, this can be solved easily starting from f1 (0) = α and f1 (1) = β. For L = 2, one obtains f2 (0, 0) = β 2 ,
f2 (1, 0) = αβ(α + β),
f2 (0, 1) = αβ,
f (1, 1) = α 2 .
(4.74)
For the special case α + β = 1, a simple closed expression for the weights can be derived: fL (n1 , n2 , . . ., nL ) = α (1 − α) N
L−N
with
N=
L
nj .
(4.75)
j=1
In this case, the steady state factorises and correlation function become simple, e.g., nj = α,
nj nk · · ·nl = nj nk · · ·nl .
(4.76)
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The first identity implies that the density profile ρ( j) = nj along the chain is f lat for all points on the line α + β = 1. From (4.73) also recursions for averages like nj or more general correlation functions can be derived. Schütz and Domany [1267] were able to solve the iteration equations exactly and calculate the exact density profiles, and so on, in the large-L-limit. Thus, the exact phase diagram in the α − β-plane could be derived (see Section 4.2.4).
4.2.3. Matrix-Product Ansatz The MPA as described in Section 2.4 has first been applied to the ASEP [293]. By inserting the Ansatz (2.42) into the master equation, the relation between the matrices E and D and the boundary vectors W | and |V have been identified as a quadratic algebra [293]. This is described in some detail in [291]. Here we use the more formal approach based on the cancellation mechanism. For the case of the ASEP, the stochastic Hamiltonian (2.6) is given explicitly by ⎛ ⎞ 0 0 0 0 ⎜0 0 −p 0⎟ ⎟ (4.77) hj,j+1 = ⎜ ⎝0 0 p 0⎠, 0 0 0 0 and the boundary interactions α 0 , h1 = −α 0
0 −β hL = . 0 β
(4.78)
From (2.13), we can see that p only rescales time (and α and β) and that it would therefore be sufficient to study p = 1. ¯ the equations reduce to the quadratic algebra [293] Taking E¯ = 1 = −D, p DE = D + E, αW |E = W |,
(4.79)
βD|V = |V . The first equation of (4.79) can be guessed intuitively: the current JL (j) (describing the flux of particles through bond j of a chain of length L) has to be constant throughout the chain. JL ( j) is given by [293] JL ( j) =
p W |C j−1 DEC L−j−1 |V , ZL
(4.80)
and we see that DE ∝ C = E + D is the simplest way to achieve a constant current.
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A more direct derivation of the algebra (4.79), which does not use the mechanism (2.53), (2.55) is given in [123, 291, 294]. It is based on the explicit identification of all processes, which lead into and out of a given configuration and their respective probabilities. One can then show that all these equations are satisfied under the conditions (4.79). How to determine physical quantities from (4.79)? For small systems, the algebra can be used to normal-order the matrices in expressions like (2.47) and (2.48). Specifically one can transform such expressions into a form where it becomes a sum of expressions of the type E nE D nD , which allows to use the boundary equations in (4.79). Some examples for results that can be obtained without the use of representations for the matrices are given in Appendix 4.9.4. For large systems, especially the thermodynamic limit, it is sometimes easier to use explicit representations [293] for D, E, W | and |V . It turns out that one finds one-dimensional representations (with E, D, and W |, |V being real numbers) if and only if α + β = p.
(4.81)
In all the other cases, the matrices are infinite-dimensional. The proof is given in Appendix 4.9.5. Some useful representations for the algebra are given in Appendix 4.9.6.
4.2.4. Exact Phase Diagram In the following, based on the exact solution, the phase diagram of the ASEP will be discussed. For simplicity, we follow the usual convention and set p = 1 which, according to (4.59), is no restriction of the general case. The exact solution obtained by the recursion relation approach and the MPA confirms the mean-field predictions for the phase diagram (Section 4.2.1): It consists of three different phases, the high-density, low-density, and maximal current phase (Fig. 4.8). β 1 MC
LD-II 1/2 LD-I HD-I 1/2
HD-II
1
α
Figure 4.8 Exact phase diagram for the ASEP for p = 1. The two high- and low-density subphases differ in the decay of the density profile to its bulk value from the boundaries. The density profile becomes flat on the thin dotted line α + β = 1.
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Their origin can be understood by a simple argument. In the stationary state, the current through the system has to be the same everywhere because of the equation of continuity. There are three different mechanisms, which in principle may limit the current, namely the efficiency of the input, the output, and the transport in the bulk. The low-density phase can also be called input-controlled phase, and high-density phase output-controlled phase. In contrast to these two boundary-controlled phases, the maximal current phase is a bulk-controlled phase. Here the current through the system has reached the maximum of the fundamental diagram of the corresponding ASEP with periodic boundary conditions. In the following, the main characteristics of the three phases are derived from the exact solution. We will focus on the dependence of the current J = nj (1 − nj+1 ) and the density profile ρj = nj on the system parameters α and β. The calculations are most conveniently performed using the results from the MPA. The current is given by Eqn (4.80), which can be rewritten by using (4.79): JL ( j) =
ZL−1 , ZL
(4.82)
In Appendix 4.9.4, it is shown that the density profile is given by L−j
ZL−n Zj−1
1 Bn,1 + BL−j,n n+1 ρj = ZL ZL n=1 β n=1 ρL =
L−j
for 1 ≤ j < L,
ZL−1 . βZL
(4.83) (4.84)
Of special interest is the bulk density far away from both boundaries where, except for the maximum current phase, the density profile becomes flat (independent of j). The behavior near the boundaries, however, can be characterized by two independent length scales7 ξα and ξβ determined by the input and output rate, respectively. Explicitly these length scales are given by [1267] ξσ = − ln[4σ (1 − σ )]
with σ = α, β.
(4.85)
There is a third length scale ξ that becomes relevant, which is given by 1 1 1 − . = ξ ξα ξβ
(4.86)
7 This is the consequence of the fact that the derivative of the density profile can be written as product two functions
[1267].
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The expressions for J and ρj can be simplified if one is only interested in the asymptotic behavior for large L. Below we will use the results, e.g., for the asymptotic behavior of ZL obtained in [294, 1267]. Reference [123] gives a detailed description of the analytical techniques to determine the asymptotic behavior of ZL and ρj . 4.2.4.1. Low-Density Phase (α < 12 , α < β) For α < 12 and α < β, the asymptotic behavior of the normalization is given by ZL (α, β)
αβ(1 − 2α) [α(1 − α)]−L−1 . (β − α)
(4.87)
Using (4.82), one finds for the current in the low-density phase JLD = α(1 − α).
(4.88)
In the bulk of the system, the density profile becomes flat with density ρLD = α.
(4.89)
This is also the density ρj ( j L) at the left (input) end of the system. For the asymptotic behavior of the density profile at the right (output) end, one has to distinguish two subphases. In subphase LD-I, where β < 12 , the density profile shows a purely exponential decay at the right boundary (if one takes the limit L → ∞ and j → ∞ at fixed distance L − j from the right boundary): ρL−j α + (1 − 2β)
α(1 − α) β(1 − β)
j+1 .
(4.90)
Thus, the decay is determined by the correlation length ξ, ρL−j − α ∝ e−j/ξ .
(4.91)
In contrast, in subphase LD-II, where β > 12 , the exponential decay is modulated by a power law: 1 4j 1 − [α(1 − α)] j+1 , (4.92) ρL−j α + √ (2β − 1)2 (2α − 1)2 π j 3/2 The decay is now governed by ξα , i.e., ρL−j − α ∝ j −3/2 e−j/ξα .
(4.93)
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The densities at the ends of the chain are [1267] ρ1 = α
and ρL =
α(1 − α) . β
(4.94)
The slope of the density profile changes sign on the line α + β = 1, where the density profile is flat throughout the whole system. It is positive for β < 1 − α and negative for β > 1 − α [1267]. 4.2.4.2. High-Density Phase (β < 12 , β < α) The asymptotic expression for ZL has the same form as in the low-density phase, but with α and β interchanged. The current is then JHD = β(1 − β).
(4.95)
The behavior of the density profile can be easily be determined from that in the low-density phase by using the particle-hole symmetry of the model. This symmetry implies ρL+1−j (α, β) = 1 − ρj (β, α)
(4.96)
because holes enter the system at the right boundary at a rate β, hop with unit rate to the left and exit at rate α. Therefore, the asymptotic behavior of the profile is given by ρj − (1 − β) ∝ e−j/ξ
(4.97)
ρj − (1 − β) ∝ j −3/2 e−j/ξβ
(4.98)
in subphase HD-I and by
in subphase HD-II. Again, the density profile becomes flat in the bulk with density ρHD = 1 − β.
(4.99)
The particle-hole symmetry implies that as in the low-density phase, two subphases HD-I and HD-II can be distinguished. The asymptotic behavior of the density profile at
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the left boundary can be obtained from (4.90) and (4.92) by using the symmetry relation (4.96). The densities at the ends of the chain are ρ1 = 1 −
β(1 − β) α
and ρL = 1 − β.
(4.100)
As in the LD phase, there is a change of sign in the slope on the line α + β = 1 in the subphase HD-II. It is positive for β < 1 − α and negative for β > 1 − α. 4.2.4.3. Maximum Current Phase (α > 12 , β > 12 ) For α > 12 and β > 12 , the asymptotics of the partition function is given by ZL (α, β) ∼ √
4L 4αβ(α + β − 1) . π (2α − 1)2 (2β − 1)2 L 3/2
(4.101)
This implies a current JMC =
1 4
(4.102)
independent of the boundary rates. This current corresponds to the maximal f low in the periodic system explaining the name of the phase. The density profile shows a power-law decay from the bulk value 1 ρMC = . 2
(4.103)
Here “bulk value” means the density in the middle of the chain at site j = L/2. The density profile does not become flat but shows a power-law behavior in the whole chain. It decays from the bulk value as 1 1 −3/2 ) . (4.104) 1 − √ + O( j ρL−j 2 πj The prefactor is independent of the system parameters, and therefore, particle-hole symmetry implies the same power-law behavior at the left boundary. The densities at the ends of the chain are [1267] ρ1 = 1 −
1 4α
and ρL =
1 . 4β
(4.105)
The density profile in the maximal current phase is an example for a quantity, which is not correctly described by mean-field theory. Although MFT correctly predicts the value 12 of the bulk density, it yields 1 for the decay exponent in (4.104) instead of 12 . The main features of the different phases are summarized in Table 4.2.
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Asymmetric Simple Exclusion Process – Exact Results
Table 4.2 Characteristic properties of the phases in the ASEP with random-sequential dynamics (p = 1). It shows the current J, the bulk density ρL/2 , the densities ρ1 , ρL at the chain ends, and the asymptotic decay of the density profile near the left (input) and right (output) end of the chain. For p = 1, the current is given by pJ(α/p, β/p) Phase
J (α, β)
ρ L/2
ρ1
LD-I
α(1 − α)
α
α
LD-II
α(1 − α)
α
α
HD-I
β(1 − β)
1−β
HD-II
β(1 − β)
1−β
MC
1/4
1/2
β(1 − β) α β(1 − β) 1− α 1 1− 4α
1−
ρL
Left end
Right end
α
e−j/ξ
α
j −3/2 e−j/ξα
1−β
e−j/ξ
1−β
1−β
j −3/2 e−j/ξβ
1−β
1 4β
1 √ 2 πj
1 − √ j −1/2 2 π
α(1 − α) β α(1 − α) β
4.2.5. Phase Transitions The three phases are separated by phase transitions characterized by the divergence of one of the length scales ξα , ξβ , or ξ defined in (4.85). 4.2.5.1. Transition between Subphases On the transition lines β = 1/2, α < 1/2, and α = 1/2, β < 1/2 separating the subphases LD-I, LD-II and HD-I, HD-II, respectively, one of the correlation lengths diverges. In the LD case, it is ξβ which remains infinite throughout the whole phase LD-II. In the HD case, it is ξα which becomes infinite in HD-II. In contrast, ξα and ξβ remain finite in LD-II and HD-II, respectively. 4.2.5.2. LD/HD-MC Transition When crossing the boundary from the LD-II to the MC phase also the correlation length ξα diverges. Because the density profile changes continuously at the boundary this corresponds to a continuous phase transition. The same scenario applies to the boundary between HD-II and MC. Here ξβ diverges. At the transition lines separating the maximal current phase from the high-density and low-density regimes, respectively, the bulk density is 12 with deviations of order j −3/2 .
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4.2.5.3. LD–HD Transition On the transition line α = β < 12 separating high-density and low-density phases, the normalization behaves as ZL ∼ LJ −L which leads to a linear density profile j ρj = α + (1 − 2α) . L
(4.106)
We will later discuss this in more detail and show that this linear profile is the consequence of a freely diffusing shock, which separates a low-density region at the left end and a high-density region at the right end of the chain. The linear profile is the average of shock profiles with random shock positions. Both correlation lengths ξα and ξβ remain finite on the coexistence line, but ξ diverges because ξα = ξβ [see Eqn (4.86)] . The density profile changes discontinuously when crossing the coexistence line. Therefore, it corresponds to a first-order transition.
4.2.6. Relation with Combinatorics The ASEP has remarkable connections to several combinatorial problems. In [147, 148], a connection between the matrix algebra and various weighted lattice paths has been established. For random-sequential dynamics, the normalization constant of the TASEP is equivalent to the partition function of one-transit walks or Dyck paths8. A different mapping has been proven in [325]. For parallel dynamics [127] and the partially asymmetric case [123, 146], relations with Motzkin paths and continued fractions [129] were derived. There is also a connection with matrix models [378]. These equivalences not only indicate the origin of the combinatorial factors in the normalization. They can also be used for simplifying the calculation of correlation functions [148] and yield a better foundation of the relevance of Lee-Yang zeroes of the normalization [127]. More recently, these results have been generalized by showing that the stationary probabilities of the PASEP can be interpreted as weight-generating functions for certain permutation tableaux (generalizations of Young tableaux) [240, 241].
4.2.7. Bethe Ansatz The Bethe Ansatz (BA) has been successfully applied in the case of periodic boundary conditions, see Section 4.1.2. For open boundary conditions, the situation becomes more complicated. In contrast to quantum spin systems, there are usually no conserved charges, but the main problems are related to the nondiagonal boundary terms. Based on results obtained in [175, 1040] for the equivalent XXZ quantum chain, de Gier and Essler 8 A Dyck path is a random walk that only visits sites j ≥ 0. For a one-transit walk, only one transit from negative to
positive j is allowed. This corresponds to two Dyck paths.
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[269, 270] have analyzed the Bethe Ansatz equations and obtained the finite-size scaling of the spectral gap. This characterizes the relaxation behavior at large times. Furthermore, they observed boundary-induced crossovers between different scaling regimes (massive, diffusive, Kardar–Parisi–Zhang [KPZ]). We will postpone the discussion of the BA to Section 4.3.2 where the general case of the PASEP will be considered.
4.2.8. Dynamical MPA The general strategy for the dynamical MPA which, in principle, allows to access the full dynamics has been discussed in Section 2.4.3. Here we will briefly present results for the case of the ASEP. The dynamical algebra can be brought into the form [1312, 1313] dD = C −1 − C −1, dt C −1 D = DC −1
(4.107) (4.108)
with the current operator = DE. The first equation can be interpreted as the operator form of the continuity equation, whereas the second equation is a constraint, which quantises the operators. The equations are automatically satisfied in the stationary limit where = C. The general procedure outlined in Section 2.4.3 leads to a form, which involves the two-particle scattering matrix S(k1 , k2 ), which is identical to that derived in [480]. Thus the known Bethe Ansatz equations S(kl , kj ) (4.109) eiLkj = l = j
are reproduced. The dynamic MPA has been used to rederive the BA equations for the symmetric case (SSEP) with open boundary conditions [1315] and the asymmetric case with periodic boundaries [1222]. The general case (ASEP with open boundaries) is treated in [1223].
4.2.9. Hydrodynamic Limit At large time and length scales where stochastic details have been smoothened out, the ASEP can be described by a suitable coarse-graining procedure (see Section 2.5.1). Due to the factorization of the stationary distribution, the hydrodynamic equation in the case of the ASEP can be derived using a mean-field approximation [1312]. Starting from the lattice continuity equation dρl = Jl−1 − Jl dt
(4.110)
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with ρl = nl and the local current Jl = pnl (1 − nl+1 ) − qnl+1(1 − nl ), one obtains in mean-field approximation dρl = pρl−1 − qρl − (p − q)ρl−1ρl − [ pρl − qρl+1 − (p − q)ρl ρl+1 ], dt
(4.111)
which is exact for the symmetric case p = q. Similar equations hold for the boundary sites l = 1 and l = L. Applying Eulerian scaling, x := la,
λ := a(p − q),
1 ν = a 2( p + q), 2
(4.112)
where a is the lattice constant and the system length L remains fixed, to equation (4.111) by taking the continuum limit a → 0 one obtains ∂ρ ∂ ∂ρ = ν − λ(ρ(1 − ρ)) , ∂t ∂x ∂x
(4.113)
which is equivalent to the noiseless Burgers equation. This equation can be linearized by the Cole-Hopf transformation h(x, t) =
ν ln u(x, t) + f (t) with λ
ρ(x, t) =
1 ∂h + . 2 ∂x
(4.114)
After another change of variable η = νt, one obtains the diffusion equation ∂u ∂ 2 u . = ∂η ∂x2
(4.115)
The relevant solutions of this diffusion equation are of the type [1312] u(x, t) =
M +1
Am eam x+am νt . 2
(4.116)
m=1
For imaginary am , one obtains wave-like solutions for ρ(x, t). Finite M and real am gives M-soliton solutions.
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4.3. PARTIALLY ASYMMETRIC EXCLUSION PROCESS As already mentioned earlier, the ASEP can be generalized to the partially asymmetric case where hopping in both directions is possible. The hopping rate to the right is p, and the rate for motion in the opposite direction is denoted by q. For obvious symmetry reasons, it is sufficient to consider the case 0 ≤ q ≤ p, which implies that the motion to the right is preferred. The boundary conditions have to be generalized by allowing particles to be inserted and removed at both ends of the chain. The new rate for removing particles at the left end will be denoted by γ , and the rate for inserting particles at the right end will be denoted by δ. In the following, we will only discuss the case of random-sequential update. An extension to the other updates is possible in a similar way with the exception of parallel dynamics. Here conflicts might occur where two particles try to move to the same site (one moving right, the other left). Therefore, the dynamics needs to be supplemented by additional rules how to treat such conflicts. Using the quantum formalism, it can be shown that after a similarity transformation L √ 1 0 , (4.117) H = pq Uμ−1 HXXZ Uμ , with Uμ = 0 μQ j−1 i=1
√
where μ is a gauge parameter and Q = q/p, the PASEP becomes equivalent to the UQ (SU (2))-invariant XXZ model with additional boundary terms [270, 354, 1210, 1223]:
1 x x y y z z σj σj+1 + σj σj+1 + σjz σj+1 HXXZ = − + h(σj+1 − σjz ) − + B1 + BL , 2 L−1 j=1
(4.118) where the interaction parameters are given by 1 1 = Q + Q −1 , h = Q − Q −1 , 2 2
Q=
The boundary terms 1 B1 = √ α + γ + (α − γ )σ1z − 2αμσ1− − 2γ μ−1 σ1+ , 2 pq
q . p
(4.119)
(4.120)
1 BL = √ β + δ − (β − δ)σLz − 2δμQ L−1 σL− − 2βμ−1 Q −L+1 σL+ (4.121) 2 pq contain nondiagonal contributions (like σ1± ) and have L-dependent coefficients.
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The spectrum of the Hamiltonian shows particle-hole and left–right symmetry: α ↔ γ,
β ↔ δ,
p ↔ q,
λ → −λ,
(4.122)
α ↔ δ,
β ↔ γ,
p ↔ q,
λ → −λ.
(4.123)
Without the boundary terms, the spectrum of the Hamiltonian is massive. The ASEP belongs to the KPZ [709] with dynamical exponent z = 3/2 or Edwards-Wilkinson universality class [338] with z = 2, depending on whether the hopping rates is asymmetric (p = q) or symmetric (p = q), respectively. Upon varying the boundary rates, crossovers in massive regions with dynamical exponent z = 0 and between massive and scaling regimes with diffusive (z = 2, symmetric case) and KPZ (z = 3/2) behavior occur.
4.3.1. MPA Solution The matrix-product solution of the TASEP can be extended to the PASEP in a straightforward way [293]. For the most general boundary conditions allowing particle insertion and removal at both ends, the MPA leads to the generalized matrix algebra pDE − qED = D + E
(4.124)
(βD − δE)|V = |V
(4.125)
W |(αE − γ D) = W |.
(4.126)
In the case q = 0, γ = δ = 0 it reduces to the previously studied TASEP case, whereas q = p corresponds to the most general form of the SSEP. In principle, one can use the same techniques as in the totally asymmetric case. Some explicit representations for the matrices are given in Appendix 4.9.6. However, the calculations become more cumbersome and involve special functions. Generically, the algebra does not have finite-dimensional representations for which the boundary vectors could have a physical interpretation. However, for certain submanifolds, such representations exist [354, 908]. These can be interpreted as superpositions of shocks diffusing through the system [91, 644, 798]. The analysis of the matrix algebra can be simplified in the case of symmetric dynamics (p = q) [287]. In this case, the weights can be calculated directly from the generalized matrix algebra. A key step is the introduction of the matrices A = βD − δE,
B = αE − γ D,
(4.127)
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which satisfy the commutation relation [A, B] = AB − BA = (αβ − γ δ)(D + E) = (α + δ)A + (β + γ )B.
(4.128)
This can be used to rearrange the matrix product such that it becomes a sum of terms of the form B b Aa , which have simple matrix elements W |B b Aa |V = W |V . Note that the case αβ = γ δ is special. In this case, an equilibrium steady state is realized [123]. Explicitly one obtains for the normalization [287, 302] W |(D + E)L |V W |V
1 1 L + + α+γ β+δ 1 = L 1 1 (ρa − ρb ) α+γ + β+δ
(4.129)
in terms of the -function. The calculation of the density profile and correlation functions for the SSEP can be done directly from the definition of the dynamics [287]. The corresponding evolution equations are closed, i.e., do not involve higher correlation functions [1269, 1273, 1315, 1316]. For the steady state, the density profile ρj = nj is given by [302, 1221] ρj =
ρa (L + 1/(β + δ) − j) + ρb( j − 1 + 1/(α + γ )) , L + 1/(α + γ )1/(β + δ) − 1
(4.130)
which in terms of the macroscopic coordinate j = L x simplifies to ρ(x) = (1 − x)ρa + xρb .
(4.131)
Here ρa and ρb are the densities of the boundary reservoirs [300, 349]: ρa =
α , α+γ
ρb =
δ . β+δ
(4.132)
The average current in the steady state is given by J=
ρa − ρb . 1 1 L + α+γ + β+δ −1
(4.133)
Asymptotically one has J (ρa − ρb )/L, i.e., the current is proportional to the gradient of the density and thus follows Fick’s law.
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4.3.2. Bethe-Ansatz Solution Although the Hamiltonian HXXZ is known to be integrable [277, 626], the determination of its spectrum is difficult due to the presence of the nondiagonal terms. So far, it has only been achieved if the rates satisfy certain additional restrictions [175, 1040], i.e., (Q L+2k − 1)(αβ − γ δQ L−2k−2 ) = 0
(4.134)
with an integer k (|k| ≤ L/2). This condition can be satisfied for generic values of the parameters by choosing k = −L/2. This Bethe Ansatz solution was applied to the PASEP case by De Gier and Essler [269]. In [269, 270], the totally asymmetric case q = γ = δ = 0 was investigated. The finite-size scaling of the spectral gap, which determines the relaxation behavior at large times, indicates boundary-induced crossovers between massive, diffusive, and KPZ scaling regimes, which are characterized by different dynamical exponents z. Related results based on Monte Carlo simulations were reported in [275]. In [271], these investigations were extended to the general PASEP case, focussing on the high-density and low-density phases. These phases can be subdivided into regimes with different relaxational behavior. In the vicinity of the coexistence line, it can be understood in terms of diffusion of domain walls. Simon [1294] has recently used the coordinate Bethe Ansatz to construct the eigenvectors of the quantum Hamiltonian HXXZ . This construction has helped to clarify the nature of the excitations and the role of the condition (4.134).
4.3.3. Phase Diagram of the PASEP Using the results from the MPA solution, the phase diagram of the PASEP can be determined similar to the TASEP case [126, 1216, 1217, 1401, 1402]. It turns out that the PASEP phase diagram is topologically identical with the TASEP diagram, but new subphases appear (Fig. 4.9). In the following, the case γ = δ = 0 is considered, i.e., particle input and output are only allowed at one end, respectively. We also set p = 1 without loss of generality. The critical rates αc , βc that separate the maximal current phase from the high-density and low-density phase depend on the hopping asymmetry: αc = βc = which reduces to αc = βc =
1 2
1−q , 2
in the totally asymmetric case q = 0.
(4.135)
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•
Maximal-current phase: α > αc , β > βc . The current is given by JMC =
•
1−q 4
(4.136)
and the density profile shows near the boundaries the same universal power-law (4.104) found in the totally asymmetric case. Low-density phase: α < αc , α < β. In the low-density phase, the current is given by JLD =
α(1 − q − α) . 1−q
(4.137)
The bulk density throughout the phase is ρLD =
α 1−q
(4.138)
and extends up to the left boundary. For the behavior near the right boundary, three subphases have to be distinguished. The boundaries between these subphases are characterized by the critical rates αs (β) =
β , q+b
βs (α) =
α , q+a
with
a=
1−q − 1, α
b=
1−q − 1, β (4.139)
and αs∗ = βs∗ =
q(1 − q) . 1+q
(4.140)
In the following, we list the main properties of these subphases. • LD-I: α < β < βs (α), β < βc . The density profile shows an exponential decay similar to (4.91) with the characteristic length β(1 − q − β) −1 ξ = ln . (4.141) α(1 − q − α) • LD-II: αs∗ < α < αc , β > βc . This phase is analogous to that in the case q = 0 with a decay as in (4.93), i.e., the power-law exponent 32 remains unchanged. The characteristic length is given by ξα−1
(1 − q)2 = ln . 4α(1 − q − α)
(4.142)
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A. Schadschneider, D. Chowdhury, and K. Nishinari
β
LD-III
LD-II
MC
βc βs*
HD-II
LD-I HD-I
HD-III
αs*
α
αc
Figure 4.9 Full phase diagram of the PASEP showing additional subphases that have different density decay forms.
• LD-III: β > βs (α), α < αs∗ . This subphase does not exist for q = 0. The density profile shows a purely exponential decay with a characteristic length ξq−1
= ln
q (q + α)2
.
(4.143)
• High-density phase: β < βc , β < α. The properties of the high-density phases are obtained from that of the low-density phases by the particle-hole symmetry ρL+1−j (α, β) = 1 − ρj (β, α). The current throughout the phase is given by JHD =
β(1 − q − β) . 1−q
(4.144)
All three subphases show an exponential decay of the density profile near the left boundary. The bulk density ρHD = 1 −
β 1−q
(4.145)
extends up to the right boundary. The above analysis applies to the case q < 1. Due to the restrictions γ = δ = 0, the case q > 1 is also interesting. Here the bias of the hopping is opposite to the bias induced by the particle input and output. Therefore, this parameter regime is sometimes called reverse-bias phase.
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The mathematical analysis in this regime becomes more complicated and so far the behavior of the density profiles has not been calculated explicitly. However, the current has been shown to vanish asymptotically as
αβ(q − 1)2 J (q − 1 + α)(q − 1 + β)
1 2
q 4−2L. 1
1
(4.146)
From heuristic arguments [123], one expects a particle-(hole-)rich region at the left (right) end of the system such that the system is typically half full. The most general case, where input and output are allowed at both ends of the chain (α, β, γ , δ = 0) has been studied in [1401, 1402]. The phase boundaries can be expressed through the parameters
1 (1 − q − α + γ ) ± (1 − q − α + γ )2 + 4αγ , 2α
1 (1 − q − β + δ) ± (1 − q − β + δ)2 + 4βδ . b± = 2β
a± =
(4.147) (4.148)
In the case q < 1, one finds the three well-known phases, i.e., a • • •
low-density phase for a+ > 1 and a+ > b+ ; high-density phase for b+ > 1 and b+ > a+ ; maximal-current phase for a+ > 1 and b+ > 1.
For q > 1, the same phase structure is found due to the symmetry (α, β, γ , δ, q) ↔ (q−1δ, q−1 γ , q−1 β, q−1α, q−1 ),
(4.149)
at least when all four boundary rates are nonzero.
4.4. EXTENSION OF THE ASEP TO OTHER UPDATE TYPES The stationary properties of the ASEP with open boundary conditions have also been determined for various discrete update types. These will briefly be discussed in the following. The most relevant case for applications is the parallel update. The main properties are collected in Table 4.3.
4.4.1. Ordered-Sequential Updates In the following, we discuss the (site)-ordered-sequential updates for open boundary conditions. As in the periodic case (Section 4.1.7), it is sufficient to consider the
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Table 4.3 Comparison of currents and bulk densities in the three phases for the random-sequential, backward-ordered sequential, and parallel update. The bulk densities for the forward-ordered sequential update are given by ρ − J in terms of the results for the backward-ordered update. The currents for the two ordered-sequential updates and the sublattice-parallel update are identical Random-sequential
Backward-ordered
Low-density
J = α(1 − α/p)
J=
α p−α p 1−α
J =α
phase
ρ=α
ρ=
α p
ρ=
High-density
J = β(1 − β/p)
J=
phase
ρ = 1−β
Maximum current J =
p 4
phase
ρL/2 =
Critical rate
αc =
1 2
p 2
p−β 1−β 1 p−β ρ= p 1−β √ 1− 1−p J= √ 1+ 1−p 1 ρL/2 = √ 1+ 1−p √ αc = 1 − 1 − p β p
Parallel
p−α p − α2
α(1 − α) p − α2
p−β p − β2 p−β ρ= p − β2 √ 1− 1−p J= 2 1 ρL/2 = 2 √ αc = 1 − 1 − p
J =β
backward-sequential case due to the particle-hole symmetry, which relates the two update directions. The matrix algebra obtained from (2.57) can be mapped onto that of a PASEP with random-sequential dynamics [1160, 1162]. This allows, in principle, to determine representations and calculate expectation values for the current and the density profile. The structure of the phase diagram is qualitatively the same as in the randomsequential case. All well-known features of the phase diagram (high-density phase, low-density phase, maximum current phase, coexistence line with linear density profile) are recovered. For (1 − α)(1 − β) = 1 − p, (4.150) a one-dimensional solution of the algebra exists.9 This equation defines a line in the phase diagram where the mean-field solution becomes exact. This line touches all phases, which allows for a simple derivation of analytic results for the current or the bulk density, which do not change inside a phase. 9 For the partially asymmetric case, this line is given by [1160, 1162] (1 − α)(1 − β)(1 − q) = 1 − p, where q is the
hopping probability to an empty site on the left.
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The intersection of the mean-field line (4.150) with the line α = β defines the endpoint αc (= βc ) of the coexistence line. This yields the critical rate αc = βc = 1 − 1 − p.
(4.151)
In the case of deterministic hopping (p = 1), the maximum current phase vanishes, and we recover the result of Hinrichsen [566]. The currents in the three phases are given by [1160] ⎧α p − α ⎪ low-density phase, ⎪ ⎪ ⎪p 1−α ⎪ ⎪ ⎨ p−β β high-density phase, (4.152) J (α, β, p) = p 1−β ⎪ ⎪ √ ⎪ ⎪ 1− 1−p ⎪ ⎪ ⎩ maximum current phase. √ 1+ 1−p The currents in the forward-sequential case are identical. The bulk densities in the three phases are
ρ(α, β, p) =
⎧α ⎪ ⎪p ⎪ ⎪ ⎪ ⎨1 p − β p 1−β ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎩ √ 1+ 1−p
low-density phase , high-density phase,
(4.153)
maximum current phase,
For better comparison with other updates, the results are collected in Table 4.3. In order to obtain the corresponding results for the forward-sequential case, one makes use of the symmetries of the system. Explicitly one finds the following symmetry of density profiles [1160]: ρ→ (α, β, p, x) = ρ← (α, β, p, x) − J (α, β, p) .
(4.154)
In addition, the particle-hole symmetry gives ρ→ (α, β, p, x) = 1 − ρ← (β, α, p, L − x + 1) .
(4.155)
By using (4.154) and (4.155), and the results for the current and ρbulk in the high-density region, the bulk density in the low-density region ρ bulk = α/p is obtained. A finite-size analysis of currents and density profiles for the forward-sequential case can be found in [149–151].
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4.4.2. Sublattice-Parallel Update The ASEP with sublattice-parallel update, first solved by Schütz [1264], can also be solved using the MPA. It leads to the same matrix algebra as the ordered-sequential updates. For p = 1 the algebra was first derived and solved by Hinrichsen [566] using a ¯ D, ¯ W |, |V . The general case, including two-dimensional representation for E, D, E, partially asymmetric hopping, was then studied by Honecker and Peschel [576]. Although the stationary states of the ordered-sequential and sublattice-parallel updates are different, they are connected through transformations. It can be shown that the density profile of the forward(backward-)ordered-sequential update corresponds to the density of the odd (even) sites produced by the sublattice-parallel update [1163]. The current is identical to that of the ordered-sequential updates.
4.4.3. Parallel Update The exact solution for the TASEP with parallel dynamics and open boundary conditions has been obtained independently by Evans, Rajewsky, and Speer [375] and de Gier and Nienhuis [272] by different types of MPA. In [375], a site-oriented Ansatz of the form L 1 W| nj D + (1 − nj )E |V P(n1, . . ., nL ) = ZL j=1
(4.156)
has been used where the matrices D and E represent particles and holes, respectively. It was shown that this Ansatz gives the steady state if the matrices satisfy the quartic algebra EDEE = (1 − p)EDE + EEE + pEE
(4.157)
EDED = EDD + EED + pED
(4.158)
DDEE = (1 − p)DDE + (1 − p)DEE + p(1 − p)DE
(4.159)
DDED = DDD + (1 − p)DED + pDD
(4.160)
plus equations for the right boundary, DDE|V = (1 − p)DE|V + DD|V ,
(4.161)
EDE|V = ED|V + EE|V , p(1 − β) DD|V = D|V , β p ED|V = E|V , β
(4.162) (4.163) (4.164)
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and the left boundary W |DEE = (1 − p)W |DE + W |EE,
(4.165)
W |DED = W |DD + (1 − p)W |ED, p(1 − α) W |EE = W |E, α p W |ED = W |D. β
(4.166) (4.167) (4.168)
The quartic algebra can be reduced to a quadratic one by making the Ansatz [375, 1481] E1 gD1 D1 0 , E= , (4.169) D= 0 0 gE1 0 which yields D1 E1 = (1 − p) D1 + E1 + p , p(1 − α) W1|, α p(1 − β) D1 |V1 = |V1 , β
W1|E1 =
(4.170) (4.171) (4.172)
where W1 |, W2 |, |V1 , and |V2 are vectors of the same dimension as D1 and E1 with W | = (W1 |, W2 |) and |V = (|V1 , |V2 )t . Although all physical quantities can be expressed in terms of E1, D1 and the boundary vectors the weights become rather complex expressions in terms of the reduced algebra. Therefore an alternative representation has been proposed in [1481]. It is inspired by the two-cluster approximation, which becomes exact in the periodic system. It has the form 0 t(01)D1 t(00)E1 0 , D= , (4.173) E= 0 t(11)D1 t(10)E1 0 where t(τ σ ) are the two-site factors of the solution for periodic conditions. The boundary vectors W | and |V are given by " ! α 1−p (4.174) W1|, |V1 , |V1 . W | = W1|, p−α 1−β Setting t(00) = t(10) = 1 as in the periodic case and t(01) = (1 − p)−1 , t(11) = 1 one recovers the algebra (4.170).
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Using this alternative representation, the weights in the stationary state have the form P(n1 , . . ., nL ) = w(n1)t(n1 , n2 ) . . .t(nL−1 , nL )v(nL ) W1 |
L
[nl D1 + (1 − nl )E1]V1 ,
l=1
(4.175) which can be interpretated as a superposition of a pair-factorized state and a matrix state. Here t(τ σ ) is defined through t(11) = (1 − p)t(01) and
t(10) = t(00).
(4.176)
and the boundary factors are w(n1) =
α t(01)n1 + t(11)n1 , p−α
v(nL ) =
1 − pnL . 1 − βnL
(4.177)
The solution by de Gier and Nienhuis [272] uses a bond-oriented matrix Ansatz P(n1 , n2 , . . ., nL ) = W (n1 )|M (n1 , n2 )M (n2 , n3 ) . . .M (nL−1 nL )|V (nL ), where the boundary vectors and matrices M (n, m) are given by # $ (1 − τ )(1 − σ )M(00) (1 − τ )σ M(01) M (τ σ ) = τ (1 − σ )M(10) τ σ M(11)
(4.178)
(4.179)
W (τ )| = ((1 − τ )W(0)|, τ W(1)|),
(4.180)
|V (τ ) = ((1 − τ )|V(0), τ |V(1))t .
(4.181)
This form can be related to the site- and bond-oriented solutions [1481] by choosing t01 = 1 and t11 = 1 − p in (4.173). Then the explicit relation between the different Ansätze is given by M(00) = M(10) = E1 , W(0)| = W1 |E1 + W2|E1 , |V(0) = |V1 ,
M(11) = (1 − p)M(01) = D1 W(1)| = W1|D1 + (1 − p)W2|D1 , |V(1) = |V2 ,
(4.182) (4.183) (4.184)
so that E = M (00) + M (10),
D = M (10) + M (11),
(W |E, 0) = W (0)|,
(0, W |D) = W (1)|,
(4.185) |V = |V (0) + |V (1).
(4.186)
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Asymmetric Simple Exclusion Process – Exact Results
4.4.3.1. Phase Diagram for Parallel Dynamics The phase diagram for parallel dynamics has topologically the same structure as that for the random-sequential case (Fig. 4.10). As found in [1160], the two-cluster approximation is exact on the line (1 − α)(1 − β) = 1 − p,
(4.187)
where the density profile becomes flat. Because this line touches all three phases, this allows to determine the currents and bulk densities from the known two-cluster results for the periodic system (see Table 4.3). The critical rate (4.188) αc = βc = 1 − 1 − p, which determines the phase boundaries is the same as in the backward-ordered case. The calculation of currents and density profiles is quite cumbersome. Explicit results can be found in [272, 375]. In [375], relations with other updates have been found. Especially useful is the connection with the forward-sequential update, which gives the relations J =
J→ , 1 + J→
ρ ( j) =
ρ( j)→ + J→ 1 + J→
(4.189)
between the currents and the density profiles in the parallel and forward-sequential updates. 1
β
LD
MC
0.5
HD 0
0.5 α
1
Figure 4.10 Exact phase diagram for the ASEP with parallel dynamics for p = 3/4. The broken line corresponds to (1 − α)(1 − β) = 1 − p, where the density profile is flat.
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4.5. BOUNDARY-INDUCED PHASE TRANSITIONS As we have seen in the previous sections, a change of the boundary rates can induce qualitative changes in the bulk of the TASEP. This is called boundary-induced phase transition and was first observed by Krug [812] in the case β = 1. In equilibrium systems, it is usually argued that boundary conditions are irrelevant in the thermodynamic limit. For short-range interactions, the contribution to the free energy due to the boundary is proportional to the surface and can be neglected because the bulk term is proportional to the volume. Similar arguments are not available in the case of nonequilibrium systems. A discussion of boundary-induced phase transitions in terms of the quantum formalism, which helps to clarify their origin, can be found in [559]. Based on hydrodynamical arguments, Krug [812] has conjectured that certain stationary properties like the bulk density can be obtained from the fundamental diagram by a maximum principle. This implies that the open system can be described in terms of quantities of the periodic system. This is also remarkable from a technical point of view because the periodic case is usually easier to study, e.g., due to its translational invariance. Later these observations have been generalized and put on a firmer footing. Again hydrodynamic arguments for coarse-grained systems have been used so that the results do not depend on the microscopic details of the interaction and are valid beyond the ASEP case. In [780], an intuitive picture in terms of the diffusion of shocks or domain walls and the drift of localized density fluctuations has been developed, which we will describe in the following.
4.5.1. Domain Wall Picture The domain wall picture allows to understand even quantitatively many aspects of driven diffusive systems with open boundaries, especially the nature and location of the phase transitions. The following arguments are valid for a rather general class of models. However, it is good to have the explicit example of the TASEP in mind. For convenience, the boundary rates α and β are replaced by appropriately chosen densities ρ− and ρ+ of the boundary reservoirs. We sketch only the main aspects, additional details can be found in [780, 1122, 1272, 1273]. The domain wall theory allows to understand the long-term dynamics of driven systems by the interplay of two characteristic velocities, the domain wall or shock velocity and the collective velocity. In nonequilibrium systems, a domain wall or shock is an object connecting two possible stationary states. A domain wall separating two stationary regions of densities ρ1 and ρ2 moves with the shock velocity vs =
J (ρ2 ) − J (ρ1) , ρ2 − ρ1
(4.190)
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Asymmetric Simple Exclusion Process – Exact Results
which follows from mass conservation (or the equation of continuity) and is also known as Rankine–Hugoniot equation. However, the center-of-mass of a local perturbation (of the density) moves in a homogeneous, stationary background of density ρ with the collective velocity vs =
d J (ρ) . dρ
(4.191)
The dynamics of the shock is mainly determined by the shock velocity, whereas the collective velocity is related to its stability. Note that both velocities can be positive and negative, depending on density. In fact, as we will see soon, the signs of the velocities are the decisive elements of the domain wall picture. One can now understand the origin of the boundary-induced phase transitions in the open system by preparing it in judiciously chosen initial states. If the system is ergodic, as the ASEP, there will be no loss of generality as it will evolve in the same stationary state. Suppose the systems is initially in a state of density ρl at the left end and ρr at the right end, separated by a domain wall. If vs > 0, the domain wall will move to the right end, leaving the whole system in a state of density ρl . For vs < 0, the shock will move to the left and the density becomes ρr . For the validity of this argument, one has to take the stability of the shock into account. In the TASEP, only upward shocks (ρl < ρr ) can be stable, whereas downward shocks (ρl > ρr ) will smear out [1273]. To see this, consider a small perturbation at the position of a upward shock. As for the TASEP vc (ρl ) > vs > vc (ρr ), the excess mass will drift back to the shock position and thus stabilize it. For the downward shock, however, the excess mass will move away from the shock and smear it out. This argument based on the fluctuations in the neighborhood of the shocks holds rather generally and leads to the stability criterion [1273] vc (ρl ) > vs > vc (ρr )
(4.192)
for a single shock. We can now understand the origin of the three phases observed in the TASEP by considering an initial state with density ρl in the left and ρr in the right part of the system, where ρl < ρr . For J (ρl ) < J (ρr ), the shock velocity will be positive. Thus the shock will move to the right boundary, and the system will have bulk density ρl . This is the lowdensity phase. In contrast, for J (ρl ) > J (ρr ), the shock velocity is negative and the system is driven into the high-density phase with bulk density ρr . For J (ρl ) = J (ρr ) < Jmax , the shock velocity is zero, where Jmax is the maximum of the current-density relation J (ρ). This corresponds to the coexistence line where the shock diffuses freely through the system.
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Within the phases, domain wall transitions occur for ρl,r = ρ ∗. The density ρ ∗ is defined by J (ρ ∗) = Jmax so that the collective velocity changes its sign. For ρl , ρr > ρ ∗ , the system is in the maximum current phase. Then vc (ρl ) < 0, and a small perturbation created at the left boundary will return to the boundary and limit further inflow. This corresponds to an overfeeding with particles and the system is effectively boundary controlled, leading to a stabilization of the maximum current. The domain wall scenario allows to make further quantitative statements [780, 1213, 1273], e.g., about the diffusion constant Ds =
1 J (ρr ) + J (ρl ) 2 ρr − ρl
(4.193)
Ds ξ∼ = |vs |
(4.194)
or the localization length
of the shock. Using J (ρ) = ρ(1 − ρ) and ρl = α, ρr = 1 − β for the TASEP one has the collective velocity vc = 1 − 2ρ
(4.195)
and the shock diffusion constant (in the high-density phase) [780] ⎧ 1 α(1 − α) + β(1 − β) ⎪ ⎪ ⎨ 2 1−α−β Ds = ⎪ 4β(1 − β) + 1 ⎪ ⎩ 4(1 − 2β)
for α, β <
1 2
1 1 for α ≥ , β < 2 2
.
(4.196)
For the TASEP, the position of the domain wall is sharp [274, 387]. The localization length is given by J (ρr ) ξ = ln . J (ρl )
(4.197)
The above arguments apply for systems with one maximum in the flow–density relation. However, they have a natural extension to more general cases [485, 486, 1122, 1273].
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4.5.2. Extremal Principle and Steady-State Selection The qualitative description of Section 4.5.1 can be summarized as an extremal principle [1122]
J (ρl , ρr ) =
⎧ ⎨maxρ∈[ρr ,ρl ] J (ρ) for ρl > ρr ⎩min ρ∈[ρl ,ρr ] J (ρ)
for ρl < ρr
(4.198)
where J (ρ) is the fundamental diagram (obtained in the periodic system) and J (ρl , ρr ) the current realized in an open system with reservoir densities ρl at the left and ρr at the right end. The number of phases realized in the open system depends only on the structure of the fundamental diagram, i.e., the number of maxima of the current. The extremal principle can also be interpreted as steady state selection principle: the state realized in an open system is selected by the boundary conditions from the steady states of the periodic system. In the simplest case of a current-density relation with only one maximum (as in the TASEP) one expects three phases, but the extremal principle also applies to systems with more complex fundamental diagrams [485, 486, 1273]. In a system where the current-density relation has two maxima (and a minimum in between), one finds an additional minimal current phase. In this case, the phase diagram generically consists of seven distinct phases. The principle has also been studied in the context of models with metastable states [36, 74, 1035] or more general traffic models [74, 1121].
4.5.3. More on Shock Dynamics We have seen that the dynamics of shocks play an important role for the understanding of the phase diagram, and so on. From physics point of view, a shock can be considered as collective dynamical mode. Its motion can be characterized by a single coordinate, the shock position. In the applications discussed in Part II of this book, macroscopically the shock position generically corresponds to the end of a traffic jam where the local density suddenly increases in space. The standard tool to determine the shock dynamics are second-class particles [30, 297, 387], which are attracted by the shocks because they follow the trajectories of density fluctuations. Roughly speaking, a second-class particle behaves like a normal particle, only when it encounters other particles it behaves like a hole (vacant site) (see also Section 4.7.1). Therefore, they tend to move toward regions of strong density variations. Different approaches have been proposed, which are useful in the cases where no second-class particle can be defined or when their number is not conserved. The
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approach of Belitsky and Schütz [91] tries to capture not only the structure but also the microscopic dynamics of shocks. The shock dynamics can not only be used to describe the stationary state but also to calculate fluctuations in the number of particles and density profiles in the transient regime [135, 1109, 1213]. Relaxation times have been determined analytically using the domain wall picture [326]:
−1 √ , τn = D + + D − − 2 D + D − cos(π n/(L + 1))
(4.199)
where D + = β(1 − β)/(1 − α − β) and D − = α(1 − α)/(1 − α − β). This prediction has been verified numerically using exact diagonalization [326] and density-matrix renormalization group methods [1030]. Surprisingly, for many processes the shocks perform rather simple random walks [64]. The occurrence of random walking shocks is closely related to the exact solvability through the MPA for systems with open boundaries. The existence of n shocks in the stationary state implies the existence of n-dimensional representations for matrix algebra [41, 640, 644, 798]. Multiple shocks, i.e., several steps in the density profile, might also occur [42, 63, 91, 641]. Micro-shocks can attract each other and form a bound state of finite width, which can be considered as a single shocks with an internal structure. In systems without particle number conservation, the shocks can become localized as has been shown in [366, 937, 1119]. Shocks also occur in multichannel [936, 1124] and multispecies systems [641, 645, 1170], as well as in systems with internal degrees of freedom [1175, 1333]. Three-site interactions can lead to the formation of double shocks [1090]. The role of the boundary conditions, e.g., the reflection of shocks, has been studied in [1114, 1115, 1125] for multispecies systems. Shocks also occur for discrete time updates, see, e.g., [642, 646, 1110]. Typically, it is assumed that the dynamics of a system is not strongly affected by the presence of a few second-class particles or, more generally, shock-tracking probes. However, this is not always the case as shown in [190, 191], especially for equilibrium systems. In the presence of a finite number of second-class particles, a long-ranged attraction between successive probe particles is induced [297, 858, 1165].
4.5.4. Fluctuations and Large Deviation Functions Apart from the averages also the fluctuations, e.g., those of the current, encode important information about the dynamics. For periodic boundary conditions, fluctuations have been calculated using mainly two approaches, namely the MPA and the Bethe Ansatz. The diffusion constant which characterizes the average deviation of the current from its mean value was first calculated using the MPA for periodic boundary conditions in
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[296] and for open boundaries in [295] and generalized to the PASEP in [304]. A derivation using the Bethe Ansatz was later given in [1146]. Higher cumulants have been obtained for various cases in [39, 288, 299, 839, 1144, 1146]. Another quantity that provides deeper insight into the nature of fluctuations is the large deviation function. We consider the integrated current Yt over all particles, i.e., the total distance covered by all particles between time 0 and time t. Yt increases by one if a particle moves to the right, and it decreases by one if a particle moves to the left. In the long-time limit, the probability to observe a value j of the current different from its mean value vanishes exponentially fast [287]: Yt (4.200) = j ∼ e−tG( j) , P t where G( j) is the large deviation function of the current. It can be calculated [288, 299] by an extension of the Bethe Ansatz method, which allows to derive exact expressions from which, e.g., the diffusion constant and higher cumulants can be obtained. The large deviation functional of the density introduced by Derrida, Lebowitz, and Speer [300] can be considered to be an extension of the notion of free energy to nonequilibrium systems [1202]. For large systems, the probability of any macroscopic density profile ρ(x) is given by Prob[ρ(x)] ∼ exp(−LF (ρ(x)),
(4.201)
where F is the large deviation functional of the density. In equilibrium, it can be expressed by the free energy [300]. For the ASEP, it gives information about density fluctuations, which are found to be non-Gaussian [290, 301–303]. A related approach is the macroscopic fluctuation theory developed by Bertini et al. [107–109]. The investigation of fluctuations, large deviations, and so on is currently a very active field where a lot of progress has been made recently which cannot be covered here due to space limitations. We refer to reviews, e.g., [109, 286, 287, 1219, 1376]. The same is true for the recent developments concerning fluctuation theorems. A nice introduction to fluctuations theorems for the class of models studied here has been given by Harris and Schütz [498] (see also [497, 1164]).
4.6. EXTENSIONS OF ASEP Apart from the PASEP (see Section 4.3), several other variants of the ASEP are of interest. In the following, we discuss some models that are either related to the ASEP in some way or which can be interpreted as more special transport processes.
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4.6.1. Quenched Disorder In the ASEP, all particles have identical properties. For many applications, this is not really realistic. Therefore, inhomogenities (or disorder) in the form of quenched (i.e., time-independent) randomness is introduced, usually in the hopping rates (see, e.g., the reviews by Krug [814] and Barma [78]). We will see that disorder is a frequent reason for phase separation in driven diffusive systems. Another reason for the interest in this problem is that several aspects of disorder in nonequilibrium systems are not well understood [1314]. One can distinguish two types of quenched disorder, depending on whether one associates the hopping rates with particles or with sites: • Particlewise disorder: Here the hopping rates can be different for different particles [355, 356, 813, 815]. One often studied situation is when a finite number (in the thermodynamic limit) of particles has a smaller hopping rate than the rest. These particles are then usually called defect particles, impurities, or simply defects. In the case of an infinite number of defects, the system is characterized by the defect density. • Sitewise disorder: Here the hopping rates are associated with the sites and are positiondependent [651, 652]. If a finite number of sites has a smaller hopping rate than the rest, one speaks of defect sites or just defects. These are localized in space in contrast to the case of particlewise disorder. A consecutive sequence of defects is sometimes called bottleneck of length . In other models, disorder can also be realized in a different way, e.g., by allowing different maximal velocities. This will be discussed in more detail in Section 8.7 of Part II. In both cases, a parameter regime exists where the global behavior of the system is controlled by the defect, which acts as a bottleneck. Generically, it induces phase separation into a high-density and a low-density region separated by a sharp discontinuity or shock. In the case of particlewise disorder with one slow particle (and no overtaking), the faster particles tend to pile up behind the slow one. This behavior has certain similarities with Bose–Einstein condensation [355, 356]. For a spatially localized defect one also finds a separation into a high-density and a low-density regime, but with the high-density region pinned to the defect. This behavior has been found in a variety of models and for different defect realizations. 4.6.1.1. Sitewise Disorder The qualitative features of the ASEP with a single defect site with hopping rate pd < p have first been studied in detail in [651, 652]. It has been found that large systems are only affected by the presence of the defect in some intermediate-density regime. This can clearly be seen in the form of the fundamental diagram (Fig. 4.11).
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0.25 Mean-field Simulation
0.2
J
0.15
0.1
0.05
0
0.2
0.4
0.6
0.8
1
ρ
Figure 4.11 Fundamental diagram of the TASEP with parallel dynamics and a single defect with pd = 0.25. The hopping probability in the bulk is given by p = 0.75. 1
0.8
ρ = 0.10 ρ = 0.50 ρ = 0.90
ρ (x)
0.6
0.4
0.2
0
500
1000
1500 x
2000
2500
3000
Figure 4.12 Density profiles for different densities ρ in the TASEP with parallel dynamics and a blockage site located at the site 1. The hopping probabilities are p = 0.75 and pd = 0.25.
What makes the problem of a single “point defect” nontrivial is that, over the interval ρ1 ≤ ρ ≤ ρ2 of the density of the vehicles, where J is maximal and independent of ρ, the localized blockage has global effects, whereby the steady state exhibits macroscopic phase segregation into high-density and low-density regions. Evidence for such macroscopic phase segregation can be obtained directly from the density profiles. Figure 4.12 implies that so long as ρ < ρ1 , the particles will not pile up. A local increase of density will
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compensate for the reduced local velocity at the blockage so that the flux around the blockage is identical to that far from it. However, if the global density exceeds ρ1 , the particles pile up during the transient period leading to the phase-segregated steady state. Because of the particle-hole symmetry, the phase segregation does not take place if the particle density exceeds ρ2 . Based on the assumption that the steady state shows phase separation, as observed in computer simulations (Fig. 4.12), a semiphenomenological theory10 for the TASEP with parallel dynamics and a single impurity site can be developed. Naturally, this theory cannot explain the underlying mechanism that gives rise to the phase-segregated structure of the steady state. But, as we shall see soon, it provides a good estimate of the flux in the phase-segregated regime. Our calculations are based on arguments suggested originally by Janowsky and Lebowitz [651, 652] for the random-sequential case. Using the exact result (4.29), the currents in the high-density and low-density regions, far from their interface, are given by 1 1 and J = 1 − 1 − 4pρ (1 − ρ) (4.202) Jh = 1 − 1 − 4pρh (1 − ρh ) 2 2 and the current across the defect bond is given by Jdef 12 1 − 1 − 4pd ρh (1 − ρ ) . Since, in the steady state, the flux is same across the entire system, we must have pρh (1 − ρh ) = pρ (1 − ρ) and, hence, ρh = ρ
or ρh = 1 − ρ.
(4.203)
The condition ρh = ρ is satisfied by the uniform density profile, whereas the condition ρh = 1 − ρ is satisfied by the phase-segregated density profile (see Fig. 4.12). Moreover, using the condition Jh = Jdef = J , we get ρh (1 − ρh ) = ρ (1 − ρ ) rρh (1 − ρ ),
(4.204)
where r = pd /p < 1 may be interpreted as the transmission probability or permeability of the blockage. From the (approximate) equations (4.204), we get ρh
1 p = r + 1 p + pd
and
ρ
r pd = r + 1 p + pd
(4.205)
and, hence, J=
1 4pr . 1− 1− 2 (1 + r)2
10 A microscopic approach for deterministic dynamics can be found in [1363].
(4.206)
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Asymmetric Simple Exclusion Process – Exact Results
The estimate (4.206) is in good agreement with the numerical data (Fig. 4.11) obtained from computer simulation [1212]. Moreover, the estimates ρ and ρh are also in good agreement with ρ and ρh , respectively, in Fig. 4.12. Note that ρ and ρh depend only on r and are independent of ρ. Moreover, the estimates (4.205) of ρ and ρh are in excellent agreement with ρ1 and ρ2 , respectively, in Fig. 4.11. At first sight, these two results may appear surprising and counter-intuitive. But, we will now show that these are related to the mechanism of the phase segregation. Conservation of the vehicles demands that ρL = ρh h + ρ = ρh h + ρ (L − h),
(4.207)
where h and = L − h are the lengths of the high-density and low-density regions, respectively. Thus, h ρ(1 + r) − r ρ − ρ = = . L ρh − ρ 1−r
(4.208)
The equation (4.208) shows that h/L → 0 as ρ → ρ and h/L → 1 as ρ → ρh . Therefore, keeping r fixed as the density is increased beyond ρ1 = ρ , the densities of the two regions remain fixed but the high-density region grows thicker at the cost of the length of the low-density region as more and more particles pile up. Eventually, at ρ = ρ2 = ρh , the low-density region occupies a vanishingly small fraction of the total length of the system signaling the disappearance of the phase segregation. The same line of argument can be applied to all updates, one only needs to replace the current function J (ρ) by that for the update under consideration. Schütz [1263] considered a TASEP with sublattice-parallel update where the motion of the particles is deterministic (i.e., p = 1) everywhere except at a defect site where they move with the probability pd < 1 (i.e., r = pd < 1). An exact solution can be found through a mapping on a six-vertex model. Later, a solution using the MPA was presented in [569]. Except for minor differences, the qualitative features of the results do not differ from the corresponding approximate results obtained for pd = 1 [651, 652]. The qualitative features of the fundamental diagram do not change significantly if the point-like defect (or, impurity) is replaced by an extended defect, i.e., bottleneck of length Ld . However, with increasing length Ld of the defect, the maximum value of the flux decreases monotonically and approaches the maximum flow of the homogeneous system, where the hopping probability associated with each of the bonds is identical to that associated with the defects. From Figs. 4.13 and 4.14, we conclude that the monotonic decrease of the flow with increasing length of the extended defects leads to a larger difference ρh − ρ between the densities of the high-density and the low-density regions of the phase-segregated steady state.
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0.3 p = 0.75 Ld = 1 Ld = 2 Ld = 4 Ld = 16 p = 0.25
0.25
J
0.2
0.15
0.1
0.05
0
0.2
0.4
0.6
0.8
1
ρ
Figure 4.13 Fundamental diagram for the TASEP with parallel dynamics and bottlenecks of different length Ld for p = 0.75 and pd = 0.25. For comparison, the fundamental diagram of the “fast” (p = 0.75) and “slow” (p = 0.25) homogeneous systems are shown.
1
0.8
Ld = 1 Ld = 2 Ld = 16
ρ
0.6
0.4
0.2
0
500
1000
1500 x
2000
2500
3000
Figure 4.14 Density profiles for different lengths of the defect in the TASEP with parallel dynamics with p = 0.75 and pd = 0.25. The average density is ρ = 0.50. The defect extends over the first Ld sites of the system.
Asymmetric Simple Exclusion Process – Exact Results
Next, instead of a single point-like or extended defect, let us consider the more general case of quenched disorder in the ASEP, where the quenched random-hopping probabilities pj, j+1 are chosen independently from some probability distribution P( p), for the hopping from the cell i to the cell i + 1 (i = 1, 2, . . . , L). For a given realization of the disordered system, every vehicle hopping from a given cell i to the next cell i + 1 must hop with the same probability pi,i+1 and a given vehicle hops across different bonds, in general, with different probabilities assigned to these bonds as it moves forward with time. The effects of finite defect density in a periodic system have been studied in detail in [393, 499, 814, 1397, 1398], mainly numerically. Tripathy and Barma [1397, 1398] have studied the TASEP with a binary distribution of defects to which we refer as disordered TASEP (DTASEP) in the following. Here p are chosen from the binary distribution P(pj,j+1 = pd ) = f and P(pj,j+1 = p) = 1 − f , i.e., f is the defect density. Tripathy and Barma have classified two different regimes: (1) a homogeneous regime with a single macroscopic density and nonvanishing current, (2) a segregated-density regime, with two distinct values of density and nonvanishing current. Considering the partially ASEP, where disorder was realized by inhomogeneous hopping bias, they also found a vanishing-current regime, which shows two distinct densities, but with a current that vanishes asymptotically for (L → ∞). They argue that phase separation can be understood by a maximum current principle: For a given mean density, the system settles in a state, which maximizes the stationary current. Thus the largest stretch of slow bonds acts as current-limiting segment. For the same system, Juhász et al. [697, 699] introduced an effective potential and determined trapping times in potential wells to investigate the vanishing of the current in a finite-size scaling. For more general distributions, a mean-field theory has been developed [1397, 1398] (see Appendix 4.9.7 for details) for computing the fundamental diagram of the DTASEP. The flux in this model has interesting symmetry properties under the operations of “charge conjugation” (which interchanges particles and holes), “parity” (which interchanges forward and backward hopping rates on each bond and “time reversal” (which reverses the direction of the current) [446, 1397, 1398]. The quenched disorder in these disordered models can be viewed as point-like impurities distributed randomly over the lattice. But, the qualitative features of the fundamental diagram of DTASEP are similar to those observed for a single point-like defect or bottleneck. Although the random distribution of the point-like impurities leads to a rough density profile for all densities, in an intermediate regime of density, phasesegregated steady states with macroscopic high-density and low-density regions have been identified. What is the underlying mechanism for the macroscopic phase segregation in these models [1397, 1398]? Let us denote the stretches of fast bonds with permeability
169
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J
0.20
J0
0.00 0
ρ1
ρ2
ρ3
ρ4
1
ρ
Figure 4.15 Origin of phase separation in DTASEP. Shown are the fundamental diagrams of two reference non-DTASEPs.
r = pj,j+1 /p = 1 by X and the stretches of bonds with permeability r = pd /p by Y . The two parabolas in Fig. 4.15 are the two steady state fundamental diagrams for the two pure reference systems consisting of all X and all Y , respectively. Because the flux must be spatially constant in the steady state, the possible densities are given by the four intersections of the line J = J0 with the two parabolas. If the average density is less (greater) than 1/2, then the two possible densities are ρ1 and ρ2 (ρ3 and ρ4 ). The variation of density between ρ1 and ρ2 (or ρ3 and ρ4 ) in the X and Y stretches is merely micro phase segregation while, on a macroscopic scale, the density remains uniform. For simplicity, we assume that the density in each stretch of like bonds is uniform. The global density of the system is approximately ρ (1 − f )ρ1,4 ( J0 ) + f ρ2,3 ( J0 ), where f is the defect density. However, as the density increases, the flux also increases till it attains the maximum allowed flux of the pure system consisting of all Y (this happens at a global density smaller than 1/2). What happens when the density increases further? According to the maximum current principle [812], no further increase of the flux is possible and the excess density is taken care of by increasing the density in some of the X stretches from ρ1 to ρ4 (or, vice versa if ρ > 1/2). This conversion takes place adjacent to the largest stretch of Y bonds where the density also changes from ρ2 to ρ3 (or, vice versa if ρ > 1/2) to accommodate the additional particles added to the system. This leads to the macroscopic phase segregation as the system consists of two macroscopic regions of two different mean densities – one with lower densities ρ1 , ρ2 in the X and Y stretches and the other with the higher densities ρ3 , ρ4 in the X and Y stretches.
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4.6.1.2. Particlewise Disorder In the case of particlewise disorder, different hopping rates pj are assigned to each particle j. In contrast to the ASEP with sitewise disorder, this case can be solved exactly by using the mapping to a ZRP or direct application of the MPA [355, 356, 815]. Before we discuss the exact solution, we will try to get a qualitative understanding of such systems. In the simplest case, all particles have equal hopping rates p except for one particle, which has the hopping rate p < p, i.e., the single impurity particle is the slowest moving one. The faster particles can be allowed to overtake the slow one at a nonzero rate [284, 906]; however, if this rate of overtaking vanishes, the slow particle will give rise to a platoon of particles behind it. This phenomenon is very similar to the formation of platoons of vehicles in a traffic behind the slow vehicles (e.g., trucks). Here, we are interested in a more general situation of quenched disorder in the form of a distribution f (p) of intrinsic hopping rates pj of the particles in the system rather than that of the single defect particle. In such situations, random initial conditions can lead to the formation of platoons if (a) slow particles are sufficiently rare and (b) if the density of particles is sufficiently low. These qualitative arguments can be made more quantitative by using the exact solution obtained by mapping onto a ZRP following the procedure outlined in Section 4.1.4. The hopping rates uj (n) = pj (for n > 0) of the ZRP then become site-dependent, but the corresponding steady state still factorizes. This implies that the gaps (headways) dj between particles are statistically independent with the distribution [103, 355, 815] v Pj (d) = (1 − αj )αjd with αj = (4.209) pj for the gap in front of particle j. v is the common average speed of the particles in the steady state. Similar expressions hold in the PASEP case, but with a more complicated expression for αj [103, 355, 815], and for other updates [356]. Now we consider probability density f (p) on [ p0 , 1] with p0 > 0. Then p0 corresponds to a minimal speed and because the particles are not allowed to pass each other, the steady-state speed cannot be larger than p0 . Using this probability density, the average headway can be calculated using (4.209). However, the average gap is given by (1 − ρ)/ρ, which yields the implicit equation [815] ⎡ ⎢ ρ = ⎣1 + v
⎤−1
(1 dp p0
f (p) ⎥ ⎦ p−v
for the density dependence of the average speed v.
(4.210)
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Now two cases have to be distinguished: • The integral diverges in the limit v → p0 : Then v(ρ) < P0 for all ρ > 0 and the gap distribution (4.209) is normalizable. In this case, the system remains homogeneous. • The integral remains finite in the limit v → p0 : Then the right-hand side of (4.210) for v = p0 defines a critical density ρc . In this case, v(ρ) = p0 in the entire interval [0, ρc ]. For the slowest particles with pj ≈ p0 , the gap distribution is no longer normalizable. Large gaps appear in front of these particles and the faster particles from platoons behind them, i.e., the system phase separates into regions of density ρ = 0, the gaps, and ρ = ρc , the platoons. The occurrence of phase separation is thus closely related to the behavior of the distribution f ( p) near p0 . In the case where f ( p) ∼ ( p − p0 )n for p → p0 phase separation occurs for n > 0. At the critical point ρ = ρc , the gap distribution then has a power law tail ∼ u−(n+2) [815]. Evans [355, 356] has pointed out that the phase transition from the low-density inhomogeneous phase (which consists of macroscopic free regions and platoons) to the high-density congested phase is, in many respects, analogous to the Bose–Einstein transition. Here f (p) plays the role of the density of states, and the slowest particle corresponds to the quantum mechanical ground state. The steady-state velocity of the particles is the analog of the fugacity of the ideal Bose gas. What makes the system interesting is the fact that the platoon appears at low density rather than at high density of the particles. The Bose–Einstein-like-condensation in the TASEP with quenched randomhopping probabilities of the individual particles survives when the random sequential updating is replaced by parallel updating [355, 356]. Finally, it is worth emphasizing that the analogy with the ideal Bose gas is only formal as the empty sites in the TASEP are not noninteracting quantum particles. The dynamics of the platoon formation has also been studied (see e.g., [78, 814] and references therein). Following their formation, starting from a random initial condition, the platoons grow through coalescence. The coarsening of the platoons has been investigated in the same spirit in which coarsening of domains (the so-called Oswald ripening) is monitored while studying spinodal decomposition in, e.g., binary alloys [155, 475]. Suppose, ξ(t) is the typical platoon size at time t. Starting from a homogeneous spatial distribution of the particles, ξ(t) can be monitored as a function of time t to find out the law of growth of the size of the platoons. In a simple model of platoon formation [95–98, 985], which was developed using the language of aggregation phenomena, it has been found that ξ(t) increases indefinitely according to the power law ξ(t) ∼ t (n+1)/(n+2) ,
(4.211)
where the exponent n characterizes the behavior of f (v) in the vicinity of the minimal velocity vmin , i.e., f (v) ∼ A(v − vmin)n as v → vmin with some positive constant A.
Asymmetric Simple Exclusion Process – Exact Results
Finally, we briefly mention that second-class particles in the ASEP can be interpreted as moving impurities that allow, in certain cases, for an exact solution using the MPA, see e.g., [297, 298, 906, 1218].
4.6.2. Disorder in Open Systems Although defects in the ASEP with open boundaries have been studied by several authors, so far no exact solutions are known and one has to rely on Monte Carlo simulations and mean-field type approximations. Kolomeisky [779] has investigated the TASEP in the presence of a single defect site deep in the bulk, applying a mean-field approach. Correlations on the defect site are neglected by dividing the system into two homogeneous ones coupled at the defect site. Comparison with Monte Carlo results shows a good agreement, at least for the high-density and low-density phases. However, deviations are observed for the density profiles near the defect site. In [484], a dynamical queueing transition at a critical defect strength was reported for the case of one defect in the middle of the chain, i.e., weak defects do not give rise to macroscopic jams. Chou and Lakatos [205] developed an analytical approach to treat a finite number of bottlenecks11 of arbitrary lengths . Their finite segment mean field theory (FSMFT) divides the system into segments that contain one or a few defect sites with arbitrary hopping rates. By determining the leading eigenvector of the corresponding transition matrix and matching the currents of the different subsystems, predictions for the current through the system are obtained. In this way, correlations near the bottlenecks are taken into account. However, due to the numerical complexity, the method is limited to segment lengths of less than 20 sites. The results for one and two defect sites in the bulk were extended by Dong et al. [317]. They also considered a single defect located near the boundary and found an edge effect. This is induced by the interaction of the defect with the boundary and leads to a dependence of the current on the position of the defect. In [464], the dependence of the current on the length of the bottleneck and its position was studied using an analytical approach, the interacting subsystem approximation (ISA), which is more efficient for longer defects than the FSMFT. The ISA reproduces the increase of the maximum current by moving defects near the boundary ( positive edge effect) and predicts a decrease of the current (negative edge effect) for lower entry rates, which is confirmed by MC simulations. For defects near the boundaries, no phase transition to a maximum current phase is observed and there is also no macroscopic phase separation. The case of two defects of arbitrary lengths and separation, where one defect is located near the boundary, can be described by introducing effective boundary 11 A sequence of consecutive defects.
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rates [464]. An analytical theory for two single-site defects which helps to understand the role of correlations has been developed in [394]. The inclusion of Langmuir kinetics (see Section 4.6.3), i.e., particle-creation and particle-annihilation in the bulk [466, 1108, 1156], leads to a rich phase diagram including novel phases. This is relevant for biological applications, e.g., intracellular transport (see Section 12.3). In this generalized model, disorder can also be introduced in the attachment and detachment rates [364, 473]. Open systems with finite defect densities, i.e., a macroscopic number of defect sites, have also been studied [350, 393, 465, 499, 699, 782, 832, 1192]. Here surprising results have been found, e.g., that the position of phase transitions depends sensitively on the sample [350]. The case of open systems with particlewise disorder turns out to be less interesting than that with sitewise disorder. The phase diagram turns out to qualitatively unchanged compared with the pure case [101, 883], although the broken particle-hole symmetry leads to slight shift of the coexistence line. This could be related to the fact that the lifetime of the particles in the open system, which of the order L, is much smaller than timescale for coarsening processes. Other subtilities of the dynamical properties have been found for the PASEP case, see [697]. Further examples for systems with various types of impurities will be discussed in Part II in the context of vehicular traffic.
4.6.3. Langmuir Kinetics For some biological application like intracellular transport (see Section 12.3), another modification of the ASEP dynamics is necessary, namely the inclusion of Langmuir kinetics. Particles from a bulk reservoir can attach to the chain with rate ωA and particles from occupied sites can detach with rate ωD into the reservoir. In the absence of diffusion, the equilibrium coverage of the chain is determined solely by the binding constant K=
ωD . ωA
(4.212)
The resulting steady state is completely uncorrelated and the density is given by the Langmuir isotherm ρL =
K . K +1
(4.213)
As an equilibrium state, this is expected to be robust against changes of the boundary condition. However, the ASEP evolves into nonequilibrium steady state, which is rather sensitive to the boundary rates. If both processes are combined on the same
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timescale, Langmuir kinetics would dominate because the bulk attachment and detachment processes occur at a large number of sites compared with the local entrance and exit processes at the ends. A nontrivial interplay between the two dynamical processes requires an appropriate scaling of the rates. Therefore, the attachment and detachment rates are rescaled with the length L of the chain as ωA =
A , L
ωD =
D . L
(4.214)
This choice leads to a competition of the two processes [1096]. The physical meaning of the choice can be understood by a look on the timescales involved. A particle on the lattice moves on a timescale, which is the inverse of the hopping rate. It will typically spend the time τ ∼ 1/ωD on the lattice before detaching. During this time, it will visit n ∼ τ sites. Therefore, for fixed ωD , the fraction n/L ∼ 1/(LωD ) would vanish in the thermodynamic limit. This is prevented by choosing the total attachment and detachment rates as in D/A = LωD/A and keep them constant for L → ∞. This leads to the ASEP with Langmuir kinetics (ASEP-LK), which has first been introduced as a market model [1467] and later as a model for molecular motor traffic [1095, 1096]. In the ASEP-LK, the usual ASEP dynamics is supplemented by attachment and detachment processes in the bulk: with rate ωa ,
(4.215)
detachment: 1 −→ 0 with rate ωd .
(4.216)
attachment:
0 −→ 1
For periodic boundary conditions, the stationary state is given by a product measure [366]. For open boundary conditions, the mean-field equations have been derived in [1095, 1096]. The bulk equation dρj = ρj−1 (1 − ρj ) − ρj (1 − ρj+1) + ωA (1 − ρj ) − ωD ρj dt
(4.217)
is supplemented by the boundary equations dρ1 = α(1 − ρ1) − ρ1(1 − ρ2), dt dρL = ρL−1 (1 − ρL ) − βρL . dt
(4.218) (4.219)
The phase diagram (Fig. 4.16) is much richer than that of the ASEP because now phase coexistence becomes possible. Besides the pure low-density (L), high-density (H), and maximum current (M) phases also the mixed phases L–M, L–H, M–H, and L–M–H are possible for certain parameter combinations.
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β
L
L−M
1/2
M
L−M−H M−H
Ω 1/2−Ω
L−H H Ω
1/2
α
Figure 4.16 Phase diagram of the ASEP with Langmuir kinetics for K = 1 and = a = d < 1/2. Besides the pure low-density (L), high-density (H), and maximum current (M) phases also the mixed phases L–M, L–H, M–H, and L–M–H are observed. For ≥ 1/2, the L and H phases disappear; for ≥ 1, the L–H phase also disappears.
The maximum current phase in the ASEP-LK is analogous to that in the standard ASEP. This bulk-controlled phase is not affected by Langmuir kinetics because equilibrium and nonequilibrium dynamics cooperate. The other phases are affected and several regimes with multiphase coexistence are observed. For certain parameter choices, even the coexistence of maximum current, high-density, and low-density regimes is possible. Seven different phases are possible, but not all phases are realized for every value of . Evans et al. [366] have derived the phase diagram from an analytical study of the continuum mean-field equations. They found strong indications that the predicted mean-field phase diagrams are exact. In contrast to the ASEP, where shocks move with constant velocity and are generally driven out of the system, the interplay with Langmuir kinetics leads to the localization of shocks √ in the bulk for certain parameter regions. The width of the shocks is of the order of 1/ L due to the fluctuations of the shock position [366, 1119]. The localization of shocks in systems without particle number conservation has been studied rather generally in [1119]. Using hydrodynamic scaling, it is investigated under which conditions the mean-field approach works. Besides the ASEP, the case of a system with correlations, the Katz–Lebowitz–Spohn (KLS) model discussed in Section 4.8.4, has been considered there, which can have localized downward shocks and localized double shocks. For a system where attachment and detachment are allowed only at one site, it was shown in [937] that the probability for the shock position is described by an overdamped
Asymmetric Simple Exclusion Process – Exact Results
Fokker–Planck equation with a V-shaped potential. If attachment and detachment are allowed at every site, the shock motion is equivalent to diffusion in a harmonic potential. A more sophisticated model coupling ASEP dynamics and Langmuir kinetics has been proposed in [879, 880]. It includes the interplay between directed motion along the chain and diffusion in the bulk reservoir explicitly and has been studied for various situations, e.g., open and periodic compartments. These models are relevant in the context of intracellular transport by molecular motors (see Section 12.3). For a twospecies extension of the ASEP, the stationary state factorizes under certain conditions. Two-lane variants have been also been investigated [669, 1175, 1447]. The dynamics of dimers under the combined influence of ASEP and Langmuir kinetics was studied in [1107]. From a physical point of view, the possible occurrence of ergodicity breaking is interesting [1166]. This can be triggered by the random motion of a shock in an effective potential. In the ergodicity-broken phase, high-density and low-density do not coexist like in the ASEP-LK, but either the one or the other phase is present. In finite systems, transition between the two states are possible, but the transition times grow exponentially with the system size.
4.6.4. Extended Particles Originally, the ASEP was introduced by MacDonald and Gibbs [892, 893] as a model for the description of protein synthesis (see Section 12.2). Here the particles correspond to ribosomes that cover = 20 . . .30 sites (or codons). This generalization of the ASEP is usually called -ASEP. The phase structure of the -ASEP is basically the same as that for the standard case = 1 [831, 1286, 1288]. For the periodic case, stationary and dynamical properties have been studied in the hydrodynamic limit [1255]. The influence of defects [316] and generalizations with applications to traffic flow [381] were also investigated. Furthermore integrable variants exist, which can be solved exactly by Bethe Ansatz [11–13, 16] and provide information about the dynamical exponent, which is z = 3/2, independent of . More details about the application of the -ASEP to protein synthesis, as well as some quantitative results, will be given in Section 12.2.1.
4.6.5. Other Boundary Conditions Often motivated by biological applications, the ASEP has been studied with other boundary conditions than periodic or open ones. The Dynamical Extending Exclusion Process (DEEP) [376, 1318, 1319] has been proposed for the description of fungal growth. In the DEEP, the boundary conditions at the output end of a TASEP are modified in such a way that the length L of the system is no longer constant, but can grow in time. A particle reaching the end of the chain will
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detach from it with rate β, but might also transform into a new lattice site with rate γ . This describes the fungal growth process, which is maintained by mass transport to the filament tip. A nontrivial aspect of the model concerns the precise definition of a steady state. It requires the choice of an appropriate frame of reference, e.g., the frame moving with the tip. The phase diagram in the case β = 0 has been determined using mean-field approaches and simulations [376, 1318, 1319]. Besides the phases known from the ASEP, a shock region exist between the high-density and low-density phases. In this region, a moving shock separates a high-density and low-density region of the chain. Another extension of the ASEP that corresponds to a lattice of varying size has been introduced in [1073]. Here the right boundary is given by a moving wall particle that cannot be passed. This particle is allowed to move to the left (backward) and right (forward) with rates w− and w+ , respectively. This model was inspired by the helicaseinduced opening of replication forks in DNA processing [111]. Another interesting type of boundary condition is realized by coupling the lattice to finite reservoirs [2]. The total number of particles Ntot on the lattice and in the reservoir is kept constant, and the entrance rate α depends on the number Np of particles in the reservoir. Therefore, this model has also been called constrained TASEP. It has natural applications to biological problems where resources (e.g., ribosomes, see Part II) are finite. It is a generalization of the parking garage problem considered in [482]. In [2], an input rate of the form αeff = αf (Np ) has been considered where f (x) is a monotonically increasing function with f (0) = 0 and f (∞) = 1. The latter condition guarantees that for sufficiently large Np , the standard TASEP is recovered. Specifically the function f (Np ) = tanh(Np /N ∗ ) has been used, in which N ∗ provides a scale for the crossover to saturation. It is chosen to be equal to the average number of particles in the standard TASEP and thus depends on the system length L. The parking garage problem corresponds to the choice f (Np ) = (Np ), in which is the Heaviside step function. The strongest influence of the finite reservoirs is observed on the coexistence line between the low-density and high-density phases. The system relaxes very slowly and the average density shows two different crossovers. This can be understood within a generalized domain wall theory [235]. It takes into account the fluctuating feedback into the entry rates, which can lead to a localization of the domain wall. In [236], this has been extended to the competition of several TASEPs that are coupled to a finite reservoir of particles. If the chains have equal length, the behavior is similar to the single chain case described earlier. For different chain lengths, however, new regimes emerge because an extra dimension is added to the feedback mechanism. Domain walls are now delocalized to an extend allowed by the lengths of the other chains [236]. The case of periodic boundary driving has been studied in [1045, 1120]. The focus in [1045] was on applications to urban traffic. Therefore a generalization of the TASEP,
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the Nagel–Schreckenberg model (see Chapter 7) was studied. To simulate the effects of a traffic light, the output rate was considered to change periodically. In [1120], a semiinfinite TASEP was studied, which has a boundary reservoir at the end with periodically changing density. Independent of the initial conditions, the density profile assumes a time-periodic sawtooth-like shape. Another related study is [649] where a periodic SSEP is investigated that has periodically varying hopping rates at two neighboring sites. In [1483], “gated” boundary conditions have been considered where exit and entry are controlled by an external component. It can bind as an “initiator” to the beginning of the track and as “receptor” to its end, thereby mediating the injection and removal of particles.
4.6.6. Long-Range Hopping Generically in the ASEP, only hopping to nearest neighbors is allowed. However, the influence of nonlocal hopping on the phase diagram has also been investigated. The models differ slightly in the hopping rates used. In [1330, 1331], the rates depend on the jump length in the form p =
−(σ +1) , ζL (σ + 1)
(4.220)
, where ζL (σ + 1) = Lj=1 j −σ −1 is the partial sum of the Riemann zeta function. For periodic systems, the model has a factorized steady state. The current is given by J = λL (σ )ρ(1 − ρ),
(4.221)
where λL (σ ) =
p =
ζL (σ ) . ζL (σ + 1)
(4.222)
The model can also be extended to the case of open boundary conditions. In finite systems of length L, a jump that would lead to a site beyond the end of the chain will end in the boundary reservoir. Effectively, this means that the boundary reservoirs can exchange particles which makes the system similar to one coupled to a bulk reservoir. Surprisingly, the phase diagram turns out to be rather robust under this modification. For σ > 1, it shows the same phase diagram as the ASEP. However, the long-range hopping has an effect on the long-range correlations at the transition lines; e.g., the algebraic decay exponents depend on σ in the region 1 < σ < 2. In contrast, no phase transitions are found in the regime σ ≤ 1 because here the current diverges in the thermodynamic limit.
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In [483, 969], the competition between local and nonlocal hopping terms was studied for generalizations of the TASEP and the SSEP. The periodic variant of the model had been studied before in [904, 905, 1159]. In the asymmetric case [483], a particle moves with probability 1 − p to an empty nearest-neighbor site. With probability p, it will move to the site immediately behind the next particle in front of it. For the totally asymmetric case with open boundary conditions, three phases are observed: (1) A reservoir-controlled phase for boundary rates α < 1/2 and small p, (2) a bulk-controlled phase for small p and α > 1/2, and (3) an empty-road phase for large p. In the empty-road phase, the average bulk density is zero, but typically finite clusters exist at the entry end and in the bulk. The transition from the reservoir-controlled phase to the empty-road phase is of second order, whereas the transition between bulk-controlled to empty-road phase is of first order [483]. In the case of symmetric hopping [969], only two phases are realized. The finite density phase is a reservoir-controlled uniform phase, which exists for small p. For strong nonlocal hopping (large p), a discontinuous phase transition into an empty road phase takes place. In another variant of the ASEP, additional long-range connections (shortcuts) between sites have been introduced in a (deterministic) hierarchical fashion [1087]. This leads to a network called HN3, which has fractional dimension. The particle dynamics is defined in such a way that first a jump along the shortcut is tried. If this is not possible, the usual nearest-neighbor hopping of the ASEP is performed. In this variant of the model, the maximal current phase is no longer observed. Because the particle-hole symmetry of the dynamics is broken by the shortcuts, the phase diagram is no longer symmetric. It consists of a high-density and low-density phases, which are separated by a first-order transition. This can be understood because the shortcuts have a different influence in both phases. In the low-density phase, they will follow the shortcuts much more often because the target site will be empty. This is not the case in the highdensity phase, where the dynamics will consists mostly of the usual nearest-neighbor jumps.
4.6.7. ASEP Beyond One Dimension To our knowledge, there are only a few works which consider higher dimensional variants of the ASEP explicitly. One reason is probably that an impressive list of results exist for the related class of driven-lattice gas models, see, e.g., the reviews by Schmittmann and Zia [1250, 1251]. Also many models used in applications can be considered higher dimensional analogues of the ASEP. Examples are the BML model of city traffic (see Section 10.2) and the floor field model for pedestrian dynamics which will be discussed in some detail in Section 11.4. Although the transition probabilities in this model are rather complex, in special limiting cases it becomes basically a two-dimensional ASEP. Also problems of granular media [693] and urban traffic (see, e.g., the BML model in Section 10.2) were studied with related models.
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As a first step in the direction of higher dimension coupled chains of ASEPs (multichannel ASEP) have been considered [500, 936, 1118, 1147–1150, 1173–1175], also with Langmuir kinetics [669, 1447]. These model show a rich phase diagram; e.g., the model with asymmetric lane-changing rules exhibits seven phases. Other effects that can have been observed are spontaneous symmetry breaking, synchronization of domain walls, and so forth. The ASEP on a network was investigated in [346].
4.7. MULTISPECIES MODELS Several multispecies generalizations have been proposed. In these processes, two or more particle species are allowed to move on the lattice, sometimes in opposite directions. These multicomponent system show having interesting properties, which cannot be understood by straightforward generalization of the results obtained for onecomponent systems. Among the new phenomena encountered in these systems are spontaneous symmetry breaking, long-range order or phase coexistence even without explicit breaking of translational invariance through defects or open boundaries. For a more detailed discussion of the generic properties of this model class, we refer to the review by Schütz [1274]. Besides this theoretical interest-related models are, e.g., used in the context of intracellular transport, especially to describe the cooperative transport by different types of molecular motors (see Section 12.3). Here also Langmuir kinetics is important, which can lead to rich behavior like continuous phase transitions, spontaneous symmetry breaking, hysteresis, or localized shocks [182, 333, 369, 643, 766, 861, 1175].
4.7.1. Models with Second-Class Particles Several generalizations of the ASEP to more than one species of particles have been proposed. A well-studied case of two-species models is that where the second species corresponds to so-called second-class particles [30, 297, 387]. These behave like normal (first-class) particles in the ASEP if their right neighbor site is empty. Then they move with rate 1 to the right. From the point of view of the first-class particles, a second-class particle behaves like a hole (empty site): if the right neighbor site of a first-class particle is occupied by a second-class particle, they will exchange their positions with rate β. In a slightly generalized model, the two particle species can have different hopping rates. The dynamics is then given by [840, 907] 1
10 −→ 01,
α
20 −→ 02,
β
12 −→ 21,
where 1, 2 denotes the two particle species and 0 denotes an empty site.
(4.223)
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For β < 1, a second-class particle will move forward in a region with a low density of first-class particles, but backwards in a region with a high density of first-class particles. Therefore, it is a useful tool to study the position and microscopic structure of shocks [135, 297, 906]. Second-class particles arise also when two ASEPs with different densities of particles are coupled [871] where they represent the excess particles in the system with higher density. The stationary state for systems with periodic boundary conditions and α = β = 1 has been obtained in [297] using the MPA. As in the single-species ASEP, it leads to a quadratic algebra for the three matrices D, A, and E corresponding to first-class, second-class particles, and holes, respectively. For the general model defined earlier, it is given by DE = D + E,
AE =
1 A, α
DA =
1 A. β
(4.224)
This algebra is closely related to (4.79). Quadratic algebras for two-species models have also been investigated in [49, 710, 747]. A rather general classification of such algebras has been provided in [627]. The phase diagram for the most general case with a single impurity “2,” where all three processes can have different rates, was investigated in [144, 906]. Phase transitions between different phases are characterized by nonanalyticities in the speed v of the impurity. Four phases are found (Fig. 4.17). • ρ < β and ρ < 1 − α: The impurity behaves like a hole with v = α − ρ. • ρ > β and ρ > 1 − α: The impurity behaves like a particle with v = 1 − β − ρ. • 1 − α < ρ < β: The defect is similar to a second-class particle with v = 1 − 2ρ and reduces the average speed of the particles. The density profile is uniform. • 1 − α > ρ > β: The impurity which moves at v = α − β. A shock profile develops because the particles cannot easily overtake. In this regime, a single defect can cause phase separation in the system. β
v = α −ρ
v = 1−2ρ
v = α −β
v = 1−β −ρ
ρ
1−ρ
α
Figure 4.17 Phase diagram of the ASEP in the presence of a single second-class particle.
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In the first three phases, the current is given by J = ρ(1 − ρ). In contrast, in the last case, it is J = ρ(α − β) + β(1 − α). The two-species model can be extended to an arbitrary number N of species with a hierarchic relation such that the jth-class particle is treated as a hole by all lth-class particles with l < j. The case N = 3 was studied in [907]. The matrix product state constructed there is based on matrices that obey more complicated relations than the quadratic algebra of the two-species case. The dynamics of this model can be related to a N -lane and queueing processes [388]. The matrix product solution has been extended to the partially asymmetric variant of this hierarchical model in [359, 1145]. A related model has been studied in [46]. The case of open boundary conditions is investigated in [44, 45, 57].
4.7.2. ABC Model The ABC model12 [367, 368] describes the dynamics of three species of α particles, labeled A, B, and C, on a one-dimensional lattice with periodic boundary conditions. The dynamical rules can be summarized in compact form by qAB
−−− AB − BA, qBA
qBC
−− BC − − CB, qCB
qCA
−− CA − − AC. qAC
(4.225)
The dynamics conserves the numbers Nα of α-particles and thus also the total number N = NA + NB + NC of all particles. An interesting special case is qAB = qBC = qCA = q and qBA = qCB = qAC = 1, i.e., cyclic rates in A, B, and C. For q = 1, the particles perform symmetric diffusion, leading to an equilibrium steady state without order. In the case q = 1, the motion of particles is biased and eventually (after a coarsening process), the system will separate into three domains of different species. This state has the form A . . .AB . . .BC . . .C.
(4.226)
In finite systems, the domains will break up again after some time. For the A domain, this time is of the order q− min{NB ,NC } . In the thermodynamic limit, these timescales for the breakup diverge and the system is expected to show phase separation. The current of the A particles is of the order of qNB − qNC and vanishes exponentially in the thermodynamic limit. The case of equal densities NA = NB = NC = 1/3 is special. Here the current is zero for any system size. In fact, it can be shown [367] that in this case the dynamics 12 If one of the species is interpreted as an empty site, it is an extension of the ASEP.
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satisfies detailed balance and that the stationary state has a Gibbs measure with effective Hamiltonian H=
N −1
N
N (Cj Bl + Aj Cl + Bj Al ) − 3
2 .
(4.227)
j=1 l=j+1
Despite the local interactions in the ABC model, this Hamiltonian has strong longrange interactions, which are independent of the distance. Such long-range interactions can lead to a rather different thermodynamical behavior compared with short-range interactions [956]. Correlation functions can be calculated [367] and one finds, e.g., A1Ar =
1 + O(r/N ). 3
(4.228)
Furthermore, the sufficient condition lim lim (A1 Ar − A1Ar ) > 0
r→∞ N →∞
(4.229)
for phase separation is satisfied. Thus, the ABC model exhibits phase separation and long-range order for all q = 1. For the weakly asymmetric case with q = exp(−β/N ), more detailed statements about fluctuations in the stationary state can be made [230]. The phase diagram has been studied in more detail in [56].
4.7.3. AHR Model Arndt, Heinzel, and Rittenberg [49, 51, 52] have studied a model (AHR model ), which has a dynamics that is very similar to the ABC model. It is a special case of the process (4.225), where now A and B correspond to positive- and negative-charged particles and C to empty sites. The allowed transitions are then given by λ
q
+ 0 −→ 0 +,
+ − −→ −+,
λ
q
0 − −→ − 0,
− + −→ + − .
(4.230) (4.231)
In the following, the densities of positive and negative particles are assumed to be identical, i.e., ρ+ = ρ− = ρ. The case of different densities was studied in [52]. Based on Monte Carlo simulations and a mean-field analysis, the structure of the steady state was determined in [51]. For 0 < q < 1, the system shows a behavior similar to that of the ABC model. It is in a pure phase, where translational invariance is
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spontaneously broken and a typical configuration consists of pure blocks of holes, + and − particles. The current vanishes exponentially with the lattice size. For 1 < q < qc (λ, ρ), the system was argued to be in a mixed phase, where a spatial condensate is formed which consists of + and − particles. The vacancies form another block, the fluid. Due to fluctuations, the condensate and fluid perform a Brownian motion so that translational invariance is not broken. Mean-field analysis predicts a simple form for the current J = (q − 1)/4 and the critical rate qc = 1 + 4λρ/(1 + 2ρ). For q > qc (λ, ρ), the system is in a disordered phase without charge segregation. Here the particles are distributed uniformly. Although Monte Carlo simulations and a mean-field approximation suggested that the AHR model exhibits a phase-separated steady state at sufficiently high densities [49, 51], it was later realized that this apparent condensed state is in fact homogeneous, with an extremely large (but finite!) average cluster size; i.e., there is no phase transition between the mixed phase and the disordered phase. This finding was based on the exact solution [1161, 1224] using the MPA in the grand-canonical ensemble, which revealed that the correlation length can become as large as 1070 in the infinite system! This is also consistent with the criterion for phase separation, which has been discussed in Section 3.5.3. The currents Jn are of the form Jn J∞ 1 + nb with b = 3/2. The physical origin of the observed sharp crossover has been considered in more detail in [703]. A nonconserving variant where positive (negative) particles can be created with rate λ (rate w) and annihilated with rate w (rate λ) has been studied in [369]. A related model with conserved bulk dynamics is known as bridge model [360, 361]. It has open boundary conditions, and particles are only allowed to move in one direction; i.e., the last process in (4.231) is not allowed. Depending on the boundary rates, the bridge model shows spontaneous symmetry breaking with phases, in which the densities for the two types of particles are asymmetric even though the dynamics is completely symmetric [360, 361, 444, 471, 478, 1468].
4.8. OTHER RELATED MODELS 4.8.1. Staggered Hopping Rates Klauck et al. [759] have proposed a variant of the ASEP with staggered hopping rates. It is related to the Uq [SU (2)]-symmetric hopping model of Sandow and Schütz [1211] and a ratchet model introduced by Kolomeisky and Widom [781]. Particles move according to a partially asymmetric dynamics with hopping rates a and 1/a (c and 1/c) to nearest neighbor sites if they occupy a site on the odd (even) sublattice. The stochastic Hamiltonian is then an alternating sum of local two-site Hamiltonians
μj hj,A j+1 + μj hj,B j+1 , (4.232) H= j∈A
j∈B
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A. Schadschneider, D. Chowdhury, and K. Nishinari
where sublattice A is formed by the odd sites and sublattice B by the even sites of the lattice. hj,A j+1 and hj,B j+1 act nontrivially on sites j and j + 1 according to ⎛
0 0 0 ⎜ 0 a − 1a hj,A j+1 = ⎜ ⎝0 −a 1 a 0 0 0
⎞ 0 0⎟ ⎟, 0⎠ 0
⎛
0 0 0 ⎜ 0 c − 1c hj,B j+1 = ⎜ ⎝0 −c 1 c 0 0 0
⎞ 0 0⎟ ⎟. 0⎠ 0
(4.233)
The μj are arbitrary positive constants that may differ from site to site. They control the activities of the bonds and influence dynamical properties only. For reflective boundary conditions, i.e., h1 = HL = 0 in equation (2.6), the model can be solved by MPA. In fact, the stationary state in this case corresponds to an optimum ground state (Section 2.7.2). The unnormalized stationary state for a system with N particles can be written as |N ∝ (B − )N |vac.
(4.234)
, with a creation operator B − of the form B − = Lj=1 bj−, where bj is a local particle creation operator. This allows to derive recursion relations for the l-point correlation functions and density profiles.
4.8.2. Two-Parameter Model In [20], a two-parameter extension of the ASEP has been proposed, which also includes the drop-push model [1268]. In this generalization, particles can push a cluster of n particles in its immediate neighborhood. The model becomes exactly solvable by Bethe Ansatz if the rates rn , which depend on the cluster size n, have a specific form. More precisely, the following processes are allowed in this model: rn
. . . 10 , 11 . . . 10 0 −→ 0 11 - ./ - ./ n+1
(4.235)
n+1
ln
. . . 10 0. 0 -11./ . . . 10 −→ 11 - ./ n+1
(4.236)
n+1
The rates of these pushing processes for clusters of n particles are given by rn =
p 1+
λ μ
+ ···+
n , λ μ
ln =
1+
μ λ
q n , + · · · + μλ
(4.237)
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Asymmetric Simple Exclusion Process – Exact Results
with 0 ≤ μ = 1 − λ ≤ 1 and p = 1 − q. The parameter p or q controls the driving and μ or λ the pushing. An asymmetry in the dynamics can thus be induced by an asymmetry in the driving (p = q) or the pushing (μ = λ). The stochastic process defined above is described by a nonlocal Hamiltonian. In the particle-hole transformed model, the particles not only move to nearest neighbor sites, but also move to any other vacant site (within the gap to the next particle) with rates that depend on the hopping distance. For μ = 1, there is only pushing to the right, and the rates are given by rn = p, l0 = q, and ln = 0 for n > 0. For q = 0, particles can only hop to the right. In this case, the model interpolates between the TASEP and the drop-push model [19]. In the drop-push model, a particle moves to the closest site to the right, which is not occupied with a rate that depends on the jump length n. It can be generalized to the case where each site can contain up to m particles. The dynamics satisfies pairwise balance (see Sections 3.4.6 and 1.5.1) that allows to determine the exact solution for the steady state, which has the form of a product measure. The master equation for the generalized model defined above can be solved on an infinite or periodic lattice using the Bethe Ansatz. In the stationary state of the periodic case, all configurations have equal weight. The correlation functions are given by (4.7) and the current-density relation by J = pJ+ − q J− ,
(4.238)
where J+ = ρ(1 − ρ)
∞
(n + 1)ρ n n , λ λ n=0 1 + μ + · · · + μ
J− = ρ(1 − ρ)
∞
(n + 1)ρ n μ n . μ n=0 1 + λ + · · · + λ (4.239)
The expression for the current simplifies in certain limits, e.g., ⎧ q ⎪ ρ p(1 − ρ) − for μ/λ 1, ⎪ 1−ρ ⎪ ⎨ for μ/λ = 1, J = (p − q)ρ ⎪ ⎪ ⎪ ⎩ρ p − q(1 − ρ) for μ/λ 1. 1−ρ
(4.240)
So, in contrast to the ASEP, there can be a nonvanishing current even for ρ = 1. In the strong pushing limit μ/λ 1, this current even diverges. The above results do not coincide with those of the PASEP because the model contains only the TASEP as special case. In the case μ/λ 1 of strongly asymmetric pushing, the current is negative for q > p. For q < p, however, pushing and driving work in opposite directions. The current
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√ is positive only for densities ρ < ρc = q/p, but becomes negative for ρ > ρc because pushing dominates. Variants with discrete time update were investigated in [1194, 1195].
4.8.3. Restricted ASEP In [35, 758], a variant of the ASEP has been studied where the hopping rate depends on the occupation of the site behind the particle. The nonvanishing transition rates correspond to the following processes, p1
110 −→ 101,
p2
010 −→ 001,
(4.241)
which reduces to the ASEP for p1 = p2 . It is also related to the model of Katz, Lebowitz, and Spohn [716] (see Section 4.8.4). This process is equivalent to another interesting one, which is obtained after a particle-hole transformation and space reflection. Then the hopping rates depend on occupation of the site directly behind the particle: p1
100 −→ 010,
p2
101 −→ 011,
(4.242)
The stationary state can be obtained in various ways, e.g., by mapping on a ZRP. In [758], a MPA solution has been presented where the representations of the matrix algebra are given by 2 × 2-matrices. In [35], it has been shown that the stationary distribution is given by the equilibrium distribution of a one-dimensional Ising model, as in the case of the ASEP with parallel dynamics and periodic boundary conditions (see Appendix 4.9.2). In fact, the stationary state is identical to that of a parallel ASEP with hopping probability 1 − p2 /p1 . The current can then be calculated using a transfer matrix technique. The fundamental equation is given by 2 1 1 − 1 − 4(1 − p2/p1 )ρ(1 − ρ) . (4.243) J (ρ) = p1 ρ 1 − 2(1 − p2 /p1 )(1 − ρ) The case of parallel dynamics has been studied using COMF in [758] and by an exact mapping onto the ZRP in [863]. For p1 < p2 , it is similar to a traffic model with slowto-start rule, which will be discussed in Part II. In the case p2 = 1, isolated particles move deterministically, similar to the cruise-control limit of traffic models (see Section 8.2). Here one finds a density regime ρc < ρ < 1/2 where both a free-flow state and a jammed state are thermodynamically stable [863]. In the presence of noise, however, the jamming transition at ρc disappears.
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Asymmetric Simple Exclusion Process – Exact Results
Open boundary conditions have also been investigated, in special cases, using the MPA [758] and more generally by Monte Carlo simulations [35]. The same three phases as in the ASEP are found. There are some differences, e.g., the phase boundary between high-density and low-density phases is no longer given by a straight line. In [84], the special limit p2 = 0, called restricted ASEP (RASEP), was used to study phase transitions between active and absorbing states. Absorbing states are the simplest possible stationary states that are “frozen” in a specific configuration. Once the system has reached such a state, it will never be able to leave it. First, an activity variable φj = nj−1 nj (1 − nj+1 ) is introduced for each site j. Sites with φj = 1 are called active because they are occupied by a particle that can move according to 110 → 101. A configuration which has at least one active site is called active, otherwise it is absorbing and no hopping can take place. L−N L absorbing configurations, and For particle densities ρ < 1/2, there are L−N N the rest is active [84]. Configurations having two or more consecutive empty sites are not allowed in the stationary state13 . After some time, the system will always reach one of the absorbing configurations and so the stationary state is absorbing. For ρ > 1/2, however, there are no absorbing configurations and the stationary state is active. Thus at ρc = 1/2, an active-absorbing-state phase transition takes place. The matrix-product solution [84, 85, 758] is based on a simplified version of the general cancellation mechanism for models with arbitrary interaction range (Appendix 2.7.3) with factorizing auxiliary matrices. The calculation can be simplified by mapping to the bond variables τj = 2nj + nj+1 . Because τj and τj+1 have a site in common, this puts some restrictions on the allowed configurations. The dynamics in terms of these variables is given by 312 −→ 213,
320 −→ 212.
(4.244)
The cancellation mechanism has a scalar solution14 that allows to study the phase transition in more detail. Critical exponents can be determined exactly and are found to be different from those of the usual directed percolation or compact directed percolation universality classes [567, 568, 1074]. An alternative solution is based on a mapping to a ZRP, which, e.g., can be used to determine the fluctuations in the number of active sites. It also allows to study certain natural generalizations of the stochastic process in a simple way. 13 Due to the dynamically rule 110 → 101, the number of empty sites between two occupied ones cannot increase with
time. 14 Some subtilities arise because one of the matrices is zero [84].
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4.8.4. KLS Model The KLS model was originally introduced as a lattice gas model of interacting charged particles that are subject to an external electrical field [715, 716] driven by Kawasaki dynamics. Such models have been used, e.g., to describe the stationary nonequilibrium states of fast ionic conductors [311]. In one dimension, the KLS model can be interpreted as an exclusion process. The dynamics of the completely asymmetric version is given by [486] 0100 −→ 0010,
with rate 1 + δ,
(4.245)
1100 −→ 1010,
with rate 1 + ,
(4.246)
0101 −→ 0011,
with rate 1 − ,
(4.247)
1101 −→ 1011,
with rate 1 − δ,
(4.248)
with |δ|, || < 1. For δ = = 0, the model reduces to the ASEP. More generally δ = corresponds (after a particle-hole transformation) to the model (4.241) of the previous section with p1 = 1 + δ and p2 = 1 − δ. A related partially asymmetric model has been studied in [537]. Similar to the ASEP, for periodic boundary conditions, the stationary state of the one-dimensional KLS model is described by an Ising measure [486, 716, 889] ⎞ ⎛ L L
1 (1 − 2nj )(1 − 2nj+1 ) − h (1 − 2nj )⎠, (4.249) P(n) = exp⎝−β Z j=1 j=1 where e4β =
1+ 1−
(4.250)
and the field h acts as chemical potential controlling the density ρ. Using transfer matrix techniques [486], one can compute averages like the density ρ and the current which is defined as J = (1 + δ)0100 + (1 + )1100 + (1 − )0101 + (1 − δ)1101.
(4.251)
Explicitly, one finds # $ e−2β sinh(h) 1 1+ ρ= 2 1 + e−4β sinh2 (h)
(4.252)
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0.4 (i) (ii) (iii) (iv)
Flow
0.3
0.2
0.1
0
0.2
0.4
0.6
0.8
1
Density
Figure 4.18 Fundamental diagram of the KLS model for (i) δ = 0, = 0, (ii) δ = 0.2, = 0.95, (iii) δ = 0.5, = 1, and (iv) δ = 0.9, = 0.9.
and √ λ [1 + δ(1 − 2ρ)] − 4ρ(1 − ρ) J= λ3
(4.253)
1 1 1 − 1/2 λ= √ . + −1+ 4ρ(1 − ρ) 1+ 4ρ(1 − ρ)
(4.254)
with
In the ASEP-limit δ = = 0, the known result J =√ρ(1 − ρ) is recovered and for δ = 0, = 1 one obtains J = x(1 − x)/(1 + x) with x = 1 − 4ρ(1 − ρ). In general, the fundamental diagram (Fig. 4.18) looks rather different from that of the ASEP [220, 486]. The parameter δ controls the particle-hole symmetry, which is broken for δ = 0. For δ > 0, the current at smaller densities is enhanced, whereas for δ < 0, the vacancy current is favored. Variation of the parameter interpolates between a fundamental diagram with a single maximum at = 0 and a double-hump structure at = 1. Strong repulsive interactions ≈ 1 suppress the current at half filling so that a current minimum develops at ρ = 1/2. For = 0, one even has J (ρ = 1/2) = 0.
4.8.5. Asymmetric Avalanche Process The asymmetric avalanche process (ASAP) was introduced in [636, 1133, 1140] and further investigated in [1134, 1135]. It is related to a model studied in the context of selforganized criticality [915]. For a periodic system of L sites, N particles are located such
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A. Schadschneider, D. Chowdhury, and K. Nishinari
that multiple occupations are allowed. During an infinitesimal time interval dt, each particle jumps with probability dt to the right. In the partially asymmetric case, the hopping rates to the left and right are pl and pr , respectively. A site j with nj > 1 is considered unstable. It relaxes immediately such that m particles are moved to site j + 1 where with probability μn nj , (4.255) m= nj − 1 with probability 1 − μn where the probabilities μn can depend on n. The relaxation process stops when nj ≤ 1 for all sites. In contrast to the ASEP, the ASAP leads to much larger reorganization of particles within the time interval dt. Because there is no exclusion, particles can always move. Nevertheless, there is a formal relation with the ASEP, which can be seen as analytical continuation of the (partially) ASAP. The parameter μ is then given by μ = −pr /pl [1133]. The model can be solved using the Bethe Ansatz (Section 2.3) [636, 1133, 1140]. This requires that the multiparticle problem can be reduced to a two-particle problem. Thus, the toppling probabilities μn have to satisfy certain conditions. For example, the probability that n particles leave a site should be equivalent to first two particles leaving (probability μ2 ) and then n − 2 particles (probability μn−2 ), and so on. In the totally asymmetric ASAP, this leads to the recursion [636] μn = μ(1 − μn−1 ) with
μ1 = 0, μ2 = μ.
(4.256)
In the limit μ → 0, the (partially) ASAP becomes a special case of the two-parameter model of Section 4.8.2. From the exact solution, other quantities can be calculated, e.g., the average velocity of the particles as function of the total density ρ = N /L: ∞
1 n v(ρ) = ρ n=1 μn+1
μρ 1−ρ
n .
(4.257)
The phase diagram of the ASAP in the ρ-μ-plane shows two different phases [1133]. For high densities, a continuous flow phase is observed, where v(ρ) = ∞. For low densities, the system is in an intermittent flow phase with v(ρ) < ∞. The two phases are separated by the critical line ρc =
1 . 1+μ
(4.258)
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Asymmetric Simple Exclusion Process – Exact Results
4.8.6. Higher Velocities For the modeling of traffic flow, especially highway traffic, an extension of the ASEP to a model where particles can move beyond nearest neighbors are necessary. The maximal hopping distance is usually identified with the maximal velocity vmax ; i.e., the ASEP corresponds to the case vmax = 1. Unfortunately, it turns out to be extremely difficult to generalize the ASEP in a way, which conserves its solvability. Therefore, there are only very few models with vmax > 1, where the stationary state is exactly known. De Gier [268] has proposed a model with deterministic bulk dynamics and parallel update. For periodic boundary conditions, it is related to the Fukui–Ishibashi model [414] (see Section 8.4.1) in which particles try to move as far as possible within the allowed interaction range of vmax sites. With open boundary conditions, particles can enter through one of the first vmax sites and can leave the system from the last vmax sites. It should be emphasized that for vmax > 1, the precise choice of the boundary conditions requires some care. They have to be chosen in such a way that the maximal possible flow can be reached and artifacts, like a strong sublattice-dependence of the density profile, are avoided. The latter could happen, e.g., if particles are only inserted on the first site, as in the case vmax = 1. This will be discussed in more detail in Section 7.4.9. In [268], the case vmax = 2 was studied. Stochasticity enters through the boundary conditions. If the first two sites are empty, a particle is inserted with probability α2 on site 2 and with probability α1 (1 − α2 ) on site 1. Thus with probability (1 − α1)(1 − α2 ), no particle is inserted. If only the first site is empty, a particle is inserted with probability α3 . Particle removal is possible from the last two sites. A particle leaves the system from the last site with probability β1 . If the last site is empty, a particle on the penultimate site will be removed with probability β2. The exact solution for the stationary state was shown to be of matrix-product form but without deriving an explicit matrix algebra. The matrices depend on three sites and are 24-dimensional. From the solution, the phase diagram, density profiles, and the correlation length are derived [268]. In [1482], a more general model with vmax = 2 and random-sequential update was studied. The dynamics is defined by the following processes: p1
100 −→ 010,
p2
100 −→ 001,
β
101 −→ 011,
(4.259)
i.e., a particle moves one or two site with rates p1 and p2 , respectively, if at least two sites in front of it are empty. If only one site is empty, it moves at rate β. A variant with parallel update was considered in [863]. The special case p1 = β was previously studied in [758]. In this case, the stationary state for periodic boundary conditions is given by a product measure. In the case p2 = 0, the stationary state is
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A. Schadschneider, D. Chowdhury, and K. Nishinari
of two-cluster form [35, 758] (see Section 4.8.3). A mapping on a mass-transport model shows that these are the only cases in which the factorization criterion of Section 3.6 is satisfied. In [1482], the case p1 = 0, which is related to the vmax = 2 model of de Gier, has been solved by an MPA of the form P(n1 , . . ., nL ) = Tr
N
Gnj ,
(4.260)
j=1
where Gnj represents a particle followed by nj holes and Tr is a special trace-like operation [1482]. In this special case, the even-odd oscillations of the headway distribution, which have been observed in the general case become strongest. The number of odd gaps decreases with time; i.e., the stationary state has either no odd gaps (for L − N even) or only one (for L − N odd). This is reflected in the matrix algebra, which contains the relation G2j+1 G2l+1 = 0. The algebra derived from the MPA can be simplified [1482] and becomes that of an ASEP with a single defect [906]. For an even number of holes, the process is completely equivalent to the ASEP. In the stationary state, all states with only even gaps have the same weight. For an odd number of holes, an odd gap exists which becomes the defect particle. It changes locally the density profile but not the headway distribution. β . A phase transition similar to the defect ASEP is observed at a critical density ρc = 2−β For ρ > ρc , the defect behaves as the other particles, whereas for ρ < ρc , it is similar to a second-class particle [297] that lowers the average speed of the other particles.
4.8.7. Reconstituting Dimers A model of directed diffusion of reconstituting dimers (DDRD) has been introduced in [80]. It is an asymmetric version of the model of reconstituting dimers introduced in [306, 928]. Its dynamics is defined by movement of dimers to the right: 110 −→ 011.
(4.261)
Alternatively, this can be interpreted as a hole moving two sites to the left if the site in between is occupied. All states linked by this dynamics are characterized by the same irreducible string (IS), a nonlocal object that can be obtained in the following way: (1) start from any state, (2) delete any pair of adjacent 1’s, and (3) repeat until no more pairs exist (e.g., 11011001111010 −→ 000010). This leads to a √ division of the configuration space into many sectors (∼ λL with the golden ratio λ = ( 5 + 1)/2).
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Asymmetric Simple Exclusion Process – Exact Results
An alternative representation of this process can be obtained by introducing new variables A = 11, B = 10, and C = 0. The dynamics is then given by the processes AB −→ BA,
and AC −→ CA,
(4.262)
which occur with the same rate. The first process involves the reconstitution of dimers (11010 −→ 01110 −→ 01011) In this representation, the IS can be obtained by deleting all A. It allows a sectordependent mapping to an ASEP by interpreting the dimers A as ASEP particles and B and C as ASEP holes. In this identification, a fixed site in the DDRD corresponds to a moving site in the ASEP (“wheeling motion” [80]). A DDRD process on a lattice of LDDRD = 2NA + 2NB + NC sites corresponds to a ASEP with LASEP = NA + NB + NC sites. The length of the IS is given by LIS = 2NB + NC . The relation with the ASEP allows to study dynamical properties of the model, which is strongly nonergodic due to the existence of conserved variables. The symmetric version has also been related to an integrable quantum spin chain in [928] with three-spin interactions, which is closely related to the Bariev model [71, 72]. In any given sector, this can be reduced to a ferromagnetic Heisenberg model where the chain length depends on the sector. Other models with a many-sector decomposable phase space are discussed in [79, 928].
4.9. APPENDICES 4.9.1. Mapping of ASEP to Surface Growth Model The PASEP can be mapped onto stochastic growth models of one-dimensional surfaces in a two-dimensional medium, the single-step model [70, 492]. To each configuration n = (n1 , . . ., nL ) of the ASEP, one can associate a unique surface profile {hj } through the relation [70, 492, 817, 818, 1353]. hj =
(1 − 2nk ).
(4.263)
j≤k
Pictorially one can interpret this mapping as shown in Fig. 4.19. The surface consists of segments of equal length, which have an angle of ±45◦ with the horizontal axis. In the mapping, each occupied site is identified with a −45◦ -segment and each unoccupied site with a +45◦ -segment (Fig. 4.19). This leads to the iteration hj+1 − hj = 1 − 2nj .
(4.264)
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A. Schadschneider, D. Chowdhury, and K. Nishinari
Figure 4.19 Schematic pictorial representation of the mapping of the PASEP onto a stochastic model of surface growth. It shows the surface profile corresponding to the ASEP configuration and its growth, which is equivalent to particle movement in the ASEP.
Particle movement to the right corresponds to local forward growth of the surface through particle deposition and particle movement to the left, which is only allowed in the PASEP, to evaporation of a particle from the surface. It is worth pointing out that any quenched disorder in the rate of hopping between two adjacent sites would correspond to columnar quenched disorder in the growth rate for the surface [1397, 1398]. The surface growth model described earlier is known to be a discrete counterpart of continuum models of growing surfaces whose dynamics are governed by the so-called KPZ equation [70, 492, 817, 818]. ∂ 2h λ ∂h 2 ∂h + η(x, t), =ν 2 + ∂t ∂x 2 ∂x
(4.265)
where η is a Gaussian white noise term. This stochastic nonlinear partial differential equation governs the shape of the surface at large length scales. Because the KPZ equation can be mapped onto the Burgers equation [167], using the Cole–Hopf transformation introduced in Section 4.2.9, it allows to make a connection with fluid-dynamical theories [1020]. This is not only interesting theoretically because it allows for new insights coming from the continuum models, which are sometimes technically easier to treat. It is also relevant for many applications. We will come back to this point in Part II, when we discuss the modeling of traffic flow. Finally, we mention another interesting mapping, which has been used to study sequence alignment [163].
4.9.2. Mapping of the ASEP to an Ising Model Introducing Ising-variables σi = ±1 instead of the lattice-gas variables ni = 0, 1 (σi = 2ni − 1), one can interpret the steady state of the periodic TASEP with parallel dynamics
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Asymmetric Simple Exclusion Process – Exact Results
as the equilibrium distribution P(σ ) ∼ e−βH(σ )
(4.266)
of a (classical) Ising model with the Hamiltonian
σj σj+1 + h σj . H = −J j
(4.267)
j
In order to determine the coupling constant J and the external field h, one interprets the local probabilities P2 (σ , σ ) as elements of a transfer matrix P = (Pσ σ )
Pσ ,σ = αe−J σ σ −h(σ +σ )/2 ,
(4.268)
where the inverse temperature β has been absorbed in the coupling constants. From Eqn (4.53) it follows directly that e4J = p or J = ln(p)/4 < 0. Therefore, the corresponding Ising model has antiferromagnetic interactions. According to Eqns (4.48) and (4.49), one has in addition 1 = 2P01 + P11 + P00,
(4.269)
ρ = P01 + P11.
(4.270)
Dividing the two expressions on both sides of the two equations gives one equation without the constant α to determine the external field as e = h
1 − 4(1 − p)ρ(1 − ρ) − (1 − 2ρ) . √ 2 p(1 − ρ)
(4.271)
h corresponds to a chemical potential that parameterizes the conserved density ρ. The steady state corresponds to an Ising model with antiferromagnetic (repulsive) interactions. Due to the conservation of the total number of particles, one has to impose the constraint of a fixed magnetization to the Hamiltonian. The normalization is then simply the partition function calculated under this constraint.
4.9.3. Solution of the Mean-Field Recursion Relations for the ASEP J
For J < 1/4, the recursion tj+1 = 1 − tj (see (4.69)) has two fixed points t± =
1 1 ± 1 − 4J , 2
(4.272)
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A. Schadschneider, D. Chowdhury, and K. Nishinari
where t+ is stable and t− is unstable. For J = 1/4, there is only one marginal fixed point t+ = t− , and for J > 1/4, no real-valued fixed point exists. The tj are then functions of t1 and J given by j−1 j−1 j j + t+ − t− t1 −t+ t− t+ − t− . tj = j−2 j−2 j−1 j−1 + t+ − t− t1 −t+ t− t+ − t−
(4.273)
The parameters α and β enter through the boundary conditions t1 − t1t2 = α(1 − t1) and βtL = tL−1 − tL−1tL . In [289], it was shown graphically that three different types of solution exist, corresponding to three different phases. For α ≤ 1/2 and β > α, the low-density phase, a consistent solution can be obtained under the conditions t1 = t− and tL < t+ . One has J = α(1 − α),
t1 = α,
tL =
α(1 − α) . β
(4.274)
For β ≤ 1/2 and β < α, the high-density phase is realized, which corresponds to tL = t+ and t1 > t− . It is characterized by J = β(1 − β),
t1 = 1 − β,
tL = 1 −
β(1 − β) . α
(4.275)
These two phases are for α = β < 1/2 separated through a coexistence line, which is described by t1 = t− + 0+ and tL = t+ + 0−. The solution contains a domain wall (discussed in more detail in Section 4.2.4) and satisfies J = α(1 − α),
t1 = α,
tL = 1 − α.
(4.276)
Finally, for α ≥ 1/2 and β ≥ 1/2, the maximal current phase is realized. Here t1 ≥ 1/2 and tL ≤ 1/2. 1 J= , 4
t1 = 1 −
1 , 4α
tL =
1 . 4β
(4.277)
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Asymmetric Simple Exclusion Process – Exact Results
4.9.4. Results Obtained from Normal-Ordering of Matrices Some explicit results can already be obtained without using an explicit representation of the algebra. By induction, it can be shown [293] that C = L
j L
j(2L − j − 1)!
L! (L − j)!
j=0
E l D j−l .
(4.278)
l=0
Using the convention W |V = 1, one obtains the explicit form of the normalization15
ZL (α, β) =
L
j(2L − j − 1)! (1/α) j+1 − (1/β) j+1
L! (L − j)!
j=0
(1/α) − (1/β)
,
(4.279)
where the combinatorial factors are known as Ballot numbers BL,j = j(2L−j−1)! L!(L−j)! . The singularity at α = β can be removed by taking the limit α → β, which yields the analytic continuation ZL (α = β) =
L
j( j + 1)(2L − 1 − j)! 1 j j=1
L! (L − j)!
α
.
(4.280)
Surprisingly, the grand-canonical partition function Z(z) =
∞
ZL zL
(4.281)
L=0
is given by the rather simple expression [128, 281] Z(z) =
4αβ √ √ . (1 − 2α − 1 − 4z)(1 − 2β − 1 − 4z)
(4.282)
In a similar way, one can determine the density profile by using the identity [293] DC j =
j−1
k=0
Bk+1,1 C j−k +
j+1
Bj,k−1 D k .
k=2
α β 15 For general p = 1, the normalization is given by Z (p) (α, β) = Z L p, p . L
(4.283)
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Using this identity in ρj = nj =
W |C j−1 DC L−j |V Zl
(4.284)
yields L−j
ZL−n Zj−1
1 Bn,1 + BL−j,p p+1 ρj = ZL ZL p=1 β n=1 ρL =
L−j
for 1 ≤ j < L,
ZL−1 . βZL
(4.285) (4.286)
For the calculation of other quantities (correlation functions) explicit representations are usually more convenient. Some useful ones are given in Appendix 4.9.6.
4.9.5. Dimension of Matrices in the MPA for the ASEP It is easy to check that the algebra (4.79) allows a one-dimensional solution (i.e., commuting real numbers) only if the condition α + β = 1 is satisfied.16 Explicitly one has E=
1 , α
D=
1 . β
(4.287)
In all other cases, the representations have to be infinite-dimensional [291, 292]. This can be seen as follows. Assuming the E and D are finite-dimensional matrices, one first can prove that E − 1 is invertible. To see this, assume that 1 is an eigenvalue and |1 the corresponding eigenvector. Then, we have D|1 = DE|1 = (D + E)|1 = D|1 + |1,
(4.288)
which would imply |1 = 0. Since the inverse of E − 1 exists, the algebra (4.79) leads to D = E(E − 1)−1 .
(4.289)
This implies that E and D commute because D is a function of E. Because a solution with commuting E and D requires the condition α + β = 1 to be satisfied, the matrices must be of infinite dimension in all other cases. 16 We have set p = 1.
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Asymmetric Simple Exclusion Process – Exact Results
4.9.6. Representations of the Matrix Algebra of the ASEP Several different representations of the in [293]. The first one is given by ⎛ 1 1 0 0 ··· ⎜ 0 1 1 0 ⎜ ⎜ 0 0 1 1 D =⎜ ⎜ 0 0 0 1 ⎜ ⎝ . .. .. .
matrix algebra (4.79) have been proposed, e.g., ⎞
⎛
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
⎜ ⎜ ⎜ E=⎜ ⎜ ⎜ ⎝
1 1 0 0 . ..
0 1 1 0
0 0 1 1
0 ··· 0 0 1 .. ⎛
⎜ ⎜ ⎜ |V = κ ⎜ ⎜ ⎜ ⎝
W | = κ(1, a, a 2 , . . .),
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
(4.290)
. 1 b b2 . . . .
⎞ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎠
(4.291)
where we have introduced the abbreviations a=
1−α α
b=
1−β . β
(4.292)
Choosing κ 2 = (α + β − 1)/αβ ensures that W |V = 1. Note that this can only be satisfied for α + β > 1, but the results can be analytically continued to the case α + β < 1 in a straightforward way. For practical calculations, one needs the eigenvalues and vectors of the matrix C = D + E, which in the above representation takes the form ⎛ ⎜ ⎜ ⎜ C=⎜ ⎜ ⎜ ⎝
2 1 0 0 . . .
1 2 1 0
0 1 2 1
0 ... 0 1 2 ..
⎞ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎠
(4.293)
.
Its eigenvalues are conveniently parametrized by 2(1 + cosθ ), where 0 < θ < 2π. The associated eigenvectors |cosθ satisfy C|cos θ = 2(1 + cosθ )|cosθ
(4.294)
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and is given by
⎛ ⎜ 1 ⎜ ⎜ |cosθ = sin θ ⎜ ⎝
sin θ sin2θ sin3θ . . .
⎞ ⎟ ⎟ ⎟. ⎟ ⎠
(4.295)
Using the resolution of the identity (2π 1 dθ sin2 θ |cosθ cosθ| Iˆ = π
(4.296)
0
and W | cosθ =
∞ κ κ n a sin(n + 1)θ = , iθ sin θ n=0 (1 − ae )(1 − ae −iθ )
(4.297)
which are both easy to prove using the orthonormality of sinnθ , the normalization is obtained as (2π 1 L dθ sin2 θ W |C L |cosθ cosθ|V ZL = W |C |V = π 0
κ2 = π
(2π dθ 0
sin2 θ [2(1 + cosθ )]L . (1 − ae iθ )(1 − ae −iθ )(1 − be iθ )(1 − be −iθ )
(4.298)
This integral representation is indeed identical to the previous one in (4.279) as can be shown using the residue calculus. For the PASEP, with p = 1, a useful representation is given by [293] ⎞ ⎛ √ c0 0 0 ··· 1+b √ ⎟ ⎜ 0 c1 0 1 + bq ⎟ ⎜ √ 2 ⎟ ⎜ 1 ⎜ 0 c2 0 1 + bq ⎟, (4.299) D= ⎟ 0 0 1 + bq3 1 − q⎜ ⎟ ⎜ 0 ⎠ ⎝ . .. . . . ⎛ ⎜ ⎜ 1 ⎜ ⎜ E= 1 − q⎜ ⎜ ⎝
1+a 0 0 0 ··· √ c0 1 + aq 0 0 √ c1 1 + aq2 0 0 √ c2 1 + aq3 0 0 . .. . . .
⎞ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎠
(4.300)
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Asymmetric Simple Exclusion Process – Exact Results
where the parameters a,b,cn are defined as a=
1−q − 1, α
b=
1−q − 1, β
cn = (1 − qn+1)(1 − abqn).
(4.301)
The boundary vectors are given by W1 | = 0|
and |V1 = |0.
(4.302)
To calculate the normalization, one can proceed analogous to the TASEP case by first diagonalizing C = E + D. This leads to a recursion involving Al-Salam-Chihara polynomials [1216] from which the normalization can be calculated. The determination of density profiles using this explicit matrix representation is more complicated. Here an alternative route using the q-deformed harmonic oscillator algebra is more suited [126, 1210, 1216, 1217]. In this approach, one writes D=
1 (Iˆ + aˆ ) 1−q
and E =
1 (Iˆ + aˆ †), 1−q
(4.303)
where the creation and annihilation operators aˆ † and aˆ satisfy a q-commutation relation aˆ aˆ † − qˆa †aˆ = 1 − q.
(4.304)
Then the matrices automatically satisfy Eqn (4.124). The operators act on basis vectors according to aˆ † |n = aˆ |n =
1 − qn+1|n + 1
(4.305)
1 − qn |n − 1,
(4.306)
which yields the explicit form of the matrices as ⎛ ⎜ ⎜ 1 ⎜ ⎜ D= 1 − q⎜ ⎜ ⎝
1 0 0 0 . ..
√
1−q 0 0 ··· 1 − q2 0 1 1 − q3 0 1 0 0 1 ..
.
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
(4.307)
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⎛ ⎜ ⎜ 1 ⎜ ⎜ E= 1 − q⎜ ⎜ ⎝
√
1 0 0 1−q 1 0 2 1−q 1 0 1 − q3 0 0 . . .
0 ··· 0 0 1 ..
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
(4.308)
.
This representation is the generalization to q = 0 of that given above for the totally asymmetric case. The boundary equations (4.125) and (4.126) for the case γ = δ = 0 are satisfied by requiring W |n = κ
an (q; q)n
and
n|V = κ
bn (q; q)n
,
(4.309)
where the parameters a and b have been defined in (4.301). The constant κ is fixed by V |W = 1 and (a; q)n :=
n−1
(1 − aqk).
(4.310)
k=0
The normalization can again be determined by methods similar to those in the TASEP case [126]. However, the main advantage of the q-deformed oscillator approach lies in the calculation of density profiles. First one observes that its gradient ρj − ρj+1 =
1 W |C j−1 [DC − CD]C N −j−2 |V ZN
(4.311)
involves the matrix DC − CD = DE − ED, which is diagonal in this representation: n|[DE − ED]|m = =
n|[ˆaaˆ † − aˆ †aˆ ]|m 1−q
(4.312)
(1 − qn+1) − (1 − qn) δn,m = qn δn,m . 1−q
(4.313)
Based on this observation, the density gradient can be expressed in terms of integrals over the q-Hermite polynomials [126]. For the partially asymmetric case also, several finite-dimensional representations exist if certain constraints on the model parameters are satisfied. These have been investigated in detail in [354, 908].
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Asymmetric Simple Exclusion Process – Exact Results
4.9.7. Mean-Field Approximation of the DTASEP A mean-field approximation scheme for the DTASEP has been developed in [1397, 1398]. The time-averaged steady-state current Jj,j+1 along the bond ( j, j + 1) is given by Jj,j+1 = pj,j+1 nj (1 − nj+1 ). In the mean-field approximation, nj (1 − nj+1 ) = nj (1 − nj+1 ) and, hence, J = Jj, j+1 = pj, j+1 ρj (1 − ρj+1),
(4.314)
where ρj = nj . In order to calculate the steady state flux J as a function of the mean density ρ of the particles, Tripathy and Barma [1397, 1398] used two different iteration schemes based on the equation (4.314). • Constant-current iteration scheme: In this scheme, for a given system length L and a fixed flux J = J0 , one starts with some value of ρ1 and, computes all the other ρj ( j > 1) using the equation (4.314), i.e., ρj+1 = 1 −
J0 pj, j+1 ρj
,
j = 1, 2, . . . , L
(4.315)
together with the periodic boundary condition ρj+L = ρj . If the iteration converges, i.e., one gets all the site densities in the physically acceptable range [0, 1], one accepts the average of these final site densities to be the global mean density of the particles corresponding to the flux J0 . • Constant-density iteration scheme: In this scheme, for a given system length L and fixed global average density c, one begins by assigning ,the site densities 0 ≤ ρj (0) ≤ 1 to the lattice sites subject to the global constraint L1 j ρj (0) = ρ. Then, the site densities are updated in parallel according to ρj (t + 1) = ρj (t) + Jj−1, j (t) − Jj, j+1(t),
j = 1, 2, . . . , L,
(4.316)
which follows from the equation (4.314). It is straightforward to verify that this iteration scheme keeps the average global density ρ unchanged at every step of updating, hence the name. After sufficient number of iterations, the set of densities converges to a set {ρj } and the flux on each bond converges to the steady-state flux J0 .
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PART TWO
Applications
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CHAPTER FIVE
Modeling of Traffic and Transport Processes Contents 5.1. Introduction 5.1.1. Some Practical Questions 5.1.2. Some Fundamental Questions
209 210 211
5.2. Classification of Models 5.2.1. Model Characteristics 5.2.2. Model Classes
212 212 214
In Part II of this book, we will try to give an overview over the most important traffic and transport systems. Despite obvious differences, there are several common principles that one encounters. The general theory exposed in Part I then provides a general framework for the modeling and investigation of traffic and transport problems. The focus will be on stochastic models although alternative approaches will also be presented to allow a direct comparison. However, we do not aim at completeness because the number of different models has become unmanageable. There is even a large number of reviews and books, e.g., [38, 419, 420, 423, 526, 587, 727, 730, 760, 857, 922, 931, 1243, 1260, 1261, 1437, 1473]1 covering various aspects of the topic which might be consulted.
5.1. INTRODUCTION In this first chapter of Part II, we want to give a brief general overview on various aspects of modelling of nonphysical systems. This is an enormous challenge because the relevant interactions in these systems are not the forces usually studied. In human and social systems, the particles show at least some intelligence which in some way has to be incorporated in the description. Therefore, it is a priori unclear whether the standard conceptual frameworks to study, e.g., classical mechanical systems can be adopted in these cases. In some sense, this makes the application of stochastic models more natural. 1 More references to reviews and books will be given in the corresponding chapters.
Stochastic Transport in Complex Systems DOI: 10.1016/B978-0-444-52853-7.00005-1
Copyright © 2011, Elsevier BV. All rights reserved.
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The stochasticity then reflects our lack of knowledge about the precise nature of the underlying interactions. In many cases, interactions are better described in terms of decisions than in terms of interactions. Here, stochastic models have a clear advantage as this aspect is usually easy to incorporate, making this approach more flexible than force-based ones. The stochasticity also nicely reflects the fact that even humans can make different decisions in situations that look very similar. How much “intelligence” of the particles is needed to generate a realistic behavior is not obvious. Based on our experience from statistical physics and many-particle systems, we expect that many fundamental phenomena encountered are based on collective effects, not requiring intelligent particles (often called agents in this context). In applications to “real” systems, often three different levels of behavior are distinguished. At the strategic level, agents decide which activities they like to perform and the order of these activities. With the choices made at the strategic level, the tactical level concerns the short-term decisions made by the agents, e.g., choosing the precise route taking into account obstacles, density of agents, and so on. Finally, the operational level describes the actual motion of agents, e.g., their immediate decisions necessary to avoid collisions, and so on. The processes at the strategic and tactical level are usually considered to be exogenous to the modelling or simulation. Here information from other disciplines, such as sociology, is required. Therefore, from a physics point of view, the operational level is the most relevant. In the following, we will mostly be concerned with this level of description, although some of the models that we are going to describe also allow to take into account certain elements of the behavior at the tactical level.
5.1.1. Some Practical Questions The aim of basic research in traffic science is to discover the fundamental laws governing traffic systems. The main aim of traffic engineering is on planning, designing, and implementing the transportation network and traffic control systems, in order to have smooth and efficient transportation and services. Statistical physicists have been contributing to traffic science by developing models of traffic and drawing general conclusions about the basic principles governing traffic phenomena by studying these models using the tools of statistical physics. By adopting and extending ideas of Part I, we will try to give an overview of the present state of the art of various aspects of traffic science. One main focus is traffic modelling, using approaches from statistical physics. These are often based on variants of the asymmetric simple exclusion process, which has been discussed in much detail in Part I. Apart from purely theoretical interest, these approaches allow to calculate several quantities that may find practical applications in traffic engineering. One example is the
Modeling of Traffic and Transport Processes
development of strategies for fast online simulation (e.g., for traffic forecasting) and traffic control so as to optimize the traffic flow.
5.1.2. Some Fundamental Questions Because of the apparent similarities between the microscopic models of traffic and systems like macroscopic samples of ionic conductors in the presence of external electric field, the tools of statistical mechanics seem to be the natural choice for studying these models. However, the actual calculation of even the steady-state properties of traffic from the microscopic models is a highly difficult problem because (apart from the human element involved) (1) the vehicles interact with each other and (2) the system is driven far from equilibrium, although it may attain a nonequilibrium steady-state. In principle, the time-independent observable properties of large pieces of matter can be calculated within the general framework of equilibrium statistical mechanics, pioneered by Maxwell, Boltzmann and Gibbs, provided the system is in thermal equilibrium (see Chapter 1). Of course, in practice, it may not be possible to carry out the calculations without making approximations because of the interactions among the constituents of the system. Some time-dependent phenomena, e.g., fluctuation and relaxation, can also be investigated using the Linear Response Theory, provided the system is not too far from equilibrium. Unfortunately, so far, there is no general theoretical formalism for dealing with systems far from equilibrium. Moreover, the condition of detailed balance does not hold although a condition of pairwise balance holds for some special systems driven far from equilibrium (see Chapter 1). The dynamical phases of systems driven far from equilibrium are counterparts of the stable phases of systems in equilibrium. Some of the fundamental questions related to the nature of these phases are as follows. 1. What are the various dynamical phases of traffic? Does traffic exhibit phase-coexistence, phase transition, criticality [445, 1308], or self-organized criticality [61, 307] and, if so, under which circumstances? 2. What is the nature of fluctuations around the steady-states of traffic? Analogous phenomenon of the fluctuations around stable states in equilibrium is by now quite well understood. 3. If the initial state is far from a stationary state of the driven system, how does it evolve with time to reach a truly steady-state? Analogous phenomena of equilibration of systems evolving from metastable or unstable initial states through nucleation (e.g., in a supersaturated vapor) or spinodal decomposition (e.g., in a binary alloy) have also been extensively studied earlier [155, 475] (see Chapter 1). 4. What are the effects of quenched (static or time-independent) disorder on the answers of the questions posed in 1–3 above?
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5.2. CLASSIFICATION OF MODELS There are different ways to classify models of traffic flow. The most important one is that between microscopic and macroscopic models. Microscopic models describe the spatiotemporal behavior of each particle or vehicle and their interactions, whereas macroscopic models describe collective properties without distinguishing individual particles. The latter model class thus assumes a large number of particles, and the approach may not be appropriate for in the low-density limit. Mesoscopic models have features of both these approaches. They do not distinguish or trace individual particles but specify the individual behavior, e.g., in terms of probability distributions. In the following, we briefly discuss the most relevant classifications of traffic-related models. To keep this discussion as general as possible, we will speak of “agents” or “particles.” These can represent vehicles and their drivers, pedestrians, molecular motors, and so forth.
5.2.1. Model Characteristics There are several characteristics that can be used to classify the modelling approaches: • microscopic versus macroscopic: In microscopic models, each agent is represented separately. Such an approach allows to introduce different types of agents with individual properties, as well as issues like route choice. In contrast, in macroscopic models different individuals cannot be distinguished. Instead, the state of the system is described by densities (aggregated quantities), usually a mass density derived from the positions of the particles and a corresponding velocity. • discrete versus continuous: Each of the three basic variables for a description of a system of particles, namely space, time, and state variable (e.g., velocities), can be either discrete (i.e., an integer number) or continuous (i.e., a real number). Here all combinations are possible. Although models with discrete variables are often an approximation, they have the advantage of being more efficient in computer simulations. However, there are also real situations in which variables are indeed discrete, e.g., for pedestrian motion on stairs, the motion of molecular motors along microtubules or models where the interactions are described by “decisions” made by the agents. • deterministic versus stochastic: The dynamics of the particles can be either deterministic or stochastic. In the first case, the behavior at a certain time is completely determined by the present state. In stochastic models, the behavior is controlled by certain probabilities such that the particles can react differently in the same situation.
Modeling of Traffic and Transport Processes
•
•
•
This is one of the lessons learnt from the theory of complex systems where it has been shown for many examples that very complex behavior can be generated through introduction of stochasticity into rather simple systems. However, the stochasticity in the models reflects our lack of knowledge of the underlying physical processes in social or human-based systems. Stochastic behavioral rules often allow to generate a rather realistic representation of complex social systems. This “intrinsic” stochasticity often reflects uncertainties in decision-making processes and should be distinguished from external noise. Sometimes external noise terms are added to the macroscopic observables, like the position or velocity. Often the main effect of these terms is to avoid certain special configurations that are considered to be unrealistic, like completely blocked states. Otherwise, the behavior is very similar to the deterministic case. For true stochasticity, however, the deterministic limit usually has very different properties from the generic case. rule-based versus force-based: Interactions between the particles or agents can be implemented in at least two different ways: In a rule-based approach, agents make “decisions” based on their current situation and that in their neighbourhood as well as their goals, and so on. It focusses on the intrinsic properties of the agents and thus the rules are often justified from psychology. In force-based models, agents “feel” a force exerted by others and the infrastructure. Therefore, they emphasize the extrinsic properties and their relevance for the motion of the agents. It is a physical approach based on the observation that the presence of other agents leads to deviations from a straight motion. In analogy to Newtonian mechanics, a force is made responsible for these accelerations. high versus low fidelity: Fidelity here refers to the apparent realism of the modelling approach. High fidelity models try to capture the complexity of decision making, actions, and so forth in a realistic way. In contrast, in the simplest models, agents are represented by particles without any intelligence. Some kind of “intelligence” can be introduced by allowing different internal states so that the particles react differently to an external stimulus, depending on the internal state. Roughly speaking, the number of parameters in a model is a good measure for fidelity in the sense introduced here, but higher fidelity does not necessarily mean that empirical observations are reproduced in a better way! Euler versus Lagrange representation: Another classification is the distinction between models based on a Lagrange representation and an Eulerian representation. The Lagrange representation is a particle-oriented description, which follows particle positions. In contrast, the Eulerian representation specifies the densities or particle numbers at each position and is thus a space- or site-oriented approach.
It should be mentioned that a clear classification according to the characteristics outlined here is not always possible.
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5.2.2. Model Classes In the following, we will describe some model classes in more detail and characterize them according to the criteria introduced in the previous subsection. • Cellular automata are models that are discrete in all variables, including the state variable [131, 202]. They are microscopic models and generically the dynamics is ruled-based and stochastic. In computer simulations, a discrete time is usually realized through the parallel or synchronous update where all particles are moved at the same time. This introduces a timescale. Sometimes a random-sequential update is used where at each step the particle or site to be updated is chosen randomly (from all particles or sites, respectively). Although this random-sequential updating is closely related to continuous time dynamics (see Section 2.1.3), it is common to call such models still “cellular automata.” • Multi-agent systems (MAS) [65, 86, 181] consist of “intelligent” particles that can have beliefs, desires, and so on. These agents make decisions and interact with other agents and their environment. Therefore, MAS can be considered as almost synonymous to “rule-based models.” They are often obtained by providing the particles in a cellular automation approach with more “intelligence.” • In coupled-map models, time is discrete whereas space and state variables are continuous. The dynamics is generically deterministic. A special case are coupled-map lattices where space is also discrete. Such dynamical systems are often used to qualitatively study the behavior of nonlinear systems based on partial differential equations, especially in the context of spatio-temporal chaos. • Microscopic forced-based models in which individual particles can be distinguished are usually continuous in space, time, and state variables. The forces acting between the particles are not necessarily related to physical forces, but they can also describe the so-called social forces. The dynamics is deterministic and governed by set of coupled Newtonian equations of motion. • In hydrodynamic or fluid-dynamic models, all variables are continuous. In these macroscopic approaches, the state variables are usually interpreted as densities. Typically, they are based on an Eulerian description with deterministic, force-based dynamics. • Gas-kinetic models are mesoscopic approaches, which do not distinguish individual particles explicitly. Instead, the individual properties are described in terms of probability distributions. The dynamics of these distributions is deterministic and given by a Boltzmann equation. Again this classification is not very strict. Sometimes, the classification is used in a loose way, like in the case of cellular automata with a random-sequential update.
CHAPTER SIX
Vehicular Traffic I: Empirical Facts Contents 6.1.
Measurement Techniques and Detectors
215
6.2.
Observables and Data Analysis
216
6.3.
Formation and Characterization of Traffic Jams 6.3.1. Jams Induced by Bottlenecks 6.3.2. Spontaneous Traffic Jams 6.3.3. Experiment on Spontaneous Jam Formation
221 222 223 224
6.4.
Fundamental Diagram
226
6.5.
Metastability and Hysteresis
228
6.6.
Phases of Traffic Flow 6.6.1. Level of Service Classification 6.6.2. Traffic Phases and Phase Transitions 6.6.3. Gas–Liquid Analogy
230 230 231 234
6.7.
Ramps, Intersections, and Other Inhomogeneities
235
6.8.
Headway Distributions
236
6.9.
Optimal-Velocity Function
238
6.10. Correlation Functions
239
6.11. Psychological Effects
240
The systematic investigations of traffic flow has a long history [457, 462, 488, 727, 922]. Although we now have a clear understanding of many aspects of real traffic, several controversial issues still remain as intellectual challenges for traffic scientists. In this chapter, we give an overview of some of the well-understood empirical findings, which are relevant for the theoretical analysis in the following chapters. These will provide benchmarks for the models since the ultimate goal is an accurate quantitative description of traffic flow. Moreover, wherever possible, we provide phenomenological explanations of these empirically observed traffic phenomena.
6.1. MEASUREMENT TECHNIQUES AND DETECTORS For several reasons, it is difficult to obtain very reliable (and reproducible) detailed empirical data on real traffic. First of all, unlike controlled experiments performed in Stochastic Transport in Complex Systems DOI: 10.1016/B978-0-444-52853-7.00006-3
Copyright © 2011, Elsevier BV. All rights reserved.
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0.8 m 0.35 m 1.5 m
1.5 m D1 1m
D2 2.5 m
Figure 6.1 Typical configurations of two-loop detectors for a short-loop detector (left) and a longloop detector (right) used on German highways.
the conventional fields of research in physical sciences, it is difficult to perform such laboratory experiments on vehicular traffic. In other words, empirical data are mostly collected through passive observations rather than active experiments. Second, unambiguous interpretation of the collected data is also often a subtle exercise because we do not have full information (e.g., due to incomplete or faulty data) and traffic states might depend on several external influences, e.g., the weather conditions. For the collection of empirical data in vehicular traffic, two different methods are employed. The most common approach uses stationary detectors based on induction loops. These make use of the fact that moving metallic object will induce an electrical current in a nearby conducting wire. This allows to determine the time tD at which a vehicle passes the detector site. Typically two-loop detectors are used (Fig. 6.1) because then also velocities can be derived from the time difference tD2 − tD1 . Floating-car data is the other main approach to obtain empirical information. Vehicles equipped with sensors (GPS, radar, IR-lasers) “float” with the traffic and record variables such as speed, position, relative speed, and distance to the next car. Although it provides accurate data, the method suffers from some intrinsic problems like the dependence on the behavior of the driver or errors if contact with the preceding car is lost which typically happens at sharp bends.
6.2. OBSERVABLES AND DATA ANALYSIS Let us first define some characteristic quantitative features of vehicular traffic. The flow J , which is sometimes also called flux or current or, in traffic engineering, traffic volume, is defined as the number of vehicles crossing a detector site per unit time [922]. The maximum possible flow is called the capacity (of the road). The distance from a selected point on the leading vehicle to the same point on the following vehicle is defined as the distance-headway (or spatial headway) [922] (Fig. 6.2). The time-headway is defined as the
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Vehicular Traffic I: Empirical Facts
d2 Δx1
ᐍ4
x2
x1
x3
x4
x
Figure 6.2 Definition of the quantities used to describe the configuration of vehicles on a onedimensional road. The cars move from left to right, and their label is increasing in driving direction (downstream). The length of car j is j , its distance-headway dj and xj = xj+1 − xj the difference of the vehicle positions xj .
time interval between the departures (or arrivals) of two successive vehicles recorded by a detector placed at a fixed position on the highway [922]. The distributions of distanceheadways and time-headways are regarded as important characteristic of traffic flow. For example, larger headways provide greater margins of safety, whereas higher capacities of the highway require smaller headways. Single-vehicle data record microscopic results for every vehicle. In the following, we will discuss how data from a two-loop detector can be used to derive quantities of interest. As mentioned earlier, the induction loops record the time when it is passed by a vehicle. In the following, we will denote these times by tD1 and tD2 . The distance between the detectors is dD . From these single-vehicle data, we can derive the following quantities: •
Flow (current, flux) J : The average flux1 during the time interval T is given by J =
N (T ) , T
(6.1)
where N (T ) is the number of vehicles passing the detector during that time interval T . The average flow is also related to the time-headways of passing vehicles: J =
1 t
with
t =
N −1 1 (tn+1 − tn ), N − 1 n=1
(6.2)
where tn is the time at which vehicle n passes the detector and t the average time between the passing of consecutive cars. 1 In the following, we will often write J instead of J to simplify the notation.
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• Velocity v: The velocity of vehicle n passing the two induction loops is given by vn =
dD . tD2 − tD1
(6.3)
Here it is assumed that the velocity does not change between the two detectors. This might not be the case in the jammed phase, e.g., for stop-and-go traffic. The average velocity in the time interval T is given by 1 vn , (6.4) v = T t≤tn
i.e., the average is taken over the velocities of all cars n which pass the detector within [t, t + T [ (at times tn ). This time mean speed has to be distinguished from the space mean speed where the average is taken at a fixed time t over the vehicles within a certain stretch of the street. In general, time mean speed overestimates the influence of faster vehicles and thus the average velocity. The error can be quite large, especially for small average velocities [770]. Another problem occurs independent of the averaging procedure because inductive loops do not detect very slow vehicles. Therefore, the calculated velocity is only the average velocity of the moving cars, not of all cars [1042]; i.e., it is usually overestimated. • Vehicle length: Some detectors allow to determine the length of the vehicle from the form of the signal. This is an effective length n , which is sometimes called electrical length. The length tD of the signal recorded at one of detectors allows to estimate the car length as n = vn tD − lD ,
(6.5)
where lD is the length of the detector. The signal form can also give information about the type of the vehicle, e.g., the number of axles. • Density ρ: Although density is one of the most important characteristics, it is difficult to measure. It is usually defined as a spatial average, i.e., the number of vehicles in a fixed stretch of the highway. However, most detectors only make local measurements and therefore a different way to estimate the density has to be used. Therefore, usually, the density is determined from the average flow and velocity which can be obtained much more easily. These quantities are related by the hydrodynamic relation J = ρv,
(6.6)
which allows to determine the density from ρ =
J , v
(6.7)
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where v = vn (t)/N is the average velocity of vehicles passing the detector in the time interval T . This method tends to underestimate the density in the congested regime because only the velocity of moving cars is detected [1042] and v is overestimated. As a consequence, sometimes the congested branch in the fundamental diagram (flow–density relation) appears to have a positive slope (see, e.g., Fig. 6.12). Another problem is that this method mixes temporal (for the flow) and spatial averages (for the density). These are, in general, different because fast cars pass a given section more frequently than slow ones. A better estimate is, therefore, obtained using the harmonic average velocity, ρ = J
1 v
,
(6.8)
which gives greater weight to small velocities [857]. However, this method is more sensitive to errors in the measurement of small velocities. Another way to estimate the density is the occupancy OD which is simply given by the fraction of time the detector D is covered by vehicles. Therefore, occupancy is usually given in percent. In terms of single-vehicle data, the occupancy during the time interval [t, t + T ] is calculated as OD (t) =
1 T
tn ,
(6.9)
t≤tn
where tn is the time the detector is occupied by car n. tn can be expressed in terms of the car length n and the velocity vn as tn = n /vn . The spatial density ρs is related to the occupancy by ρs = OD · ρmax ,
•
(6.10)
estimated by ρmax = where ρmax is the maximal density in a dense jam. ρmax can be 1 ¯ ¯ ¯ 1/, where is the average length of vehicles, i.e., = N (T ) t≤tn
n−1 vn−1
(6.11)
and dn = vn (tn − tn−1) − n−1 ,
(6.12)
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assuming that vn and vn−1 are constant. Sometimes, also the gross headways tn = tn − tn−1,
xn = vn (tn − tn−1 )
(6.13)
are used which do not require the knowledge of the car length . The flux is equal to the inverse of the average time-headway, see (6.2). Therefore, the full time-headway distribution contains much more information than the flux J alone. In a similar way, the distribution of spatial headways is related to the density ρ. It can be estimated as 1 = x, ρ
(6.14)
where x is the gross spatial headway. • Correlation functions: More detailed information about the spatio-temporal structure of traffic states can be obtained from correlation functions. They can detect and quantify coupling effects between variables. The autocorrelation function acx (τ ) of an observable X (e.g., density or flow) is defined as acX (τ ) =
X(t)X(t + τ ) − X(t)X(t + τ ) . X 2 (t) − X(t)2
(6.15)
It provides information about the temporal evolution of X. Analogous autocorrelation can also be defined as function of the spatial position x. The brackets . . . indicate the average over a time interval T or a sequence of N vehicles. In a similar way, one defines the crosscorrelation function ccX ,Y (τ ) of two observables X and Y as ccX ,Y (τ ) =
X (t)Y (t + τ ) − X(t)Y (t + τ ) . X 2 (t) − X(t)2 Y 2 (t) − Y (t)2
(6.16)
This correlation tells us how strongly the value of the observable Y at time t + τ depends on the value of the observable X at time t. Analogous correlation functions can also be defined by replacing time t by the vehicle index n. In this way, e.g., correlations between consecutive vehicles can be studied. Single-vehicle data provide very detailed information about the local traffic state at the detector site [776, 1042, 1362]. However, especially older detector types do not store the signal times tn but only averaged data for observables like the flux or velocity. Typically, averages are taken over short time periods T of 1–5 min.
Vehicular Traffic I: Empirical Facts
6.3. FORMATION AND CHARACTERIZATION OF TRAFFIC JAMS In everyday life, traffic jams are an annoyance. Considering the fact that all drivers try to move as fast as possible, it is surprising that traffic jams occur at all! For a physicist, this is an indication that here interesting collective phenomena, complex dynamics, emergence, and so forth play a decisive role. Therefore, it is not surprising that traffic jams are the most extensively studied traffic phenomenon. There is no general precise definition of a jam. Typically one has in mind a region where the average velocity is much smaller than in the rest of the system. In traffic, one usually thinks of vehicles standing for most of the time, moving only sporadically for a short distance. Jams are typically associated with large densities, usually close to the maximal possible density in the system. In large jams, frequently blanks are observed [730, 741], i.e., regions with no vehicles. These can move upstream2 due to the vehicle motion within the jam (moving blanks). Therefore, it can become difficult to decide whether a region consists of one or several jams. In traffic engineering, the terminology congestion is also frequently used. It is more general than jam and includes all states of high density and small average velocity. A jam is then a state where the density is almost maximal and the velocity is close to 0. Jams can emerge because of various different reasons. In principle, one can distinguish two major types of jams: • Jams at bottlenecks: Bottlenecks are areas in space where locally the transport capacity is reduced. If the inflow into such an area is larger than this capacity, jams emerge. Typically, the jam front of bottleneck-induced jams is stationary and located at the beginning of the bottleneck. The length of the jam fluctuates according to the inflow. The dynamics of the jam length is governed by the equation of continuity. • Spontaneous jams: Spontaneous jams occur without any (obvious) external reason even in homogeneous systems. Typically, the origin are local fluctuations. From the physicist’s point of view, these are emergent phenomena that indicate the relevance of collective behavior. This is just a rough classification, and the real behavior can be more complex. According to [548], congested traffic occurs due a combination of three ingredients: 1. high traffic volume, 2. spatial inhomogeneities (ramps, bottlenecks, gradients,...), 3. temporary perturbations of traffic, e.g., due to lane changes or overtaking trucks. However, the experiment described in Section 6.3.3 shows that jams can indeed form spontaneously under certain circumstances even without any of these conditions to be 2 Upstream means against the direction of vehicle motion (backward), downstream means the direction of vehicle motion
(forward).
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satisfied. In real life, it is often difficult to distinguish, e.g., between the effects of an temporary external perturbation (like in (3)), fluctuations in driving behavior, and so on. Therefore, the true origin of a jam can usually not be determined without a doubt.
6.3.1. Jams Induced by Bottlenecks
Flow A
Speed
Speed
Speed
Bottlenecks are the most frequent reason for the formation of jams. Bottleneck-induced jams occur when the flow becomes larger than the capacity of the bottleneck. This typically happens due to lane-reductions, merges, or accidents [257]. Also ramps can act effectively as bottlenecks, as will be discussed in more detail later. At bottlenecks, the capacity of the road is locally reduced, thereby leading to the formation of jams in upstream traffic. Downstream the bottleneck, typically, a free-flow region is observed. Figure 6.3 shows as an example empirical data of the velocity at a three-lane highway close to an on- and off-ramp [488]. The data show that downstream of the bottleneck (at detector C), no slow vehicle has been recorded. In contrast, in the merging regime near detector A, vehicles often have to move slowly. In between the on- and off-ramps, the vehicles move with larger velocities compared with those in location A, although the number of vehicles passing detector B is maximal. Therefore, the on-ramp causes a local reduction of the capacity of the highway. Although the basic mechanism is rather simple, the precise nature of traffic breakdowns at complex bottlenecks (e.g., ramps) is still discussed controversially [730, 1257]. Other factors that can act effectively as bottlenecks are weather, accidents, slopes, and so on. U.S. authorities have estimated that 15% of congestions are caused by weather [173]. A theoretical investigation of heavy bottlenecks induced by bad weather or
Flow B
Flow C
Figure 6.3 Experimental flow-speed diagrams near on- and off-ramps of a three-lane highway. The upper part of the figure shows the empirical results. Each dot represents a 5-min average of the local measurement. The lines serve merely as a guide to the eyes. The lower part of the figure shows the location of the detectors (from [488]).
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accidents has been presented in [729]. The influence of the slope of the highway has been studied in [783].
6.3.2. Spontaneous Traffic Jams Perhaps, what makes the study of traffic jams so interesting is that jams often seem to appear from nowhere (apparently without obvious reasons) suddenly on crowded highways. These so-called phantom jams are formed by spontaneous fluctuations in an otherwise streamlined flow. Direct empirical evidence for this spontaneous formation of jams was presented by Treiterer [1391] by analyzing a series of aerial photographs of a multilane highway. In Fig. 6.4, the picture from [1391] is redrawn. Each line represents the trajectory of an individual vehicle on one lane of the highway. The space-time plot (i.e., the trajectories x(t) of the vehicles) shows the formation and propagation of a traffic jam. In the beginning of the analysed time, vehicles are well separated from each other. Then, due to fluctuations, a dense region appears which, finally, leads to the formation of a jam. The jam remains stable for a certain period of time but, then, disappears again without any obvious reason. This figure clearly establishes not only the spontaneous formation of traffic jam, but also shows that such jams can propagate upstream (i.e., backwards, opposite to the direction of flow of the vehicles). Moreover, it is possible that two or more jams coexist on a highway. A more detailed analysis of traffic jams in absence of hindrances has been given by Kerner and Rehborn [725, 740, 741], who pointed out characteristic features of wide jams. They found that the upstream velocity and, therefore, the outflow (mean values of flux, density and velocity) from a jam is approximately constant. The outflow from x
t
Figure 6.4 Trajectories of individual vehicles on a single lane of a multilane highway. The trajectories were drawn from aerial photographs. During the analyzed time-interval, the spontaneous formation, propagation, and dissolution of a jam have been observed (modified from [1391]). The trajectories have been determined from aerial photographs which give the vehicles positions at certain times (horizontal lines). The diagonal line indicates the jam front, which moves at constant speed opposite to the driving direction of the vehicles. The discontinuous trajectories correspond to vehicles changing the lane.
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a jam and the velocity of the jam fronts are now regarded as two important empirical parameters of highway traffic, which can be used for calibrating theoretical models. Daganzo [256, 257] argues that spontaneous jam formation does not occur and that jams are always induced by bottlenecks. Usually, these are road inhomogeneities such as ramps or sharp bends, or temporary disturbances through lane changes, and so on. Once a jam has occurred due to such a (temporary) bottleneck, it might not dissipate immediately, causing most drivers to wonder about the reason for the congestion.
6.3.3. Experiment on Spontaneous Jam Formation In [1032, 1320, 1321], it is shown by a laboratory experiment that the emergence of a traffic jam occurs spontaneously if the average vehicle-density exceeds a certain critical value. The experiment was performed on a circular road in a homogeneous lane condition on flat ground. A snapshot of the experiment is shown in Fig. 6.5. A sufficient number of vehicles are put onto the circular road so that the density satisfies the condition for the onset of the instability of free flow, which is estimated by the optimal velocity model [69]. In the actual experiment, the length of circumference was 230 m with 22 vehicles. The drivers were requested to cruise at about 30 km/h, and they were only instructed to follow the vehicle ahead safely in addition to trying to keep the velocity of cruising. In the initial stage, the free flow has been maintained for a while. After some time, small fluctuations appeared in the headway distances. Due to this disturbance of free flow, the vehicles could not move homogeneously any more. Then fluctuations started to grow enough to induce the breakdown of free flow. Finally, several vehicles were forced to stop completely for a moment. A clear stop-and-go wave was observed which propagated in the opposite direction of the movement of vehicles with −20 km/h, in good agreement with measurements on real highways [1390]. The jam cluster moved as a solitary wave and maintained its size and velocity, as shown in Fig. 6.6(a). The time-series of the average and standard deviation of velocities of all vehicles is shown in Fig. 6.6(b). We can observe the existence of three different stages divided
Figure 6.5 Snapshot of the experiment on a circular road. The length of circumference is 230 m and the number of vehicles is 22.
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Vehicular Traffic I: Empirical Facts 250
8
200
0.8
7
150 100
6
Flow (1/s)
Velocity (m/s)
Time (s)
1
9
5 4 3 2
50
0.6 0.4 0.2
1 0
50 100 150 200 Position (m) (a)
0
50
100 150 Time (s) (b)
200
250
0
50
100 150 Time (s) (c)
200
250
Figure 6.6 (a) Space-time diagram for a run of 22 vehicles. Data for 250 s around the time when a jam appears are shown. In the graph, the 0-m point is identical to the 230-m point. (b) The red and green lines represent the average and standard deviation of the velocity data, respectively. Blue lines at t = 85s and t = 145s divide the whole period into three stages. Note that the average headway is constant. (c) Red represents the section flow rate during 40 s. For example, the data point at t = 20s shows the flow rate for t = 0 − 40s. Green represents the average of flow rates calculated from velocity and headway of each vehicle.
by vertical lines. The early stage is characterized as the state where the fluctuations of velocity and headway are relatively large. This state continues for about 10 min from the start of run. The data for the last 85 s of this stage are shown. In the middle stage, the average velocity increases and the average headway is constant. The standard deviations of velocity and headway decrease. This state can be considered to be a homogeneous flow with large velocity. The flow quickly transits to the jammed flow. Therefore, this process indicates that the state of the homogeneous flow with large velocity is metastable. In the final stage, the state of the jammed flow is characterized by a small velocity and large standard deviation. The change of flow rate in the process of a jam formation is also shown in Fig. 6.6(c). Flow is defined by two ways: First is the number of vehicles passing through the observational point during 40 s. This duration corresponds to the period in which a jam cluster moves around the circuit once. The other is the average of the product of velocity and the inverse of headway over all vehicles. The flow rates by both definitions become large at the middle stage, when the metastable homogeneous flow is realized. The flow rates decrease in the final stage, when the jammed flow appears. The existence of the metastable homogeneous flow is understood also in the change of the flow rate. The increase of the flow rate means that the homogeneous flow in the middle stage has a large average velocity. The fact that the state with high flow rate soon transits to the jammed flow means the state is metastable. The physical mechanism of forming a jam is summarized as follows. Small fluctuations always exist in the movement of vehicles in a traffic flow. If the vehicle density is low, such a fluctuation disappears and a free flow is maintained. However, if the density
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exceeds the critical value, the fluctuation cannot disappear but instead grows steadily and eventually free flow breaks down. After relaxation, a jam is spontaneously created and travels along the circuit without decaying. From this experiment, it is shown that instability of free flow on the circle road and the emergence of a traffic jam are observed with neither obstacle nor bottleneck under laboratory conditions. It should be emphasized that this effect is not restricted to Japanese drivers. Similar experiments have, at least qualitatively, been performed with German drivers (e.g., [1236]) with the same result.
6.4. FUNDAMENTAL DIAGRAM The fundamental diagram is the most important quantitative characteristic of traffic flow3 . It is defined as the relation J (ρ) between the traffic density ρ and the traffic flow J . Due to the hydrodynamic relation J = ρv [see (6.6)] there are three different equivalent forms of the fundamental diagram, namely J (ρ) or v(ρ) or v( J ). The generic shape of the fundamental diagram is shown in Fig. 6.7. Obviously, traffic flow phenomena strongly depend on the occupancy of the road. What type of variation of flux and average velocity v with density ρ can one expect on the basis of intuitive arguments? So long as ρ is sufficiently small, the average speed v is practically independent of ρ as the vehicles are too far apart to interact mutually. Therefore, at sufficiently low density of vehicles, practically free flow takes place. However, from the practical experience that vehicles have to move slower with increasing density, one expects that at intermediate densities, dv ≤ 0, (6.17) dρ when the forward movement of the vehicles is strongly hindered by others because of the reduction in the average separation between them. A faster-than-linear monotonic decrease of v with increasing ρ can lead to a maximum [922] in the flux J = ρv at ρ = ρ ∗ . For ρ < ρ ∗, increasing ρ would lead to increasing J , whereas for ρ > ρ ∗ , sharp decrease of v with increase of ρ would lead to the overall decrease of J . However, v
J
ρ
v
ρ
J
Figure 6.7 Schematic forms of the fundamental diagram in different representations related through the hydrodynamic relation J = ρv. 3 For a discussion of historical and practical aspects, see, e.g., [824].
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2200 2000 1800
Flow (PCU/h)
1600 1400 1200 1000 800 600 400 200 0
10
20
30 40 50 Occupancy (x)
60
70
Figure 6.8 Empirical data for flow and occupancy. The data have been collected by counting loops on a Canadian highway. Both the occupancy and the flow have been directly measured by the detector. Each point in the diagram corresponds to an average over a time interval of 5 min (from [489]).
contrary to this naive expectation, in recent years, some nontrivial variation of flux with density has been observed. This will be discussed in the following sections. Figure 6.8 shows typical time-averaged local measurements of the density and flow that have been obtained from the Queen Elizabeth Way in Ontario (Canada) [489]. At low densities, the data indicate a linear dependence of the flow on the density. In contrast, strong fluctuations of the flow at large densities exist, which prevents a direct evaluation of the functional form at high densities. Figure 6.8 also indicates typical values for the main features of the fundamental diagram. The maximal flow can be of the order of 2500 vehicles/h. For the left lane, it can even larger. The maximum is reached at densities of around 30 vehicles/km, which correspond to an occupancy of about 20%. The typical outflow from a large jam is about Jout = 1800 vehicles/km. The overall structure of the fundamental diagram will be discussed in detail in Section 6.6. It should be noted that certain details of the fundamental diagram can depend strongly on the specific road network and regulations like speed limits, e.g., the free-flow branch can be both linear and nonlinear [777]. Different analytical forms for the fundamental diagram have been suggested. The simplest fundamental diagram, which captures the basic features consist of two linear branches and has triangular shape as shown in Fig. 6.7. Historically, one of the oldest forms is the Greenshields model [462] ρ , J (ρ) = vfree ρ 1 − ρmax
(6.18)
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which corresponds to the linear velocity–density relation v(ρ) = vfree (1 − ρ/ρmax). Del Castillo and Benitez [278, 279] have proposed more complex speed–density relations of the general form
v(ρ) = vfree 1 − f (λ) ,
(6.19)
where f (λ) is called generating function and λ the equivalent spacing which is defined as λ=
cJ vfree
ρJ −1 , ρ
(6.20)
where cJ is the kinematic wave speed at jam density and ρJ the jam density. Typical values for cJ are from 15 km/h to 25 km/h and 150 vehicles/km for ρJ . A generic form for the generating function is
λ n f (λ) = exp 1 − 1 + , n
(6.21)
where especially the cases n = 1, the exponential curve, and n = ∞, the maximum sensitivity curve are interesting. It should be mentioned that Kerner [727, 730] uses the terminology “fundamental diagram” in a stricter sense where it implies the existence of a unique relation between density and flow. Therefore, for him the occurrence of synchronized traffic is a phenomenon, which can not be understood by a fundamental diagram approach. We will not use such a strict convention here and consider “fundamental diagram” mostly to be equivalent with “flow-density relation.” This includes cases in which this relation is not unique, as in the presence of metastable states or synchronized traffic.
6.5. METASTABILITY AND HYSTERESIS In several situations, it has been observed that J does not depend uniquely on ρ in an intermediate regime of density; it indicates the existence of hysteresis effects and metastable states. In the context of traffic flow, hysteresis effects have the following meaning: if a measurement starts in the free-flow regime, an increase of the density leads to an increase of the flow. However, beyond a certain density, a further increase of the density leads to a discontinuous reduction of the stationary flow (capacity drop) and jams emerge. The corresponding fundamental diagram (Fig. 6.9) then has the so-called inverse-λ form. Figure 6.10 shows an experimental verification of a hysteresis loop [489] at a transition from a free flow to a congested state.
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J(ρ)
J(ρ2)
J(ρ1)
ρ1
ρ
ρ2
Figure 6.9 Inverse λ-form of the fundamental diagram. The high-flow states in the density regime ρ1 < ρ < ρ2 are metastable.
C
2200 2200 1800
Flow (PCU/h)
1600 1400 1200 1000 800 600 400 200 0
10
20
30 40 50 Occupancy (x)
60
70
Figure 6.10 Time-traced measurements of the flow (from [489]). Each data point corresponds to a time-averaged value of the flow and density obtained from a local measurement. The average was performed over 5-min intervals.
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The fundamental diagram of metastable systems is characterized by a density regime ρ1 < ρ < ρ2 , where the flow is not uniquely determined by the density. State of high flow are not stable and decay under perturbations. We will later see the occurrence of metastable states is related to the fact that the outflow from jams, which is given by J (ρ1 ), is smaller than the maximal possible flow.
6.6. PHASES OF TRAFFIC FLOW Traffic is intrinsically a nonequilibrium process. As discussed in Section 1.8, the notion of “phases” can be quite subtle in systems far from equilibrium. Therefore, it should not always be interpreted in a strictly physical sense when this terminology is used in connection with traffic. Some authors are aware of these subtleties and so sometimes a somewhat unusual terminology is used, like “local phase transitions” [740]4 . Therefore, a better terminology would be traffic pattern, i.e., a state with a typical spatio-temporal appearance [548]. Kerner emphasizes the difference among traffic phases, traffic states, and traffic patterns. A traffic phase is characterized by its spatiotemporal features (Section 6.6.2), i.e., there can be different traffic states of the same traffic phase. A traffic pattern, however, is defined as a distribution of traffic flow variables in space and time and can consists of different traffic phases [730, 734].
6.6.1. Level of Service Classification In traffic engineering, the Level of Service (LOS) concept is frequently used. It is a quantitative classification of traffic states according to the density [1381]. LOS A (0–11 vehicles/mile/lane): Free flow. Vehicles can move at their desired speeds; density is low and speeds are high. LOS B (11–18 vehicles/mile/lane): Reasonably free flow. Vehicles do not influence each other much, most vehicles can move at their desired speeds without the need to change lanes. LOS C (18–26 vehicles/mile/lane): Stable flow. Interactions between vehicles become relevant, desired speeds are restricted, lane changes are no readily possible. LOS D (26–35 vehicles/mile/lane): Approaching unstable flow. Restrictions on the attainable speeds, which starts to decrease. LOS E (33–45 vehicles/mile/lane): Unstable flow. Stop-and-go traffic, i.e., sequences of short jams, and traffic waves and queues appear, traffic is operating near the maximum of capacity. 4 This indicates that not the whole system is affected by the transition.
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LOS F (>45 vehicles/mile/lane): Forced or breakdown flow. The speeds are low and density is high. Traffic jams with complete stops appear. With regard to perturbations, LOS A-C are stable, E and F are unstable, D is at the verge of instability. It can be tolerated for short time periods only and will then lead into the regime of unstable traffic. Note that this classification might differ slightly in other countries (see, e.g., [824] for the classification in Germany).
6.6.2. Traffic Phases and Phase Transitions The empirical results for the flux–density relation suggest the existence of at least two different dynamical phases of vehicular traffic on highways, namely a free-flow phase and a congested phase. In the free-flow regime, all vehicles can move with high speed close to the speed limit. Flow increases monotonically with increasing density. In the congested regime, however, flow decreases with increasing density. Extensive empirical observations in the recent years indicate the existence of two different congested phases, namely, the synchronized phase and jammed phase [724–726, 741, 742, 789]. Currently it is widely accepted (at least among physicists) that – following Kerner [727, 730] – one can distinguish three different traffic phases (although some points remain controversial, see, e.g., [547, 1257]): • Free flow: In this phase, interactions between vehicles are rare. Every car moves with its desired velocity corresponding, e.g., to a speed limit. Therefore, the flow increases linearly with the density of cars. The free flow branch F can clearly be seen in Fig. 6.11. The part of the branch with flows larger than Jout is called metastable
Flow (free)
Jmax
(syn)
Jmax
Jout F
J
ρout
(free)
ρmax
ρmax
Density
Figure 6.11 Schematical form of the flow-density relation in three-phase traffic theory. F denotes the free flow branch and the line J is determined by the properties of wide moving jams.
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branch. It corresponds to a region where the flow is not uniquely determined by the density. All states not of free flow type are called congested states. They are characterized by an average velocity that is smaller than the “desired” velocity of the drivers. Two congested phases can be distinguished. • Wide moving jams: Jams can form spontaneously, i.e., without any obvious external reason (see Section 6.3). Wide (moving) jams are regions of very high density and negligible average velocity and flow. Their width is much larger than the fronts at both ends where the speed of vehicles changes sharply. The jam front moves upstream (i.e., opposite to the driving direction) at a typical velocity vjam ≈ 15 km/h [740]. Other characteristic properties of wide jams are its density and the outflow Jout , which does not depend on the inflow into the jam [727]. The velocity vjam and the corresponding flow rate is only determined by the density inside a wide jam and the delay-time between two vehicles leaving the jam [740]. Frequently moving blanks are observed in large jams [730, 741], i.e., regions without vehicles that move upstream due to the vehicle motion. • Synchronized flow: In synchronized flow [727] the average velocity is significantly lower than in free flow, but the flow can be much larger than in wide moving jams. The main characteristic is the apparent absence of a functional flow-density form, i.e., the corresponding data points are spread irregularly over a large two-dimensional area (Fig. 6.12). In time-series of flow-density measurements, the flow can increase or decrease with increasing density, in sharp contrast to the free-flow (jammed) phase where the flow always increases (decreases). This irregularity of the time-series can be quantified by using the crosscorrelation function ccρ, J (τ ) ∝ ρ(t)J (t + τ ) − ρ(t) J (t + τ ) between density ρ and flow J [1042]. In free flow, ccρ, J (τ ) is very close to 1, but in synchronized flow, it is almost 0 [1042]. Here flow and density are independent of each other. Furthermore, on a multilane highway, the time-series of measurements on different lanes are highly correlated, i.e., synchronized [789]. This was the motivation for denoting this traffic state as synchronized traffic. Another characteristic property that distinguishes synchronized traffic from wide moving jams is the behavior at bottlenecks. Wide moving jams propagate through bottlenecks maintaining the mean velocity of the downstream front. This is not the case for synchronized flow, which typically has a downstream front that is fixed at the bottleneck. Above we have only given the most important properties of the phases. A more detailed characterization of the three-phase traffic theory based on a wealth of empirical data and a discussion of the related spatio-temporal organization of traffic states can be found in [727, 730] (and references therein). According to Kerner [727, 730], the transition F → J does not occur spontaneously in real traffic. It can be observed only when there is a large enough perturbation. Instead,
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3000
2500
Flow (vehicles/h)
2000
1500
1000
500
0
20
40 60 Occupancy (%)
80
100
Figure 6.12 Empirical flow–density relation. One can clearly distinguish the three phases. The differences to the schematical form are mainly due to the use of local measurements leading to the problems discussed in Section 6.2. V F
S
J ρ
Figure 6.13 Transitions between the phases of the three-phase traffic theory in the velocity–density plane.
traffic breakdown is governed by a first-order phase transition F → S from free flow to synchronized flow. Narrow moving jams then emerge spontaneously through the pinch effect, a self-compression of synchronized flow. Some of these narrow moving jams can grow into wide moving jams. This transition usually occurs at a different location than the F → S transition. Figure 6.13 illustrates the phases and transitions in the velocity– density plane.
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Originally, Kerner has distinguished three different types of synchronized traffic by the time-dependent behavior of the density and flow [725–727, 741]: • Stationary and homogeneous states: Constant values of density and flow can be observed during a long period of time (for a few minutes). • Homogeneous-in-speed states: Patterns of density and flow look quite similar to the upper part of the free flow states. However, compared with free flow, the average velocity is reduced and the speeds on different lanes are aligned. • Nonstationary and nonhomogeneous states: Irregular patterns of time-traced measurements of the flow have been found in synchronized traffic. Typically, the occurrence of synchronized traffic is related to inhomogeneities like ramps. It has been proposed to characterize the structure of the traffic states by different traffic patterns. However, this classification, which will be discussed in Section 6.7, is still controversial. Furthermore, other potential origins of the observed scattering of flow–density data in the congested regime have been suggested, e.g., mixtures of different vehicle types [1383] or variations of the net time-headways, which lead to time-dependent variations of the jam velocity [1068].
6.6.3. Gas–Liquid Analogy Several attempts have been made to understand traffic phases and the transitions between them in a larger, consistent framework. Especially the analogies with the gas–liquid transition has been discussed for many years [1028, 1142, 1178]. In this framework [1028], free-flow (laminar) traffic corresponds to the gas phase, jammed traffic to the liquid phase and stop-and-go traffic to the coexistence of gas and liquid (Fig. 6.14). At low densities (ρ < ρ1 ), only the gas phase exists, corresponding to T
Gas
One phase
T > Tc
Density
Gas
T = Tc
Free−flow
Coexistence Liquid
Gas
Mixed
Jammed
T < Tc ρ1(T)
ρ4(T)
Liquid ρ
(a) (b) Figure 6.14 Illustration of the gas–liquid analogy: schematic view (a) and phase diagram (b) (adopted from [1028]).
Vehicular Traffic I: Empirical Facts
free flow. At higher densities ρ1 < ρ < ρ2 droplets can form, which can be interpreted as jams. These are only stable if they are large enough. Droplets will grow until the density outside has reached ρ1 . In the regime ρ2 < ρ < ρ3 droplets are formed immediately and will coagulate until only a single droplet is left (spinodal decomposition). For ρ3 < ρ < ρ4 , the behavior is analogous to the regime ρ1 < ρ < ρ2 , but now the gas phase, which has density ρ1 , expands until the liquid phase has reached the density ρ4 . Finally, for densities above ρ4 , only the liquid phase exists. In gas–liquid systems, the densities ρj depend on temperature. At the critical point, they all coincide and the difference between the gas and liquid phases vanishes (Fig. 6.14). For some deterministic models, this scenario appears to be realized at least approximately. For stochastic problems, it is usually difficult to identify the different phases because the gas–liquid phases are characterized by a density difference where density is coarse-grained quantity, not a particle property. Another problem is the intrinsic asymmetry of traffic system. As a consequence, one interface (e.g., the front of a jam) between phases can be stable, whereas the other is unstable. For a more detailed discussion of phase transitions in traffic flow models, we refer to [694, 900, 1025, 1028, 1431] and Section 7.4.8.
6.7. RAMPS, INTERSECTIONS, AND OTHER INHOMOGENEITIES The phase diagram near inhomogeneities like ramps shows a rich structure. Typically these inhomogeneities act as bottlenecks by locally restricting the maximal possible flow. Several proposals for a classification of the elementary traffic patterns associated with such inhomogeneities have been made, but several aspects are still controversial. On highways, interactions between bottlenecks become relevant. A given bottleneck interacts with downstream bottlenecks because its outflow is the inflow for the next bottleneck. However, the interaction with upstream bottlenecks is due to propagation of congested patterns. The structure of traffic states can be characterized by different traffic patterns (see Section 6.6). According to Kerner, at (isolated) bottlenecks, two main types of patterns are observed [728]: • synchronized flow pattern (SP): upstream of the bottleneck only synchronized flow is observed, no wide moving jams emerge • general pattern (GP): wide moving jams and synchronized flow coexist. The SP is further subdivided into three different subpatterns: the localized SP where the spatial width is limited over time, the widening SP, which is continuously growing over time and the moving SP which propagates. In the typical scenario for emergence of the GP, first a F → S occurs at the bottleneck. Then small moving jams emerge in synchronized region that propagate upstream and grow into wide moving jams.
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On highways, typically, many bottlenecks (e.g., ramps) are found in close succession. Then their interactions become relevant and further patterns emerge, e.g., the expanded congested pattern where the synchronized region extends over several bottlenecks and which again comprises several different special cases. Helbing and collaborators have proposed a slightly different classification by identifying six different elementary traffic states [525, 1256, 1257, 1387]. Besides free traffic which is characterized by an average speed larger than some threshold vth , two classes of congested states are distinguished. In a localized cluster (LC), the average speed is below vth only over a short subsection, otherwise one talks about spatially extended congestion states. The more details classification leads to the following six elementary traffic states: • free traffic: the average speed exceeds a certain threshold vth • pinned localized clusters: LC which stay at a fixed position over a longer time period • moving localized clusters: propagate upstream with a characteristic speed vjam and correspond to wide moving jams • stop-and-go waves: a sequence of several moving LC • oscillating congested traffic: show oscillating speeds in the congested range • homogeneous congested traffic: the speeds are in the congested range over a spatially extended area, but not oscillating. They argue that the majority of congested traffic states can be interpreted as a spatial combination of these six states. In this scenario, one main jam-generating mechanism is the boomerang effect [521, 1256]: A small perturbation travels downstream and grows in this process. When the perturbation exceeds a certain amplitude, given by the drop in the vehicle speed, it changes its propagation direction and moves as a wide jam upstream. It has been suggested that the boomerang effect is related to overtaking maneuvers by trucks which cause moving bottlenecks [427]. The main source of the controversy is the question: which are the elementary patterns observed at bottlenecks? Partially the conflicting interpretations of Kerner and Helbing could be attributed to different methods of data analysis. They use different methods of data processing and representation. Kerner’s measurements were performed at discrete crosssections of the highway, whereas Helbing et al. studied averages along freeway sections as function of time [1256]. A more detailed discussion of the controversial points can be found, e.g., in [547, 727, 730, 1257].
6.8. HEADWAY DISTRIBUTIONS We have already seen in Section 6.2 that the distributions of the temporal and spatial headways are related to the average flux and density, respectively. However, both
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distributions contain more detailed information on traffic flow than that available from the flux and density alone. In general, the shape of the distributions of temporal and spatial headway depends on the density. May [922] distinguishes three different classes: (1) random headway states for free flow, (2) intermediate headway states for light congestion, and (3) constant headway states for heavy congestion. For small densities, the headways are uncorrelated. The probability density of the time-headway distribution of such a random-headway state is given by the negative exponential distribution5 [922] P(t) =
1 −t/t , e t
(6.22)
where t is the average time headway. At high densities, all vehicles are interacting. Most headways are small and tend to become equal, e.g., in a jam, but with some variation due to fluctuations and driver errors. These constant headway states are described by a normal distribution. Between these two limits, the distribution is more complex. The intermediate headway states are described by the Pearson-type-III distribution6 P(t) =
t − a b
k−1
1 −(t−a)/b . e b(k)
(6.23)
It includes the negative exponential distribution as a special case (k = 1, a = 0). For special situations like tunnels or bridges also the log-normal distribution has been found to good results [337]. The distribution of the spatial headway looks very similar to the time-headway distribution because it is simply given by x = vt if the velocities of the vehicles is uniform. Krbalek et al. [796] have pointed out a possible connection of the (spatial) headway distribution with random matrix theory, especially the Dyson gas. A similar approach relates it with superstatistics [1]. The Dyson gas describes a system of confined particles interacting through a repulsive Coulomb potential V = ln |xi − xj |. For the nearest neighbor Dyson gas, this leads to the following heuristic form of the headway distribution [796]: P(t) =
(β + 1)β+1 (t)β exp[−(β + 1)t] . (β + 1)
5 The corresponding count distribution is the Poisson distribution. 6 For a = 0, this is also known as Erlang distribution.
(6.24)
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1.2
1.2 ρ = 0−12 vehicles/km ρ = 12−24 vehicles/km ρ = 24−36 vehicles/km
1.0
0.8 P (th)
P (th)
0.8 0.6
0.6
0.4
0.4
0.2
0.2
0
ρ = 24−36 vehicles/km ρ = 36−48 vehicles/km ρ = 48−60 vehicles/km
1.0
1
2
3 th (s)
4
5
6
0
(a)
1
2
3 th (s)
4
5
6
(b)
Figure 6.15 Distribution of time-headways in the free flow (a) and the synchronized regime (b) obtained empirically on German highways (after [776]).
This distribution has been scaled so that the average headway is one: P(s)ds = 1. In the Dyson gas picture β corresponds to the inverse temperature β = 1/kB T . In the high temperature limit β → 0 the distribution becomes Poissonian with P(t) = exp(t). The distribution (6.24) was found to provide a good description of the empirical data even in the synchronized phase [796]. Through the availability of empirical single-vehicle data, the calculation of timeheadway distributions has become possible [776, 796, 1042]. The empirical distributions show a surprisingly large fraction of headways that are much shorter than allowed by (German) legal regulations (1.8 sec) [776, 1042]. This is a strong indication that anticipation (Section 6.11) plays an important role for the behavior of the drivers. The distributions in the three phases differ, e.g., it becomes much broader in the synchronized phase (Fig. 6.15) than in free flow. In the outflow region of wide jams, a typical headway of about 2 s is observed, which is consistent with a jam velocity of vjam ≈ 15 km/h.
6.9. OPTIMAL-VELOCITY FUNCTION Another important quantity characterizing the microscopic states is the dependence of the velocity of individual vehicles on the distance headway. It is the most relevant information for the adjustment of the speed by the driver. This function is
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30 Free, ρ = 0..10 vehicles/km
s =1 .8
0.8 s
Cong., ρ = 30..40 vehicles/km Cong., ρ = 50..60 vehicles/km
Δt
20
Cong., ρ = 10..20 vehicles/km
Δt =
Speed v (m/s)
25
15 10 5 0
20
40 60 Headway Δx (m)
80
100
Figure 6.16 Speed–distance relation in free and congested traffic. The typical velocities at large distance headways strongly depend on the traffic state (from [1042]).
also of great importance for theoretical approaches; e.g., it is used as input for the socalled optimal velocity model (Section 9.4.2). It is therefore usually called optimal-velocity function. Figure 6.16 shows typical empirical optimal-velocity functions for different densities. One finds that the speed does not only depend on the headway x, but also on the local density ρ; e.g., in dense traffic, low velocities of the vehicles are also observed even when large distance headways are available. In the free-flow regime, the asymptotic velocity is reached already for small distance-headways. The slope of the velocity function is much larger than that in the congested regime. Further information about the optimal-velocity function and other characteristics of car-following can be obtained using floating-car data, see, e.g., [396, 1043].
6.10. CORRELATION FUNCTIONS Various correlation functions have been used to obtained quantitative information about coupling effects between variables. In [1042], the autocorrelations of the local density, flow, and average velocity have been analyzed. In free flow, the average speeds are only correlated on short timescales. In contrast, long-ranged correlations are present in the time series of local density and flow that vary systematically on much longer timescales up to the order of magnitude of hours. This long-range signal reflects the daily variation of the traffic loads. This behavior of the autocorrelation is clearly contrasted with the behavior found in synchronized traffic, where all temporal correlations are short-ranged irrespective of the chosen observable.
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Both results show that longer time-scales are only apparent in slow variations of the density during a day, while the other time-series reveal a noisy behavior. The crosscorrelation ccρ, J (τ = 0) indicates the strong coupling between flow and density in the free-flow regime [1042]. Variations of the flow are mainly controlled by density fluctuations, while the average velocity is almost constant. Again, the results for synchronized states differ strongly. Here all combinations of flow, density, and average velocity lead to small values of the crosscorrelation ccρ, J (τ = 0) ≈ 0 also supporting the existence of irregular patterns in the fundamental diagram. Furthermore, the correlation between different lanes in the synchronized phase leads to large values of ccvi ,vj , where vi is the speed on lane i [1042]. Similar to free-flow states, stop-and-go traffic is characterized by strong correlations between density and flow (ccρ, J (0) ≈ 1) Beyond that also the autocorrelation function shows an oscillating structure for all three quantities of interest. The period of these oscillations is given by ≈ 10 min. This result is in accordance with measurements by Kühne [823], who found oscillating structures in stop-and-go traffic with similar periods. The stability of platoons can be evaluated using the crosscorrelations between velocities at different locations [776]. The lifetime of such platoons can have large values, and even in free flow, moving structures can be identified. Velocity correlations between non-neighbor vehicles have been studied by Appert-Rolland [37], who found strong correlations even for platoons of up to eight vehicles. Of special interest from a physics point of view are correlations exhibiting power laws because this is a possible indication for critical behavior. 1/f noise in real traffic has been discovered by Musha and Higuchi [964, 965]. They recorded transit times of vehicles passing underneath a bridge. The corresponding power spectral density of the flow fluctuations shows 1/f behavior at low frequencies. This was confirmed by Wagner and Peinke [1433] who studied the probability distribution of velocity differences. Tadaki et al. [1338] have observed power-law fluctuations with three crossover times of 1 h, 1 day, and 15 days in empirical time series from Japanese expressways. Helbing and Tilch [543] found a power law for the lifetimes of high-flow states. A possible explanation are vehicle platoons which are caused by long-lasting overtaking maneuvers of trucks and other slow vehicles.
6.11. PSYCHOLOGICAL EFFECTS Naturally, one would expect that psychology, or human factors [786], plays an important role for the behavior of the drivers and thus traffic. However, we will later see that in most models, these effects are mostly neglected. They appear to be more relevant for the “fine-structure” of traffic, whereas the broad picture is dominated by many-particle interactions.
Vehicular Traffic I: Empirical Facts
There are many studies of a wide variety of such effects that we can not really cover here. Therefore, we restrict ourselves to a few aspects. For an entertaining discussion of psychological effects, we refer to the recent book [1412]. One of the most relevant psychological factors in traffic is the reaction time of the drivers. It is the sum of three times [857]: (1) the perception time needed to recognise an event, (2) the decision time during which a driver decides about his/her response, and (3) the application time needed to take action. The reaction time can be reduced by anticipation, e.g., when the driver tries to “guess” what the preceding driver will do in the next moment. Indeed, this is relevant in various situations and certain traffic states like the metastable high-flow branch would not be possible without anticipation. The empirical car-following behavior indicates that human drivers apply a noisy control mechanism [1430]. The applied control is not continuous because drivers react only at certain moments in time, the so-called action points [1365]. In general, because of the different human temperaments and driving habits, different drivers react slightly differently to the same conditions on a highway, even when no other vehicle influences its motion [180]. Consequently, even on an empty stretch of a highway, a driver can neither maintain a constant desired speed nor accelerate in a smooth fashion. The root-mean-square deviation of the acceleration of the vehicles is a measure of the so-called acceleration noise. The distributions of the accelerations have been measured [572, 1470], and they depend on the density of vehicles and the type of highway. The fact that drivers have different personalities has been demonstrated for the headway selection [180]. After a perturbation, drivers have the tendency to return to their original headway before the perturbation, an effect called driver memory by Cassidy and Windover [180]. The memory effects reported in [1384] describe the adaption of drivers to surrounding traffic on timescales of a few minutes. Frustration effects cause changes in the driving behavior, e.g., an increase in the preferred time headways after standing in a jam for a longer time [1068, 1384]. An interesting effect connected to the perception of the average velocities on different lanes has been found in [1172]. Usually drivers tend to overestimate the speed in the neighbouring lanes. This illusion could be related to the fact that a group of closely packed vehicles which move slowly can be overtaken in a short time. However, if these cars accelerate they spread out and it takes a much longer time to be overtaken by the same vehicles.
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CHAPTER SEVEN
Vehicular Traffic II: The Nagel–Schreckenberg Model Contents 7.1. Definition of the Model 7.1.1. Update Rules 7.1.2. Relation with Other Models
244 244 247
7.2. Fundamental Diagram and Limiting Cases of the NaSch Model 7.2.1. Fundamental Diagram 7.2.2. NaSch Model in the Deterministic Limit p = 0 7.2.3. NaSch Model in the Deterministic Limit p = 1 7.2.4. NaSch Model with vmax = 1 7.2.5. NaSch Model in the Limit vmax = ∞
248 248 250 251 251 253
7.3. Analytical Theories for NaSch Model with vmax > 1 7.3.1. SOMF Theory for the NaSch Model 7.3.2. Cluster-Approximations for the NaSch Model 7.3.3. pMF Theory of the NaSch Model 7.3.4. Car-Oriented Mean-Field Theory of the NaSch Model
255 255 256 257 259
7.4. Spatio-Temporal Organization of Vehicles 7.4.1. Microscopic Structure of the Stationary State 7.4.2. Spatial Correlations 7.4.3. Headway Distributions 7.4.4. Distributions of Jam Sizes and Gaps between Jams 7.4.5. Distribution of Lifetimes of Jams 7.4.6. Temporal Correlations and Relaxation Time 7.4.7. Structure Factor 7.4.8. Phase Transition 7.4.9. Boundary-Induced Phase Transitions
260 260 261 262 263 265 266 267 268 270
7.5. Appendices 7.5.1. Details of SOMF for NaSch 7.5.2. Details of PMF for NaSch 7.5.3. Details of COMF for NaSch
272 272 276 277
After giving an overview on the most relevant empirical aspects, we will now discuss the various modeling approaches that have been used to explain these findings. Because the focus of this book is on stochastic systems, we will start with CA models. Due Stochastic Transport in Complex Systems DOI: 10.1016/B978-0-444-52853-7.00007-5
Copyright © 2011, Elsevier BV. All rights reserved.
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to their simplicity and flexibility, this class of models has become very popular since the early 1990s [214–216, 895, 1231, 1233, 1235] when the basic model, now known as Nagel–Schreckenberg (NaSch) model, was introduced.
7.1. DEFINITION OF THE MODEL As we have seen in Section 5.2, cellular automata (CA) are idealizations of physical systems in which both space and time are assumed to be discrete and each of the interacting units can have only a finite number of discrete states. Because usually CA models have a rule-based dynamics, they are well-suited for modeling interdisciplinary problems. This might partially explain the enormous success of this approach, especially in modeling traffic systems. To our knowledge, the first CA model for vehicular traffic was introduced by Cremer and Ludwig [243], but the now fundamental model was introduced in 1992 by Nagel and Schreckenberg [1027]. In the CA models of traffic, the position, speed, acceleration, as well as time are treated as discrete variables. In this approach, a lane is represented by a one-dimensional lattice. Each of the lattice sites represents a cell, which can be either empty or occupied by at most one vehicle at a given instant of time (see Fig. 7.1). At each discrete time step t → t + 1, the state of the system is updated following a well-defined prescription, see Section 2.1. The computational efficiency of the discrete CA models is the main advantage of this approach over the car-following and coupled-map lattice approaches.
7.1.1. Update Rules We consider a road divided into L cells, which can contain at most one vehicle. Until not stated otherwise, we will assume periodic boundary conditions for simplicity. On the street, N vehicles are moving so that their density is given by ρ = N /L. In the Nagel–Schreckenberg (NaSch) model [1027], the speed v of each vehicle can take one of the vmax + 1 allowed integer values v = 0, 1, . . . , vmax . Suppose, xn and vn denote the position and speed, respectively, of the n-th vehicle. Then, dn = xn+1 − xn − 1, is the (spatial) headway of the n-th vehicle at time t, i.e., the number of empty cells in front of this car 1
0
1
2
7, 5 meter
Figure 7.1 A typical configuration in the NaSch model. The number in the upper right corner is the speed of the vehicle.
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(see Fig. 6.2) At each time step t → t + 1, the arrangement of the N vehicles on a finite lattice of length L is updated in parallel according to the following rules: NaSch1: Acceleration. If vn < vmax , the speed of the n-th vehicle is increased by one, but vn remains unaltered if vn = vmax , i.e., vn → min(vn + 1, vmax ) NaSch2: Deceleration (due to other vehicles). If vn > dn , the speed of the n-th vehicle is reduced to dn , i.e., vn → min(vn , dn ) NaSch3: Randomization. If vn > 0, the speed of the n-th vehicle is decreased randomly by unity with probability p but vn does not change if vn = 0, i.e., vn → max(vn − 1, 0)
with probability p
NaSch4: Vehicle movement. Each vehicle is moved forward according to its new velocity determined in (NaSch1)–(NaSch3), i.e., xn → xn + vn The rules can be summarized in a more formal and concise way: xn (t + 1) = xn (t) + max[0, min {vmax , vn (t) + 1, dn (t)} − ηn (t)] .
(7.1)
Here ηn (t) is a Boolean random variable, which takes the value 1 with probability p and 0 with probability 1 − p and accounts for the randomization step. The term given by the maximum is the new velocity vn (t + 1). Note that the rules can be expressed solely in terms of the positions xn by using dn (t) = xn+1 (t) − xn (t) − 1 and vn (t) = xn (t) − xn (t − 1). This also shows that the configuration at time t + 1 depends both on that at time t and t − 1. The NaSch model is a minimal model in the sense that all the four steps are necessary to reproduce the basic features of real traffic. However, additional rules need to be formulated to capture more complex situations. The rule (NaSch1) reflects the general tendency of the drivers to drive as fast as possible, if allowed to do so, without crossing the maximum speed limit. The rule (NaSch2) is intended to avoid collision between the vehicles. The randomization in (NaSch3) takes into account the different behavioral patterns of the individual drivers, especially, nondeterministic acceleration, as well as overreaction while slowing down; this is crucially important for the spontaneous
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formation of traffic jams. Even changing the precise order of the steps of the update rules stated above would change the properties of the model [1233]. For example, after changing the order of rules (NaSch2) and (NaSch3), there will be no overreactions at braking and thus no spontaneous formation of jams. Further consequences of changing the order of the rules are discussed in [1486]. The NaSch model may be regarded as stochastic CA [1476–1478]. In the special case vmax = 1, the NaSch model reduces to the totally asymmetric simple exclusion process (TASEP) with parallel dynamics (see Chapter 4). The deterministic limit p = 0 is then equivalent to the CA rule 184 in Wolfram’s notation [1476, 1477] and some abstract extensions of this CA-184 rules [55] have been studied in the more general context of complex dynamics and particle flow. Why should the updating be done in parallel, rather than in random sequential manner, in traffic models like the NaSch model? In contrast to a random sequential update, parallel update can lead to a chain of overreactions. Suppose, a vehicle slows down due to the randomization step. If the density of vehicles is large enough, this might force the following vehicle also to brake in the deceleration step. In addition, if p is larger than zero, it might brake even further due to rule (NaSch3). Eventually, this can lead to the stopping of a vehicle, thus creating a jam. This mechanism of spontaneous jam formation is rather realistic and cannot be modeled by the random sequential update. The update scheme of the NaSch model is illustrated with a simple example in Fig. 7.2. (a) Acceleration 2
2
2
1
2
0
1
2
0
1
(b) Braking 1
(c) Randomization (p = 1/3) 0
(d) Driving (= configuration at time t + 1) 0
2
0
1
Figure 7.2 Step-by-step example for the application of the update rules. We have assumed vmax = 2 and p = 1/3. Therefore on an average, one-third of the cars qualifying will slow down in the randomization step.
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x
t
x
t
(a)
(b)
Figure 7.3 Typical space-time diagrams of the NaSch model with vmax = 5 and (a) p = 0.25, ρ = 0.20, (b) p = 0, ρ = 0.5. Each horizontal row of dots represents the instantaneous positions of the vehicles moving toward right while the successive rows of dots represent the positions of the same vehicles at the successive time steps.
Space-time diagrams showing the time evolutions of the NaSch model demonstrate that no jam is present at sufficiently low densities, but spontaneous fluctuations give rise to traffic jams at higher densities (Fig. 7.3(a)). From the Fig. 7.3(b), it should be obvious that the intrinsic stochasticity of the dynamics [1027], arising from nonzero p, is essential for triggering the jams [1024, 1027]. For a realistic description of highway traffic [1027], the typical length of each cell should be about 7.5 m, which is the space occupied by a vehicle in a dense jam. When vmax = 5, each time step should correspond to approximately 1 s of real time, which is of the order of the shortest relevant timescale in real traffic, namely the reaction time of the drivers. Almost all the models of traffic considered in this book, including the NaSch model, have been formulated in such a way that no accident between successive vehicles is possible. However, accidents become possible if the condition for safe driving is relaxed. This will be discussed in more detail in Section 8.9.
7.1.2. Relation with Other Models The NaSch model can be considered as an extension of the TASEP discussed in the first part of the book (see Chapter 4). In the TASEP, particles are only allowed to move forward by one site, whereas in the NaSch model, the hopping distance can be up to vmax sites and depends on the current velocity of the particle.
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In the limit vmax = 1, the NaSch model reduces to the TASEP with parallel dynamics because one can neglect the possibility of having velocities v = 0 and v = 1 in the NaSch model. After the acceleration step of the update rules, all particles have velocity v = 1, and at this stage, the two models are fully equivalent. Note, however, that the hopping rate pASEP and the randomization probability pNaSch of the NaSch model are related by pNaSch = 1 − pASEP.
(7.2)
Because both are generically denoted by p this can be a constant source of confusion! As we have seen in Section 4.9.1, it is possible to map the asymmetric simple exclusion process (ASEP), and thus the NaSch model with vmax = 1, onto a stochastic growth model of one-dimensional surfaces in a two-dimensional medium, the single-step model [70, 492]. It is the discrete counterpart of continuum models of growing surfaces whose dynamics are governed by the so-called Kardar–Parisi–Zhang (KPZ) equation [70, 817]. Because the KPZ equation can be mapped onto the Burgers equation [167] using the Cole-Hopf transformation [70], it is not surprising that several features of vehicular traffic are described by the NaSch model at the microscopic level and by the noisy Burgers equation for the coarse-grained continuum of the fluid-dynamical theory [1020].
7.2. FUNDAMENTAL DIAGRAM AND LIMITING CASES OF THE NaSch MODEL In contrast to the ASEP, the NaSch model is not exactly solvable in general, except for some limiting cases. Despite the fact that the deterministic limits p = 0 and p = 1 of the NaSch model do not capture some of the most essential features of vehicular traffic, it is instructive to examine these limits to gain insight into the features of this simpler scenario. Another limiting case that exhibits a surprisingly complex behavior is the case vmax = ∞.
7.2.1. Fundamental Diagram Before we discuss some limiting cases in more detail, we present numerical results for the fundamental diagrams for generic parameter values. Our focus will be on the variation of the shape in dependence of the randomization p and the maximal velocity vmax . For vmax = 1, the flux is invariant under charge conjugation ( particle-hole transformation), i.e., under the operation ρ → 1 − ρ, which interchanges particles and holes. Therefore, the fundamental diagram is symmetric about ρ = 1/2 when vmax = 1 (see Fig. 7.6). This symmetry breaks down for all vmax > 1 (Fig. 7.4), and the corresponding fundamental diagrams become more realistic. Moreover, for given p, the magnitude of ρ ∗, the density at which the flow becomes maximal, decreases with increasing vmax as larger vmax implies
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vmax = 1 vmax = 2 vmax = 3 vmax = 5
0.5
J
0.4 0.3 0.2 0.1
0
0.2
0.4
0.6
0.8
1
ρ
Figure 7.4 Fundamental diagram in the NaSch model for vmax = 1, 2, 3, 5 and p = 0.25 obtained through computer simulations.
p=0 p = 0.25 p = 0.5 p = 0.75
0.8
J
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
1
ρ
Figure 7.5 Fundamental diagram for vmax = 5 and different values of p.
a longer effective range of interaction of the vehicles (Fig. 7.4). Furthermore, for vmax = 1, flux merely decreases with increasing p (Eqn (7.10)), but remains symmetric about ρ = 1/2 = ρ ∗. However, for all vmax > 1, increasing p not only leads to smaller flux but also lowers ρ ∗ (Fig. 7.5). For vmax > 1, the system is no longer particle-hole symmetric and the fundamental diagram is no longer symmetric around ρ = 1/2. This can already be seen from the limiting behavior above. In fact, the maximum is shifted to smaller densities so that the fundamental diagram becomes more realistic. For vmax > 1, no exact solutions for the NaSch model at general densities ρ and randomization p are known. Apart from the deterministic limits p = 0 and p = 1 discussed
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below, the asymptotic behavior of the fundamental diagram J (ρ) is known exactly for arbitrary vmax and p in the two limits ρ → 0 and ρ → 1: J (vmax − p)ρ
for ρ → 0,
(7.3)
J (1 − p)(1 − ρ)
for ρ → 1.
(7.4)
Here vF = vmax − p is the free-flow velocity. An isolated vehicle moves with probability p with velocity vmax − 1 and with probability 1 − p with velocity vmax so that vF = p(vmax − 1) + (1 − p)vmax. For ρ → 1, the flow is determined by holes moving backward at a speed 1 − p, which explains J (1 − p)(1 − ρ).
7.2.2. NaSch Model in the Deterministic Limit p = 0 The NaSch model becomes a deterministic CA in the limit p = 0. In this special case, the deterministic dynamical update rules of the model can be written as vn (t + 1) = min[vmax, vn (t) + 1, dn (t)],
(7.5)
xn (t + 1) = xn (t) + vn(t + 1),
(7.6)
which can lead to two types of steady states depending on the density ρ [1024]. At low densities, the system can self-organize so that dn ≥ vmax for all n and, therefore, every vehicle can move with vmax , i.e., vn (t) = vmax , giving rise to the corresponding flux ρvmax . This steady-state is, however, possible only if enough empty cells are available in ∗ with front of every vehicle, i.e., for ρ ≤ ρdet ∗ ρdet =
1 vmax + 1
(7.7)
vmax . vmax + 1
(7.8)
and the corresponding maximum flux is max = Jdet
∗ , d (t) ≤ min[v (t) + 1, v However, for ρ > ρdet n n max] and, therefore, the relevant steadystates are characterized by vn (t) = dn (t), i.e., the f low is limited by the density of holes. Because the average distance-headway is 1/ρ − 1, the fundamental diagram in the deterministic limit p = 0 of the NaSch model (for arbitrary vmax) is given by the exact expression
Jdet = min {ρvmax, 1 − ρ}.
(7.9)
Note that the result vn = 1/ρ − 1 is identical with Greenshields’ Ansatz v = 1/ρ − 1/ρjam (see Section 9.1.3) if we identify ρjam = 1.
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7.2.3. NaSch Model in the Deterministic Limit p = 1 The other natural deterministic limit of the NaSch model, namely p = 1, is a bit more subtle than one might expect. First of all, it is not identical to the same NaSch model but just maximum allowed speed vmax − 1, although this expectation may seem to be consistent with the fact that J = 0 for all ρ in the special case vmax = 1, i.e., the ASEP. An important observation is that a vehicle with vn (t) = 0 will never move again. Even if it accelerates in step (NaSch1), it will decelerate again in the randomization step (NaSch3). This leads to subtle features of the deterministic limit p = 1 in the case vmax > 1. Taking, for example, vmax = 2, then, for ρ > 1/3, all stationary states correspond to J = 0 because at least one vehicle will have only one empty cell in front (i.e., dn = 1) and it will never succeed in moving forward. For vmax = 2 and p = 1, although there are stationary states corresponding to J = 0 for all ρ ≤ 1/3, such states are metastable in the sense that any local external perturbation leads to complete breakdown of the flow. If the initial state is random, such metastable states cannot lead to nonzero J because they have a vanishing weight in the thermodynamic limit. Hence, if p = 1, all random initial states lead to J = 0 in the stationary state of the NaSch model irrespective of vmax and ρ !
7.2.4. NaSch Model with vmax = 1 For vmax = 1, the NaSch model is equivalent to the TASEP with parallel dynamics. However, in principle, in the NaSch formulation, the particles still have an internal degree of freedom, the velocity vn = 0, 1 which corresponds to the number of sites the particle has moved in the previous timestep. Changing the update ordering slightly to (NaSch2)-(NaSch3)-(NaSch4)-(NaSch1) eliminates particles with velocity vn = 0, and thus, we have full equivalence with the TASEP case. Then the exact results derived in Chapter 4 can be used. There it has been shown that the fundamental diagram for parallel dynamics is given by1
J (ρ, p) =
1 1 − 1 − 4(1 − p)ρ(1 − ρ) . 2
(7.10)
This result has been derived by various approaches, i.e., the two-cluster approximation, car-oriented mean-field theories (COMF) and paradisical mean-field (pMF) theory. It is shown in Fig. 7.6 where for comparison also the (site-oriented) mean-field result is presented. The flow only depends on ρ(1 − ρ), which is a consequence of the particlehole symmetry ρ ↔ 1 − ρ. 1 We remind the reader that now p denotes the randomization parameter, which is related to the hopping probability
pASEP by p = 1 − pASEP .
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0.3 Exact Mean-field Simulation
0.25
J
0.2
0.15
0.1
0.05
0
0.2
0.4
0.6
0.8
1
ρ
Figure 7.6 The fundamental diagram in the NaSch model for vmax = 1 and p = 0.25.
We give the exact two-cluster and gap probabilities here because they are useful for the calculation of other expectation values: P(0, 0) = 1 − ρ − P(1, 0), P(1, 1) = ρ − P(1, 0), P(1, 0) = P(0, 1) =
1 2(1 − p)
1 − 1 − 4(1 − p)ρ(1 − ρ) ,
(7.11)
and 1 2(1 − p)ρ − 1 + 1 − 4(1 − p)ρ(1 − ρ) , 2(1 − p)ρ n (p(1 − P0) P0 Pn = (n ≥ 1). p P0 + p(1 − P0)
P0 =
(7.12)
The exact solution yields valuable information about the correlations. In Appendix 7.5.1, the fundamental diagram within the simple site-oriented mean-field (SOMF) approach is derived: JMF (ρ) = (1 − p)ρ(1 − ρ).
(7.13)
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This result can easily be understood since the SOMF assumes a completely random distribution of particles. Then a contribution to the current along a bond comes from configurations where the left site is occupied (probability ρ) and the right site is empty (probability 1 − ρ). In this situation, the particle will move with probability 1 − p. This SOMF result (7.13) considerably underestimates the flow [1244, 1258] (Fig. 7.6). This implies that correlations play an important role. These correlations can be characterized by a particle-hole attraction, i.e., the probability to find an empty cell (hole) in front of an occupied cell (particle) is enhanced compared with a completely random distribution of particles and holes (as assumed in the mean-field approach). More explicitly the particle-hole attraction implies P(0)P(1) = (1 − ρ)ρ ≤ P(0, 1).
(7.14)
Similar correlations also exist for vmax > 1. For vmax = 1, however, the origin of these correlations is solely due to the existence of Garden of Eden states (see Section 4.1.5). This will be discussed in more detail in Section 7.3.3.
7.2.5. NaSch Model in the Limit vmax = ∞ The limit vmax = ∞ can be introduced in several possible ways because only finite systems of length L can be treated in computer simulations. In [1226], the case vmax = L has been investigated2 , but other limiting procedures are also possible, e.g., vmax ∝ L α with α > 0 or even vmax = ∞ independent of the system size. In principle, these different limiting procedures could lead to different results, but up to now, no systematic study has been performed. Surprisingly one finds that the fundamental diagram has a form quite different from that for finite vmax [1226]. The flow does not vanish in the limit ρ → 0 since already one single car produces a finite value of the f low, J (ρ → 0) = 1. Due to the hindrance effect of other cars, J (ρ) is a monotonically decreasing function of the density ρ (see Fig. 7.7). Another characteristic feature of the fundamental diagram is the existence of a plateau at flow JP , where the value JP depends on the randomization p, but not on the system size L. The length of the plateau, however, increases with L. What is the microscopic structure of the stationary state leading to such a fundamental diagram? At low densities, where flow J is larger than the plateau value JP , the cars tend to be uniformly distributed just as in the deterministic case p = 0 (see Section 7.2.2). For densities in the plateau regime, however, one jam exists in the system, whereas for higher densities, there is more than one jam. In the thermodynamic limit, one expects a 2 See also [1024], where the case p = 0 was studied.
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1.0 L = 1000
0.8
J
0.6 0.4 0.2
0.0
0.2
0.4
0.6
0.8
1.0
ρ
Figure 7.7 Fundamental diagram of the NaSch model with vmax = L for a system size of L = 1000 (from [1226]).
phase transition at ρ = 0 between a jamless phase with J = 1 and a phase with one jam and f low JP [1226]. Increasing the density further, more jams develop and the plateau ceases. Note that this behavior is completely different from the prediction of meanfield theory in that limit [1244, 1258] (see Section 7.3.1) showing the importance of correlations. The low-density limit of the NaSch model for high velocities has also been analyzed in [609]. Here three different regions have been identified with increasing density: (1) linear increase of the flow with density, (2) abrupt decrease of the flow, and (3) a plateau. Beyond region (3), traffic jams start to dominate the behavior which becomes similar to that for small vmax . Similar observations have been made in a related toy model [152] based on a statistical model for dimers, which is exactly solvable. It can also be mapped to a five-vertex model [154]. In a certain parameter regime, the flow in the low-density limit does not vanish. The model has a CA analog, which is basically an ASEP, where particles can make an unrestricted number of steps per timestep. Knospe et al. [777] have studied the NaSch model in the limit vmax → ∞ with vmax / constant, where is the length of a car measured in cells.3 This leads to a reduction of the acceleration step so that velocity fluctuations and vehicle interactions in the free-flow regime are reduced. Metastable high-flow states are observed for homogeneous initial states. In the congested regime, the reduced cell length has only little influence. The model behaves similar to a NaSch model with car length = 1 and randomization p/. The density of jammed cars η = L1 N j=1 δvj ,0 shows a behavior, which is very similar to that of the VDR model. It vanishes below a finite critical density ρc and increases 3 This corresponds to a reduced cell length 1/.
Vehicular Traffic II: The Nagel–Schreckenberg Model
linearly for ρ > ρc . With increasing system size, the high-flow states become unstable and the jump in the order parameter at ρc vanishes.
7.3. ANALYTICAL THEORIES FOR NaSch MODEL WITH Vmax > 1 For vmax > 1, no exact solutions for the NaSch model at general densities ρ and randomizations p are known. Apart from the deterministic limits p = 0 and p = 1 discussed earlier, one can determine the asymptotic behavior of the fundamental diagram J (ρ) is known exactly for arbitrary vmax and p in the two limits ρ → 0 and ρ → 1 (see Section 7.2.1). However, the analytical methods developed in the first part of the book still provide important information even though they are only approximations.
7.3.1. SOMF Theory for the NaSch Model In the site-oriented theories, one describes the state of the finite system of length L by completely specifying the state of each site, i.e., by the set (σ1 , σ2 , . . ., σL ) where σj ( j = 1, 2, . . ., L) can, in principle, take vmax + 2 values one of which represents an empty site while the remaining vmax + 1 correspond to the vmax + 1 possible values of the speed of the vehicle occupying the site j. In some of the analytical calculations of steady-state properties of the NaSch model one follows, for convenience, the sequence 2 − 3 − 4 − 1, instead of 1 − 2 − 3 − 4 of the stages of updating [1244, 1258] as this merely shifts the starting step and, therefore, does not influence the steady-state properties of the model. The advantage of this new sequence is that, in a site-oriented theory, the variable σj can now take vmax + 1 values as none of the vehicles can have a speed v = 0 at the end of the acceleration stage of the updating. Let us introduce the lattice gas variables n(i; t) through the following definition: n(i; t) = 0 if the site labeled by i is empty and n(i; t) = 1 if it is occupied by a vehicle (irrespective ofthe speed). Obviously, the space-average of n(i; t) is the density of the vehicles, i.e., i n(i; t)/L = ρ. Suppose, cv (i; t) is the probability that there is a vehicle with vmaxspeed v (v = 0, 1, 2, . . . , vmax ) at the site i at the time step t. Obviously, c(i; t) = v=0 cv (i; t) is the probability that the site i is occupied by a vehicle at the time step t and d(i; t) = 1 − c(i; t) is the corresponding probability that the site i is empty at the time step t. In the naive SOMF approximation (see Section 2.2.1) for the NaSch model, one writes down the equations relating cv (i; t + 1) (v = 1, . . ., vmax ) with the corresponding probabilities at time t and, then, solves the equations in the steady-state. These calculations are presented for arbitrary vmax in Appendix 7.5.1. For all values of vmax and the randomization parameter p, SOMF considerably underestimates the f low. Only for small (ρ 1) and large (1 − ρ 1) densities it becomes
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asymptotically exact. This shows the importance of the correlations in the stationary state of the NaSch model.
7.3.2. Cluster-Approximations for the NaSch Model The cluster theoretic approach has been described in detail in Section 2.2.1. In the case vmax = 1 corresponding to the TASEP with parallel dynamics, we have seen in Section 4.1.3 that the two-cluster approximation yields the exact solution for the stationary state [1244, 1258]. For vmax > 1, the cluster approximation does not become exact, but it leads to a systematic improvement of the naive SOMF theory, especially for higher order cluster calculations [1230, 1244, 1258]. For vmax = 2, the fundamental diagrams obtained from the n-cluster approximation (n = 1, 2, .., 5) are compared in Fig. 7.8 with Monte Carlo (MC) data. This comparison clearly establishes a rapid convergence with increasing n, and already for n = 4, the difference between the cluster calculation and MC data is extremely small. In [1230], the cluster probabilities for vmax = 2 have been obtained from computer simulations. The results suggest that the n-cluster approximation for n ≥ 3 becomes asymptotically exact in the limit p → 0. It should be noted that for the higher order cluster approximation, so far, no closed analytical expressions for the cluster probabilities have been derived for the NaSch model with vmax > 1. For the n-cluster approximation, one has to distinguish (vmax + 1)n different cluster probabilities, which are related by nonlinear equations. Although the number of independent probabilities can be reduced, e.g., using the Kolomogorov consistency conditions, the 0.25 1-,2-, . . .,5-cluster Simulation
0.2
J
0.15
0.1
0.05
0
0.2
0.4
0.6
0.8
1
ρ
Figure 7.8 Fundamental diagrams for vmax = 2 in the n-cluster approximation (n = 1, 2, .., 5).
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resulting equations have to be solved numerically, in general. However, this still has certain advantages over computer simulations, e.g., no averaging over different realizations of the stochastic process is necessary. Furthermore, it is possible to treat certain limiting cases exactly.
7.3.3. pMF Theory of the NaSch Model The general idea behind the pMF theory has already been explained in Section 4.1.5. It requires identifying those configurations, called Garden-of-Eden (GoE) states or paradisical states, which can not be reached by the dynamics and eliminate them from the mean-field equations. For example, the configuration shown in Fig. 7.9 is a GoE state4 because it could occur at time t only if the two vehicles occupied the same cell simultaneously at time t − 1. The naive SOMF theory does not exclude the GoE states. Results of the pMF theory are derived by repeating the SOMF calculations, but excluding all the GoE states from consideration. For the NaSch model with vmax = 1, the exact expression for the f lux in the steady state is recovered (see Appendix 7.5.2 for details), thereby indicating that the only source of correlation in this case is the parallel updating [1247]. In the case of random-sequential dynamics, no GoE states exist. For vmax = 1, the stationary state is uncorrelated and therefore already SOMF is exact. But, for vmax > 1, there are dynamical correlations so that exclusion of the GoE states merely improves the naive SOMF estimate of the flux (Fig. 7.10), but does not yield exact results [1230, 1247]. For vmax = 1 the question, whether a state is a GoE state or not, can be decided locally by investigating just nearest-neighbor configurations. By analysing the update rules, one finds that all states containing the local configurations (0, 1) or (1, 1), i.e., configurations where a moving vehicle is directly followed by another car, are GoE states. This is not possible as can be seen by looking at the previous configurations. The momentary velocity gives the number of cells that the car moved in the previous timestep. In both configurations the first car moved one cell. Therefore, it is immediately clear that (0, 1) is a GoE state because otherwise there would have been a doubly occupied cell before the last timestep. The configuration (1, 1) is also not possible since both cars must 1
Figure 7.9 A GoE state for the NaSch model with vmax ≥ 2. 4 The configuration shown in Fig. 7.1 is also a GoE state!
2
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0.5 0.4 J (ρ)
J (ρ)
0.2
0.1
0.3 0.2 0.1
0
0.2
0.4
0.6
0.8
0
1
0.2
0.4
ρ
0.6
0.8
1
ρ
(a)
(b)
Figure 7.10 Fundamental diagram for vmax = 2 and p = 0.5 (a) and p = 0.1 (b). Comparison of paradisical MFT (dotted line) with results from computer simulations (full line) and the naive MFT (broken line).
have occupied neighboring cells before the last timestep too. Therefore, according to (NaSch2), the second car could not move. In the case vmax = 2, more GoE states exist. In order to identify these, it is helpful to note that the rules (NaSch1) – (NaSch4) imply dj (t) = dj (t − 1) + vj+1(t) − vj (t), where dj (t) and vj (t) are headway and velocity of car j at time t respectively, and therefore dj (t) ≥ vj+1 (t) − vj (t),
(7.15)
vj (t) ≤ dj (t − 1).
(7.16)
The second inequality (7.16) is a consequence of (NaSch2). A detailed analysis [1247] gives the following elementary GoE states, i.e., the local configurations, which are dynamically forbidden: (0, 1), (1, 1),
(2, 1),
(0, 2), (2, 2),
(1, 2),
(0, •, 2),
(7.17)
(1, •, 2),
(2, •, 2),
(7.18)
(0, •, •, 2).
(7.19)
Numbers give the velocity of a vehicle in an occupied cell and • denotes an empty cell. The elementary GoE states in (7.17) violate the inequality (7.15), and the configurations in (7.18) violate (7.16). The state in (7.19) is a second-order GoE state. Going one step back in time leads to a first-order GoE state because (0, •, •, 2) must have evolved from (0, v) (with v = 1 or v = 2). Details of the derivation of the fundamental diagram within the pMF theory are given in Appendix 7.5.2 for the cases vmax = 1 and vmax = 2. As mentioned earlier for vmax = 1, the exact solution is recovered. For vmax = 2, the pMF is a considerable improvement over the naive SOMF theory (Fig. 7.10).
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The existence of GoE states gives a simple criterion for the quality of an approximation: it should be able to account for all GoE states. This partly explains why for vmax = 2 the four-cluster approximation and COMF are able to reproduce the fundamental fairly well. It has also been attempted to combine the paradisical MFT with the cluster approximation [607, 608]. But this turns out to be technically difficult due to the Kolmogorov consistency conditions [1248].
7.3.4. Car-Oriented Mean-Field Theory of the NaSch Model In the car-oriented theories, the state of the traffic system is described by specifying the headways and speeds of all the N vehicles in the system [1245]. Therefore, the central quantity is Pn (v, t), the probability to find at time t exactly n empty sites immediately in front of a vehicle n, which moves with velocity v. The essence of the car-oriented mean-field (COMF) approximation [1245] is to neglect the correlations between the gaps in front of the successive cars (see Section 2.2.1). The equations describing the time evolution of the probabilities Pn (v, t), under this approximation, can be solved in the steady state using a generating function technique [1245]. Details for the case vmax = 1 have been presented in Section 4.1.3 for the ASEP, where COMF reproduces the exact results. The corresponding stationary headway distribution has been given in Eqn (7.12). It allows to calculate other quantities, e.g., the flux. For vmax = 2, one has to distinguish between Pn (v = 1) and Pn (v = 2). Moreover, one has to generalize the quantity g to gα , the probability that the vehicle moves α cells (α = 1, 2) in the next time step. The derivation of the dynamical equations and their solution in the stationary state using generating functions can be found in Appendix 7.5.3. In contrast to the case vmax = 1, COMF only yields approximate results [1245]. Interestingly, finite size of the system affects the equations for vmax = 2 in a much more dramatic way [1230] than those for vmax = 1, thereby revealing the intrinsic qualitative differences in the nature of correlations in the NaSch model for vmax = 1 and vmax > 1. For p = 0.1, the COMF result shows an excellent agreement with computer simulations. For p = 0.5, the agreement is still excellent for small (ρ < 0.2) and high densities (ρ > 0.5), only near the maximum there are deviations. The COMF result is much better than the two-cluster result and comparable to the three-cluster approximation. COMF tends to overestimate the flux although it is not systematically larger than the simulation results. In contrast, the n-cluster approximation yields a lower bound for the flux [1258]. For small densities, the average distance between the cars is large. Therefore, correlations between cars and neighboring empty sites are much more important than those between two cars. These correlations are better described by the COMF, which is the reason why it is superior to the cluster approach for small cluster sizes in this regime.
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Comparisons with MC simulations show that, in contrast to the three-cluster approximation, COMF does not become asymptotically exact in the limit p → 0 for vmax = 2 [1230]. This implies that even in this limit, correlations between the headways are not negligible. It is interesting, however, that for the fundamental diagram, one finds an excellent agreement between MC simulations and the predictions of COMF [1245] for p → 0. The reason is that in the deterministic limit, many configurations exist which produce the same flow. COMF is not able to identify the dominating structures correctly, but nevertheless can predict the correct current.
7.4. SPATIO-TEMPORAL ORGANIZATION OF VEHICLES In the following, we will discuss a few quantities that allow to characterize the microscopic structure of the states in the NaSch model in more detail. One important question is: Is there a qualitative difference between free flow and congested states, i.e., are these regimes separated by a phase transition?
7.4.1. Microscopic Structure of the Stationary State SOMF underestimates the flow in the stationary state of the NaSch model considerably. Deviations become larger for higher velocities vmax . This shows the importance of correlations, e.g., a particle-hole attraction. For vmax = 1, all improvements of SOMF (two-cluster, COMF, and pMFT) are exact. Here only correlations between neighboring cells are important. All correlations in the case vmax = 1 have their origin in the existence of GoE states, which exist due to the parallel updating. The situation changes for higher velocities vmax > 1. Here pMFT is no longer exact, not even for random-sequential dynamics. Therefore “true” correlations exist. This corresponds to the observation made in [1244, 1258] that the NaSch model shows a qualitatively different behavior for vmax = 1 and vmax > 1. Furthermore, it explains why so far the exact determination of the stationary state for vmax > 1 has not been possible. It is interesting to investigate how the microscopic structure of the stationary state depends on the randomization p. For p = 0, we have seen in Section 7.2.2 that for densities ρ ≤ 1/(vmax + 1), the vehicles arrange themselves in such a way that all headways are at least vmax . This is no longer possible for larger densities, but still the vehicles have the tendency to maximize their headway. Furthermore, for p = 0, no spontaneous formation of jams exists because overreactions are not possible. The behavior in this limit can be interpreted as coming from a kind of “repulsive interaction” between the vehicles. The behavior for p = 1 is a little bit different. Here we have seen in Section 7.2.3 that metastable states with finite flow exist for ρ ≤ 1/3 and vmax > 1.
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For 0 < p < 1, the microscopic structure interpolates between these two limiting cases. This can be seen by analysing the three-cluster probabilities obtained from MC simulations [1230]. For small p, the microscopic structure of the stationary state is determined by the “repulsive interactions” between vehicles. With increasing p one finds a tendency toward phase separation into jammed and free-flow regions. A standing vehicle is able to induce a jam even at low densities because the restart probability is small. The jams formed are typically not compact, but of the form5 “.0.0.0.” because a vehicle approaching the jam slows down in the randomization step with a rather high probability. Concluding one might say that the microscopic structure for 0 < p < 1 is determined by the competition of the two “fixed points” p = 0 and p = 1. In the following, we discuss the behavior of some observables in more detail. A particular focus is the region near the maximum of fundamental diagram, especially the question whether free-flow and congested regime are separated by a true phase transition.
7.4.2. Spatial Correlations A striking feature of second-order phase transitions is the occurrence of a diverging length scale at criticality and a corresponding algebraic decay of the correlation function [445, 1308]. In terms of the occupation number nj , the equal-time density–density correlation function is defined by 1 1 nj nj+r − ρ 2, G(r) = T L t=1 j=1 T
L
(7.20)
which measures the correlations in the density fluctuations that occur at the same time at two different points in space separated by a distance r. In the deterministic case p = 0, there are exactly vmax empty sites in front of each ∗ . The correlation function at ρ = ρ ∗ is then given by vehicle at ρ = ρdet det G(r) =
∗ ∗ ) ρdet (1 − ρdet ∗ )2 −(ρdet
for r ≡ 0 mod(vmax + 1),
(7.21)
else.
For all vmax , the correlation function for small nonzero p has essentially the same structure as that for p = 0, but the amplitude decays exponentially [341] for all ρ. In the general case of nonvanishing p, the asymptotic behavior (r → ∞) of the correlation length ξ can be obtained analytically [1230] only for vmax = 1. It turns out that, for given p, ξ is maximum at ρ = 1/2 = ρ ∗ and that ξ(ρ = 1/2) ∝ p−1/2 . Thus, for vmax = 1, ξ 5 Here “.” denotes an empty site and “0” a site with a stopped car.
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A. Schadschneider, D. Chowdhury, and K. Nishinari 35
11 10
vmax = 2
p = 1/128 p = 1/64 p = 1/32 p = 1/16
9 8
25
ξmax
ξ
7 6 5 4
20 15 10
3
5
2 1
vmax = 2 vmax = 3
30
0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4 0.42 ρ
0
0.02
(a)
0.04
0.06
ρ
0.08
0.1
0.12
0.14
(b)
Figure 7.11 The dependence of (a) ξ(ρ, p) on the density ρ for four different values of p (vmax = 2) and (b) ξmax on p (vmax = 2, 3).
diverges only for p = 0, but remains finite for all nonzero p. For vmax > 1, the trend of variation of ξ with ρ (Fig. 7.11(a)) in the vicinity of ρ ∗ is the same as that for vmax = 1 [341]. Moreover, for vmax > 1, the maximum value of the correlation length, ξmax plotted against p (Fig. 7.11(b)), is also consistent with the corresponding trend of variation for vmax = 1. Thus, the correlation function G(r) gives a strong indication that the NaSch ∗ , only for p = 0 but this tranmodel exhibits a second order phase transition, at ρ = ρdet sition is smeared out if p= 0. This noise-induced smearing of the phase transition in the NaSch model is very similar to the smearing of critical phenomena by finite-size effects.
7.4.3. Headway Distributions Information about the spatial organization of the vehicles can be obtained from the distance-headway distribution Pdh by following either a site-oriented approach [209] or a car-oriented approach [1245]. Stated precisely, Pdh (k) is the conditional probability of finding a string of k empty sites in front of a site, which is given to be occupied by a vehicle. A comparison between the naive mean-field expression MFA ( j) = ρ(1 − ρ)j Pdh
(7.22)
for the distance-headway distribution in the NaSch model with vmax = 1 and the corresponding MC data [209] reveals the inadequacy of equation (7.22) at very short distances, which indicates the existence of strong short-range correlations in the NaSch model that are neglected by the mean-field treatment. This is consistent with the short-range particle-hole attraction observed in the NaSch model with vmax = 1.
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0.5 ρ = 0.1 ρ = 0.5 ρ = 0.7
0.6 0.5
0.3
Pn
0.4
Pn
ρ = 0.05 ρ = 0.1 ρ = 0.3
0.4
0.3
0.2
0.2 0.1
0.1 0
2
4
6
8
10
0
n (a)
5
10 n
15
20
(b)
Figure 7.12 Distributions of distance-headways in the NaSch model for p = 0.5 and different densities for vmax = 1 (a) and vmax = 5 (b). n denotes the number of empty cells in front of a vehicle and is related to the distance-headway by n = x − 1.
The exact distance-headway distribution [209, 1230, 1245] in this case is given by the COMF result, i.e., Pdh ( j) = Pj
(7.23)
with Pj as defined in (7.12). For all vmax > 1, at moderately high densities, Pdh ( x) exhibits two peaks [209, 215, 792, 793], in contrast to a single peak in the distance-headway distributions for vmax = 1 at all densities (Fig. 7.12). The peak at x = 1 is caused by the jammed vehicles while that at a larger x corresponds to the most probable distance-headway in the freeflowing regions. At first sight, the simultaneous existence of free-flowing and jammed regions may appear analogous to the coexistence of gaseous and liquid phases of matter in equilibrium. However, the two-peak structure cannot be interpreted as a manifestation of the coexistence of two dynamical phases [213], namely the free-flowing phase and the jammed phase, because the analog of the gas–liquid interfacial tension is zero in the NaSch model. Time-headway distributions have also been determined within various approximation schemes [213, 435] and computer simulations [215]. For vmax = 1, an approximate analytical expression can be obtained [213, 435], which is plotted in Fig. 7.13(a) for a few typical values of ρ for a given p. A few typical time-headway distributions in the NaSch model for vmax > 1, obtained through computer simulation [215], are shown in Fig. 7.13(b).
7.4.4. Distributions of Jam Sizes and Gaps between Jams One can identify a string of k successive stopped vehicles as a jam of length k (by definition, such jams are compact). Similarly, when there are k lattice sites between two
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A. Schadschneider, D. Chowdhury, and K. Nishinari
0.6
0.4 ρ = 0.10 ρ = 0.25 ρ = 0.50
0.5
0.3 0.25
0.3
Pth
Pth
0.4
ρ = 0.10 ρ = 0.25 ρ = 0.50
0.35
0.2 0.15
0.2
0.1 0.1 0
0.05 5
10
15 τ (a)
20
25
30
0
5
10
15 τ (b)
20
25
30
Figure 7.13 Time-headway distribution in the NaSch model with (a) vmax = 1 and (b) vmax = 5.
successive jams, each occupied by a moving vehicle or is vacant then we say that there is a gap of length k between the two successive jams. Analytical expressions for the distributions of the jam sizes as well as of the gaps between jams can be calculated for the NaSch model (and some of its extensions) using the two-cluster approximation or COMF [209, 217, 1228, 1230]. The expressions are exact in the case vmax = 1 with periodic boundary conditions. For higher velocities, the results are only approximative. In COMF, the probability Ck to find a jam of length k is given by Ck(COMF) = (1 − P0)P0k−1 ,
(7.24)
whereas in the two-cluster approach one finds (2)
Ck = (2) C1
1 P( 0|1)P( 1|1)k−2 P( 1|1)P( 1|0) NJ
vmax 1 = P( 0|v)P( v|0), NJ v=1
(k ≥ 2), (7.25)
max where NJ = vv=1 P1|1) is the
number of jams and the conditional probabilities are defined by P(a|b) = P(a, b)/ c P(a, c) . For the n−cluster approximation, similar expressions can be derived. Both distributions (7.24) and (7.25) decay exponentially for large jam sizes. COMF (COMF) . In contrast, always predicts a monotonous distribution with Ck(COMF) ≥ Ck+1 the jam size distribution in the n−cluster approximation can, in principle, exhibit a maximum at small jam sizes 1 ≤ k ≤ n.
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7.4.5. Distribution of Lifetimes of Jams Another quantity which should be able to give information about the nature of the transition from free-flow regime to the jammed regime is the distribution of lifetimes of jams. Following Nagel [1019] each vehicle which has a velocity less than vmax before the randomization step will be considered jammed. This definition is motivated by the cruise-control limit (see Section 8.2), where it is more natural than in the NaSch model. One expects, however, that the long-time behavior of the lifetime distribution is independent of the exact definition of a jam. The short-time behavior, however, might differ strongly, e.g., for “compact jams” where a jam is defined as a series of consecutive standing vehicles without any empty cells in between. Figure 7.14 shows the results of MC simulations for the lifetime distribution in the NaSch model for different densities near the transition region, ρ ≈ ρm = 0.085 ± 0.005 (for vmax = 5, p = 0.5), where ρm is the density where the flow is maximal. The most interesting feature of the lifetime distribution is the existence of a cutoff near τc = 10 000. It has been shown [1019] that this cutoff is neither a finite-size nor a finite-time effect. For times smaller than τc , a scaling regime exists where the distribution decays algebraically. Gerwinski and Krug [434] gave an intuitive criterion, which allows the distinction of free-flow and jammed phases. It is based on the investigation of jam-dissolution times. Starting from a megajam configuration, i.e., a block of N consecutive cells occupied by vehicles with the remaining L − N cells being empty, they determined the time until the jam6 has dissolved completely. 10 ρ = 0.10, L = 100 000 ρ = 0.08, L = 100 000 ρ = 0.06, L = 100 000 slope –0.54 ρ = 0.10, L = 10 000 ρ = 0.08, L = 10 000 ρ = 0.06, L = 10 000
1 0.1
T * P (T )
0.01 0.001 0.0001 1e–05 1e–06 1e–07 1e–08
1
10
100
1000 T
10 000
100 000
1e + 06
Figure 7.14 Lifetime distribution in the NaSch model for vmax = 5 and p = 0.5 and various densities below and above ρ ∗ ≈ 0.085. 6 In [434], the same definition of a jam as in Section 7.4.5 (see [1019]) has been used.
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A simple estimate gives the density at which the lifetime is expected to become infinite. Suppose that the jam dissolves with velocity vJ . Because the first vehicle moves freely with an average velocity vF = vmax − p, it will reach the end of the jam at the same time as the dissolution wave if the condition (L − N )/vF = N /vJ is satisfied. The corresponding density is then given by ρ∗ =
vJ vJ = . vJ + vF vJ + vmax − p
(7.26)
For vmax = 1, vehicles accelerate immediately to vmax . In this case, one has vJ = q = 1 − p. For higher velocities, q = 1 − p is only an upper bound for vJ . Inserting vJ = 1 − p into (7.26), one therefore obtains an upper bound for the density ρ ∗ . Taking into account interactions between vehicles in the outflow region of the jam, one can derive an effective acceleration rate qeff , and thus the jam dissolution velocity vJ = qeff , as a function of p [434]. Computer simulations show a sharp increase of the lifetime near the density ρ ∗ . It becomes infinite, i.e., the jam does not dissolve within the measurement time, only at a higher density ρ1∗, which is considerably larger than the density ρ ∗ of maximum flow. At intermediate densities ρ ∗ < ρ < ρ1∗ , the jam does not dissolve during the first lap, but later due to fluctuations of the two ends of the jam. During this time, other jams have usually formed. All results found in [434] are consistent with the measurements of the lifetime distributions presented in the previous subsection.
7.4.6. Temporal Correlations and Relaxation Time In order to probe the spatio-temporal correlations in the fluctuations of the occupation of the cells, space-time correlation functions like ni (t)nj+r (t + τ ) have been studied. According to [198], three different regimes can be distinguished: (1) free flow, (2) jamming, where free flow and jams with finite lifetime coexist, and (3) superjamming, where the whole system is congested. Neubert et al. [1041] have introduced a special autocorrelation function of the density in order to study the velocity of jams. They have determined jam velocities for several variants of the NaSch model. A characteristic feature of a second-order phase transition is the divergence of the relaxation time at the transition point. For p = 0, this has been studied first by Nagel and Herrmann [1024]. They found a maximum τmax of the relaxation time of the velocity at the density ρ = 1/(vmax + 1). In the thermodynamic limit L → ∞, it diverges as τmax ∝ L. For finite p, the behavior of the relaxation time is more complicated. Csányi and Kertész [247] have studied the relaxation of the average velocity for general p by introducing a parameter that related to the corresponding relaxation time. It shows an interesting behavior [247, 341, 1226], which is difficult to interpret in terms
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of critical point phenomena. One finds a maximum of the relaxation parameter near, but below, the density of maximal flow. This maximum value increases with system size, but the width of the transition region does not seem to shrink [1226].
7.4.7. Structure Factor Structure factors are known to give valuable information about driven systems [1250, 1251, 1287]. For the NaSch model, the static structure factor
L
1
ikj
(7.27) nj e , S(k) = L
j=1
which is related to the Fourier transform of the density–density correlation function (7.20), has been investigated in [888]. For all densities, S(k) exhibits a maximum at k0 ≈ 0.72, which corresponds to the characteristic wavelength λ0 = 2π/k0 of the density fluctuations in the free-flow regime. For vmax > 1, one finds k0 (vmax + 1) = const. In [1197, 1198], these investigations have been extended to the dynamical structure factor in velocity-space,
2
1
i(kn−ωt) (7.28) v (t) e Sv (k, ω) =
, n
NT n,t
with k = 2π mk /N , ω = 2π mω /T , where N is the number of vehicles and mk and mω are integers. vn (t) is the velocity of the n-th vehicle at time t. Compared with the dynamical structure factor in real space, (7.28) has the advantage that the free-flow regime only contributes white noise, Sv (k, ω)|free-flow = const. Therefore, it is easier to study jamming properties. It is found in [1197] that Sv (k, ω) exhibits one ridge with negative slope, corresponding to backward-moving jams. The velocity of the jams is a function of the randomization parameter p only. It is independent of the density ρ and the maximal velocity vmax [1197]. This is consistent with results from a direct study of the autocorrelation function [1041]. Measurements of the jam dissolution speed in [434], however, show a decrease with increasing vmax and saturation for large vmax .
Above a transition density, an algebraic behavior Sv (k, ω) ω/k=v ∼ k−γ of the strucj ture factor is found. This has been interpreted as an indication of critical behavior in [1197, 1198]. However, due to the difficulties involved in the calculation of (7.28) only relatively short times T ≤ 2048 have been considered in [1197]. This is much smaller than the cutoff found in the lifetime of jams (see the discussion in Sec. 7.4.5) and lies well in the region where an algebraic decay is found [211]. Therefore, the results for
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the dynamical structure factor (7.28) and the lifetime measurements are consistent, but the algebraic decay is not to be interpreted as an evidence for the existence of a critical point in the NaSch model. In order to see the cutoff, times T > 104 would have to be considered [211].
7.4.8. Phase Transition For a proper description of a phase transition, one should introduce an appropriate order parameter, which can distinguish the two phases because of its different qualitative behavior within the two phases [445, 1308]. In Landau’s theory, the order parameter is related to the symmetry breaking at the phase transition and distinguishes the states in the ordered phase (e.g., a ferromagnet).7 So far, no order parameter has been found for the NaSch model with general vmax because it is not clear which symmetry is broken. For the aggressive-driver model, a variant of the NaSch model that will be discussed in Section 8.4.1, it has been argued that the random-braking parameter p can be considered as a symmetry-breaking field, which is conjugated to the order parameter m = vmax − v. A similar proposal has been made for the NaSch model with p = 0 in [276]. Here, the broken symmetry in the jammed phase is characterized by the order parameter m = 1−
J (ρ) , ρvmax
(7.29)
i.e., basically in terms of the fundamental diagram J (ρ). In [276] also the critical exponents for the phase transition at p = 0 have been determined. A natural candidate [1419] for an order parameter for the NaSch model with p > 0 would be the average fraction of vehicles at rest, i.e., with instantaneous speed v = 0. In the deterministic limit p = 0, this, indeed, serves the purpose of the order parameter ∗ from the free-flowing dynamical phase to the congested for the sharp transition at ρdet dynamical phase. But, in the general case of nonzero p, there is a nonvanishing probability that a vehicle comes to an instantaneous rest merely because of random braking even at extremely low density ρ. This probability is p for vmax = 1 and decreases with increasing ρ. Another candidate [341] is the density of nearest-neighbor pairs in the stationary state 1 1 nj nj+1 , m= T L t=1 j=1 T
L
(7.30)
7 Note that according to Landau’s theory, an order parameter is not just a quantity which is zero in the disordered and
nonzero in the ordered phase.
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0.9
vmax = 1 vmax = 2
0.5
m
m
0.7
0.3 0.1 −0.1
0
0.2
0.4
0.6 ρ (a)
0.8
1
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3
vmax = 2, p = 0.5
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ρ (b)
Figure 7.15 Density-dependence of the order parameter m in (a) the deterministic limit p = 0 (vmax = 1, 2) and (b) for vmax = 2, p = 0.5.
where, as defined earlier, nj = 0 for an empty cell and nj = 1 for a cell occupied by a vehicle (irrespective of its velocity). Because of the deceleration rule (NaSch2), m gives the space-time-averaged density of those vehicles with velocity 0, which had to brake ∗ if p = 0. due to the next vehicle ahead. Figure 7.15(a) shows that m vanishes at ρdet ∗ However, if p = 0, m does not vanish even for ρ < ρ although m becomes rather small at small densities (see Fig. 7.15(b)). We now present a heuristic argument to point out why any quantity related to the fraction of jammed vehicles is nonzero at any density ρ > 0 and, hence, inadequate to serve as an order parameter [434]. To slow down to vmax − 2, a vehicle must be hindered by one randomly braking vehicle in front. Similarly, to reach a speed vmax − 3, a vehicle must find two vehicles within the range of interaction, and so on. The probability for n vehicles to be found in the close vicinity of a given vehicle is proportional to ρ n . Therefore, the probability Pv (ρ) of finding a vehicle with speed v < vmax − 1 is proportional to ρ vmax−1−v and, hence, even for v = 0, Pv (ρ) is, in general, nonzero for all ρ → 0. The previous subsections have shown that there is strong evidence excluding a phase transition in the NaSch model with periodic boundary conditions for p > 0. Instead, the free-flow and jammed regime are related by crossover which, especially for small p, shows remnants of the phase transition that only occurs in the deterministic limit p = 0. These results are further supported by general considerations, e.g., based on the condensation criterion discussed in Section 3.5.1 (see, e.g., [862]) and analogies with the fluid–gas transition (Section 6.6.3). For a more general discussion of phase transitions in various traffic models, we refer to the work of Nagel, Wagner, and collaborators [694, 1025, 1028, 1431] which partly has been presented in Section 6.6.3.
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7.4.9. Boundary-Induced Phase Transitions Although the NaSch model does not exhibit a bulk phase transition, boundary-induced phase transitions occur. The scenario is very similar to that in the ASEP, which has been discussed in detail in Chapter 4. Figure 7.16 shows the phase diagram for an open system obtained from computer simulations [1121]. The input and removal rates at the boundaries induce effective boundary densities ρ + and ρ − . It has the same structure with a free-f low (low-density) phase, a congested (high-density) phase, and a maximal current phase. One important difference is missing symmetry of the phase diagram, which is a consequence of the breaking of the particle-hole symmetry in the NaSch model with vmax > 1. These results are also consistent with the prediction of the extremal principle discussed in Section 4.5.2. Because the NaSch model with vmax > 1 has not been solved for the case of periodic boundary conditions, it is not surprising that for the open case also no general exact solution is known, although in the case of deterministic bulk dynamics and stochastic boundary conditions, some analytical results can be obtained. For details, we refer to [597, 659, 660, 1045, 1046].
1 Theory Simulation
0.8
0.6
ρ+
CT
0.4 FF 0.2
MF 0
0.2
0.4
ρ−
0.6
0.8
1
Figure 7.16 Phase diagram of the NaSch model with open boundaries for p = 0.25 and vmax = 4 coupled to reservoirs, which induce the upstream-density ρ − and the downstream-density ρ + . The phases are free-flow (FF), congested traffic (CT), and maximal flow (MF) phase. The solid (dashed) curve is the theoretical prediction for the first (second) order transition lines obtained from the extremal principle.
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Contrary to above-mentioned findings, large differences between the phase diagrams of the NaSch model and the ASEP were reported in [199, 200, 596], e.g., the absence of the maximal current phase. These deviations are an artefact of the special boundary conditions considered, which do not allow to generate all bulk states [74]. Inserting vehicles with velocity vmax at a single boundary site leads to what is called injectionproduced slowdown in [659]. Therefore, it has been suggested by Barlovic et al. [74] to use a small system of vmax + 1 cells as left boundary which, together with an appropriate input strategy into this minisystem, allows to generate all possible bulk states. In [659, 660, 1045, 1046], the inflow resulting from the different realizations of the injection rule has been studied analytically for deterministic bulk dynamics ( p = 0). The standard injection rule uses a reservoir of just one site j = 0. If site j = 1 is empty, a particle with velocity vmax moves with probability α from the reservoir into the system according to the NaSch rules. For vmax = 2 and vmax = 5, this leads to the inflow [659] (2) Jin (α) =
α(1 + α) , 1 + α + α2
(5) Jin (α)
α 1 + α − α3 + α4 = . 1 + α − α3 + α4 + α5
(7.31)
One finds that this injection rule can lead to a nonmonotonic dependence of the inflow on the insertion probability α. Furthermore, the maximal inflow is smaller than the maximum of the fundamental diagram so that the maximal current phase can not be reached. These problems are avoided for the extended reservoir with vmax + 1 cells. Here the update rules are as follows: First, the reservoir is emptied. Second, a particle with velocity vmax is created with probability α. Its position has to satisfy the following conditions: (i) the resulting headway d must be at least vmax cells, and (ii) the distance to the main system has to be as small as possible. The corresponding inflow has been determined in [1045, 1046]: Jin (α) =
α(1 − α vmax ) . 1 − α vmax + α vmax+1 − α vmax+2
(7.32)
For large vmax , this is well approximated by Jin (α) ≈ α − α vmax+1 / 1 − α vmax+1 [74, 1046]. Besides reservoirs, other types of open boundaries have been considered, e.g., traffic light boundary conditions where the boundaries close and open periodically [1044, 1045, 1120]. These are not only relevant for certain applications, but also give rise to a variety of new effects.
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7.5. APPENDICES 7.5.1. Details of SOMF for NaSch We denote the probability to find a vehicle with speed v (v = 0, 1, 2, . . ., vmax ) at time t at site i by cv (i, t). Then c(i, t) =
vmax
cj (i, t)
(7.33)
j=0
gives the probability that site i is occupied by at time t and ¯c (i, t) = 1 − c(i, t) the probability that it is empty. Expressing the flux J (ρ, p) =
vmax
vcv
(7.34)
v=1
in terms of the cv then gives the fundamental diagram in mean-field approximation. By applying the update rules of the NaSch model with arbitrary vmax step-by-step, we can determine the probabilities cv explicitly [1258]: Step 1. Acceleration stage (t → t1 ) c0 (i, t1 ) = 0,
(7.35)
cv (i, t1 ) = cv−1 (i, t),
(0 < v < vmax )
cvmax (i, t1 ) = cvmax (i, t) + cvmax−1 (i, t)
(7.36) (7.37)
Step 2. Deceleration (t1 → t2 ) c0 (i, t2) = c0 (i, t1) + c(i + 1, t1)
vmax
cv (i, t1 )
(7.38)
v=1
cv (i, t2) = c(i + v + 1, t1 )
v
¯c (i + j, t1 )
j=1
+ cv (i, t1)
v
c¯(i + j, t1 ),
vmax
cv (i, t1 )
v =v+1
(0 < v < vmax )
(7.39)
j=1
cvmax (i, t2) =
v max j=1
¯c (i + j, t1)cvmax (i, t1)
(7.40)
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Step 3. Randomization (t2 → t3 ) c0 (i, t3) = c0 (i, t2) + pc1 (i, t2) cv (i, t3) = qcv (i, t2 ) + pcv+1 (i, t2),
(7.41) (0 < v < vmax )
cvmax (i, t3) = qcvmax (i, t2 )
(7.42) (7.43)
Step 4. Movement (t3 → t + 1) cv (i, t + 1) = cv (i − v, t3),
(0 ≤ v ≤ vmax )
(7.44)
These equations are nonlinear and an exact analytical solution has not been obtained up to now. However, considering the steady state leads to considerable simplifications. Then the cv (i, t) are independent of t and, due to the translational invariance in the case of periodic boundary conditions, also independent of i, i.e., cv (i, t) = cv . In the following, we will present results for the stationary state in a periodic system. Details of the calculations can be found in [1258]. 7.5.1.1. The Case vmax = 1 In the case vmax = 1, the equations (7.35–7.44) reduce to c0 (i, t + 1) = c(i, t)c(i + 1, t) + pc(i, t)¯c(i + 1, t),
(7.45)
c1 (i, t + 1) = qc(i − 1, t)¯c (i, t),
(7.46)
which due to c(i) = ρ and ¯c (i) = 1 − ρ = ρ¯ simplifies in the stationary limit to c0 = ρ + pρ ρ, ¯
(7.47)
c1 = qρ ρ. ¯
(7.48)
Using the flux J = c1 , this yields the mean-field fundamental diagram JMF (ρ) = (1 − p)ρ(1 − ρ),
(7.49)
which had already been derived for the ASEP in Section 4.1.3. 7.5.1.2. The Case vmax = 2 In the case of vmax = 2, the stationary mean-field equations are c0 = [ρ + pρ]c ¯ 0 + [1 + pρ]ρ[c ¯ 1 + c2 ],
(7.50)
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c1 = qρc ¯ 0 + ρ[qρ ¯ + pρ][c ¯ 1 + c2 ],
(7.51)
c2 = qρ¯ [c1 + c2 ].
(7.52)
2
These are solved by [1258] c0 = ρ 2
1 + pρ¯ , 1 − pρ¯
c1 = qρ ρ¯
1 − qρ¯ 2 , q + pρ
c2 =
q2 ρ ρ¯ 3 . q + pρ
(7.53)
Here again ρ¯ = 1 − ρ. The mean-field flux is thus given by J (ρ) = c1 + 2c2 = qρ ρ¯
1 + qρρ¯ 2 . q + pρ
(7.54)
7.5.1.3. General vmax For general vmax , the stationary mean-field equations can be rewritten in the form ¯ 0 + (1 + pρ)ρ ¯ c0 = (ρ + pρ)c
vmax
cβ ,
(7.55)
β=1
⎡
cα = ρ¯ α ⎣qcα−1 + (qρ + pρ)c ¯ α + (q + pρ)ρ ¯
vmax
⎤ cβ ⎦ (0 < α < vmax − 1), (7.56)
β=α+1
cvmax −1 = ρ¯
vmax −1
[qcvmax −2 + (qρ + pρ)(c ¯ vmax −1 + cvmax )],
cvmax = qρ¯ vmax [cvmax −1 + cvmax ].
(7.57) (7.58)
For fixed density of ρ of cars, the mean-field equations are linear in the probabilities cv and can be recast in matrix form Ac = c . The matrix A can be read off from the meanfield equations, and c is the vector with elements cα (α = 0, . . ., vmax ). For small vmax , it is possible to invert A to find the densities cα explicitly. For large values of vmax , it is more convenient to write down a recursion relation in order to obtain the steady-state solution. From (7.55) one can determine c0 directly without the knowledge of the other cα : c0 = ρ 2
1 + pρ¯ . 1 − pρ¯
(7.59)
Using this result and (7.56) for α = 1, one also obtains an explicit expression for c1 c1 = qρ 2 d
2 − ρ + pρ¯ 2 . (1 − pρ¯ 3)(1 − pρ¯ 2)
(7.60)
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For α > 1, a recursion relation can be derived calculating cα − ρc ¯ α−1 using again (7.56) cα =
1 + (q − p)ρ¯ α qρ¯ α ρc ¯ − cα−2 α−1 1 − pρ¯ α+2 1 − pρ¯ α+2
(7.61)
for α = 2, 3, . . ., vmax − 2. Therefore, starting with the explicit expressions for c0 and c1 , one can determine c2 , c3 , . . ., cvmax −2 recursively. These results do not depend on the actual value of vmax and thus are valid generally (provided vmax > 2). Finally, from (7.57) and (7.58) one gets 1 − qρ¯ vmax qρ¯ vmax−1 cvmax −2 1 − d vmax −1 (q + pρ) ¯ qρ¯ vmax cv −1 . cvmax = 1 − qρ¯ vmax max
cvmax −1 =
(7.62) (7.63)
The vmax -dependence only occurs in these two quantities. The densities of the fast cars go to zero rapidly because one expects an exponentially fast decay from the recursion relations. 7.5.1.4. The Limit vmax → ∞ For vmax → ∞, one can analyze the iteration by introducing the generating function g(z) =
∞
cn zn .
(7.64)
n=1
The iteration is then equivalent to the functional equation [1258] ¯ − z). g(z)[1 − ρz] ¯ − g(ρz) ¯ ρ¯ 2 (1 − z)( p + qz) = ρ 2 + pρ 2ρ(1
(7.65)
Because vmax is infinite, one can neglect the contribution of the upper boundary equations for cvmax −1 and cvmax . However, since the generating function g occurs with two different arguments z and ρz, ¯ an explicit solution of this (linear) equation for g(z) is difficult. The fundamental diagram can be obtained directly from the generating function because J (ρ) = g (z = 1). One has ¯ − pρ) − g(ρ) ¯ g (1) = ρ(1
ρ¯ 2 ρ
(7.66)
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A. Schadschneider, D. Chowdhury, and K. Nishinari
An expression for g(ρ) ¯ can be found by successive application of equation (7.65) with z = ρ¯ n . The final result is an asymptotic expression for g (1) = J (ρ): J (ρ) = qρ ρ¯ 1 +
∞
ρ¯ 2n
n=1
n−1
( p + qρ¯ l ) .
(7.67)
l=0
7.5.2. Details of PMF for NaSch In the case vmax = 1, the MFT equations (7.45) and (7.46) have to be modified to take into account the existence of GoE states. In general, one has to follow the procedure outlined in Appendix 7.5.1. A quicker way to derive the pMF equations is to analyze the MF equations (7.45) and (7.46). In (7.45), the contribution c(i; t)c(i + 1; t) appears. Because we know that site i + 1 can never be occupied by a car with velocity 1 if site i is not empty, this contribution has to be modified to c(i; t)c0 (i + 1; t) in pMFT. All other contributions are left unchanged compared with MFT. Due to this modification and the corresponding reduction of the configuration space, the normalization c0 + c1 = ρ is no longer satisfied automatically. Therefore, a normalization constant N has to be introduced. The final equations for a homogeneous stationary state are then given by ¯ c0 = N (c0 + pρ)ρ,
(7.68)
c1 = N qρ ρ, ¯
(7.69)
with ρ¯ = 1 − ρ and the normalization N=
1 . c0 + ρ¯
(7.70)
Because c0 + c1 = ρ, we have only one independent variable for fixed density ρ, e.g., c1 . Solving (7.68), (7.69) for c1 , we obtain c1 =
1 1 − 1 − 4qρρ ¯ . 2
(7.71)
The flow is given by J (ρ) = c1 , and we recover the exact solution for the case vmax = 1. The case vmax = 2 can be treated in a similar way by modifying the method for the derivation of the MFT. Taking into account only the first-order GoE states (7.17) and (7.18), one obtains the following pMF equations: ¯ 0 + c1 ρ) , c0 = N c0 c + pρ(c
(7.72)
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Vehicular Traffic II: The Nagel–Schreckenberg Model
c1 = N pρ¯ 2(c1 + c2 ) + q¯c (c0 + c1 ρ) ,
(7.73)
c2 = N qρ¯ (c1 + c2 ).
(7.74)
2
The normalization N ensures c0 + c1 + c2 = ρ and is given explicitly by N=
1 1 = . c0 + ρc ¯ 1 + ρ¯ 2c2 c0 + ρ(1 ¯ − c2 )
(7.75)
c2 can be expressed through c0 and ρ by c2 =
1 c0 + ρ¯ − (c0 + ρ) ¯ 2 − 4qρ¯ 3(ρ − c0 ) . 2ρ¯
(7.76)
After inserting into (7.72), we obtain a cubic equation that determines c0 in terms of the parameters ρ and p [1247]. Results for different values of p are shown in Fig. 7.10. They are only slightly modified when also the second-order GoE state is taken into account [1247].
7.5.3. Details of COMF for NaSch The COMF treatment of the case vmax = 1 has been discussed in Section 7.3.4. We give here the COMF equations for the case vmax = 2 [1245], which can be treated in a similar way as vmax = 1. However, it is now necessary to introduce two different functions Pn (t) and Bn (t) describing the probabilities to find exactly n empty sites in front of a car with velocity 1 and 2, respectively. For the stationary state, the probabilities obey the equations P0 = g0 [P0 + B0] ,
(7.77)
P1 = g1 [P0 + B0] + pg0 [P1 + B1 ] ,
(7.78)
P2 = g2 [P0 + B0] + pg1 [P1 + B1 ] + pg0 P2 ,
(7.79)
P3 = pg2 [P1 + B1] + pg1P2 + pg0P3 ,
(7.80)
Pn = pg2 Pn−2 + pg1Pn−1 + pg0Pn ,
(n ≥ 4)
(7.81)
and B0 = qg0 [P1 + B1 + B2] ,
(7.82)
B1 = qg1 [P1 + B1 + B2] + g0 qP2 + pB2 + qB3 , (7.83) B2 = qg2 [P1 + B1 + B2] + g1 qP2 + pB2 + qB3 + g0 qP3 + pB3 + qB4 , (7.84) Bn = g2 qPn−1 + pBn−1 + qBn + g1 qPn + pBn + qBn+1 (7.85) + g0 qPn+1 + pBn+1 + qBn+2 . (n ≥ 3).
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The probabilities gα that a car moves α sites (α = 0, 1, 2) in the next timestep are g0 = P0 + B0 + p g1 = q
Pn + pB1,
n≥1
Pn + qB1 + p
n≥1
g2 = q
Bn ,
(7.86)
n≥2
Bn .
n≥2
From (7.77–7.86), one obtains g0 = lization
n≥0 Pn
and g1 + g2 =
n≥0 Bn . Using the norma-
[Pn + Bn ] = 1,
(7.87)
n≥0
we have g0 + g1 + g2 = 1. The conservation of density leads to the constraint 1 (n + 1) [Pn + Bn ] = . ρ n≥0
(7.88)
We introduce the generating functions, P(z) =
∞ n=0
Pn zn+1
and B(z) =
∞
Bn zn+1 ,
(7.89)
n=0
which satisfy P(1) + B(1) = 1 and P (1) + B (1) = ρ1 due to (7.87) and (7.88), respectively. After multiplication with zn+1 and summation over all equations (7.77–7.85) one finds g(z) (7.90) (qP0 + B0 )z + pB1z2 , 1 − pg(z) zg(z) qP(z) − qB0 − (qP0 + pB0 + qB1 )z + (q − p)B1z2 , (7.91) B(z) = 2 z − (q + pz)g(z) P(z) =
where we have introduced the function g(z) = g0 + g1z + g2z2 . Note that this function satisfies g(1) = 1 and g (1) = ρ1 J (ρ, p) is just the average velocity of the vehicles.
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Vehicular Traffic II: The Nagel–Schreckenberg Model
If one expresses the sums appearing in (7.86) by P(z = 1) and B(z = 1), one obtains the following relations q 1 B1 = g0 − (B0 + qP0), p p p p g1 = p(1 − P0) + q − B0 + 1 − B1 , q q g2 = q(1 − P0) − (1 + q)B0 − B1.
(7.92) (7.93) (7.94)
With these relations also the normalization condition P(1) + B(1) = 1 is satisfied. Now we can express the generating function completely in terms of the two probabilities P0 0 and B0 only, since g0 = P0P+B from (7.77). 0 At this point, it is surprising that we are still left with two unknowns P0 and B0 because we only have one free parameter, the density ρ. The missing relation between P0 and B0 can be obtained from analytic properties of B(z). The denominator of B(z) has three zeroes, z = 1, and two nontrivial ones. Two of these are located in the unit circle. These zeroes have to be cancelled by corresponding zeroes of the numerator because B(z) has to be analytic in the unit circle (otherwise, one would not have limn→∞ Bn = 0). It is easy to see that the numerator indeed has a zero at z = 1. Demanding that also the second zero in the unit circle is cancelled by a corresponding zero of the numerator gives the missing relation between P0 and B0 , which can be found in [1245].
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CHAPTER EIGHT
Vehicular Traffic III: Other CA Models Contents 8.1. Slow-to-Start Rules, Metastability, and Hysteresis 8.1.1. General Remarks 8.1.2. The Velocity-Dependent-Randomization Model 8.1.3. Takayasu–Takayasu Slow-to-Start Rule 8.1.4. The BJH Model of Slow-to-Start Rule 8.1.5. Other Slow-to-Start Rules 8.1.6. Flow Optimization and Metastable States
282 282 283 288 289 289 290
8.2. Cruise-Control Limit
291
8.3. CA Models of Synchronized Traffic 8.3.1. Brake-Light or Comfortable Driving Model 8.3.2. Kerner–Klenov–Wolf Model 8.3.3. Mechanical Restrictions Model of Lee et al.
294 295 299 301
8.4. Other CA Models 8.4.1. Fukui–Ishibashi Model 8.4.2. Velocity-Dependent Braking Model 8.4.3. Time-Oriented CA Model 8.4.4. Models with Anticipation 8.4.5. Galilei-Invariant Model 8.4.6. Car-Following CA
304 304 305 306 307 309 311
8.5. CA from Ultradiscrete Method 8.5.1. Generalizations of BCA 8.5.2. Euler–Lagrange Transformation 8.5.3. Traffic Models in Lagrange Form
313 314 315 316
8.6. CA Models of Multilane Traffic 8.6.1. Classification of Lane Changing Rules 8.6.2. CA Models of Bidirectional Traffic
319 319 322
8.7. Effects of Quenched Disorder 8.7.1. Randomness in the Braking Probability 8.7.2. Random vmax 8.7.3. Randomly Placed Bottlenecks 8.7.4. Ramps
324 324 326 326 328
8.8. Bus-Route Model
329
8.9. Accidents
332
Stochastic Transport in Complex Systems DOI: 10.1016/B978-0-444-52853-7.00008-7
Copyright © 2011, Elsevier BV. All rights reserved.
281
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A. Schadschneider, D. Chowdhury, and K. Nishinari
As we have seen in the previous chapter, the Nagel–Schreckenberg (NaSch) model is able to reproduce the basic properties of highway traffic in many respects. However, when compared with the empirical results presented in Chapter 6, it is not able to capture many aspects of the “fine-structure” of traffic dynamics, especially the existence of metastable states or synchronized traffic. Therefore, over the recent years, a plethora of variants of the NaSch model has been proposed. It is impossible to discussed them all here; therefore, we have limited ourselves to the most popular and successful ones, but also included some models, which have introduced other interesting aspects into the dynamics.
8.1. SLOW-TO-START RULES, METASTABILITY, AND HYSTERESIS In the next few subsections, we shall demonstrate the rich variety of traffic phenomena that can be observed by appropriate modifications of the random braking and how they lead to empirically observed metastability and the related hysteresis effects (see Section 6.5). A common feature of these models is a so-called slow-to-start rule. This refers to variants of the acceleration step, where the acceleration of standing or slow vehicles is in some sense delayed compared with that of moving cars. This can be implemented in various ways, but it will turn out that the observed behavior is rather universal. The slow-to-start rules not only lead to metastability and, consequently, hysteresis, but also to phase-separated states at high densities. They are implemented to capture, e.g., the loss of attention of drivers in a jam, the slow pick-up of the motor of standing cars, and so on. Empirical evidence for the slow-to-start effect has been provided, e.g., in [1428].
8.1.1. General Remarks Before we begin our discussions on specific generalized versions of the NaSch model, which exhibit metastability, we make some general remarks. In the schematic stationary fundamental diagram of Fig. 6.9, which is of inverse λ-form, the low-density branch corresponds to homogeneous free-flow states, while the high-density branch corresponds to configurations, where jammed states are present. Obviously, at densities ρ1 < ρ < ρ2 , the flow depends nonuniquely on the global density. In order to establish the existence of metastable states one can follow two basic strategies. In the first method, the density of vehicles is changed adiabatically by adding or removing vehicles from the stationary state at a certain density. Starting in the jamming phase (large densities) and removing vehicles, one obtains the lower branch of the hysteresis curve. However, by adding vehicles to a free-flowing state (low densities), one
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Vehicular Traffic III: Other CA Models
obtains the upper branch. This method is closely related to the experimental situation, where the occupancy of the road varies continuously. The second method does not require changing the density. Instead one starts from two different initial conditions: the mega-jam and the homogeneous state. The mega-jam consists of one large, compact cluster of standing vehicles. In the homogeneous state, vehicles are distributed periodically with same constant gap between successive vehicles (with one larger gap for incommensurate densities). Then, for ρ > ρ1 , the homogeneous initialization leads to a free-flow state, whereas the mega-jam initialization leads to the jammed high-density states.
8.1.2. The Velocity-Dependent-Randomization Model Although the NaSch model does not exhibit metastable states and hysteresis, a simple generalization exists, which is able to reproduce these effects. It is the so-called velocitydependent-randomization (VDR) model [76]. Here, in contrast to the original NaSch model, the randomization parameter depends on the velocity of the vehicle, p = p(v). The rules (see Section 7.1.1) are supplemented by a new rule, VDR0: Determination of the randomization parameter. The randomization parameter for the n-th vehicle is given by p = p(vn (t)). This new step has to be carried out before the acceleration step (NaSch1) of the NaSch model. The rest of the NaSch rules remains unchanged. The randomization parameter used in step (NaSch3) now depends on the velocity vn (t) after the previous timestep. In order to implement a simple slow-to-start rule, one chooses [76] p0 p(v) = p
for v = 0, for v > 0,
(8.1)
with p0 > p. This means that vehicles, which have been standing in the previous timestep have a higher probability p0 of braking in the randomization step than the moving vehicles. Typical fundamental diagrams look like the one shown in Fig. 8.1 where, over a certain interval of ρ, J (ρ) can take one of the two values depending on the initial state and, therefore, exhibit metastability. Moreover, typical space-time diagrams of the VDR model (see Fig. 8.2) clearly demonstrate that metastable homogeneous states have a lifetime after which their decay leads to a phase-separated steady state. The microscopic structure of these phase-separated high-density states is typical for models with slow-to-start rules and differs drastically from that found in the NaSch model. It is instructive to compare the fundamental diagram of the VDR model with those of the corresponding NaSch models. It can be derived on the basis of heuristic arguments utilizing the observed structures of the steady states. For small densities ρ 1,
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0.8 hom jam NaSch ( p = 1/64) NaSch ( p = 0.75)
0.7 0.6
J
0.5 0.4 0.3 0.2 0.1 0
0.2
0.4
0.6
0.8
1
ρ
Figure 8.1 Fundamental diagram of the VDR model for vmax = 5, p0 = 0.75, p = 1/64 obtained using two different initial conditions, namely, a completely jammed state (jam) and a homogeneous state (hom). For comparison, the fundamental diagrams for the NaSch models with p = 1/64 and p = p0 = 0.75 are also given.
92920.0
92970.0
Time
93020.0
93070.0
93120.0
93170.0
93220.0 0.0
100.0
200.0 Space
300.0
400.0
Figure 8.2 Typical space-time diagram of the VDR model with vmax = 5 and ρ = 0.20, p = 0.01 and p0 = 0.75. One can see the spontaneous formation of a jam through breakdown of the metastable states of high flow.
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Vehicular Traffic III: Other CA Models
there are no slow vehicles in the VDR model because interactions between vehicles are extremely rare. In this regime, every vehicle can move with the free-flow velocity vf = (1 − p)vmax + p(vmax − 1) = vmax − p and, therefore, the flux is given by Jhom (ρ) = ρ(vmax − p),
(8.2)
which is identical to the NaSch model with randomization p. However, for densities close to ρ = 1, the vehicles are likely to have velocities v = 0 or v = 1 only. Therefore, the random braking is dominated by p0 , rather than p, while the flow is determined by the movement of the holes. Hence, for large densities (1 − ρ 1), the flow is given by J (ρ) ≈ (1 − p0 )(1 − ρ), which corresponds to a NaSch model with randomization p0 . This expression for the flux in the high-density regime can also be derived as follows. In the phase-separated state, the vehicles are expected to move with the velocity vf = vmax − p in the free-flow region. Neglecting interactions between vehicles (which is justified because of the corresponding low density), the average distance of two consecutive vehicles in the free-flow region is given by x = 1/ρf = Tw vf + 1, where the average waiting time Tw of the first vehicle at the head of the megajam is given by Tw = 1/(1 − p0 ). In other words, the density ρf in the free-flow regime is determined by the average waiting time Tw and vf . Now suppose that NJ and NF are the number of vehicles in the megajam and free-flowing regions, respectively. Using the normalization L = NJ + NF x, we find that for the density ρ = (NF + NJ )/L, the flux Jsep (ρ) is given by Jsep (ρ) = (vmax − p)NF /L and, hence, Jsep (ρ) = (1 − p0)(1 − ρ).
(8.3)
Obviously, ρf is precisely the lower branching density ρ1 because for densities below ρf , the jam-length is zero. It should be noted that the heuristic arguments presented above remain valid for p0 p and vmax > 1. The condition p 1 guarantees that the jams are compact in that limit. In the case vmax = 1, vehicles can stop spontaneously, even in the free-flow regime and these vehicles might initiate a jam. This is the basic reason why hysteresis is usually not observed for vmax = 1. Phase separation can be directly identified using the results of the jam-gap distribution (see Section 7.4.4), which shows a peak at macroscopic distances. The size of the freeflow regime is found to be proportional to the system size [217]. The heuristic arguments and results of computer simulations presented above are valid for periodic systems of finite length. Now, it is self-evident to ask what kind of stationary states are realised in the thermodynamic limit L → ∞. The simulation results indicate that ρ = ρ1 − ρ2 decreases with larger system sizes and is expected to vanish for L → ∞; i.e., the jammed branch is stable in that limit. This is readily understood if one analyzes the typical configurations, which lead to an emerging jam or, vice versa,
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A. Schadschneider, D. Chowdhury, and K. Nishinari
the mechanism of the dissolution of a jam. Jams emerge if overreactions of drivers lead to a chain reaction. This is possible in dense regions of the free-flow state, where the gap between the vehicles is not larger than vmax . Obviously, the probability to find large platoons of vehicles driving with small spatial headways increases with the system size (for fixed density). Additionally, the jammed states are phase separated; i.e., the size of the jam is of the order of the system size. During a simulation run, the size of the jam fluctuates due to the stochastic movement and acceleration of the vehicles. Jams can dissolve if the amplitude of these fluctuations is of the order of the length of the jam, which is impossible in the thermodynamic limit. Therefore, the nonunique behavior of the fundamental diagram is only observable if finite system sizes are considered or if the vehicles move deterministically in the freeflow regime. Nevertheless, the results discussed earlier are highly relevant for practical purposes because the hysteresis effects have been observed at realistic system sizes (e.g., L = 10000 corresponds to a highway of length 75 km). The characteristic properties of the wide jams have been investigated using various approaches. Using higher order density correlation functions, the velocity of wide jams can be obtained [1041]. A random-walk description [77] allows to determine some characteristics analytically. The resolving probability of a jam of initial size n0 is given by α ∗ α ∗ (1 − β ∗) n0 −1 , n0 = ∗ ∗ β β (1 − α ∗)
(8.4)
where α ∗ = 1 − p0 is the outflow from the jam and β ∗ is the inflow into the jam which depends on the boundary conditions and the free-flow randomization p. Besides, the resolving probability also the average lifetime Tn0 of a jam of initial length n0 can be determined based on random walk arguments [77]. Jams can not only occur due to velocity fluctuations in the metastable branch, but can also be induced by perturbations. In the VDR model, a perturbation induces jam formation only with some probability, which depends on the density and the perturbation strength. In contrast in hydrodynamic models, jams are formed deterministically if the perturbation exceeds a critical size, which depends on the density. The properties of the VDR model with open boundary conditions differ from those of the NaSch model [74]. The phase diagram as a function of the insertion and removal probabilities qin and 1 − qout has the same basic structure, but the jam phase is split into two phases with different microscopic structure of the states (Fig. 8.3). Both phases have a striped structure, i.e., compact jam clusters alternating with free flow regions, but show different structures near the input boundary. These phases have also been observed in a simpler model, which corresponds to the case vmax = 1 [36]. The maximum current phase is replaced by a new phase with a nonstationary oscillating density pattern and very high flows in finite systems. In this jam outflow ( JO) phase,
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Vehicular Traffic III: Other CA Models
1 (b) Jam I
q*out
qout
(c) Jam II
(a) Free Flow
(OJ + SJ) (d) JO
0
1
q*in qin
Figure 8.3 Phase diagram of the VDR model with p > 0. qin is the insertion probability and qout the probability that the output reservoir is occupied. In contrast to the NaSch model, the VDR model shows two different jam phases (b) and (c). The JO phase (d) is characterized by very wide continuously growing jams. In this phase, very-high flows can be observed in finite systems. For relatively small qin, the microscopic pattern is dominated by one large jam marked by “OJ” (one jam), whereas for larger qin, in addition spontaneous jams “SJ” occur at erratic positions most likely near the left boundary. In the deterministic limit p = 0, these spontaneous jams are not observed.
the system is dominated by one large jam in the thermodynamic limit. For increasing inflows, this microscopic structure transforms into the striped pattern of the jam phase. A surprising feature is a nonmonotonic dependence of the current on the input probability qin . For large inflows, the current is reduced due to overfeeding and is then controlled by the outflow from large jams occurring mostly at the left boundary, but also spontaneously at erratic positions in the system [74]. These observed behavior indicates potential applications in traffic flow optimization (Section 8.1.6). One surprising result in this context is that a restriction of the inflow can lead to an improvement of the total flow through a bottleneck [74]. Localized defects lead to competition of two different mechanisms for phase separation. The dynamics of the VDR model leads to a phase separation at high densities into a large moving jam and a free-flow region. In contrast, a localized defect triggers the formation of a high-density region pinned at the defect. The defects are implemented by locally increasing the deceleration probability to max{p(v), pd }, where pd is the strength of the defect. Three different system states (phases) can be observed as the defect noise pd is varied [1127]. Small defect noises pd reduce the lifetimes of the metastable states. In the phase-separated state of the system, vehicles can pass nearly undisturbed through the defect. Because there is almost no difference to the jammed state of the VDR model without defect, this phase is denoted as VDR phase. In contrast, for a large pd , a pinned high-density region is formed at the defect limiting the overall
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system flow. In this high-density small compact jams occur that are separated by small free-flow regions. This phase is called stop-and-go phase as the jam pattern shows strong similarities to stop-and-go traffic. An important point is that stop-and-go traffic cannot be found in the VDR model without a lattice defect. Furthermore, for an intermediate defect noise pd , a crossover behavior is observed. A wide jam moves backwards through the system with additional small jams of limited lifetime being formed at the defect. Maerivoet and De Moor [894] have studied the behavior of the VDR model in the limit p0 = 0, i.e., fast-to-start rules. They observed the jam velocity becomes positive at approximately p = 0.5 so that for large values of p, jams propagate forward. The fundamental diagrams show an unusual shape with two maxima and a local minimum so that four different regimes can be distinguished characterized as (i) free flow, (ii) dilutely congested, (iii) densely advancing, and (iv) heavily congested traffic. The two-lane variant of the VDR model also exhibits metastable states. However, these no longer exist in the presence of aggressive drivers who do not look back before changing lanes [951].
8.1.3. Takayasu–Takayasu Slow-to-Start Rule Takayasu and Takayasu (TT) [1347] were the first to suggest a cellular automata (CA) model with a slow-to-start rule. Here, we discuss the generalization suggested in [1246]. According to this generalized model, a standing vehicle (vn = 0) with exactly one empty cell in front (dn = 1) accelerates with probability qt = 1 − pt , while all other vehicles accelerate deterministically. The other steps of the update rule (NaSch2)-(NaSch4) of the NaSch model are kept unchanged. TT1: Acceleration. (a) If vn = 0 and dn = 1, then vn = 1 with probability 1 − pt ; (b) If vn = 0 and dn > 1, then vn = 1; (c) If vn > 0, then vn → min(vn + 1, vmax). It is instructive to consider first the deterministic limits of the TT model [415, 1347]. The TT model reduces to the NaSch model in the limit pt = 0. What happens in the other deterministic limit, namely, pt = 1? In this case, a stopped vehicle can move only if there are at least two empty cells in front [1347]. Obviously, completely blocked states exist for densities ρ ≥ 0.5, when the number of empty cells in front of each vehicle is smaller than two. In the region 0.5 ≤ ρ 0.66, the number of blocked configurations is very small compared with the total number of configurations and states with a finite flow exist. Precisely at ρ = 0.5, there are only two blocked states and the time to reach these states diverges exponentially with the system size. The fundamental diagram for the TT model with vmax = 1 has been derived analytically by (approximate) two-cluster calculations [1246]. It should be noted that the particle-hole symmetry is broken in the TT model even in the limit vmax = 1.
Vehicular Traffic III: Other CA Models
Correlations functions and the jam velocity were studied in [1041]. Fundamental diagrams for vmax > 1 have been obtained so far only numerically by computer simulations and look very similar to that of the VDR model, including metastability and hysteresis. Note, however, that the mechanism for metastability in the case pt = 1 is different from that for the metastability observed for 0 < pt < 1.
8.1.4. The BJH Model of Slow-to-Start Rule Benjamin, Johnson, and Hui (BJH) [102] introduced a different slow-to-start rule: Vehicles that had to brake due to the next vehicle ahead will move on the next opportunity only with probability 1 − ps . The steps of the update rules can be stated as follows: BJH1: Acceleration. vn → min(vn + 1, vmax ). BJH2: Slow-to-start rule: If f = 1, then vn → 0 with probability ps . BJH3: Blockage (due to other vehicles). vn → min(vn , dn − 1) and, then, f = 1 if vn = 0, else f = 0. BJH4: Randomization. vn → max(vn − 1, 0) with probability p. BJH5: Vehicle movement. xn → xn + vn . Here f is a flag (label) distinguishing vehicles which have to obey the slow-to-start rule (f = 1) from those which do not have to (f = 0). Obviously, for ps = 0, the above rules reduce to those of the NaSch model. The slow-to-start rule of the TT model is a “spatial” rule. In contrast, the BJH slow-to-start rule requires “memory”; i.e., it is a “temporal” rule, depending on the number of trials and not on the free space available in front of the vehicle. The fundamental diagram of the BJH model shows the existence of metastable states and hysteresis effects [217]. But, in the special case of vmax = 1, for which approximate analytical calculations can be carried out [1246], no metastable states exist. Since for all vmax > 1 in the BJH model, just as in the VDR and TT models, the outflow from a jam is smaller than the maximal flow, the phase-separated steady states at high global densities consist of a macroscopic jam and a macroscopic free-flow regime both of which coexist simultaneously [217]. Moussa [948] has found that for vmax > 1 above a threshold probability p0 ≈ 0.08, no hysteresis occurs. However, the macroscopic jam is not compact. The typical size of the macroscopic free-flow regime can be estimated by measuring the distribution of the gaps between the successive jams [217]. A peak occurs in this distribution for headways of the order of the system size. The position of the peak indicates the typical size of the macroscopic free-flow regime.
8.1.5. Other Slow-to-Start Rules Similar effects as in the VDR model can be observed in headway-distance effect models, where the randomization depends on the density or headway [593, 948, 1522]. Other
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forms of state-dependent randomization have also been suggested, e.g., a velocitydifference dependent randomization [594]. A systematic study of various types of state-dependent randomization has been performed in [454]. Huang [602] has studied a variant of the NaSch model where the slow-to-start rule replaces the randomization step (NaSch3). In this new step, the velocity of a car, which had velocity 0 in the previous timestep and has now velocity vn = 1 after steps (NaSch1) and (NaSch2) will reduce its velocity to vn = 0 again with some probability p0 . The fundamental diagrams derived from systems with open boundaries show a very unusual behavior. A stochastic version of a model proposed by Nishinari, Fukui, and Schadschneider [1059] was investigated in [1206]. This model is a combination of the Burgers CA and the NaSch model. Besides a slow-to-start rule, it also includes anticipation. A generalization of the asymmetric simple exclusion process (ASEP) that contains a slow-to-start rule has been proposed by Gray and Griffeath [456]. Cars can move forward by from site n to the empty site n + 1 with some probability, depending on the occupation of the sites n − 1 and n + 2. A model that can be studied by queueing theory has been introduced in [172]. The trajectories of the vehicles are related to a M /M /1 queue by identifying space in the traffic model with time in the queuing model. This model allows to derive several analytical results.
8.1.6. Flow Optimization and Metastable States Hysteresis effects and metastable states are not only of theoretical interest, but also have interesting applications. From the previous discussion of the slow-to-start models, it is evident that one can optimize the maximum flow if the homogeneous state is stabilized by controlling the density so that it never exceeds ρ2 . This strategy was followed in minimizing frequent jams in the Lincoln- and the Holland-Tunnels in New York [458] by installing traffic lights at the entrance of the tunnels. Before, jams used to form spontaneously within the tunnel because (1) the vehicle density used to be sufficiently high and (2) the drivers used to drive more carefully inside the tunnel thereby giving rise to stronger fluctuations that caused the jams. But, the traffic lights installed at the entrance of the tunnels are controlled in such a way that the density does not exceed ρ2 . Consequently, no jams are formed spontaneously by the decay of any metastable high-density state. One can mimic this situation using CA models with slow-to-start rules [74, 76, 216]. The tunnel is considered as part of the road, where the braking probabilities pt , p0t are higher compared with the remaining part of the lattice. Therefore, if one allows for an uncontrolled inflow of the vehicles, jams typically appear inside the tunnel and the system capacity is governed by p0t . The situation differs drastically if traffic lights are implemented [76, 216]. A considerable increase of the maximum capacity can be achieved for an
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optimal combination of the red-/green-signal periods, which is of the same order as for the realistic examples [458]. A slightly different explanation based on the observed strong finite-size effects in the VDR model has been given in [74].
8.2. CRUISE-CONTROL LIMIT In the cruise-control limit of the NaSch model [1026, 1089], vehicles moving with their desired velocity vmax are not subject to noise. This is exactly the effect of a cruisecontrol, which automatically keeps the velocity constant at a desired value. In contrast to the NaSch model, the randomization step (NaSch3) is now applied only to vehicles that have a velocity v < vmax after step (NaSch2) of the update rule. We can express this more formally by recasting the randomization stage of the NaSch model as follows: CC3: Randomization. vn → max(0, vn − 1)
with probability p =
pvmax p
if vn = vmax , if vn < vmax ,
(8.5)
where vn is the velocity of the vehicle at the end of the deceleration step (NaSch2) of the update rule. In the original formulation of the NaSch model, pvmax = p. However, the cruise-control limit corresponds to pvmax = 0 and p = 0. Note the difference between this cruise-control limit of the NaSch model and the VDR model. The vehicles with velocity vn = vmax (at the end of the step 2) are treated deterministically in the cruise-control limit. In contrast, in the VDR model, all cars with velocity vn > 0 (just before step 3) are updated stochastically, but using different values of the braking parameter. The fundamental diagram for the cruise-control limit is shown in Fig. 8.4. For pvmax 1, at sufficiently low densities, all the vehicles move deterministically with the velocity vmax . If this deterministic motion is by small perturbations, the system gets enough time to relax back to the state corresponding to the deterministic algorithm before it is perturbed again. This effectively and completely separates the timescales for perturbing the system and the response of the system and leads to the flow J = vmax ρ for low densities (Fig. 8.4). In the cruise-control limit, it is natural to define a jam as a string of vehicles all of which have instantaneous velocities smaller than vmax . For periodic boundary conditions, a sharp transition from free flow to the congested phase takes place at a critical concentra∗ = 1/(v tion ρc (p, vmax ). For all p = 0, ρc is smaller than ρdet max + 1). For a given vmax , ρc ∗ in the deterministic limit p → 0. Unlike the increases with decreasing p with ρc → ρdet deterministic limit p = 0 of the NaSch model, the cruise-control limit exhibits metasta∗ . In this interval, on appropriate initialization, the bility in the interval ρc < ρ < ρdet
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0.8 0.7 0.6
J
0.5 0.4 0.3 0.2 0.1 0
0.2
0.4
0.6
0.8
1
ρ
Figure 8.4 Fundamental diagram in the cruise-control limit for vmax = 5 and p = 0.5.
system can reach steady states, where no jam appears and where the fluxes are higher than J (ρc ). However, perturbations of such a metastable state creates long-lived jams, thereby reducing the flux to a level consistent with the stable branch of the fundamental diagram. At all ρ > ρc , jams present in the initial configuration never disappear completely and, in this density regime, the stable steady state is a mixture of laminar flow regions and jams. The long-lived jams lower the flux beyond ρc , and the flux decreases linearly with density (Fig. 8.4). Let us assume that at densities slightly more than ρc , only one jam of length Ljam containing Njam vehicles exists in the system. Then, because of the periodic boundary conditions, the total number of vehicles N is conserved and, hence, N = ρjam Ljam + ρout (L − Ljam),
(8.6)
where ρjam = Njam /Ljam and ρout = (N − Njam)/(L − Ljam) are the densities of the vehicles in the jam and in the outflow region, respectively. Dividing both sides of (8.6) by L, we get Ljam Ljam ρ = ρjam + ρout 1 − . (8.7) L L Since in the cruise-control limit of the NaSch model Ljam must vanish as ρ → ρc , we conclude [1026, 1089] that we must have ρout = ρc ; i.e., the average density in the outflow region of a jam is equal to the critical density ρc .
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In order to study the traffic at the critical point of the cruise-control limit, Nagel and Paczuski [1026, 1089] used a special boundary condition, which consists of an infinite jam at the left boundary while the right boundary is open. Such a system was found to select automatically the state of maximum throughput [1026, 1089]; i.e, it exhibits self-organized criticality [61, 308, 1400]. Characteristic quantities like the distributions of the sizes of the jams, lifetimes of jams,and so on do, indeed, exhibit power-laws, which are hallmark of self-organized criticality. For example, the branching behavior of the jams gives rise to intermittent dynamics with a 1/f power law spectrum [1026, 1089] (see Section 6.10). The exponents associated with the various power laws in the cruise-control limit of the NaSch model can be calculated analytically, at least for vmax = 1, by utilizing a formal relation with an unbiased random walk. If vmax = 1, all the vehicles in the jams have velocity v = 0. Moreover, the jams are compact so that the number of vehicles in a jam is identical to its spatial extent. The probability distribution P(n; t) for the number of vehicles n in such a jam, at time t, is determined by the following equation: P(n; t + 1) = (1 − rin − rout )P(n; t) + rinP(n − 1; t) + routP(n + 1; t),
(8.8)
where the phenomenological parameters rin and rout are the rates of incoming and outgoing vehicles, respectively. Of course, rin depends on the density of the vehicles behind the jam. For large n and t, taking the continuum limit of the equation (8.8) and expanding, we get ∂P rin + rout ∂ 2P ∂P . = (rout − rin ) + ∂t ∂n 2 ∂n2
(8.9)
If rin > rout , the jams would grow for ever. However, the jams would shrink, and eventually disappear if rin < rout . If rin = rout , the first term on the right-hand side of the equation (8.9) vanishes, and the resulting equation governing the time evolution of P(n; t) is identical to that of the probability of finding, at time t, an unbiased one-dimensional random walker at a distance n, which was initially at the origin. Thus, when rin = rout , the jams exhibit large (critical) fluctuations, which can be characterized by critical exponents. Using this formal mapping onto an unbiased random walk, one can make use of the well-known results from the theory of random walks [385, 1455] to derive the critical exponents for the mean jam size and the lifetime distribution of jams [1026, 1089]: n(t) ∝ t 1/2 ,
and
−3/2
P(τlife ) ∝ τlife .
(8.10)
The power-laws are not restricted merely to the special case vmax = 1 of the cruisecontrol limit, but appear also for arbitrary vmax . The power-law distribution of P(τlife ) is
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in sharp contrast with the exponential distribution observed in the NaSch model [1019]. In the cruise-control limit, the large jams are fractal [909] in the sense that there are smaller subjams inside larger jams, ad infinitum. In other words, in between subjams, there are holes of all sizes. The cruise-control limit has further been studied in [456] for a larger class of models.
8.3. CA MODELS OF SYNCHRONIZED TRAFFIC Although the VDR model shows metastable states and related phenomena like hysteresis, it is not able to reproduce the characteristic properties of synchronized traffic. Furthermore, the agreement with empirical data on a microscopic level (single-vehicle data) is not very satisfactory [777]. This indicates that an important ingredient is still missing. So far, the dynamics was mainly based on the avoidance of accidents and the wish to move at the desired velocity. For an understanding of the “fine-structure” of traffic flow, this is insufficient. Therefore, it has been suggested that the desire of the drivers for smooth and comfortable driving has to be taken into account as a possible origin of synchronized traffic [774]. The avoidance of accidents implies that only interactions with the next car ahead are important and any information about the velocity or the velocity difference to the preceding car is not taken into account. As we know from our own experience, usually one tries to “anticipate” the behavior of the predecessor. Velocity anticipation means that drivers estimate the future velocity of the preceding vehicle. If its headway is rather large, then an abrupt braking manoeuver is very unlikely. Therefore, one is willing to accept a much smaller headway. Thus anticipation allows for a much smoother driving. Furthermore, it provides a mechanism for the moving platoons observed in synchronized traffic. Thus, the three observed traffic phases correspond to different driving strategies. In free flow, drivers try to drive as fast as possible and interactions are rare. In the jammed phase, the avoidance of accidents determines the behavior, and in synchronized traffic, it is the desire to drive in a smooth and comfortable way. Drivers try to avoid abrupt velocity changes, which requires to observe the behavior of the surrounding traffic in a more detailed way than in the other two phases. There are several aspects that appear to be important for realistic models, but are not included in the NaSch or VDR models: 1. Velocity anticipation: The empirically observed very short temporal headways (Section 6.8) can only be explained by strong anticipation effects. Furthermore, at inhomogeneities, e.g., created by ramps, the anticipation of the leaders velocity avoids abrupt braking of the traffic behind and, therefore, reduces the probability to form jams.
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2. Delayed acceleration: Comfortable driving also implies that cars do not accelerate immediately in case of a larger gap ahead if they observe slow downstream traffic. 3. Timely braking: Finally, timely braking suppresses another mechanism of jam formation: When the velocity adjustment is only based on the distance to the next car ahead, jams often emerge in the layer between free-flow and synchronized traffic. In these models, the jam formation arises from cars approaching a slow-moving cluster with high speed, which leads to a compactified region. This artificial mechanism of jam formation can be avoided by allowing the drivers to adjust their speed to the vehicles ahead. An important feature for realizing (1)–(3) is anticipation of the actions of other drivers in the next timestep. Thus, more information about the next car n + 1 ahead is needed, not just its distance dn as in the NaSch model. From its velocity vn+1 (t) and the headway dn+1 (t), an anticipated velocity vanti = min dn+1 (t), vn+1(t) − 1
(8.11)
can be estimated such that vn+1 (t + 1) ≤ vanti . This allows to determine a safe velocity vn (t + 1) not leading to a collision. Before we discuss some models in more detail, some remarks concerning the terminology as used by Kerner should be made. He argues that no fundamental diagram-based approach can explain the occurrence of synchronized traffic. By this, he means models based on a fundamental diagram with a unique flow–density relation (up to the metastable high-flow branch). More specifically he requires the deterministic limit of such model to show an extended nonunique flow–density relation [728]. This restriction appears to be much too strong, especially because we have seen from most previous examples that the deterministic limit of most stochastic models is nongeneric or even singular.
8.3.1. Brake-Light or Comfortable Driving Model The brake-light model [773, 775, 777], also known as comfortable driving model, tries to incorporate effects that go beyond simple accident avoidance. In NaSch-type models, this leads to sharp braking maneuvers, especially if fast cars approach the end of a jam. In real life, drivers usually want to drive more comfortably, which also means avoiding quick velocity changes. This requires some sort of anticipation and, as has been argued earlier, might lead to the formation of synchronized traffic. Apart from velocity-anticipation of the type (8.11), the brake-light model includes braking anticipation as a new feature. As in real life, this is realized through the introduction of brake lights that transmit information about braking processes immediately to the
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following car(s). The brake light allows a smoother approach at the end of jams. The state of the brake light of the n-th vehicle is characterized by the variable 0 bn = 1
brake light off brake light on.
(8.12)
Other important aspects, which are incorporated in the model are (1) delayed acceleration of standing vehicles, which is realized by a slow-to-start rule, and (2) delayed deceleration after braking. This leads to a suboptimal usage of gaps in traffic so that cars move at velocities, which are smaller than their headway would allow. However, larger gaps in a dense region reduce the interactions and cut-off the chain reactions of braking overreactions, which are responsible for spontaneous jam formation. For the model dynamics, three different regimes can be distinguished: • For large headways, drivers will drive (apart from fluctuations) with their desired velocities vmax . • For intermediate headways, drivers foremost react to velocity changes of their predecessor. These are transmitted, e.g., through the brake lights. • For small headways, accident avoidance is the dominating strategy for adjusting the velocity. To allow for a more realistic modeling of the acceleration process (a ≈ 1 m/s), one also introduces smaller cells of length 1.5 m (instead 7.5 m). This implies that now vehicles have an effective length of five cells, instead of just one as in the previous models. For the specification of the update rules, two different times are of relevance, namely th =
dn (t) vn (t)
and
ts = min(vn (t), h).
(8.13)
th is the time to reach the current position of the preceding car when moving with constant velocity vn (t). In traffic engineering, th is known as time-to-collision. ts introduces a cut-off h, which prevents drivers to react to brake lights of vehicles that are too far ahead. h is the interaction range of the brake lights. Psychological studies1 have indicated that the relevant interaction horizon is velocity-dependent so that h corresponds to a temporal headway, which is of the order of 6–11 s. All of these effects are combined in the dynamics of the brake-light model [773, 775, 777]. Its full dynamics is then given by the following set of update rules. 1 See the references given in [773].
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BL0: Determination of randomization parameter: ⎧ ⎪ ⎨pb p = p(vn (t), bn+1(t), th , ts ) = p0 ⎪ ⎩ pd
for bn+1 = 1 and th < ts for vn = 0 else
(8.14)
and bn (t + 1) = 0. BL1: Acceleration: If bn+1 (t) = bn (t) = 0 or th ≥ ts , then vn = min(vn (t) + 1, vmax). BL2: NaSch braking rule:
(i): vn = min dn(eff ) , vn (ii): If (vn < vn (t)), then bn (t + 1) = 1 Here dn(eff ) = dn + max(vanti − gapsafety , 0) is the effective gap with vanti from (8.11) and gapsafety ≥ 1, in order to avoid accidents. BL3: Randomization, braking: If rand() < p,
then
vn (t + 1) = max(vn − 1, 0) if p = pb , then bn (t + 1) = 1.
BL4: Car motion: xn (t + 1) = xn (t) + vn(t + 1). Here rand() is a random number drawn from a uniform distribution in [0, 1]. The relevant parameters of the brake-light model are summarized in Table 8.1. These rules have simple interpretations. Step (BL0) determines the new randomization parameter. For standing vehicles, it is given by p0 according to the usual slow-to-start rule. p0 determines the jam velocity. If the brake light of the preceding car is on Table 8.1 Summary of variables (left column) and parameters (right column) of the brake-light model Variable
vanti d eff d b p th ts
Anticipated velocity Effective gap Gap State of brake light Randomization variable Temporal headway Temporal horizon
Parameter
vmax
p0 pb pd gap safety h
Maximal velocity (Effective) car length Slow-to-start parameter Randomization parameter Randomization parameter Effectiveness of anticipation Range of brake lights
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(bn+1 = 1) – and this car is within the interaction horizon – then p = pb . In all other cases, one has p = pd . Furthermore, the brake light will be switched off in this step. Step (BL1) prevents a car from accelerating if its own brake light was on in the previous step or that of a its predecessor (provided it is within the interaction horizon). In Step (BL2), the velocity is adjusted according to the effective gap to the preceding car, i.e., taking into account velocity anticipation. In Step (BL3), the brake light will be switched on if a car had to slow down due to the brake light of its predecessor. This leads to the propagation of the brake lights. The brake-light model shows good agreement with empirical results even on a more detailed microscopic level [773, 777]. Suitable parameter values are vmax = 20, pd = 0.1, pb = 0.94, p0 = 0.5, h = 6, = 5 cells, and gapsafety = 7. For a cell length of 1.5 m, each timestep corresponds to about 1 s real time, as in the NaSch model. The properties of the brake-light model have been studied in detail. It is able to reproduce all three phases of traffic [773], as well as macroscopic phenomena like the “tunneling” of a wide jam through a region of synchronized traffic [774] near ramps. Furthermore, the agreement of the microscopic properties with empirical single-vehicle data is very good [777]. The model reproduces the synchronization of lanes and other multilane phenomena [775]. The behavior of the brake-light model near ramps has also been studied in [735], where two different characteristic congested patterns at an on-ramp were found: (1) the pinned layer, which can either be widening or localized, and (2) oscillating moving jams, which are characterized by the emergence and dissolution of narrow moving jams on a short timescale. A variant of the brake-light model, usually called modified comfortable driving model was introduced by Jiang and Wu [674] to include so-called light synchronized flow, which has a larger speed (> 60 km/h) than the velocities in synchronized phase of the original model. The basic structure of the rules is the same as for the brake-light model. The main modifications concern the acceleration properties and the use of the brake lights: • In the acceleration step, the acceleration capacity of a stopped car is 1, whereas that of a moving car is 2. • The brake light is only switched on if the velocity decreases, i.e., for vn (t + 1) < vn (t). • The brake light will not be switched off unless the car begins to accelerate. Furthermore, the slow-to-start rule is also modified so that it becomes effective only if a car has stopped longer than some time tc . Further modifications were introduced later mostly in order to describe certain aspects in more detail [677, 679, 1448], e.g., open boundaries [676] and ramps [678]. In [835], the brake-light model has been modified by including rules that take into account the limited deceleration properties. This avoids unrealistic abrupt braking maneuvers when fast cars approach jams or obstacles. This is implemented by interpolating the
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speed between two timesteps linearly when calculating a safe speed in a similar way to [793] (see Section 9.5.2) and [841] (see Section 8.3.3).
8.3.2. Kerner–Klenov–Wolf Model The Kerner–Klenov–Wolf (KKW) model [735] is a fully discretized version of the spacecontinuous microscopic model introduced by Kerner and Klenov [731], which will be discussed in Section 9.4.5. It combines elements of car-following theory with the standard distance-dependent interactions. The update rule consists of a deterministic and a stochastic part. The deterministic rule, which reads (8.15) KKW1: vn (t1 ) = max 0, min{vmax, vsafe (t), vdes (t)} , (t < t1 < t + 1), is applied first.2 The three velocities appearing are the free flow or maximal speed of the cars vmax , the safe velocity vsafe (t), and finally the desired velocity vdes (t). vsafe (t) is the velocity that guarantees collision-free motion and is simply the gap to the preceding car, vsafe (t) = dn (t). It is the introduction of vdes (t), which makes the difference to the NaSch model. The velocity vdes (t) is given by v (t) + a for dn > D(vn (t)) − , (8.16) vdes (t) = n vn (t) + (t) for dn ≤ D(vn (t)) − . The calculation of vdes (t) replaces the acceleration step of the NaSch model by a more complex rule. Here is the length of the vehicles and D(v) a synchronization distance for the velocity-dependent interaction range, which is assumed to be linear or quadratic, i.e., D(v) = D0 + kv
or
D(v) = D0 + v + βv 2 .
(8.17)
The parameter D0 is typically chosen to correspond to the vehicle length [735]. Furthermore, the results for the linear and quadratic D(v) functions (8.17) agree at least qualitatively. The interaction range has been introduced as a synchronization radius, i.e., D(v) is the distance that separates free driving cars from cars that had already adjusted their velocity according to the vehicle ahead. For large distances to the vehicle ahead, dn > D(vn (t)) − l, the calculation of vdes is equivalent to the acceleration step of the NaSch model. Inside the enlarged interaction radius, however, vdes depends on the velocity of the leading car. Explicitly (t) is given by ⎧ ⎨−b if vn (t) > vn+1 (t) (8.18) (t) = 0 if vn (t) = vn+1 (t) ⎩ a if vn (t) < vn+1 (t), 2 In the following, we choose the time discretization as δt = 1.
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This means that within the interaction radius, drivers tend to adapt their velocity to the vehicle ahead. The second update rule is stochastic. It is given by KKW2: vn (t + 1) = max 0, min{vn (t1 ) + aη, vn(t1 ) + a, vfree, dn } ,
(8.19)
The stochasticity is included in the term vn (t1 ) + η, while the others guarantee that the new velocity is below the speed limit, leads to no collisions and is in accordance with the acceleration capacity a of the cars. The stochastic variable η can take the following values: ⎧ ⎨−1 with probability pb , with probability pa , η= 1 ⎩ 0 with probability 1 − pa − pb .
(8.20)
Both probabilities pa and pb introduced here are velocity dependent. One has p if v = 0 pb (v) = 0 p if v > 0.
(8.21)
with p0 > p. The stochastic braking is analogous to the slow-to-start rule known from the VDR model. On the contrary, the stochastic acceleration is a new feature of the model, which weakens the synchronization of speeds as it applies to cars that have reduced or kept their velocity although safe driving would have allowed a larger velocity. The function pa (vn ) is explicitly given by p if v < vp pa (v) = a1 pa2 if v ≥ vp ,
(8.22)
where vp , pa1 , and pa2 < pa1 are adjustable parameters of the model. The different probabilities have to be chosen such that pa + pb ≤ 1 is fulfilled for any velocity. The velocity update is completed by this second stochastic rule and is followed by a parallel update of the positions. The definition of the dynamics of the KKW model is quite complex and governed by many parameters. The parameter values proposed in [735] are shown in Table 8.2. For further illustration of the update rules, we compare them briefly to the BL model. Both models include the update rules of the VDR model and enlarge the interaction radius of the drivers within a velocity-dependent interaction range. The driving strategy within this larger interaction range is, however, different. While the BL model introduces an event-driven interaction model, the KKW is more car-following like. Another
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Table 8.2 Typical parameter values proposed for the KKW model [735]. The discretizations of space, time, velocity, and acceleration are δx = 0.5 m, δt = 1 s, δv = 1.8 km/h, and δa = 0.5 m/s2 . This leads to characteristic values of = 7.5 m, vmax = 108 km/h, vjam = −15.5 km/h, and Jout = 1810 veh/h
Parameter
vmax
a=b
k
p
p0
pa1
pa2
vp
Value
15
60
1
2.55
0.04
0.425
0.2
0.052
28
important difference is that the velocity anticipation is not included in the approach of [735], although such an extension is possible [732]. In the KKW model, there is no unique flow–density relationship for steady states. Because all accelerations vanish in the steady state, this implies either D(v) < d +
and v = vmax or D(v) ≥ d + and v ≤ min{vmax , d}. These conditions define a twodimensional region in the flow–density plane where homogeneous steady states can exist. In the upper boundary, the flow is determined by the safe speed, J = ρvsafe = 1 − ρ , whereas the lower boundary is determined by the synchronization distance D(v). Its shape depends on the choice for the function D(v). The KKW model shows three distinguishable traffic states, which have properties that are consistent with three-phase traffic theory. Kerner and his collaborators have studied the model for various situations in detail and compared its predictions with empirical data. We refer to his books [727, 730] and references therein for further information. The microscopic properties of the KKW model have been investigated in detail in [777] and compared with measurements of single-vehicle data. Indeed the three phases are reproduced, and also the agreement of the fundamental diagram and time-headway distributions is satisfactory. Only the optimal velocity (OV) function does not agree too well with empirical data.
8.3.3. Mechanical Restrictions Model of Lee et al. Lee et al. [841, 842] have pointed out the conflict between human overreaction and limited mechanical capabilities as possible origin of congested traffic states. Their model takes the limited acceleration and deceleration capabilities of the vehicles into account. Therefore, it will be called mechanical restrictions (MR) model in the following. In most models discussed so far, deceleration is not limited to avoid accidents. Furthermore, in the MR model, different driving strategies are distinguished, depending on the local traffic situation. Optimistic driving controls the behavior in free flow, where drivers accept “unsafe” gaps, which are too small to react to an emergency braking of the leading vehicle. Pessimistic driving occurs at high densities, where interactions between the cars are strong and braking is likely so that drivers have to keep a sufficiently large headways.
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The driver’s state depends on the local traffic situation and is characterized by the two-state variable γn (t) =
0 for vn (t) ≤ vn+1 (t) ≤ vn+2 (t) or 1 otherwise
vn+2(t) ≥ vfast ,
(8.23)
where the constant vfast is slightly below vmax , e.g., vfast = vmax − 1. γn (t) = 0 corresponds to an optimistic state because either the driver is in a platoon of vehicles that are at least as fast as he or the next, but one vehicle is speeding away. In such situations, it is relatively unlikely to be forced to brake hard. However, γn (t) = 1 corresponds to a defensive or pessimistic state, where the vehicle will start to slow down. The core of the model is an inequality that defines a velocity cn (t + 1), which is considered to be safe by the driver: τf (cn (t+1))
xn (t) + +
i=0
(cn (t + 1) − Di) ≤ xn+1 (t) +
τl (v n+1 (t))
(vn+1 (t) − Di) .
(8.24)
i=1
xn (t) and vn (t) are position and velocity of vehicle n, respectively, and represents the minimum gap between the vehicles and is at least the length l of the leading vehicle. Each summation in (8.24) denotes a deceleration cascade with maximum braking capability D. As long as both τf (v) and τl (v) are set to v/D and = l, the deceleration would end in a bumper-to-bumper configuration. But this is weakened if the human factor is introduced. Because cnt+1 is not uniquely determined by (8.24), usually the upper limit is used. In [841], functions τf (v), τl (v), and D have been proposed that lead to realistic behavior: = L + γn (t) max 0, min gadd , vn (t) − gadd , v τf (v) = γn (t) + (1 − γn (t)) max {0, min {v/D, tsafe } − 1} , D v τl (v) = γn (t) + (1 − γn (t)) min {v/D, tsafe }. D
(8.25)
Here gadd is introduced for an additional security gap in the pessimistic state (γn (t) = 1) and tsafe is a maximal timestep during which the follower observes his/her own safety in the optimistic state. The additional −1 for τf (v) compensates for the surplus time step due to the follower’s response time only when γn (t) = 0, and thus the role of tsafe is properly implemented. For γn (t) = 0, the τf ,l can be smaller than those (v/D) necessary for complete stops while returns to L. Consequently, lower safety is required
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compared to that needed by strict collision-free dynamics, and thus a faster ˜cn (t + 1) can be chosen. However, for γn (t) = 1, can be larger than L while τf ,l (v) return to v/D, which implies over safety. In this way, a lower ˜cn (t + 1) can be assigned. The update rules of the MR model can be written in the following form: MR1: p = max pd , p0 − vn (t)(p0 − pd )/vslow . MR2: ˜cn (t + 1) = max {cn (t + 1)|cn (t + 1) satisfies Eqn (8.24)}. MR3: v˜n (t + 1) = min {vmax , vn (t) + a, max{0, vn (t) − D, ˜cn(t + 1)}}. MR4: vn (t + 1) = max {0, vn (t) − D, ˜vn(t + 1) − η}, where η = 1 if rand( )< p, or 0 otherwise. MR5: xn (t + 1) = xn (t) + vn(t + 1). The randomization p interpolates linearly between p0 and pd if vn (t) is smaller than vslow (step (MR1)). For p0 > pd , step (MR1) realizes a slow-to-start rule (Section 8.1). Step (MR3) guarantees that the updated velocity is consistent with the mechanical restriction (acceleration a and braking capability D) and traffic regulations. The length of one cell is chosen to be x = 1.5 m, and the unit time is set to t = 1 s. The following model parameters are motivated by empirical facts: a = 1, D = 2, L = 5, vfast = 19, tsafe = 3, gadd = 4, p0 = 0.32, pd = 0.11, vslow = 5, and vmax = 20. It turns out the model is not intrinsically accident-free, although accidents are rare [1128]. Usually collisions are avoided by introducing a strict hardcore repulsion between the individual cars. However, this typically leads to processes with a very large deceleration. Therefore, it is rather difficult to define a model that, at the same time, captures the finite deceleration capabilities of vehicles and is accident-free. In the presence of limited deceleration capabilities, crashes have to be avoided by choosing the dynamics appropriately. Therefore, in [1128, 1129], a slight modification of the model has been suggested that appears to be accident-free and, at the same time, keeps the realistic behavior of the original model. A key ingredient is again brake lights, which provides a way to communicate the presence of a hindrance and therefore a possible change of the driving behavior (from optimistic to pessimistic) to the following cars. This modification is also essential for an extension of the model to multilane traffic [1128, 1129]. The MR model is able to reproduce the three phases of traffic [841, 842], as well as the typical patterns observed near inhomogeneities like ramps [1129]. Furthermore, it exhibits the pinch effect and the peak at headways of 1 s observed empirically in the time-headway distribution of free flow. Another way of implementing accident-free rules together with limited deceleration properties was proposed in [835] by replacing the discrete speed variation through a piecewise-linear function. The speed of the vehicles then changes continuously to the new value during each timestep.
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8.4. OTHER CA MODELS 8.4.1. Fukui–Ishibashi Model Fukui and Ishibashi [414] have introduced a simplified version of the NaSch model. It can be interpreted as an extension of the rule-184 CA to a larger interaction range. The main difference to the NaSch model is the absence of a velocity memory. All vehicles have an intrinsic velocity vmax . In each timestep, all drivers try to move at the maximum velocity vmax ; i.e., they accelerate to vmax instantaneously. The stochastic variant of the Fukui–Ishibashi (FI) model [414] is then defined by the following set of rules: FI1: Acceleration: vn = min(vmax , dn ). FI2: Randomization: If vn = vmax , then vn → vmax − 1 with probability p. FI3: Vehicle movement: xn → xn + vn . As usual, xn and vn denote the position and speed, respectively, of the nth vehicle and dn = xn+1 − xn − 1, i.e., as the number of empty cells in front of this car is the (distance) headway. The rules have a simple interpretation. A car that has at least vmax empty sites in front will move vmax cells with probability 1 − p or vmax − 1 cells with probability p. However, if only d < vmax sites in front of the n-th vehicle are empty at time t, then it moves by d sites in the next timestep. Here the randomization step is not applied. Therefore, fluctuations occur only at high speeds, which is just the opposite of the cruise-control limit (Section 8.2). For vmax = 1, the FI model and the NaSch model are identical and reduce to the rule-184 CA in the deterministic limit (p = 0). For general vmax , the FI model differs from the NaSch model in two respects: (1) the increase of speed of the vehicles is not necessarily gradual and (2) the stochastic delay applies only to high-speed vehicles. Due to these modifications, no overreactions at braking occur and therefore the FI model does not exhibit spontaneous jamming. The fundamental diagram of the deterministic FI model (p = 0) is identical to that of the deterministic NaSch model (see Eqn (7.9)) [409, 1439, 1441] JFI (p = 0) = min(ρvmax , 1 − ρ) ,
(8.26)
with two linear branches corresponding to the free-flow and jammed phase separated by a critical density ρc = 1/(vmax + 1). For arbitrary vmax and p, a site-oriented mean-field theory [1438, 1441] and a car-oriented mean-field theory [1439, 1442] have been developed. The fundamental diagram again looks qualitatively similar to that of the corresponding NaSch model.
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Several generalizations of the FI model have been proposed. Extensions of the deterministic limit p = 0 of the FI model were considered by Fuks and Boccara [409, 410]. They introduced anticipation effects and also studied the occurrence of accidents. An exact solution for some cases was given in [825]. Anticipation effects have also been incorporated in [847]. These lead not only to higher values of the flux, but also to the appearance of a “synchronized regime” at intermediate densities. In this regime, all vehicles are moving, but at low speed. In [132, 828, 1228], an alternative high-acceleration variant of the NaSch model has been proposed, the aggressive driving model (ADM). Here only the acceleration step (NaSch1) of the NaSch model is changed to ADM1: Acceleration.
vn = vmax ,
i.e., all vehicles accelerate immediately to the maximal possible velocity vmax . The other update steps of the NaSch model are left unchanged. In contrast to the FI model, all vehicles are subject to the randomization step. The behavior of this variant is therefore similar to that of the NaSch model, e.g., one finds spontaneous jam formation. The ADM has been studied extensively by Kunwar et al. [828], using various analytical methods and computer simulations for both periodic and open boundary conditions. It can also be used to describe certain in aspects of intracellular transport (see Chapter 12). The behavior of a molecular motor called dynein, which moves along microtubules is well described by the ADM with additional Langmuir kinetics [828]. Wang et al. [1445] have introduced a model, which interpolates between NaSch and FI. It is defined by the following update steps: R1: Acceleration. vn (t1 ) = min{dn (t), vmax}. R2: Randomization.If 0 < dn (t) ≤ vmax, then vn (t + 1) = vn (t1 ) − 1 with probability p, else vn (t + 1) = vn (t1 ). R3: Vehicle movement. xn (t + 1) = xn (t) + vn(t + 1). The main difference to the NaSch model is that vehicles with headways dn larger than their maximal velocity vmax are not affected by the randomization rule (R2). Therefore, the flux for ρ ≤ 1/(vmax + 2) is given by J = vmax ρ.
8.4.2. Velocity-Dependent Braking Model Levine et al. [862] have proposed the velocity-dependent braking model. The vehicle dynamics is defined by the velocity update min vj (t) + 1, vmax, dj (t) vj (t + 1) = 0
with probability 1 − p(vj (t)) with probability p(vj (t)),
(8.27)
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which is applied to all cars in parallel. The braking probability is given by p(v) =
p q
for v < vmax, for v = vmax.
(8.28)
The implementation of the randomization is different from that in the NaSch or VDR models because, e.g., even a car moving unhindered at maximum speed can stop suddenly with probability p(vmax). The case q = 0 corresponds to a cruise-control limit similar to that in the NaSch model. For densities ρ ≤ ρmax = 1/(vmax + 1), free-flow states with J (ρ) = vmax ρ exist. At higher densities local jams are formed. A phase transition between these states occurs at some density ρc ≤ ρmax ; i.e., in the region ρc ≤ ρ ≤ ρmax , the flow–density relation is not unique. The phase transition is related to the fact that for q = 0, the free-flow state is an absorbing state, which has no dynamics. The cruise-control limit q = 0 can be analyzed in more detail for vmax = 1 [862, 863]. The current in the jammed state (ρ > ρc = (1 − p)/(2 − p)) is given by J (ρ) = ρc (1 − ρ)/(1 − ρc ). In the regime ρc ≤ ρ ≤ ρmax = 1/2, both free-flow and jammed state coexist in the thermodynamic limit. For q > 0, the free-flow state is no longer absorbing and therefore the phase transition turns into a crossover. For vmax > 1, the microscopic dynamics of free-flow domains in the regime ρ > ρc can effectively be described by a chipping model (see Section 3.7.3) [862, 863]. This analogy supports the view that no phase transitions occur for q > 0. The mapping to chipping models can be extended to a larger class of models [456, 862, 863].
8.4.3. Time-Oriented CA Model The time-oriented CA model (TOCA) introduced by Brilon and Wu [157, 158] increases the interaction horizon of the NaSch model (where cars interact only for d ≤ v) and therefore changes the car-following behavior. Compared with the NaSch model, the acceleration step is modified; i.e., a car accelerates only if its temporal headway th = d(t)/v(t) is larger than some safe time-headway ts . Even for sufficiently large headways, the acceleration of a vehicle is not deterministic, but is applied with probability pac. As a second modification also the randomization step is modified; i.e., it is performed only for cars moving with short time-headways th < ts . The limited interaction radius of this third step leads, for a given value of pdec , to a reduction of the spontaneous jam formation. The update rules of the TOCA then read as follows (t < t1 < t2 < t + 1): TOCA1: if (th > ts ), then vn (t1 ) = min{vn (t) + 1, vmax} with probability pac . TOCA2: vn (t2 ) = min{vn (t1 ), dn (t)}.
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TOCA3: if (th < ts ), then vn (t + 1) = max{vn (t2 ) − 1, 0} with probability pdec . TOCA4: xn (t + 1) = xn (t) + vn(t + 1). with ts = 1.2, pac = 0.9, and pdec = 0.9 [157, 158]. With this choice of ts , the update rules can be simplified for vmax ≤ 4 because of the discrete nature of the model [777]: TOCA1∗ : vn (t1 ) = min{vn (t) + 1, vmax} with probability pac . TOCA2∗ : vn (t2 ) = min{vn (t1 ), dn (t)}. TOCA3∗ : if (vn (t + 1) ≤ dn (t)) then vn (t + 1) = max{dn (t) − 1, 0} with probability pdec TOCA4∗ : xn (t + 1) = xn (t) + vn(t + 1) For this parametrization of the model, one obtains results for the fundamental diagram that are similar to the NaSch model. In congested traffic, the time-headway distribution has two maxima, one corresponding to the typical time-headway in free-flow traffic and the other corresponding to the typical temporal distance in the outflow region of a jam. A detailed discussion of the microscopic properties of the model and a comparison with empirical single-vehicle data can be found in [777].
8.4.4. Models with Anticipation The occurrence of very small headways that was found empirically in the distribution of time-headways (see Section 6.8) indicates the relevance of anticipation in real traffic. It also plays an important role in multilane situations or some driver assistance systems. Usually “anticipation” means velocity anticipation, i.e., the estimation of the velocity of nearby cars in future timesteps. In physics terminology, anticipation corresponds to nextnearest or multiple-particle interactions. We have already seen that anticipation might be an important factor in the formation of synchronized traffic. Therefore, anticipation effects are an essential part of many models of three-phase traffic theory, like the brake-light model (Section 8.3.1). But there are also other models that try to elucidate the consequences of anticipation. Larraga et al. [838] have introduced a anticipatory driving parameter in the deceleration process to estimate the velocity of the preceding vehicle. This estimation, plus the real spatial distance to the leading vehicle, establishes a safe distance among vehicles. The dynamics of the model is then defined by the following rules: A1: Acceleration. If vn < vmax , the velocity of car n is increased by one, i.e., vn → min(vn + 1, vmax). A2: Randomization. If vn > 0, the velocity of car i is decreased randomly by one unit with probability p, i.e., vn → max(vn − 1, 0) with probability p.
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A3: Deceleration. If dns < vn , where (with a parameter 0 ≤ α ≤ 1) 1 dns = dn + (1 − α) · vn+1 + , 2 the velocity of car n is reduced to dns . [x] denotes the integer part of x, i.e. [x + 12 ] corresponds to rounding x to the next integer value. The new velocity of the vehicle n is therefore vn → min(vn , dns ). A4: Vehicle movement. Each car moves forward according to the new velocity determined in steps A1–A3: xn → xn + vn. The order of the substeps is different compared with the NaSch model because (A2) is applied before (A3). If (A2) would be applied after (A3), cars are unable to adjust to the randomization-reduced velocities of the traffic in front and the model would no longer be automatically collision-free. Rule A3 is the main modification to the original NaSch model. In this rule, the distances between the ith and (i + 1)th vehicles, and their corresponding velocities are considered. Knowledge of the preceding vehicle’s velocity is incorporated through the anticipatory driving parameter α with range 0 ≤ α ≤ 1. Through a variation of the parameter α, different driving strategies can be modeled. For α = 1, the speed of the preceding vehicle is not taken into account (no anticipation). For α = 0, vehicle may move with the same speed without additional safety distance. This case occurs with either a very aggressive driver or when vehicles can obtain information about the velocity of vehicles ahead, e.g., for Automated Highway Systems or vehicles equipped with appropriate sensors [1188]. There is a price to pay with this modification that limits deceleration values. It could be the case that a deceleration of a vehicle also implies decelerations of following vehicles. Step A3 which assures that collisions are avoided is then applied sequentially to take into account the limited deceleration capability. The final configuration is independent of the starting point of this sequential updating. To determine vn consistently for all cars, (A3) has to be iterated at most vmax times in systems with periodic boundary conditions [838]. Anticipation can lead to substantial changes of the behavior, e.g., forward (downstream) moving jams. As expected, flow values increase with increasing anticipation parameter α (Fig. 8.5). For intermediate values of α (0.13 < α < 0.5), the fundamental
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2.5
α = 0.0, 0.10, 0.12 α = 0.13, 0.15, 0.16 α = 0.18, 0.20, 0.25 α = 0.30, 0.40, 0.50 α = 0.60, 0.70, 0.80, 1.0
2.0
Flow
1.5
1.0
0.5
0.0
0.0
0.2
0.4
0.6 Density
0.8
1.0
Figure 8.5 Fundamental diagram of the model of Larraga et al. for different values of the anticipation parameter α and p = 0.2. The legend indicates the effect of rounding in the estimation of the velocity of the preceding car, defined in rule A3.
diagram consists of three branches. Besides the free-flow and jammed branch, a mixed regime appears at intermediate densities ρ1 < ρ < ρ2 , where ρ1 =
1−p vf − v + (1 − p)
and
ρ2 =
(1 − p)2 p(v + p − 2) + 1
(8.29)
with the free-flow velocity vf = vmax − p. In the mixed regime, the system exhibits phase separation into a free-flow region and v-platoons. In these dense platoons, vehicles move with the same velocity v and vanishing headway. The velocity v depends on the strength of anticipation α and the randomization. These states have certain similarities with the homogeneous-in-speed states (Section 6.6.2) that have been observed in synchronized traffic. Other studies of anticipation effects in CA models can be found in [130, 772, 847, 869, 1253]. Anticipation effects have also been studied for car-following models like the optimal-velocity model [855], the SK model [344], and the intelligent-driver model [1389].
8.4.5. Galilei-Invariant Model In the NaSch model, the gap between a pair of successive vehicles is adjusted according to the velocity vn+1 of the leading vehicle alone. In contrast, often in real traffic, drivers
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tend to adjust the gap in front, taking into account the difference between the velocity vn of their own vehicle and that of the leading vehicle. The latter aspect of real traffic is captured by a model developed by Werth, Froese, and Wolf (WFW) [1460, 1474]. If both the following vehicle n and the leading vehicle n + 1 move with constant acceleration b, then a collision between the two can be avoided provided v¯ gap ≥ vn τr + (vn − vn+1), b
(8.30)
where τr is the reaction time of the following vehicle and v¯ = (vn + vn+1)/2 is the average velocity of the pair of vehicles under consideration. In the limiting case b → ∞, the condition (8.30) reduces to gap ≥ vn τr , which is identical to the form of vehicle–vehicle interactions in the NaSch model if one chooses τr as the unit of time. In the opposite limit τ = 0, a sufficient condition for avoiding a collision is 0 for vn ≤ vn+1 , gap ≥ vmax (8.31) for vn > vn+1. b (vn − vn+1) Because this type of vehicle–vehicle interaction involves the difference of the velocities vn − vn+1, it is clearly invariant under a Galilean transformation and, hence, the name Galilei-invariant model. The vehicle–vehicle interactions in real traffic may be somewhere in between the two limiting cases of NaSch model and the Galilei-invariant model. The update rules for the Galilei-invariant model are then as follows [1460]: GI1: GI2: GI3: GI4: GI5:
Acceleration. vn(1) = min(vn + 1, vmax ). Deceleration (due to other vehicles). vn(2) = min(vn(1) , dn − 1 + vn+1). p Randomization. vn(3) = max(vn(2) − 1, 0) with probability p. (4) Deceleration (due to other vehicles). vn(4) = min(vn(3) , dn − 1 + vn+1). Vehicle movement. xn = xn + vn(4) and vn = vn(4) .
Note that the rule for deceleration (due to other vehicles) is applied twice. Step GI4 makes sure (4) that collisions are avoided. Since also the new velocity vn+1 of the preceding car enters, this step can not be performed in parallel for all cars. Instead, it is performed sequentially, but the final configuration is independent of the starting point of this sequential updating. Step GI4 has then to be applied twice in order to determine all velocities vn(4) consistently. The rules as given above define the retarded version of the Galilei-invariant model. (2) In the nonretarded version, vn+1 in step GI2 is replaced by the new velocity vn+1 . To determine vn(2) consistently for all cars, step GI2 has then to be iterated vmax − 1 times. The most interesting feature of the Galilei-invariant model is that its fundamental diagram has a metastable branch, although its update scheme involve neither cruisecontrol nor slow-to-start rules. The mechanism leading to the existence of metastable
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states is different from the models with slow-to-start rules (see Section 8.1). The outflow from jams is the same as in the NaSch model because it is independent of the interaction between vehicles. However, due to the inclusion of anticipation effects free-flow is less sensitive to fluctuations.
8.4.6. Car-Following CA One of the oldest microscopic approaches to traffic modeling are so-called car-following models, which will be discussed later in more detail (see Section 9.4). These models are characterized by specifying equations of motion, analogous to Newton’s equation, which determined the motion of each vehicle. The role of the force is taken by a stimulus, which captures the interactions between the vehicles. Typically, this stimulus is the velocity difference to the proceeding car or, in the OV models (Section 9.4.2), the velocity difference to some desired velocity which depends on the headway. Several CA analogs of car-following models have been proposed. However, it turns out that many suffer from generic problems, especially the occurrence of accidents or backward motion [216]. Therefore, the rather simple rules of these models have to be supplemented by mechanisms to avoid these problems, which unfortunately often are not given explicitly by the authors. These additional rules, e.g., an explicit exclusion principle, can have a significant influence on the original dynamics [216]. As we will see later, these problems also occur in most of the original car-following approaches, at least for realistic reaction times [1389]. Here, however, vehicles are point-like particles moving in continuous space so that accidents can be interpreted as overtaking maneuvers. In the following, we will briefly describe OV-type CA models. In the OV model, the control of velocity is given by the control of acceleration through the OV function, which gives the optimal velocity for the current distance-headway. In principle, the NaSch model is also an OV model, but with a linear OV function, v(d) = min[d − 1, vmax ]. An early attempt to generalize this relation is due to Emmerich and Rank (ER) [348]. The main idea of the ER model is to generalize step (NaSch2) of the NaSch rules, the deceleration due to other vehicles, by introducing a matrix Mjl , which determines the new velocity according to ER2: vn → Mdn −1,vn
if dn − 1 ≤ vmax ,
i.e., a vehicle with velocity j and i empty cells in front of it (i.e., a gap dn = i + 1) reduces its velocity to Mij (0 ≤ i, j ≤ vmax ). A general matrix Mij has to satisfy certain conditions (e.g., Mij ≤ min(i, j) and Mij ≤ Mik for j ≤ k) to guarantee the absence of collisions in the model. The model proposed in by ER [348] assumes that faster vehicles keep a relatively larger headway to the preceding vehicle. This is implemented by reducing the values of Mii for i close to vmax compared
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with the NaSch model, which has Mii = i. In another variant of the model, the velocity of the vehicles is reduced to a value v < vmax even for gaps larger than vmax [348]. Using a parallel update scheme, the model shows unrealistic behavior in the freeflow regime, especially in the deterministic limit p 1, leading to a nonmonotonic fundamental diagram [216]. In order to circumvent this problem, a special kind of ordered-sequential update had to be introduced where first the vehicle with the largest gap ahead is updated. Then, the position of the next vehicle upstream is updated, and so on, using periodic boundary conditions. ER also investigated more general rules where even for gaps larger than vmax , the velocity of the vehicles is reduced to a value v < vmax . Later, a similar model has been proposed by Helbing and Schreckenberg [540]. It is closer to the spirit of the original optimal-velocity model (see Section 9.4.2). The main modification is a new acceleration step HS: vn (t + 1) = vn (t) + λ[Vopt (dn (t)) − vn(t)] , where y denotes the floor function, i.e., the largest integer i ≤ y. In [540], various optimal velocity functions Vopt (d) have been used in order to fit experimental data. The simplest, but unrealistic, choice was Vopt (d) = min(d, vmax ), where d is the distanceheadway. The naive discretization of the OV function produces some undesirable features of the model, e.g., the flow corresponding to the OV function is nonmonotonic in the free-flow region. Furthermore one finds, for certain initial conditions, a breakdown of the flow at a finite density ρc < 1. For λ < 1, the model is not intrinsically collision-free [216, 777]. Problems occur e.g., when fast vehicles approach the end of a jam. For λ > 1, however, backward motion of vehicles is possible. The stochastic optimal velocity model (SOV) [706, 707] is defined by the evolution equation vn (t + 1) = (1 − a)vn(t) + aVopt(xn (t)),
(8.32)
where 0 ≤ a ≤ 1 is a sensitivity parameter and the optimal velocity function Vopt takes values in the interval [0, 1]. The velocity vn (t) is interpreted as a probability so that the position update of the SOV model is given by xn (t) + 1 xn (t + 1) = xn (t)
with probability vn (t + 1), with probability 1 − vn (t + 1),
(8.33)
i.e., in terms of expectation values one has xn (t + 1) = xn (t) + vn(t + 1). For a = 0, the SOV model reduces to the ASEP, and for a = 1 to the zero-range process. The fundamental diagram exhibits three different branches. Besides the free-flow phase, a
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congested stop-and-go phase characterized by a mixture of small clusters and free vehicles and a jam phase with one large stable jam are found. Six different density regimes have been identified where also two or three different branches can coexist. There is even a tristable regime, where all three branches are stable [706]. Another CA variant of the OV model can be derived using the ultradiscretization method [705]. Due to the close relation of the OV model with the modified KdV equation, this ultradiscrete OV model (udOV) inherits its solitonic properties. It exhibits three distinct regimes: jammed states which is absolutely stable, oscillatory stop-and-go states, and free flow which is found to be convectively unstable. Nagatani [983] has suggested a CA model that combines the OV idea with the totally asymmetric simple exclusion process (TASEP). It is the discrete analog of the simplified OV model presented in Section 9.4.2. Here the n-th vehicle moves ahead with probability vn (t), where vn (t) is interpreted as velocity. This velocity is obtained by integration of the OV function a for xn (t) ≥ xc (8.34) v˙n (t) = −a for xn (t) < xc , where a > 0, xn (t) = xn+1 (t) − xn (t), and xc is a safety distance. Furthermore, the velocity is restricted to the interval 0 ≤ vn (t) ≤ vmax ≤ 1. Another model which combines elements of car-following and CA has been proposed by Bham and Benekohal [112]. Their approach focusses on a more realistic description of the acceleration and deceleration processes. The Markov-process inspired model [1444] distinguished four different driving modes (stopped, free-driving, slowingdown, and car-following mode). It describes the variation of the gap between consecutive vehicles as a Markov process. The model can be considered as a weighted gap extension of the NaSch model. In contrast to the latter, vehicles will not try keeping the gap as small as possible, but a random gap whose length follows a certain distribution.
8.5. CA FROM ULTRADISCRETE METHOD In Section 2.5.4, we have derived the Burgers cellular automation (BCA) t t , L − Ujt ) − min(M , Ujt , L − Uj+1 ). Ujt+1 = Ujt + min(M , Uj−1
(8.35)
from the Burgers equation (2.74) through the ultradiscrete method. This model can be interpreted as a particle-hopping model where each site can hold L particles at most. Ujt denotes the number of particles at site j and time t. The maximum number of movable particles is M . In each timestep, as many as possible particles at site j can move to
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vacant space at site j + 1. M represents a bottleneck if one sets M < L. According to the t ). above rule, the number of movable particles at site j and time t is min(M , Ujt , L − Uj+1 Therefore, Ujt+1 is calculated by (8.35). We can easily see from the above rule that the total number of particles is conserved. The BCA can be considered as a simple traffic model. The physical meaning of this model may be interpreted in three different ways: (1) The road is L-lane freeway in a coarse sense, and effect of lane changes of cars is not considered explicitly. (2) Considering a single-lane freeway, then Ujt /L represents the probability that site j is occupied by a car at time t. If we choose large L, Ujt /L can take a fine value between 0 and 1. In the latter case, the number Ujt itself no longer represents the real number of cars at site j. (3) A site might represent a longer segment of the expressway capable of accommodating a maximum of L cars. If we set L = 1 and M > L, then these interpretations coincide and BCA becomes the rule-184 CA. Another possible application is bicycle traffic where multiple occupation of a cell is quite natural because several bicycles can move alongside [658, 668].
8.5.1. Generalizations of BCA In [1066, 1067], some generalizations of the BCA have been suggested. One of the examples is a high-speed generalization of BCA, which is given by the following two successive steps at each timestep: 1. Cars move to the next site according to BCA if the site is not fully occupied. 2. Only cars that moved in step (1) can move to another site if their next site is not fully occupied after step (1). The number of moving cars at site j and time t in step (1) is given by bjt ≡ min(Ujt , L − t ). Here, we neglect the parameter M for simplicity. In step (2), the number of movUj+1 t − bt t ing cars at site j + 1 becomes min(bjt , L − Uj+2 j+1 + bj+2), where the second term in min represents vacant spaces at site j + 2 after the first step (1). Therefore, considering a total number of cars entering into and escaping from site j, evolution equation is given by t t t − bjt + min bj−2 , L − Ujt − bj−1 + bjt Ujt+1 = Ujt + bj−1 t t t . , L − Uj+1 − bjt + bj+1 − min bj−1
(8.36)
This rule with L = 1 is equivalent to rule–3372206272 CA after Wolfram’s terminology [1476], which differs from the FI model [414] (Section 8.4.1). Note that this model is a combination of BCA and slow-to-start rules. If we write each procedure of above
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explicitly using an intermediate time step, we obtain t+1/2
t = Ujt + bj−1 − bjt . t+1/2 t+1/2 t+1/2 t t , L − Uj Ujt+1 = Uj + min (Uj−1 − Uj−1 − bj−1 t+1/2 t+1/2 , − Ujt − bjt , L − Uj+1 − min (Uj
Uj
(8.37)
(8.38)
t+1/2
where Uj denotes number of cars at site j just after step (1). Equation (8.37) is nothing but BCA, and (8.38) corresponds to the TT model (see in Section 8.1.3) [1067]. The fundamental diagram of the model (8.36) is also given in [1067]. It exhibits multiple metastable states around the critical traffic density.
8.5.2. Euler–Lagrange Transformation As discussed in Section 5.2.1, in analogy with fluid dynamics, there are two ways to represent traffic flow: the Euler and Lagrange form [1057]. A road is considered as a field, which becomes the dependent variable in the Euler form. Each particle (vehicle) is not distinguished, and the number of particles at each cell is stored by the dependent variable. BCA without M in Euler form is given by t t . (8.39) , L − Ujt − min Ujt , L − Uj+1 Ujt+1 = Ujt + min Uj−1 We have extended the BCA to the case of maximum velocity 2 in (8.36). However, the extension to general velocities is found to be difficult in the Euler form because it becomes complex in general when the number of neighboring sites becomes large. The Lagrange form becomes suitable when the interaction range is large. The dependent variable in this form is the position of each particle. Thus, we can follow each particle individually, while particles are not distinguished in the Euler form. In order to obtain the Lagrange form of the BCA, a Euler–Lagrange transformation has been proposed in [919]. First, we define G by the equation t t Ujt = GL( j+1) − GLj .
(8.40)
Substituting (8.40) into (8.35), we obtain the evolution equation of G as t+1 t t = max(GL( GLj j−1) , GL( j+1) − L).
(8.41)
Here we put Gjt =
N −1 i=0
H ( j − xti − 1),
(8.42)
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where H (x) is the Heaviside step function defined by H (x) = 1 if x > 0 and H (x) = 0 otherwise, and N is the total number of cars. Inserting (8.42) into (8.41), one obtains N −1 N −1 N −1 H ( jL − xt+1 − 1) = max H ( jL − xti − L − 1), H ( jL − xti + L − 1) − L . i i=0
i=0
i=0
(8.43) In order to simplify (8.43), we introduce a new identity n n n H ( j − min(ak , bk )) = max H ( j − ak ), H ( j − bk ) k=1
k=1
(8.44)
k=1
assuming a1 < a2 < · · · < an , b1 < b2 < · · · < bn . Another identity we use here is t H ( j − xi ) − L, 0 = H ( j − xti+L ). (8.45) max i
i
By using (8.44) and (8.45), since xti < xti+1 for any i, (8.43) can be transformed into
H ( jL − xt+1 − 1) = max i
i
=
H ( jL − xti − L − 1),
i
H ( jL − xti+L + L − 1)
i
H ( jL − min(xti + L + 1, xti+L − L + 1)).
(8.46)
i
Thus comparing both sides, we finally obtain xt+1 = min(xti + L + 1, xti+L − L + 1) − 1 i = xti + min(L, xti+L − xti − L),
(8.47)
which is the Lagrange form of the BCA.
8.5.3. Traffic Models in Lagrange Form We generalize (8.47) as = xti + min(V , xti+S − xti − S). xt+1 i
(8.48)
The BCA corresponds to the case V = S = L, but other important models are in this model class. The choice V = S = 1 gives the rule-184 CA and the FI model with the
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maximum velocity V is obtained for S = 1. In the model, cars can move forward at most V sites per unit time if their front sites are empty. S represents a perspective of drivers because it represents the number of cars that a driver can see in front. Next, we comment on a relationship between the OV model [69] (see Section 9.4.2) and the FI model. The Lagrange form of the FI model can be written as − 2xti + xti−1 = min(V , hit ) − (xti − xt−1 xt+1 i i ),
(8.49)
where hit = xti+1 − xti − 1 is the headway. Taking the continuous limit t → 0 and after appropriate scalings, we obtain dxi d 2 xi = min(V , hit ) − . 2 dt dt
(8.50)
This is nothing but an OV model with the piecewise linear OV function min(V , hjt ) [1031]. The TT slow-to-start model (Section 8.1.3) can be written in Lagrange form as t−1 = xti + min(1, xti+1 − xti − 1, xt−1 − 1). xt+1 i i+1 − xi
(8.51)
The velocity of cars depends not only on the present headway but also on the past headway. Next, the NaSch model is written as
= xti + max 0, min V , xti+1 − xti − 1, xti − xt−1 + 1 − ηit , (8.52) xt+1 i i where ηit = 1 with probability p and ηit = 0 with 1 − p. Finally, an extended model of (8.48) in order to include the TT slow-to-start effect is given as [1059] k t t (8.53) = xtj + min Vjt , min hj+i + Vj+k xt+1 j k=1, ..., S−1
where
i=0
t−1 Vjt = min V , xtj+S − xtj − S, xt−1 − x − S . j+S j
(8.54)
The term mink=1,...,S−1 (· · · ) in the right-hand side of (8.53) plays an important role in avoiding collisions of cars. This is because the condition that a car does not overtake all the viewable cars in its front is given by k i=0
t t hj+i + Vj+k ≥ Vjt ,
(k = 1, . . ., S − 1).
(8.55)
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1.2 A
1 0.8 B J
0.6 0.4 −S/2
V
0.2
0.2
0.4
0.6
0.8
1
C 1
0 ρ
(a)
(b)
Figure 8.6 (a) A fundamental diagram of the Lagrange model (8.53) obtained by computer simulation with periodic boundary conditions for V = 5 and S = 2. (b) A schematic fundamental diagram of the model. The line O − A represents free flow and the line B − C corresponds to the jammed phase.
The fundamental diagram is given in Fig. 8.6. It shows a complex phase transition from free to congested traffic. There are many metastable branches in the diagram, as seen in observed data on highways. The schematic diagram is given in Fig. 8.6. In state A, the unstable stationary pattern · · 000 11 · · 11 000 · · S
(8.56)
V
is observed, and its density and average velocity become S/(V + S) and V , respectively. Also at B, from the stationary pattern · · 000 11 · · 11 000 · · S
(8.57)
2V
the density and velocity become S/(2V + S) and V , respectively. The branch A − B represents overdense free flow. It corresponds to metastable states that will change into a congested state due to perturbation. There exist both metastable weak jams (in the dotted region) and strong jams ( line B − C) in the congested state. A stochastic extension of this model is proposed in [1206].
Vehicular Traffic III: Other CA Models
8.6. CA MODELS OF MULTILANE TRAFFIC For a realistic description of traffic on highways, the idealized single-lane models must be generalized to develop models of multilane traffic. Lane-changing maneuvers are often an initial perturbation, which lead to the formation of a jam, especially near bottlenecks such as on- and off-ramps. Therefore, it is important to understand their impact on stability, capacity, and so on. Here models are particularly useful because only few empirical results exist, see, e.g., [188, 490, 1303, 1360]. The main ingredient required for the generalization of single-lane CA model to multilane situations are lane-changing rules. This has been a the main focus of most early articles, which are reviewed and discussed in more detail in [1029] and [216]. In the following, we will focus on two-lane traffic, although situations with more than two lanes have also been considered, e.g., in [197]. The lane changing rules for two-lane traffic can be symmetric or asymmetric with respect to the lanes. For symmetric lane-changing rules, overtaking is allowed on both lanes. However, for asymmetric lane-changing rules, overtaking is forbidden on one lane, e.g., on the right lane in many European countries. Besides this lane (a)symmetry, the rules can also be symmetric or asymmetric with respect to the vehicles (e.g., cars and trucks).
8.6.1. Classification of Lane Changing Rules Generically, the decision of drivers to change lane is based on two criteria: • Incentive criterion: Drivers determine whether a lane change improves the individual traffic situation, e.g., to move at their desired velocity. • Safety criterion:The traffic situation in the target lane is checked, especially the available gap for a lane change, to estimate the effect on upstream vehicles. A lane-change is then only performed if both criteria are satisfied. In general, the update in the two-lane models is divided into two substeps: in one substep, the vehicles may change lanes in parallel following the lane-changing rules and in the other substep, each vehicle may move forward effectively as in the single-lane NaSch model. Drivers must find some incentive in changing the lane. Two obvious incentives are (1) the situation on the other lane is more convenient for driving, and (2) the need to make a turn in near future. Two general prerequisites have to be fulfilled in order to initiate a lane change: first, there must be an incentive and second, the safety rules must be fulfilled [1303]. Lane-changing rules according to this scheme have been introduced by Rickert et al. [1186]. They suggested that vehicles are allowed to change the lane if the following four criteria are satisfied: LC1: Incentive criterion 1: gap(i) < l, LC2: Incentive criterion 2: gapo (i) > lo ,
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LC3: Safety criterion: gapo,back (i) > lo,back , LC4: Randomization: rand() < pc . Here gap(i) and gapo (i) are the gaps in front of vehicle i on the own lane and the other lane3 , respectively. gapo,back (i) is the gap on the other lane to the next vehicle behind. l, lo , lo,back , and pc are parameters specifying the rule and rand() is a random number in the interval [0, 1]. The first rule LC1 represents the incentive criterion, i.e., if the gap, gap(i), in front of the vehicle is not sufficiently large vehicles want to change the lane. Typical choices of the parameter l are given by l = min(v + 1, vmax). This choice of the minimal headway ensures that vehicles driving in a slow platoon try to change the lane if possible. In the next rule LC2, it is checked if the situation on the other lane is indeed more convenient. This motivates the choice l = lo . The third rule LC3 avoids too small distances of following vehicles on the other lane. Rickert and coworkers suggested lo,back = vmax . It is also important to perform lane changing stochastically. Even if the incentive and safety criteria are fulfilled, a lane change is performed only with probability pc (LC4). This avoids, at least partially, so called ping-pong lane changes, i.e., multiple lane-changes of vehicles in consecutive timesteps. This artifact of the parallel update was pointed out by Nagatani [976, 979, 981, 986], who simulated a two-lane system with vmax = 1. Already implementations of the NaSch model using the basic lane-changing rules revealed quite realistic results. Nevertheless several variants of the basic rules have been developed in order to improve the realism. A large number of lane changing rules considered in the literature have been tabulated and compared by Nagel et al. [1029] (see also [216]). The lane changing rules for two-lane traffic can be symmetric or asymmetric with respect to the lanes [1186]. If symmetric lane-changing rules are applied, the rules do not depend on the direction of the lane-changing maneuver. In contrast, also asymmetric lane-changing rules have been considered. Lane-changing rules can be asymmetric in two ways. First, it is possible that it is preferred to drive on the right lane at low densities. This behavior can be implemented simply by leaving out the first rule for a change from the left to the right lane. Second, it is also possible that it is even forbidden to overtake a vehicle on the right lane, e.g., on German highways. Then the single-lane dynamics on the right lane depends on the configuration on the left lane. These examples show the flexibility of the CA approaches. Moreover, the simulations also show that the details of the lane-changing rules may lead to considerable changes of the model results [1029, 1432]. In multilane traffic, it is of particular interest to investigate systems with different types of vehicles. For CA models, this has been done first by Chowdhury et al. [221] 3 The gap on the other lane is defined in the same way as the gap on the own lane by imagining that the vehicle occupies
the site parallel to its current position.
Vehicular Traffic III: Other CA Models
who simulated a periodic two-lane system with slow and fast vehicles, i.e., vehicles with different vmax. The simulation results have shown that already for small densities, the fast vehicles take on the average free-flow velocity of the slow vehicles, even if only a small fraction of slow vehicles is present. Analogous results have been obtained by Helbing and Huberman [527] who used a different CA model for the in-lane update. Nagel et al. [1029] have shown that for a suitable choice of the lane-changing rules and different types of vehicles even the phenomenon of lane inversion, which has been observed at German highways can be reproduced. On German highways, the left lane is considered for overtaking vehicles only. Therefore, at low densities, the right lane is used more often. Surprisingly, at higher densities not simply a balancing of the lane usage has been observed, but for densities close to the optimum flow, the left lane is even higher frequented. The results discussed so far show the strong influence of slow vehicles in multilane systems. They fit fairly well the empirical results, which show an alignment of the speeds on different lanes and of different types of vehicles. Nevertheless, simulation results of Knospe et al. [772] indicate that the influence of slow vehicles is overestimated by the multilane variants of the NaSch model. In particular for symmetric lane-changing rules even a single slow vehicle can dominate the dynamics close to the optimal value of the flow. This effect can be weakened by introducing anticipation [772]. Another interesting quantity to look at is the frequency of lane changes at different densities. Here the simulation results show that close to the density of maximal flow, the number of lane-changing maneuvers drastically decreases if the small values of the braking noise are considered in CA models, where the velocities of vehicles are solely determined by the distance to the vehicle ahead. This is due to the fact that for homogeneous states at high densities, no sufficiently large gaps exist. For larger values of the braking noise, large density fluctuations are observable. Therefore, the local minimum of the lane-changing frequency is not found for larger values of p. In general, the simulation results show that some generic multilane effects can be pointed out. First of all, the maximal performance of multilane systems is slightly increased compared with corresponding single-lane network. In addition, slow vehicles lead to an alignment of velocities of different type of vehicles already at low densities, which is confirmed by empirical observation. This effect is quite robust for different choices of the CA model, as well as for different lane-changing rules. It can be weakened most efficiently if anticipation effects are applied. The details of the lane-changing rules, however, may have strong influence on the fundamental diagram, lane usage characteristics, and so on, see, e.g., [54, 599, 954]. The basic principles described earlier have been realized in most multilane extensions of CA models. These extensions, which are particularly relevant in the context of synchronized traffic, are discussed in the sections devoted to specific models. A toy model based on the CA184 has been studied by Belitsky et al. [89]. It allows to study certain
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fundamental aspects in an analytical way. The same is true for a formulation of two-lane traffic as a reaction-diffusion process [395]. A two-lane extension of the ASEP including disorder in the hopping rates and open boundary conditions has been investigated by Harris and Stinchcombe [500]. Finally, we mention that Kesting et al. [745] have proposed a general decision model called MOBIL (minimizing overall braking induced by lane changes) for lane changes, which is especially suited for car-following models. It uses accelerations as utility measure for the attractiveness and risk of lanes changes. In addition, a politeness factor takes into account the cooperativeness among the drivers.
8.6.2. CA Models of Bidirectional Traffic Simon and Gutowitz [1295] have introduced a two-lane CA model, where the vehicles move in opposite directions. Passing may be allowed on one or on both lanes. It is only attempted if there is a chance to complete the pass. Therefore, drivers measure the local density, i.e., the density of vehicles in front that have to be passed. If it is sufficiently low, a pass will be attempted. This means that at high global densities, the lanes are effectively decoupled because only very few passes will occur. In principle, three types of jams can occur on a bidirectional road: (1) Spontaneous jamming and start-and-stop waves on one of the lanes; (2) jams caused by drivers who try to pass but can not return to their home lane because there is not enough space; and (3) “super jams” when an adjacent pair of drivers tries to pass simultaneously. These super jams halt traffic on both lanes and can be prevented by breaking the symmetry between the lanes. The precise rules of the CA are in the same spirit as the rules for multilane traffic described in Section 8.6.1. First, the situation on the own lane is examined. If the motion is hindered by another vehicle (moving in the same direction), a pass is attempted. This will only be initiated if the safety criteria are satisfied: (1) The gap on the other lane has to be sufficiently large, and (2) the number of vehicles to be passed has to be small. Even if these criteria are satisfied, a lane change occurs only with probability pchange . After this lane-changing step, the vehicles move forward similar to the dynamics of the NaSch model. There are, however, important differences. Passing vehicles never decelerate randomly. In order to break the symmetry between the two lanes, moving vehicles which are on their home lane and see oncoming traffic decelerate deterministically by one unit. This rule prevents the occurrence of a super jam. The results of [1295] show the expected behavior that passing makes traffic more fluid. Start-stop waves are suppressed if the density is not too large. The improvement of the flow on one lane compared with the one-lane model depends on the density of vehicles on the other lane. It is maximal for very small densities (ρ → 0) on the passing
Vehicular Traffic III: Other CA Models
lane. If the density on the other lane is small (ρ < 0.25), the flow may be lower than in the one-lane model because passing oncoming vehicles create an additional hindrance. For large densities on at least one of the lanes, there is little difference between the oneand two-lane models. Moussa [952] has studied the congested patterns in more detail. He proposed modified lane-changing rules to make the model more realistic by avoiding the occurrence of wide jams on both lanes in the case of large density differences between the lanes. Lee et al. [840] have proposed a toy model for bidirectional traffic based on a multispecies generalization of the ASEP. Here no passing is allowed. Instead oncoming traffic on the opposite lane reduces the hopping rates of the vehicles. The dynamics on each lane is given by that of the ASEP with random-sequential update4 and vmax = 1, but the hopping rate from an occupied cell j to an empty cell j + 1 on lane 1 depends on the occupancy of cell j + 1 on the opposite lane (lane 2). When this cell is empty, vehicles hop with rate 1, otherwise with rate 1/β. On lane 2, vehicles move in the opposite direction and the hopping rate from cell j + 1 to cell j depends on the occupancy of cell j on lane 1. It is given by γ when this cell is empty and by γ /β if it is occupied. For γ < 1, the uninfluenced hopping rate on lane 2 is smaller than that of lane 1. The vehicles on lane 2 might, therefore, be interpreted as trucks. The interlane interaction parameter β can be interpreted as a kind of road narrowness. For β = 1, vehicles are not slowed down by oncoming traffic. This corresponds to a highway with divider. The case β → 0 corresponds to a narrow road being completely blocked by the oncoming traffic. The behavior of the model with only one truck is rather similar to that of the NaSch model with quenched disorder (see Section 8.7). For β > βc , the system segregates into two phases, a high-density phase in front of the truck and a low-density phase behind it. By forbidding trucks and cars to occupy parallel cell j, simultaneously, the model can be mapped onto an exactly solvable two-species variant of the ASEP. Using the matrixproduct Ansatz (see Section 2.4), many steady-state properties for the single-truck case can be obtained exactly. Two phase can be distinguished: A low-density phase for ρβ < 1 and a jammed phase for ρβ > 1, where ρ is the density of vehicles on lane 1. In contrast to the case of a fixed defect site (see Section 8.7.3), only one critical density ρcrit = 1/β exists because the particle-hole symmetry is broken. In [953], conditions for collisions in the bidirectional model have been investigated. Three different accidents can be distinguished: (1) head-on collisions, (2) rear-end collisions, and (3) lane-changing collisions. It has been found that the collision risk is important when the density in one lane is small, but high in the other lane. Another relevant factor is the fraction of slow vehicles (trucks). Under the conditions investigated 4 Other updates have been studied in [397].
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in [953], the dominating accident-type are rear-end collision, which usually happen due to jam formation. The other accident-types are more dangerous for the health of the drivers and occur typically when faster cars overtake slower vehicles. A two-way traffic model based on two coupled five-vertex models was investigated in [1117]. It can be solved by reducing it to the model introduced by Brankov et al. [152].
8.7. EFFECTS OF QUENCHED DISORDER In the following, we briefly discuss the effects of quenched disorder, i.e., timeindependent inhomogeneities in various parameters. General aspects have been described in more detail in Section 4.6.1 of Part I, mainly for the ASEP. These remain valid in the more general cases discussed here, with mostly only minor modifications. However, there are now different natural forms of particlewise and sitewise disorder, which will be considered separately in the following.
8.7.1. Randomness in the Braking Probability We have seen how modifications of the rule(s) for random braking in the NaSch model can give rise to a rich variety of physical phenomena, e.g., self-organized criticality, metastability, and hysteresis, and so on. Now we consider the effects of quenched randomness in the random braking probability p; i.e., we study the effects of assigning randomly different time-independent braking probabilities pn to different drivers n in the NaSch model. Such quenched (i.e., time-independent) randomness in the random braking in the NaSch model can lead to exotic phenomena [772, 819], which are reminiscent of Bose-Einstein-like condensation in the TASEP where particle-hopping rates are quenched random variables [355, 356, 815]. Various aspects of these phenomena have been presented in Section 4.6.1 and, therefore, we will restrict our discussion to only the essential points. The qualitative features of the dynamical phases and phase transitions observed in the NaSch model with random braking probabilities are very similar to those of the TASEP with random hopping probabilities [819]. Typical snapshots of the system at three different stages of evolution from a random initial state are shown in Fig. 8.7. The typical size of the platoons ξ(t) can be determined directly [819] by computing the equal-time density–density correlation function and identifying the separation r = R0 of the first zero-crossing of this correlation as ξ(t). Following this procedure, it has been observed that ξ(t) follows the power law (4.211), when the
n distribution of the random braking probabilities is given by P( p) = 2n (n + 1) 12 − p .
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← Time
Space (road) →
Figure 8.7 Typical space-time diagram of the NaSch model with random braking probabilities at three different stages of evolution from a random initial state.
It is also possible to include random acceleration by modifying the acceleration rule (NaSch1) to [402] vn → min{vn + an , vmax } with
an = [rn dn ] + 1,
(8.58)
where [x] denotes the integer part of x and rn is a quenched variable drawn from some probability distribution in the interval [c, 1]. In [402], also random braking was introduced by randomly decreasing the velocity by fn . This is achieved by replacing max{0, vn − 1} in rule (NaSch3) with max{0, vn − fn } where fn = [qn min{vmax , dn }] + 1
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with a quenched variable qn drawn from [c, 1]. These modifications of the dynamics lead to several new features, e.g., an intermediate regime where the current is quasi-constant.
8.7.2. Random vmax The two important parameters of the NaSch model are p and vmax . In the preceding subsection, we have seen the effects of randomizing p assigning the same vmax to all the vehicles. In this subsection, however, we investigate the effects of randomizing vmax , assigning a nonrandom constant p to every driver. The effects of the quenched randomness in vmax on the steady states of traffic-like models have been studied for a simple model of platoon formation [95–98, 985], which was developed using the language of aggregation phenomena. In this model, an initial velocity vj is assigned to each vehicle j, drawn randomly from a continuous probability density f (v). The particles then move ballistically along a line and coalesce whenever a faster vehicle catches up with a slower one in front. It has been found that ξ(t), the typical platoon size at time t, increases indefinitely according to the power law ξ(t) ∼ t (n+1)/(n+2) where the exponent n characterizes the behavior of f (v) in the vicinity of the minimal velocity vmin , i.e., f (v) ∼ A(v − vmin )n as v → vmin with some positive constant A. An attempt has been made to develop a coarse-grained description of this phenomenon [1015]. In order to model traffic consisting of two different types of vehicles, say, for example, cars and trucks, of which a fraction f fast are intrinsically fast (say, cars) while the remaining fraction 1 − f fast are intrinsically slow (say, trucks), Chowdhury et al. [221] assigned a higher vmax (e.g., vmax = 5) to a fraction f fast of vehicles chosen randomly while the remaining fraction 1 − f fast were assigned a lower vmax (e.g., vmax = 3). As the density of the vehicles increases, the vehicles with higher vmax find it more difficult to change lane in order pass a vehicle with lower vmax ahead of it in the same lane. This leads to the formation of coherent moving blocks of vehicles each of which is led by a vehicle of lower vmax [527]. Two main causes of traffic accidents, namely, differences in vehicles speeds and lane changes, are reduced considerably in this state thereby making this state of traffic much safer. It is worth mentioning that even a small number of slow vehicles in two-lane models, where overtaking is possible, can have a drastic effect. For details, we refer to [772] and the discussion in Section 8.6. The effect of trucks in the NaSch model has further been studied in [335]. In [654], the NaSch model with fast and slow vehicles of different lengths has been investigated. Surprisingly a higher fraction of long vehicles can improve flow in certain situations.
8.7.3. Randomly Placed Bottlenecks So far we have investigated the effects of two different types of quenched randomness both of which were associated with the vehicles (i.e., particles). We now consider the
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effects of yet another type of quenched randomness, which is associated with the road (i.e., lattice). In order to anticipate the effects of such randomness associated with the highway, let us begin with the simplest possible caricature of traffic with a “point defect” [225]: a single defect site in the deterministic limit p = 0 of the NaSch model with vmax = 1. In this model, vehicles move forward, in parallel, by one lattice spacing if the corresponding site in front is empty; each vehicle takes Timp (> 1) timesteps to cross the defect site but only one timestep to cross a normal site when the next site is empty. The defect sites acts like a blockage for all Timp > 1. As explained in Section 7.2.2, in the absence of the defect, J = ρ for 0 < ρ ≤ 1/2 and J = 1 − ρ for 1/2 < ρ ≤ 1. Note that, if the defect is present, 1/Timp vehicle passes through the defect site per unit time. Therefore, the bottleneck created by the defect introduces an upper cut-off of the flux, viz., 1/Timp . Obviously, J = ρ < 1/Timp so long as ρ < ρ1 = 1/Timp . Similarly, J = 1 − ρ < 1/Timp for ρ > ρ2 , where ρ2 = 1 − ρ1. In the density interval ρ1 < ρ < 1 − ρ1, the bottleneck at the defect is the flow-limiting factor and, hence, in this regime, J = 1/Timp is independent of ρ. Thus, in the simple caricature of traffic under consideration the flux varies with density following the relation ⎧ ⎪ ⎨ρ J = 1/Timp ⎪ ⎩ 1−ρ
if 0 < ρ ≤ ρ1 , if ρ1 < ρ ≤ ρ2 , if ρ2 < ρ ≤ 1,
(8.59)
where ρ1 =
1 1 + (t)imp
and ρ2 =
(t)imp , 1 + (t)imp
(8.60)
and Timp = 1 + (t)imp such that (t)imp = 0 for the normal sites but (t)imp > 0 for the defect site. Some alternative parameterizations of the defect sites in the NaSch model for arbitrary vmax have also been suggested. In [771, 1212], localized defects have been investigated where the randomization parameter pd is larger than in the rest of the system. Csahok and Vicsek [246] have considered the blockages as sites with a permeability smaller than unity, whereas the permeability of all the other sites is unity. This effectively reduces vmax while the vehicle is at a blockage. However, ER [213, 347] considered a model of where the velocity of every vehicle in the region occupied by the blockage (or, more appropriately, hindrance) at the time step t + 1 is half of that at time t, i.e., vn (t + 1) = vn (t)/2 is the n-th vehicle is located within the hindrance region. Some effects of static hindrances on vehicular traffic have also been investigated following alternative approaches, e.g., car-following theory [989].
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Defects in the FI model have been studied in [630]. The defects were realized as gates which open and close with certain probability. Overall a similar behavior as described above was observed. A mean-field approach was used to derive analytical results for vmax > 1, which were found to be in good agreement with simulations. The influence of tollbooths on the traffic has also been investigated [195, 605, 653, 667, 784]. Drivers have to stop for a certain waiting time at the tollbooth to pay the toll. This has similar effects as the bottlenecks discussed earlier. When the vehicle density reaches a critical value, a first-order phase transition into a queuing phase occurs and the flow saturates [605]. The corresponding fundamental diagrams are, therefore, characterized by a plateau at intermediate densities, as in the case of simple bottlenecks. To increase the capacity in the tollbooth area often single-lane highway is expanded into two (or more) branches [605, 667]. This can indeed lead to a substantial increase of the capacity, depending on the vehicle density and details of the merging section behind the tollbooth. Another important scenario is a reduction in the number of lanes, e.g., when lane is closed due to road construction. This has been studied, e.g., in [657] and [334]. It was found that the capacity of the bottleneck is slightly smaller than that of a single-lane road. Furthermore, the density distribution upstream of the closing depends on the regulations at the closing, especially the length of the merging area.
8.7.4. Ramps The most important type of quenched disorder in real traffic are on- and off-ramps where vehicles can enter and leave a highway. The implementation of such ramps follows a similar logic to that of multilane traffic by specifying incentive and safety criteria. However, the ramp lane either has an open boundary, which is characterized by an inflow Jin in the case of an on-ramp or acts as a reservoir for vehicles trying to enter the highway. The effects of ramps on the dynamics in the NaSch model and its generalizations have been studied in detail in various works. In one of the earliest studies, Diedrich et al. [310] have shown that ramps have effects very similar to those of a static defect at the position of the ramp. In a certain density regime ρl < ρ < ρh , one observes a plateau in the fundamental diagram. Here the ramp induces phase separation into a free-flow and a jammed regime. The propagation of jams can be interrupted, and they become localized at the ramp. Similar conclusions were reported in [174], where the role of on-ramps as nucleation points was found and in [598]. These fundamental results have been extended in various ways. In [614, 671, 672, 681], the influence of the main road on the ramp was considered. Four different phases were found. The order of the phase transitions depends on the length of the interaction section, and increasing this length can lead to a higher capacity. Further studies of the
Vehicular Traffic III: Other CA Models
influence of acceleration and deceleration lanes can be found in [655, 656]. In [666], the influence of the randomization p in the NaSch model on the phase diagram has been compared for a one-lane and two-lane highway. More detailed rules for on-ramps were proposed in [1098]. Nassab et al. [1036] have shown that in the NaSch and VDR model, the spacing between on- and off-ramp can have a strong influence on the behavior. For the deterministic NaSch and the brake-light model, methods for flow optimization in the presence of several ramps were investigated in [227, 1446]. Introducing ramp metering by a traffic signal can reduce the mutual perturbations of the flows on the main road and the ramp considerably [865]. The influence of speed limits for a two-lane road near a ramp has been studied in [866]. Popkov et al. [1121] have shown that real traffic data close to on-ramps can be interpreted in terms of boundary-induced phase transitions. Furthermore, simplified models, mostly based on the ASEP or in deterministic limits, have been investigated to obtain analytical results. Systems with ramps of just one cell and open boundary conditions were studied in [603] for the deterministic NaSch model and in [380]. Ramps in the ASEP were considered in [600, 601]. In [90], a series of dynamical phase transitions inside the jammed phase has been found in the approach to stationary state in a deterministic model with stochastic inflows. Each dynamical phase is characterized by a fixed number of relaxation cycles. Further investigations of the effect of ramps have been performed in the context of synchronized traffic (see Section 8.3). In these studies, more realistic models are used. Typically, the focus is on the classification of traffic patterns that occur in the vicinity of the ramps and their dynamics.
8.8. BUS-ROUTE MODEL A kind of dynamical disorder can be induced, e.g., by the interaction of different types of traffic. One simple example is the bus-route model (BRM) [1082, 1083]. It describes the motion of buses that can pick up passengers at bus stops, which are represented by the sites i = 1, 2, ..., L of a one-dimensional lattice with periodic boundary conditions. N buses are moving from one bus stop to the next picking up passengers. Two binary variables σi and τi are assigned to each cell i: 1 cell i is occupied by a bus 1 passengers are waiting cell i σi = , τi = . (8.61) 0 otherwise 0 otherwise Because a cell cannot have simultaneously a bus and waiting passengers, let us impose the condition that a cell cannot have both σi = 1 and τi = 1, simultaneously. Each bus is assumed to hop from one stop to the next.
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αb
βb λ
Figure 8.8 Definition of the bus-route model.
Next, let us specify the update rules (Fig. 8.8): a cell i is picked up at random. Then, BRM1: If σi = 0 and τi = 0 (i.e, cell i contains neither a bus nor waiting passengers), then τ → 1 with probability λ, where λ is the probability of arrival of passenger(s) at the bus stop. BRM2: If σi = 1 (i.e., there is a bus at the cell i) and σi+1 = 0, then the hopping rate μ of the bus is defined as follows: αb μ= βb
for τi+1 = 0, for τi+1 = 1.
(8.62)
BRM3: The bus moves with hopping rate μ to the next site. Where αb is the hopping rate of a bus onto a stop that has no waiting passengers and βb is the hopping rate onto a stop with waiting passenger(s). Generally, βb < αb , which reflects the fact that a bus has to slow down when it has to pick up passengers. We can set αb = 1 without loss of generality. When a bus hops onto a stop i with waiting passengers, τi is reset to zero as the bus takes all the passengers. Note that the density of buses ρ = N /L in a conserved quantity, whereas that of the passengers is not. An ideal situation in this bus-route model would be one where the buses are evenly distributed over the route so that each bus picks up roughly the same number of passengers. However, because of some fluctuation, a bus may be delayed and, consequently, the gap between it and its predecessor will be larger than the average gap. Therefore, this bus has to pick up more passengers than what a bus would do on the average and will get further delayed. However, the following bus has to pick up fewer passengers than what a bus would do on the average and, therefore, it would catch up with the delayed bus from behind. The slowly moving delayed bus would slow down the buses behind it thereby, eventually, creating a jam. In other words, once a larger-than-average gap opens up between two successive buses, the gap is likely to grow further, and the steady state in a finite system would consist of a single jam of buses and one large gap. This is very similar to the Bose-Einstein-condensation-like phenomenon we have observed earlier in particle-hopping models with slow impurities. On the basis of heuristic arguments and mean-field approximation, it has been argued [1082, 1083] that this model exhibits
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a true phase transition from an inhomogeneous low-density phase to a homogeneous (but congested) high-density phase only in the limit λ → 0. The BRM with parallel dynamics has been studied in [208], where also its connection with the NaSch model has been elucidated. A model that takes into account the finite capacity Nmax of the buses was introduced in [663]. The phase diagram in Nmax − λ-plane consists of five phases: (1) insufficient transport capacity where the number of waiting passengers increases in time; (2) bus bunching, similar to the standard BRM; (3) phase separation into bunched and freeflow regions; (4) almost periodic oscillations between bunching and phase separation, and (5) bistability where either the bunching or phase-separating state is realized. This nontrivial phase diagram also allows to conclude that the efficiency of the bus-route system can not always be improved through increasing the number of buses. An extended model, called public conveyance model has been proposed by Tomoeda et al. [1372]. Here, there a S equispaced bus stops distributed over a system of length L. For S = L, this includes a hail-and-ride system like the BRM. In each timestep, a passenger arrives with probability f in the system and is assigned with equal probability 1/S to one of the stops. For buses with finite passenger capacity Nmax , the hopping rates for the buses are chosen as H=
Q , min(Nj , Nmax ) + 1
(8.63)
where Nj is the number of passengers waiting at stop j. Furthermore, buses can be provided with information about the number of buses in the segment to the next stop. This allows the implementation of an information-based traffic control system. More details of the update rules can be found [1372]. Computer simulations and mean-field analysis show that the system works most efficiently in a region of moderate density of buses. If there are too many buses, efficiency is reduced due the mutual hindrance of buses. If there are too few buses, passengers are kept waiting at the stops or the buses are slowed down because they need to pick up a large number of passengers. The information-based traffic control system can improve the efficiency in a certain density region, but not in all possible situations. The bus-route problem has also been studied with other types of models, e.g., carfollowing models [565, 624, 996, 997, 1001] or nonlinear maps [998]. Empirical results for the headway statistics in public transport systems have analyzed in [794] and found to be in good agreement with the Gaussian unitary ensemble (GUE) of random matrix theory. Models have been proposed in [60, 795]. A problem that has certain similarities with the BRM is elevator traffic [999, 1002, 1003, 1126], where the role of the bus stops is taken by the different floors. In this case, it is natural to have different inflow rates, especially for the ground floor.
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8.9. ACCIDENTS An important aspect of real traffic that is usually neglected in modeling approaches is the occurrence of accidents although these are responsible for a considerable fraction of jams, either directly or indirectly (e.g., through passing drivers that slow down). With regard to accidents, traffic models can be classified into two classes: (1) models that are intrinsically accident-free, and (2) models which are not. “Intrinsically” means that the dynamics is defined in such a way that accidents are strictly avoided (for every initial condition). A typical example are CA like the NaSch model where Step 2 in the dynamics (and the exclusion principle) enforces the absence of accidents. However, this can lead to unrealistically large decelerations. Extreme braking manoeuver might have a strong effect on the dynamics and are a possible origin of jams. In principle, two types of accidents in real traffic can be distinguished, namely those due to careless drivers and those related to technical failures (blown tyre etc.). The latter are difficult to implement explicitly in simple models and could be treated as random events. Careless and aggressive drivers, however, can be taken into account, e.g., by modification of the dynamical rules in CA models. One general problem with models that are not intrinsically accident-free is the need to specify how to deal with these accidents if they occur in the simulations. This is an important issue because it might have a strong influence on the dynamics, especially for microscopic quantities.5 However, in most investigations, accidents are only occurring virtually and the simulated dynamics is always accident-free. Therefore, it is better to speak about dangerous situations instead of accidents. Because accidents do not occur during the simulations, but are replaced by a “safe” dynamics, quantities like the accident probability are just special correlation functions of the original model. In fact, it is closely related to the spatial headway distribution. Some studies have tried to elucidate the importance of accidents in CA models. In these approaches, the braking rule is modified such that with a certain probability q, drivers do not respect the safety distance in Step (NaSch2), i.e., with some probability pc for careless driving, the velocity after this step is vn(2) = dn + 1. This leads to an accident, if the car in front of the careless driver will not move in the same timestep. The first CA model including dangerous situations (accidents) has been proposed by Boccara et al. [133]. They replaced the update rule of the NaSch model by the rule if vn+1 (t) > 0,
then xn (t + 1) = xn (t) + vn(t + 1) + v,
(8.64)
5 A critical discussion about the treatment of accidents in some articles and how it can lead to artifacts in the dynamics
can be found in [1237].
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where v is a Bernoulli random variable which takes the value 1 with probability pc and zero with the probability 1 − pc. If vn+1 (t + 1) = 0, i.e., the preceding car stops, then careless driving will result in an accident. These modified rules can be rephrased as follows: An accident occurs with probability pc if the following three accident criteria are satisfied [133]: (1): dn (t) ≤ vmax ,
(2): vn+1 (t) > 0 ,
(3): vn+1(t + 1) = 0 ,
(8.65)
where dn is the number of empty cells in front of vehicle n. This modification has first been studied for the deterministic limit of the NaSch model in [133]. Accidents only occur on the jammed branch of the fundamental diagram, i.e., for densities ρ > ρc = 1/(vmax + 1). The probability Pac of an accident (per car and timestep) is a nonmonotonic function of the vehicle density ρ and exhibits a maximum at some density ρ ∗ > ρc [133, 613]. These investigations have later been generalized to other models, like the probabilistic version of the NaSch model [610, 611, 670, 947, 1496, 1500, 1501], the VDR model [1498], models of synchronized flow [665], two-lane models [950] and other models and situations [664, 1501, 1519], e.g., in the presence of bottlenecks and other types of quenched randomness [1497, 1499]. In the original definition, dangerous situations only occur with stopped cars and thus only at high densities. Therefore modifications of the accident criteria (8.65) have been suggested, e.g., in [949]. In reality, accidents due to careless driving more generally involve vehicles with large velocity differences. Another source of dangerous situations are abrupt velocity changes and small safety gaps. This poses the question how relevant these investigations are for real traffic accidents. It appears that the probability of dangerous situations is systematically overestimated due to the correlations induced by the fact that accidents are not explicitly modeled. If an “accident” has occurred it is rather likely that another one, involving the same vehicles, will occur shortly after that because the conditions are usually still fulfilled. A first step in the direction of a more realistic investigation of traffic accidents has been made in [40]. A model is proposed that describes the dynamics of a platoon of vehicles undergoing emergency braking, but taking into account reaction times and braking capabilities. It can be applied to real platoons obtained from empirical studies, which takes into the effects of short headways (Section 6.8) and allows to investigate the impact of legal regulations (speed limits or minimum headways).
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CHAPTER NINE
Vehicular Traffic IV: Non-CA Approaches Contents 9.1. Fluid-Dynamical Theories 9.1.1. Lighthill–Whitham–Richards Theory and Kinematic Waves 9.1.2. Diffusion Term in LWR Theory and Its Effects 9.1.3. Greenshields Model and Burgers Equation
336 337 340 341
9.2. Second-Order Fluid Dynamical Theories 9.2.1. Special Models 9.2.2. Instabilities and Jam Formation 9.2.3. Problems with Second-Order Models 9.2.4. Aw–Rascle Model 9.2.5. Fluid-Dynamical Models and Synchronized Traffic 9.2.6. Fluid-Dynamical Theories of Traffic on Multilane Highways and in Cities
342 344 345 348 348 349 350
9.3. Gas-Kinetic Models 9.3.1. Prigogine Model 9.3.2. Paveri-Fontana Model 9.3.3. Derivation of Fluid-Dynamical Equations from Gas-Kinetic Equations
351 351 353 356
9.4. Car-Following Models 9.4.1. Follow-the-Leader Model 9.4.2. Optimal Velocity Model and Its Extensions 9.4.3. Generalized Force Models 9.4.4. Intelligent Driver Model 9.4.5. Kerner–Klenov Model 9.4.6. Inertial Car-Following Model
357 358 360 364 366 368 369
9.5. Coupled-Map Models 9.5.1. Gipps Model 9.5.2. Krauss Model (SK Model) 9.5.3. Yukawa-Kikuchi Model 9.5.4. Nagel-Herrmann Model
371 372 373 375 376
9.6. Other Approaches 9.6.1. Probabilistic Traffic Flow Theory 9.6.2. Cell Transmission Model 9.6.3. Queueing Models
377 377 379 381
Stochastic Transport in Complex Systems DOI: 10.1016/B978-0-444-52853-7.00009-9
Copyright © 2011, Elsevier BV. All rights reserved.
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Cellular automata (CA) modeling of traffic flow as described in the previous chapters has become popular in the 1990s. In the following, we discuss some of the “classical” approaches to traffic modeling that were partly introduced already in the 1950s1 [185, 423, 425, 426, 512, 899, 1001, 1023, 1028, 1142]. Of course, we will also describe further developments of these theories and try to make a comparison with the CA approaches wherever possible.
9.1. FLUID-DYNAMICAL THEORIES When viewed from a long distance, say, an aircraft, flow of fairly heavy traffic appears like a stream of a fluid. Therefore, a macroscopic theory of traffic can be developed, in analogy with the hydrodynamic theory of fluids, by treating traffic as an effectively one-dimensional compressible fluid (a continuum) [579, 822, 1028, 1106]. Suppose that ρ(x; t) and J (x; t) are the coarse-grained density and flux at the location x and time t. The equation of continuity for the fluid representing traffic is in out ∂ρ(x; t) ∂J (x; t) αi (x − xi ; t) − βj (x − xj ; t), + = ∂t ∂x
N
N
i=1
j=1
(9.1)
where the first and the second terms on the right-hand side take care of the sources and sinks, respectively, at the Nin on-ramps situated at xi (i = 1, 2, . . . , Nin ) and Nout off-ramps situated at xj (j = 1, 2, . . . , Nout ). We can write αi (x − xi ; t) and βj (x − xj ; t) as αi (x − xi ; t) = αi0 (t)φi (x − xi )
and βj (x − xj ; t) = βj0 (t)φj (x − xj ),
(9.2)
where φi (x − xi ) and φj (x − xj ) describe the spatial distribution of the incoming and outgoing flux, respectively, while αi0 (t) and βj0 (t) account for the corresponding temporal variations. In the following, we mostly consider a stretch of highway with no entries or exits. In such special situations, the equation of continuity reduces to the simpler form [875] ∂ρ(x; t) ∂J (x; t) + = 0. ∂t ∂x
(9.3)
One cannot get two unknowns, namely, ρ(x; t) and J (x; t) [or, equivalently, v(x; t)], by solving only one equation, namely (9.3), unless they are related to each other. In order 1 For a personal account of the history of traffic modeling in that era, see “Memoirs on highway traffic flow theory in
the 1950’s” [1050] by Newell, one of the pioneers.
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to proceed further, one needs another independent equation, say, for v(x; t); we shall write down such an equation later in Section 9.2. An alternative possibility, which Lighthill and Whitham [875] and independently Richards [1184] adopted in their pioneering work, is to assume that J (x; t) is determined primarily by the local density ρ(x; t) so that J (x; t) can be treated as a function of only ρ(x; t). Consequently, the number of unknown variables is reduced to one as, according to this assumption, the two unknowns ρ(x; t) and J (x; t) are not independent of each other.
9.1.1. Lighthill–Whitham–Richards Theory and Kinematic Waves The Lighthill–Whitham–Richards (LWR) theory is based on the assumption that J (x; t) = j(ρ(x; t)),
(9.4)
where j(ρ) is a function of ρ. The functional relation (9.4) between density and flux cannot be calculated within the framework of the fluid-dynamical theory. It must be either taken as a phenomenological relation extracted from empirical data or derived from more microscopic considerations. In general, the flux-density curve implied by equation (9.4) need not be identical with the fundamental diagram in the steady state. Here, we should make some remarks concerning the terminology commonly used in traffic engineering. Generically j(ρ) is called equilibrium fundamental diagram. In contrast to the terminology used in physics, “equilibrium” here means “stationary.” Sometimes, it even includes spatial homogeneity, i.e., an equilibrium traffic state satisfies [1514] the conditions dv = 0 and dt
∂ρ =0 ∂x
(9.5)
for all x and t. If at least one of the conditions is not satisfied, one speaks of nonequilibrium traffic. Obviously, this has been a source of confusion, and so we will avoid this terminology. Under the assumption (9.4), the x-dependence of the local flux J (x; t) arises only from the x-dependence of ρ(x; t). Alternatively since J (x; t) = ρ(x; t)v(x; t), assuming v(x; t) to depend only on ρ(x; t), the x-dependence of v(x; t) arises only from the x-dependence of ρ(x; t). Using (9.4), the equation of continuity (9.3) can be expressed as ∂ρ(x; t) ∂ρ(x; t) + vg =0 ∂t ∂x
(9.6)
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where vg =
dJ dv = v(x; t) + ρ(x; t) . dρ dρ
(9.7)
Note that the equations (9.4) and (9.6) form the complete system of dynamical equations governing traffic flow in this first approximation. However, the equation (9.6) is nonlinear because, in general, vg depends on ρ. If vg were a constant v0 , independent of ρ, equation (9.6) would become linear and the general solution would be of the form ρ(x; t) = f (x − v0t), where f is an arbitrary function of its argument. In that case, the solution of any particular problem would be found by merely matching the function f to the corresponding given initial and boundary conditions. Such a solution describes a density wave motion, as an initial density profile would get translated by a distance v0t in a time interval t without any change in its shape. However, the nonlinearity of the equation (9.6) gives rise to subtleties, which are essential to capture at least some aspects of real traffic. The solution of the nonlinear equation (9.6) is of the general form ρ(x; t) = F(x − vg t),
(9.8)
where F is an arbitrary function of its arguments.2 If we define a wave to be “recognizable signal that is transferred from one part of a medium to another with a recognizable velocity of propagation” [1463, 1464], then the solutions of the form (9.8) can be regarded as a density wave. There are several similarities between the density wave and the more commonly encountered waves like, e.g., acoustic or elastic waves. But the acoustic or elastic waves are solutions of linearized partial differential equations, whereas the equation (9.6) is nonlinear, and hence, vg is ρ-dependent. Waves of the type (9.8) are called kinematic waves [875, 1184, 1463, 1464] to emphasize their purely kinematic origin, in contrast to the dynamic origin of the acoustic and elastic waves. From the initial given density profile ρ(x; 0), the profile ρ(x; t) at time t can be obtained by moving each point on the initial profile a distance vg (ρ)t to the right; obviously, the distance moved is different for different values of ρ. The time evolution of the density profile can be shown graphically [875, 1463, 1464] on the space-time diagram (i.e., the x − t plane) where an arbitrary point x0 on the t = 0 axis moves along a straight line of slope vg (ρ) if the initial density at x0 is ρ. These straight lines are referred to as characteristics and correspond to the curves along which information from the initial traffic conditions is transported. Generically different characteristics corresponding to different ρ have different slopes vg (ρ). The speed vg (ρ) of the density wave should not be confused with v(ρ), the actual speed of the continuum fluid representing traffic. In fact, at any instant of time, v(x; t) 2 For given initial conditions, a formal solution in terms of an implicit equation can be found, e.g., in [1256].
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can be obtained from the corresponding density profile ρ(x; t) by using the relation dv(ρ) v(x; t) = j(ρ(x; t))/ρ(x; t). Moreover, since vg = v(ρ) + ρ dv(ρ) dρ and dρ < 0, the speed of the density wave is less than that of the fluid. Therefore, the density wave propagates backward relative to the traffic, and the drivers are thereby warned of density fluctuations ahead downstream. Furthermore, the density wave moves forward or backward relative to the road, depending on whether ρ < ρ ∗ or ρ > ρ ∗ where ρ ∗ corresponds to the maximum in the function j(ρ). When J (ρ) is convex, i.e., d2 J /dρ 2 < 0, we have dvg /dρ < 0. Consequently, higher values of ρ propagate slower than lower values thereby distorting the initial density profile. On the other hand, when dvg /dρ > 0, higher values of ρ propagate faster, and the distortion has the opposite tendency as compared to the case of dvg /dρ < 0. In both situations, the distortion of the initial density profile is caused by the ρ-dependence of vg which arises from the nonlinearity of equation (9.6). The distortion of the density profile with time can also be followed on the space-time diagram. If dvg /dρ < 0, in regions of decreasing density (i.e., ρ(x1 ) > ρ(x2 ) for x1 < x2 ), the characteristics move away from each other, whereas in regions of increasing density, the characteristics move toward each other. When two characteristic lines on the space-time diagram intersect, the density would be double-valued at the point of intersection. We can avoid this apparently impossible scenario by the following interpretation. When two characteristic lines intersect, a shock wave is generated. By definition, a shock represents a mathematical discontinuity in ρ and, hence, also in v. The speed of a shock wave is given by vs =
J (ρ2 ) − J (ρ1) , ρ2 − ρ1
(9.9)
where ρ2 and ρ1 are, respectively, the densities immediately in front (downstream) and behind (upstream) the shock, while J (ρ2 ) and J (ρ1 ) represent the corresponding downstream and upstream fluxes, respectively. Thus, the shock velocity can be determined directly from the fundamental diagram (Fig. 9.1). Note that the shock wave moves downstream (upstream) if vs is positive (negative). Often the shock is weak in the sense that the relative discontinuity (ρ2 − ρ1)/ρ1 is small, and in such cases, the shock wave speed tends to vg = dJ /dρ. As a shock separates a section of high and low densities, it corresponds to a section of a highway where a free-flow and a congested regime is present. In particular for large differences between ρ1 and ρ2 , the velocity of the shock can be interpreted as the velocity of a backwards moving jam. Newell [1047] has proposed a simplified theory of kinematic waves and applied it to analyze more complex situations, e.g., due to the presence of bottlenecks or (timedependent) on-ramp and off-ramp flows [1048, 1049].
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ρ
J J2
ρ2
J1
vs
ρ1
J2
J1
xs
x
ρ1
(a)
ρ2
ρ
(b)
Figure 9.1 Illustration of a shock at position xs connecting two regions of density ρ1 and ρ2 , respectively (a). The shock velocity vs is determined by the slope of the line connecting the points (ρ1 , J1 ) and (ρ2 , J2 ) in the fundamental diagram (b).
One advantage of the kinematic approach over any dynamic approach is that the dynamical equation, which will be given in Section 9.2, is difficult to derive from first principles. It usually involves quite a few phenomenological parameters and even a phenomenological function. On the other hand, the only input needed for the kinematic approach is the phenomenological function J (ρ), which can be obtained from empirical data.
9.1.2. Diffusion Term in LWR Theory and Its Effects An improvement over the original LWR theory can be made if one assumes that the local flux J (x; t) is determined not only by the local density ρ(x; t) but also by the gradient of the density. In other words, we replace the assumption (9.4) by J (ρ) = j(ρ) − D
∂ρ , ∂x
(9.10)
where D is a positive constant. Note that for fixed ρ(x; t) (and, hence, fixed j(ρ)), a positive (negative) density gradient leads to a lower (higher) flux as the drivers are expected to reduce (increase) the speed of their vehicles depending on whether approaching a more (less) congested region. Using (9.10) in the equation of continuity (9.3), we now get ∂ρ(x; t) ∂ρ(x; t) ∂ 2ρ(x; t) , + vg =D ∂t ∂x ∂x2
(9.11)
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where vg (ρ) =
dj(ρ) dρ .
The equation (9.11) reduces to (9.6) when D = 0. The non-
linearity and diffusion have opposite effects: the term vg (ρ) ∂ρ ∂x leads to “steepening” and ultimate “breaking” of the wave, whereas the diffusion term D∂ 2ρ/∂x2 smoothens out the profile. Nonvanishing D also leads to a nonzero width of the shock wave.
9.1.3. Greenshields Model and Burgers Equation So far in the preceding subsections we have not considered any specific form of the function j(ρ) relating flux with density. One can start with the simplest (differentiable) approximation capturing the basic form of the fundamental diagram, J = vmax ρ(1 − ρ).
(9.12)
Note that vmax in (9.12) is a phenomenological parameter, and it is interpreted to be the maximum average speed for ρ → 0. In traffic science and engineering, one usually uses 1 − ρ/ρjam instead of 1 − ρ in the equation (9.12), and the corresponding form of the relation between J and ρ is known as the Greenshields model [462]. Substituting (9.12) into the equation (9.11), we get ∂ρ(x; t) ∂ρ(x; t) ∂ρ(x; t) ∂ 2ρ + vmax − 2vmax ρ =D 2. ∂t ∂x ∂x ∂x
(9.13)
Introducing the linear transformation of variables x = vmaxt − x
and t = t ,
(9.14)
one gets the (deterministic) Burgers equation [167, 1463, 1464] ∂ρ(x; t) ∂ρ(x; t) ∂ 2ρ + 2v ρ = D . max ∂t ∂x ∂x 2
(9.15)
Note that the transformation (9.14) takes one from the space-fixed coordinate system (x, t) to a coordinate system (x , t ) that moves with uniform speed vmax; so, vehicles moving with speed vmax with respect to the coordinate system (x, t) do not move at all with respect to the coordinate system (x , t ). The advantage of this route to traffic flow theory is that the Burgers equation (9.15) can be transformed further into a diffusion equation, thereby getting rid of the nonlinearity, through a nonlinear Cole–Hopf transformation [1463, 1464] (see Section 2.5.4). Since it is straightforward to write down the formal solution to the diffusion equation, one can see clearly the role of the coefficient D and the nature of the solutions in the limit D → 0.
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If equation (9.6) is assumed to be the only equation governing traffic flow, then an inhomogeneous initial state can lead to a shock wave, but the amplitude of the shock wave decreases with time, and eventually the shock wave fades out leading to a homogeneous steady state in the limit t → ∞. In [853], the time evolution of a random initial distribution of steps in the density profile was studied. No traffic jam forms spontaneously from a state of uniform density at this level of sophistication of the fluid-dynamical approach.
9.2. SECOND-ORDER FLUID DYNAMICAL THEORIES The LWR theory is the prototype of a first-order model since it is defined by a first-order partial differential equation (PDE). It has several serious deficiencies [1514]: • It assumes that the stationary fundamental diagram also holds in nonstationary situations (because the fundamental diagram is the only manifestation of collective effects due to interactions!). This leads to unsatisfactory results for most dynamic traffic situations. • Since density and flow are always “in equilibrium,” there is no mechanism for spontaneous jam formation (shocks are stable objects).3 There is no density regime where small density fluctuations grow. Typical solutions are piecewise smooth, separated by (sharp) shocks. • Near large density gradients (shocks) acceleration and deceleration diverge. Therefore, it appears to be natural to include higher-order terms (like the diffusion term in Section 9.1.2), which account for effects like anticipation, interactions, etc. This leads to so-called second-order models, which include second-order derivatives in the constitutive equations. A general strategy is to consider an additional momentum equation, similar to the Navier–Stokes theory for fluids: ∂v ∂v F dv := +v = . dt ∂t ∂x m
(9.16)
∂v It describes the change in momentum of a point particle by the force F. The term v ∂x is called transport or convection term and describes motion of the velocity profile with the vehicles, i.e., changes in the mean velocity due to inflowing and outflowing vehicles. In the case of traffic, the force F is related in some way to the dynamics of the vehicle. Depending on the choice of F, different models are obtained. 3 However, the LWR model has a dissipation mechanism (rarefaction waves), so jams can disappear.
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The general structure of most second-order fluid dynamical traffic models is then given by the equation of continuity ∂ρ ∂(ρv) ∂ 2ρ + =D 2 ∂t ∂x ∂x
(9.17)
and a Navier–Stokes-type momentum or velocity equation ∂v ∂v Ve (ρ) − v ∂ 2v 1 ∂P +v = + ν(ρ) 2 − . ∂t ∂x τ (ρ) ∂x ρ ∂x
(9.18)
In the continuity equation, the diffusion term has been included although in most models D = 0 is chosen. If not stated otherwise, this will be assumed in the following. The first force term in the velocity equation describes the (exponential) relaxation of the local velocity v(x, t) toward a density-dependent equilibrium velocity Ve (ρ), which is a phenomenological function. Ve (ρ) might be interpreted as a kind of desired velocity of the drivers at a certain density. Typically, it decreases monotonically with increasing density. This relaxation is characterized by a typical timescale τ (ρ), which in general depends on the density as well. ∂ 2v The second term, which is proportional to ∂x 2 , tends to reduce spatial inhomogeneities of the velocity field in a similar way to the diffusion term in the continuity equation. It is usually interpreted as the analog of the viscous dissipation term in the Navier–Stokes equation. In the last term, P is usually interpreted as traffic pressure. Depending on its specific form, it takes into account various effects. To include anticipation effects, one chooses P ∝ ρ. The term ∂ρ ∂x takes into account the natural tendency of the drivers to accelerate (decelerate) if the density gradient is negative (positive), i.e., if the density in front becomes smaller (larger). The specific form of the velocity equation (9.18) can also be justified in a different way [515, 1196]. V (ρ) is then a “safe” velocity, which only depends on the density ρ. Drivers adjust their velocities such that the average velocity v relaxes on a timescale τ to V (ρ): v(x + vτ , t + τ ) = V (ρ(x + x)),
(9.19)
where x is the average distance between vehicles. A first-order Taylor expansion gives
v(x, t) + vτ
∂v dV ∂ρ ∂v +τ = V (ρ) + x + O (x)2 . ∂x ∂t dρ ∂x
(9.20)
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With x = 1/ρ, one then obtains ∂v ∂v V (ρ) − v 1 dV ∂ρ +v = + . ∂t ∂x τ ρτ dρ ∂x Introducing the abbreviation c02 := − τ1 dV dρ ≥ 0, where we have assumed equation is obtained
(9.21) dV dρ
≤ 0, Payne’s
∂v ∂v V (ρ) − v c02 ∂ρ +v = − , ∂t ∂x τ ρ ∂x
(9.22)
which is of the form (9.18) with ν = 0 and P ∝ ρ. Although the structure of the traffic equations (9.1) and (9.18) is very similar to classical hydrodynamics, there are important differences. Most notably it is the relaxation term which leads, e.g., to decreasing velocities at bottlenecks. In contrast to fluid particles, there is in general no conservation of momentum in car traffic. It is in some sense replaced by other factors like anticipation, which try to describe how an average driver reacts to density variations in space. In general, the fluid-dynamical models have to be studied numerically by discretizing the partial differential equations (9.1) and (9.18) together with appropriate initial and boundary conditions. An overview of the numerical techniques involved are given in [546]. As in fluid dynamics, one can even go further and include higher order equations. The next step would be a variance equation [508], which describes the dynamics of the velocity variance (x, t).
9.2.1. Special Models Depending on the specific choice of the functions P, ν, etc. in (9.17), (9.18), various models are obtained. Some of the most important ones are as follows: • The LWR model is obtained in the limit τ → 0 since then v(x, t) = Ve (ρ) to fulfill the velocity equation. • The Payne model corresponds to the choice ν(ρ) = 0,
P(ρ) = −c02 ρ .
• The Burgers equation corresponds to the limit τ → 0 in the Payne model. • The Phillips model is obtained for ρ , ν(ρ) = 0, P(ρ) = ρ 0 1 − ρmax
(9.23)
(9.24)
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•
i.e., it assumes a traffic pressure that can be written in terms of a density-dependent velocity variance 0 (1 − ρ/ρmax ). The Kühne model is characterized by τ (ρ) = const.,
•
P(ρ) = ρ 0,
ν(ρ) = const.,
(9.25)
where 0 is a constant velocity variance. The Kerner–Konhäuser model is obtained for τ (ρ) = const.,
P(ρ) = ρ 0,
ν(ρ) =
η0 , ρ
(9.26)
i.e., velocity variance, viscosity coefficient, and relaxation time are constant. Sometimes, the models of Kühne and Kerner–Konhäuser are subsumed as Kühne– Kerner–Konhäuser (K3 ) model. Another model, gas-kinetic based traffic (GKT) model, has been derived by Treiber et al. [523, 524, 544, 1386] starting from the gas-kinetic approach, which will be discussed in detail in Section 9.3. It takes into account the space requirements of vehicles and correlations of successive vehicle velocities. This leads to a velocity equation with a nonlocal equilibrium velocity Ve (ρ) = V0 − b(ρ, ρ , V , V , , ) .
(9.27)
In the braking term b, the quantities ρ , V , and have to be evaluated at the advanced interaction point x = x + s with some safety distance s. The nonlocal velocity term has similar smoothing properties as the viscosity term. However, these effects are anisotropic (no smoothing in forward direction) and thus more realistic.
9.2.2. Instabilities and Jam Formation In the following, we discuss the basic principles of the instabilities and mechanisms of jam formation in fluid dynamical models. Starting point is the continuity equation (9.17) with D = 0 and the velocity equation (9.18). We assume that the traffic pressure P is proportional to the velocity variance : P ∝ . Then, the two equations have the stationary homogeneous solution ρ(x, t) = ρ0 for which = 0 and thus P = 0.
and
v(x, t) = Ve (ρ0 )
(9.28)
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The next step is a linear stability analysis to test the stability of these solutions against small perturbations δρ(x, t) and δv(x, t). One finds that the homogeneous solution becomes unstable if the condition [511] dV
> dP 1 + τ (ρ0)ν(ρ0)k2 ρ0 dρ dρ
(9.29)
is satisfied. For small densities ρ0 → 0, the left side goes to 0, and the condition cannot be satisfied, i.e., the homogeneous solution is stable. If dV dρ decreases sufficiently fast with increasing ρ0 , then the homogeneous solution becomes instable for larger densities. The instability mechanism typical works in the following way. Let us consider a local increase of the density ρ(x) > 0 at some location x. Since dVe (ρ0 )/dρ < 0, the local increase of the density leads to a decrease of Ve (ρ). This decrease in the safe velocity forces drivers to reduce their average velocity v sharply if |Ve (ρ0 )/dρ| is large enough. On the other hand, it follows from the equation of continuity that the local decrease of v gives rise to further increase of ρ around x and, consequently, further subsequent decrease of v(x) in this location. This avalanche-like process, which tends to increase the amplitude of the local fluctuation of the density around the homogeneous state, competes against other processes, like diffusion and viscous dissipation, which tend to decrease inhomogeneities. More specifically, we have for some of the models discussed previously: dV > 1 Payne model: ρ0 dρ 2ρ0 τ dV
3 > (ρ) 1 + τ (ρ0)ν(ρ0 )k2 . Phillips-, K − model: ρ0 dρ
(9.30) (9.31)
However, for strong nonlinearities, a linear stability analysis is not really reliable. One effect of nonlinearities is that the waves are no longer periodic. A harmonic perturbation can change shape and even become localized. This is called local breakdown effect [737, 738]. Then, the perturbation starts to grow and finally a wide jam is formed, which propagates upstream and is surrounded by free-flowing traffic. This has been called local cluster effect [562, 736, 737]. A detailed stability analysis [738, 739, 1204] yields five different density regimes (Fig. 9.2): ρ < ρc1 : all perturbations eventually disappear (stable) ρc1 < ρ < ρc2 : for perturbations that are strong enough a wide jam forms, i.e., the critical amplitude is finite (metastable) ρc2 < ρ < ρc3 : a sequence of jams is formed (stop-and-go) (linearly unstable)
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Metastable
Unstable
Metastable
Stable
Δρc
Stable
ρc
ρc
1
2
ρ
ρc
ρc
3
4
ρmax
Figure 9.2 Stability of homogeneous traffic in generic fluid-dynamic models. ρc is the critical amplitude above which localized density fluctuations grow.
ρc3 < ρ < ρc4: for sufficiently large perturbations an anticluster or dipol layer [739] is formed; it consists of a low- and a high-density region following downstream after another (metastable) ρc4 < ρ: all disturbances disappear (stable) The critical densities ρcj depend on the pressure function P, the relaxation time τ , and the desired velocity V (ρ). We see that in the medium density regime, traffic flow is unstable with respect to small perturbations. The reason is a delayed adaption of the vehicle speed to the local situation, which then lead to the formation of phantom traffic jams [736, 737]. Narrow jams tend to grow and stabilize in wide moving jams, which can be characterized by their propagation speed vjam and the outflow Jout . The motion of jams can be given more explicitly. The jam front moves with velocity vjam opposite to the direction of motion of the vehicles. Inserting ρ(x, t) = ρ(x − vjamt, 0)
and
J (x, t) = J (x − vjamt, 0)
(9.32)
into the equation of continuity gives 0=
∂ρ(˜x, 0) ∂J (˜x, 0) ∂ρ(x, t) ∂J (x, t) + = −vjam + ∂t ∂x ∂ x˜ ∂ x˜
(9.33)
with x˜ := x − vs t. The solution of this equation is J (˜x, 0) = J0 + vjamρ(˜x, 0) =: Js (ρ(˜x, 0)).
(9.34)
This is a linear flow-density relation corresponding to the jam line J in the fundamental diagram (Fig. 6.11). The characteristics and mechanism of jam formation described above appear to be typical for models with a deterministic instability mechanism. Most stochastic models
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have different instability mechanism, but show a similar phase separation between free flow and jammed traffic.
9.2.3. Problems with Second-Order Models The second-order models were introduced to overcome the problems encountered in the original LWR model. Despite being an improvement over the first-order approaches, several problems were encountered in second-order models. As an example, the Phillips model can produce unphysical results at high densities when the density gradient of the traffic pressure is negative, vehicles accelerate into traffic jams. For certain model parameters, the Kerner–Klenov (KK) model can produce densities exceeding the maximal densities and negative velocities. Since the viscosity is introduced in an ad hoc way and is not a directly measurable quantity, the choice of the traffic pressure is hard to justify. Further discussions of potential inconsistencies can be found in [505, 1515]. Daganzo has pointed out several generic problems in a “requiem” [255] on secondorder models. • Isotropy: In contrast to fluid particles, vehicles in traffic flow should be mainly influenced by the situation in front, not in their back. This is not taken into account in generic second-order models. The origin of the isotropy problem is the existence of a characteristic speed, which is larger than the average velocity. Therefore, the future conditions of the flow are influenced by the traffic conditions behind. • Wrong-way travel: Terms that will smoothen shocks necessarily lead to negative velocities, i.e., vehicles moving backwards, or accidents. • Jam fronts: Because of the viscosity term, the jam fronts in second-order models are typically extended. This is in contrast to empirical observations where it consists of a few cars only. • Driver characteristics: The basic models do not take into account the heterogeneity, e.g., through different driving styles. Some of these problems are less serious and can be solved rather easily. For example, the heterogeneity of drivers has been incorporated in multiclass models [578, 1292]. But other problems, especially those related to the (an-)isotropy, are much deeper conceptual issues.
9.2.4. Aw–Rascle Model As we have seen, the LWR and related models are isotropic. This results from the wave propagation being faster than the traffic movement and leads to wrong-way traffic. In order to solve this problem, Aw and Rascle [53] introduced a pressure term in which the acceleration is determined by the convective derivative of P(ρ). Then, the model
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becomes anisotropic, and for mathematical reasons, a second conservation law (besides mass conservation) has to be introduced. The Aw–Rascle model [53] is then defined by the equation of continuity (9.17) with D = 0 and the velocity equation4 ∂v ∂v ∂P ∂P +v =− −v . ∂t ∂x ∂t ∂x
(9.35)
The pressure is assumed to be a smooth function which increases with density, e.g., of the form P(ρ) = cρ γ with γ > 0. The essential point now is that instead of the partial derivative ∂P/∂x, the convective derivative of the pressure appears. Introducing a generalized flux ρw := ρ(v + P), the velocity equation (9.35) can be interpreted as conservation equation ∂ ∂ (ρw) + (ρwv) = 0. ∂t ∂x
(9.36)
The characteristic speeds of model are then given by vc(1) = v,
vc(2) = v = v − ρP (ρ).
(9.37)
Both are now no greater than the traffic speed so that the problem of forward propagation of information is avoided. A similar solution of the problem of wrong-way travel has been proposed in [683] where a continuum version of the full velocity difference model (see Section 9.4.3) was derived. Here, the density gradient appearing in the anticipation term of the equation of motion is replaced by the speed gradient ∂v/∂x. This modification also leads to characteristic speeds, which are always less or equal to the average velocity. Other solutions have been proposed, e.g., introducing nonlocal terms [515], a more careful stability analysis, or a quick decay of modes corresponding to unphysical characteristic speeds [530, 695]. However, these proposals remain controversial [516, 1516].
9.2.5. Fluid-Dynamical Models and Synchronized Traffic To our knowledge, the first attempt to understand the physical mechanism of synchronized traffic within the framework of the fluid-dynamical formalism was made by 4 An extension that takes into account the desire of the drivers to reach a maximal velocity has been proposed in [459].
Another modification of the model was studied in [106].
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Lee et al. [844, 845].5 They considered a finite stretch of highway with an on- and off-ramp where the spatial distribution of the external flux φ(x) in equation (9.2) ˆ was assumed to be Gaussian. They also assumed the form vsafe (ρ) = v0 (1 − ρ/ρ)/ ˆ First, the sys[1 + E(ρ/ρ) ˆ θ ] for the safe velocity with adjustable parameters v0, E, θ , ρ. tem was allowed to reach a steady free-flow state where homogeneous regions with different densities are separated from one another by narrow transition layers near the ramps. Then, they applied a pulse of additional flux δq at the on-ramp for a short duration δt. After a transient period, the system was found to settle in a limit cycle in which the local density and local flux oscillate periodically, and the oscillations are localized near the on-ramp. The discontinuous change of the spatiotemporally averaged velocity induced by the localized perturbations of finite amplitude, associated hysteresis effects, and the stability of the limit cycle were found to be qualitatively similar to some of the empirically observed characteristics of synchronized flow in real traffic. Therefore, Lee et al. [844, 845] identified the limit cycle observed in their theoretical investigation as the synchronized state of vehicular traffic. They drew analogy between this state and a “self-excited oscillator” [844, 845]. Similar results have been obtained for a GKT model [544], also using on- and offramps in order to explain the transition from free-flow to synchronized states. In [525, 846], phase diagrams were calculated, which depends on the on-ramp activity and the flow on the highway. The GKT model introduced in Section 9.2.1 has also been shown to provide a good description of synchronized traffic and the patterns observed near bottlenecks [523]. Further suggestions for hydrodynamic approaches to synchronized flow are based on a modified LWR model [1039] or the balanced vehicular traffic model, [1293], which is an extension of the Aw–Rascle model. The latter has a congested regime which is unstable against the formation of synchronized traffic, triggered, e.g., by a localized velocity perturbation, at intermediate densities.
9.2.6. Fluid-Dynamical Theories of Traffic on Multilane Highways and in Cities One can describe the traffic on two-lane highways [843] by two equations each of the same form (9.1) and where the source term in the equation for lane 1 (lane 2) takes into account the vehicles that enter into it from the lane 2 (lane 1), while the sink term takes into account those vehicles entering the lane 2 (lane 1) from the lane 1 (lane 2). A lattice hydrodynamic theory for city traffic has been formulated in [994]. This fluiddynamical model is motivated by the CA model, developed by Biham et al. [115], which will be discussed in detail later in this review. Instead of generalizing the Navier–Stokes equation (9.18), a simpler form of the velocity equation has been assumed. 5 A brief summary of that work can be found in [545].
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9.3. GAS-KINETIC MODELS In the kinetic theory, traffic is treated as a gas of interacting particles where each particle represents a vehicle. The various different versions of the kinetic theory of vehicular traffic have been developed by modifying the kinetic theory of gases. Recall that in the kinetic theory of gases [616], f ( r , p; t)d 3rd 3 p denotes the number of molecules which, at time t, have positions lying within a volume element d 3 r about r and momenta lying within the momentum-space element d 3 p about p. The Boltzmann equation, which describes the time evolution of the distribution f (x, v; t), is given by
∂f ∂f p , + · ∇r + F · ∇p f ( r , p; t) = ∂t m ∂t coll
(9.38)
where the symbols ∇r and ∇p denote gradient operators with respect to r and p, respec ∂f tively, while F is the external force. The term ∂t coll represents the rate of change of f , with time, which is caused by the mutual collisions of the molecules. In the first of the following two subsections, we present the earliest version of the kinetic theory of vehicular traffic, which was introduced by Prigogine and coworkers [1141, 1142] by modifying some of the key concepts in the kinetic theory of gases and by writing down an equation analogous to the Boltzmann equation (9.38). In the subsequent subsection, we discuss the kinetic theory developed later by PaveriFontana [1097] to cure the defects from which the Prigogine theory was found to suffer.
9.3.1. Prigogine Model Suppose that f (x, v; t)dxdv denotes the number of vehicles, at time t, between x and x + dx, having actual velocity between v and v + dv. In addition, Prigogine and coworkers [1141, 1142] introduced a desired distribution fdes (x, v), which is a mathematical idealization of the goals that the population of the drivers collectively strives to achieve. The actual distribution may deviate from the desired distribution because of various possible influences, e.g., road conditions, weather conditions, or interaction with other vehicles. They also argued that some of these influences cease after some time, while the interactions with the other vehicles persist for ever. For example, only a short stretch of the road surface may be icy, and strong winds or rain may stop after a short duration; in such situations, f can relax to fdes over a relaxation time τrel provided mutual interactions of the vehicles are negligibly small. On the basis of these arguments, Prigogine and coworkers [1141, 1142] suggested that the analog of the Boltzmann equation for traffic
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should have the form ∂f ∂f +v = ∂t ∂x
∂f ∂t
∂f + ∂t rel
,
(9.39)
int
∂f where ∂t rel accounts for the relaxation of f toward fdes in the absence of mutual inter ∂f actions of the vehicles, while ∂t int accounts for the changes of f arising from mutual ∂f interactions among the vehicles. Note that the term ∂t int on the right-hand side of (9.39) ∂f may be interpreted as the analog of the term ∂t coll in the equation (9.38), whereas ∂f the term ∂t rel in equation (9.39) may be interpreted as the counterpart of the term F · ∇p f ( r , p; t) in the equation (9.38). ∂f Prigogine and coworkers wrote down an explicit form for the term ∂t int by gener ∂f alizing that for the term ∂t coll in the kinetic theory of gases. We shall consider this term in the next subsection. In order to write down a simple explicit form of the relaxation term in the equation (9.39), they assumed that • the collective relaxation, which would cause the actual distribution to tend toward the desired distribution, involves only a single relaxation time τrel so that
∂f ∂t
=− rel
f − fdes τrel
(9.40)
• the desired speed distribution Fdes (v) remains independent of the local concentration ρ(x; t) so that fdes (x, v; t) = ρ(x; t)Fdes (v).
(9.41)
Therefore, a more explicit form of the Boltzmann-like equation (9.39) in the Prigogine theory is given by ∂f ∂f ∂f f (x, v; t) − ρ(x; t)Fdes(v) + . +v =− ∂t ∂x τrel ∂t int
(9.42)
Note that in the absence of mutual interactions of the vehicles, the distribution f (x, v; t) would relax exponentially with time. The concept of desired distribution fdes (x, v; t) and this scenario of collective relaxation of f toward fdes have subsequently come under severe criticism [1097]. Analyzing a set of “ideal experiments” in the light of the Prigogine theory, Paveri-Fontana [1097] showed that the results obtained from the Boltzmann-like equation (9.42) are physically unsatisfactory.
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Lehmann [851] has reformulated the Prigogine approach as a semiphenomenological theory where the distribution f (x, v; t) is assumed to follow the simpler form ∂f ∂f f − fdes (v, ρ) , +v =− ∂t ∂x τrel
(9.43)
and the effects of the interactions are taken into account implicitly through a densitydependent desired distribution function fdes (v, ρ), which has to be determined empirically. By discretizing time and phase space, one obtains lattice Boltzmann models, which can be investigated numerically. The macroscopic dynamics can be derived from Taylor and Chapman-Enskog expansions [927]. The lattice Boltzmann approach allows to reproduce metastability as well as stop-and-go waves.
9.3.2. Paveri-Fontana Model In order to remove the conceptual and mathematical drawbacks of the Prigogine model of the kinetic theory of vehicular traffic, Paveri-Fontana [1097] argued that each vehicle, in contrast to the molecules in a gas, has a desired velocity toward which its actual velocity tends to relax in the absence of interaction with other vehicles. Thus, the Paveri-Fontana model is based on a scenario of relaxation of the velocities of the individual vehicles rather than a collective relaxation of the distribution of the velocities. In mathematical language, Paveri-Fontana introduced an additional phase-space coordinate, namely, the desired velocity. Suppose that g(x, v, vdes ; t)dxdvdvdes denotes the number of vehicles at time t between x and x + dx, having actual velocity between v and v + dv and desired velocity between vdes and vdes + dvdes . The one-vehicle actual velocity distribution function f (x, v; t) =
dvdes g(x, v, vdes ; t)
(9.44)
describes the probability of finding a vehicle between x and x + dx having actual velocity between v and v + dv at time t. Similarly, the one-vehicle desired velocity distribution function f0 (x, vdes ; t) =
dvg(x, v, vdes ; t)
(9.45)
describes the probability of finding a vehicle between x and x + dx having desired velocity between vdes and vdes + dvdes . The local density of the vehicles ρ(x; t) at the position x
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at time t can be obtained from ∞ ∞ ρ(x; t) = dvdes dvg(x, v, vdes ; t). 0
0
Similarly, the corresponding average actual speed v(x; t) and the average desired speed vdes (x; t) are defined as ∞ v(x; t) =
0
∞
dvvg(x, v, vdes ; t)
0
ρ(x; t) ∞
vdes (x; t) =
dvdes
0
dvdes
∞
.
dvvdes g(x, v, vdes ; t)
0
ρ(x; t)
.
Finally, the local flux J (x; t) is defined as J (x; t) = ρ(x; t) v(x; t) . Now, let us assume that the desired velocity of each individual driver is independent of time, i.e., dvdes /dt = 0. Of course, the drivers may also adapt to the changing traffic environment, and their desired velocities may change accordingly. In principle, these features can be incorporated at the cost of increasing complexity of the formalism. Next, let us also assume that in the absence of interaction with other vehicles, an arbitrary vehicle reaches the desired velocity exponentially with time, i.e., dv/dt = (vdes − v)/τ where τ is a relaxation time. The Boltzmann-like kinetic equation for g(x, v, vdes ; t) can be written as
∂ ∂g ∂ ∂ vdes − v (9.46) +v g+ g = ∂t ∂x ∂v τ ∂t int In order to write down an explicit form of the interaction term, we have to model the interactions among the vehicles. First, we model the vehicles as point-like objects. We consider the scenario where a fast vehicle, when hindered by a slow leading vehicle, either passes or slows down to the velocity of the lead vehicle. Let us now make some further simplifying assumptions: i. The slowing down takes place with a probability 1 − Ppass where Ppass is the probability of passing. ii. If the fast vehicle passes the slower leading vehicle, its own velocity remains unchanged. iii. The velocity of the slower leading vehicle remains unchanged, irrespective of whether the faster following vehicle passes or slows down.
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iv. The slowing down process is instantaneous, i.e., the braking time is negligibly small. v. It is adequate to consider only two-vehicle interactions; there is no need to consider three-vehicle (or multivehicle) interactions. vi. The postulate of vehicular chaos, which is the analog of the postulate of molecular chaos in the kinetic theory of gases, holds so that the two-vehicle distribution func ; t) can be approximated as a product of two one-particle tion g2 (x, v, vdes , x , v , vdes ; t), i.e., distributions g(x, v, vdes ; t) and g(x , v , vdes ; t) g(x, v, vdes ; t)g(x , v , vdes ; t) . g2 (x, v, vdes , x , v , vdes
(9.47)
Thus, equation (9.46) can be written explicitly as
∂ ∂ ∂ vdes − v +v g+ g ∂t ∂x ∂v τ ∞ = f (x, v; t) dv (1 − Ppass )(v − v)g(x, v , vdes ; t) v
v
− g(x, v, vdes ; t)
dv (1 − Ppass )(v − v )f (x, v ; t) .
0
(9.48) The form of the interaction term on the right-hand side of the equation (9.3.2) follows from the assumptions (i)–(vi) above. The first term on the right-hand side of (9.3.2) describes the gain of probability g(x, v, vdes ; t) from the interaction of vehicles of actual velocity v with slower leading vehicle of actual velocity v, while the second term describes the loss of the probability g(x, v, vdes ; t) arising from the interaction of vehicles of actual velocity v with even slower leading vehicle of actual velocity v . The stationary homogeneous solution g(v, vdes ) is, by definition, independent of x and t. But to our knowledge, so far, it has not been possible to get even this solution of the Boltzmann-like integro-differential equation (9.3.2) by solving it analytically even for the simplest possible choice of the desired distribution function. However, numerical solutions [988, 990, 1425–1427] provide some insights into the regimes of validity of equation (9.3.2) and give indications as to the directions of further improvements of the Paveri-Fontana model. For example, the finite sizes of the vehicles must be taken into account at high densities [510]. Besides, the assumption (iv) of instantaneous relaxation has also been relaxed in a more recent extension [1425, 1426]. Normally passing would require more than one lane on the highway. Therefore, the models discussed so far in the context of the kinetic theory may be regarded, more
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appropriately, as quasi-one-dimensional. These neither deal explicitly with gi (x, v, vdes ; t) for the individual lanes (labeled by i) nor take into account the process of lane-changing. Besides, all the vehicles were assumed to be of the same type. Now, in principle, we can generalize the formalism of the kinetic theory of traffic to deal with different types of vehicles on multilane highways. Suppose that gia (x, v, vdes ; t) is the distribution for vehicles of type a on the i-th lane of the highway. Obviously, the Boltzmann-like equations for the different lanes are coupled to one another. However, one needs additional postulates to model the lane-changing rules [509, 522, 988, 990, 1292]. Also the relation with the observed hysteretic phase transitions to synchronized traffic has been studied within the gas-kinetic approach [544]. Very little work has been done so far on developing kinetic theories of twodimensional traffic flow which would represent, e.g., traffic in cities. Boltzmann-like equations governing the time evolutions of these distributions are given in [216, 984].
9.3.3. Derivation of Fluid-Dynamical Equations from Gas-Kinetic Equations Next, we discuss the results of the attempts to derive the phenomenological equations of traffic flow in the macroscopic fluid-dynamical theories from the microscopic gas-kinetic models. Several attempts have been made so far to derive the equation of continuity and the Navier–Stokes-like equation for traffic from the corresponding Boltzmannlike equation in the same spirit in which the derivations of the equation of continuity and Navier–Stokes equation for viscous fluids from the Boltzmann-equation have been carried out. However, because of the postulate of vehicular chaos, equation (9.3.2) is expected to be valid only at very low densities where correlations between the vehicles are negligibly small, whereas traffic is better approximated as a continuum fluid at higher densities! In analogy with classical fluid-dynamics, the connection between the two descriptions can be made using the method of moments [1105]. Schematically this approach can be summarized as gas-kinetic. (9.49) macroscopic = v vdes
To be more specific, let us define the moments mk, (x; t) = dv dvdes v k vdes g(x, v, vdes ; t).
(9.50)
Note that ρ = m0,0 , v = m1,0 . Integrating the Boltzmann-like equation (9.3.2) over the actual velocities, we get ∂ ∂ f0 (x, vdes ; t) + [¯v (x, vdes ; t)f0 (x, vdes ; t)] = 0, ∂t ∂x
(9.51)
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where v¯ (x, vdes ; t) is defined as v¯ (x, vdes ; t) =
dv v g(x, v, vdes ; t) . f0 (x, vdes ; t)
(9.52)
Equation (9.51) is an equation of continuity for each desired speed vdes separately. It is a consequence of the assumption that dvdes /dt = 0, i.e., no driver changes the desired speed. Using the Boltzmann-like equation (9.3.2) and the definition (9.50), we can get separate partial differential equations for the moments of v, moments of vdes , and the mixed moments of v and vdes . Unfortunately, these lead to a hierarchy of moment equations where each evolution equation for moments of a given order involves also moments of the next higher order. To close this system of equations, one needs to make appropriate justifiable assumptions. This program can in principle be carried out for arbitrary orders k. For k = 0, it leads to the continuity equation, i.e., a first-order traffic. In the next order, k = 1, the velocity (momentum) equation of the second-order models is obtained. Furthermore, a new macroscopic variable, the product of density and variance, which is interpreted as traffic pressure, is introduced. To close the system, assumptions about the traffic pressure are needed, e.g., its density and speed dependence. k = 2 leads to an equation for the velocity variance, which is sometimes called energy equation. Furthermore, new variables are introduced, which can, e.g., be related to the skewness of the speed distribution.
9.4. CAR-FOLLOWING MODELS In the car-following theories [425, 560, 1199], one writes, for each individual vehicle, an equation of motion. This is analogous to the Newtonian description of a classical system of interacting particles. In Newtonian mechanics, the acceleration may be regarded as the response of the particle to the stimulus it receives in the form of force, which includes both the external force and those arising from its interaction with all the other particles in the system. Therefore, the basic philosophy of the car-following theories [425, 560, 1199] can be summarized by the equation [Response]n ∝ [Stimulus]n
(9.53)
for the n-th vehicle (n = 1, 2, . . .). Each driver can respond to the surrounding traffic conditions only by accelerating or decelerating the vehicle. Different forms of the equations of motion of the vehicles in the different versions of the car-following models arise from the differences in their postulates regarding the nature of the stimulus
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(i.e., behavioral force or a generalized force [542]). The stimulus may be composed of the speed of the vehicle, the difference in the speeds of the vehicle under consideration and its lead vehicle, the distance headway, etc., and therefore, in general, x¨ n = fsti (vn , xn , vn ),
(9.54)
where the function fsti represents the stimulus received by the n-th vehicle. Different versions of the car-following models model the function fsti differently. In the next two subsections, we discuss two different conceptual frameworks for modeling fsti .
9.4.1. Follow-the-Leader Model In the earliest car-following models [1111, 1179, 1180], the difference in the velocities of the n-th and (n + 1)-th vehicles was assumed to be the stimulus for the n-th vehicle.6 In other words, it was assumed that every driver tends to move with the same speed as that of the corresponding leading vehicle so that x¨ n (t) =
1 x˙ n+1 (t) − x˙ n(t) , τ
(9.55)
where τ is a parameter that sets the time scale of the model. Note that 1/τ in the equation (9.55) can be interpreted as a measure of the sensitivity coefficient S of the driver; it indicates how strongly the driver responds to unit stimulus. According to such models (and their generalizations proposed in the 1950s and 1960s), the driving strategy is to follow the leader and, therefore, such car-following models are collectively referred to as the follow-the-leader model. Pipes [1111] derived the equation (9.55) by differentiating, with respect to time, both sides of the equation xn (t) = xn+1 (t) − xn (t) = (x)safe + τ x˙ n (t),
(9.56)
which encapsulates his basic assumption that (1) the higher is the speed of the vehicle, the larger should be the distance headway, and (2) in order to avoid collision with the leading vehicle, each driver must maintain a safe distance (x)safe from the leading vehicle. It has been argued [187] that for a more realistic description, the strength of the response of a driver at time t should depend on the stimulus received from the other vehicles at time t − T where T is a response time lag. Therefore, generalizing the equation (9.55), one would get [187] x¨ n (t + T ) = S[˙xn+1 (t) − x˙ n (t)]
(9.57)
where the sensitivity coefficient S is a constant independent of n. 6 In the following, we label the vehicles in driving direction such that the (n + 1)-th vehicle is in front of the n-th vehicle.
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According to the equations (9.55) and (9.57), a vehicle would accelerate or decelerate to acquire the same speed as that of its leading vehicle. This implies that, as if, slower following vehicle are dragged by their faster leading vehicle. In these linear dynamical models, the acceleration response of a driver is completely independent of the distance headway. Therefore, this oversimplified equation fails to account for the clustering of the vehicles observed in real traffic. Moreover, since there is no density-dependence in this dynamical equation, the fundamental relation cannot be derived from this dynamics. In order to make the model more realistic, we now assume [428] that the closer is the n-th vehicle to the (n + 1)-th, the higher is the sensitivity of the driver of the n-th car. In this case, the dynamical equation (9.57) is further generalized to x¨ n (t + T ) =
κ [˙xn+1 (t) − x˙ n (t)], [xn+1 (t) − xn (t)]
(9.58)
where κ is a constant. An even further generalization of the model can be achieved [429, 436] by expressing the sensitivity factor for the n-th driver as Sn =
κ[vn (t + τ )]m , [xn+1 (t) − xn(t)]
(9.59)
where and m are phenomenological parameters to be fixed by comparison with empirical data. These generalized follow-the-leader models lead to coupled nonlinear differential equations for xn . Thus, in this microscopic theoretical approach, the problem of traffic flow reduces to problems of nonlinear dynamics. So far, as the stability analysis is concerned, there are two types of analyses that are usually carried out. The local stability analysis gives information on the nature of the response offered by the following vehicle to a fluctuation in the motion of its leading vehicle. On the other hand, the manner in which a fluctuation in the motion of any vehicle is propagated over a long distance through a sequence of vehicles can be obtained from an asymptotic stability analysis. From experience with real traffic, we know that drivers often observe not only the leading vehicle but also a few other vehicles ahead of the leading vehicle. For example, the effect of the leading vehicle can be incorporated in the same spirit as the effect of nextnearest-neighbors in various lattice models in statistical mechanics. A linear dynamical equation, which takes into account this next-nearest-neighbor within the framework of the follow-the-leader model, can be written as [1199] x¨ n (t + T ) = S (1) [˙xn+1 (t) − x˙ n (t)] + S (2)[˙xn+2 (t) − x˙ n(t)], where S (1) and S (2) are two phenomenological response coefficients.
(9.60)
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The weakest point of these theories is that these involve several phenomenological parameters that are determined through calibration, i.e., by fitting some predictions of the theory with corresponding empirical data [744, 1429]. Besides, an extension to multilane traffic is difficult since every driver is satisfied if he or she can attain the desired speed!
9.4.2. Optimal Velocity Model and Its Extensions Some general principles should be considered while formulating the dynamical equations for updating the velocities and positions of vehicles in any microscopic theory: • In the absence of any disturbance from the road conditions and interactions with other vehicles, a driver tends to drive with a desired velocity V des ; if the actual current velocity of the vehicle v(t) is smaller (larger) than V des , the vehicle accelerates (decelerates) so as to approach V des . • In freely-flowing traffic, even when a driver succeeds in attaining the desired velocity V des , the velocity of the vehicle fluctuates around V des rather than remaining constant in time. • The interactions between a pair of successive vehicles in a lane cannot be neglected if the gap between them is short in relation to V des ; in such situations, the following vehicle must decelerate so as to avoid collision with the leading vehicle. In the car-following models, such driving strategy is expressed mathematically as7 v˙n (t) =
1 des Vn (t) − vn(t) , τ
(9.61)
where Vndes (t) is the desired speed of the n-th driver at time t. In all follow-the-leader models mentioned above, the driver maintains a safe distance from the leading vehicle by choosing the speed of the leading vehicle as his or her own desired speed, i.e., Vndes (t) = vn+1 . An alternative possibility has been explored in works based on the car-following approach. This formulation is based on the assumption that Vndes depends on the distance headway of the n-th vehicle, i.e., Vndes (t) = V opt (xn (t)) so that v˙n (t) =
1 opt V (xn (t)) − vn(t) , τ
(9.62)
where the so-called optimal-velocity (OV) function V opt (xn ) depends on the corresponding instantaneous distance headway xn (t) = xn+1 (t) − xn (t). In other words, 7 Fluctuations can in principle be included by an additional noise term.
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according to this alternative driving strategy, the n-th vehicle tends to maintain a safe speed that depends on the relative position, rather than relative velocity, of the n-th vehicle. In general, V opt (x) → 0 as x → 0 and must be bounded for x → ∞. For explicit calculations, one has to postulate a specific functional form of V opt (x). Car-following models along this line of approach have been introduced by Bando et al. [67–69]. For obvious reasons, these models are usually referred to as OV models. Since the equations of motion in the follow-the-leader models involve only the velocities, and not positions, of the vehicles, these can be formulated as essentially firstorder differential equations (for velocities) with respect to time. In contrast, since the equations of motion in the OV model involve the positions of the vehicles explicitly, the theoretical problems of this model are formulated mathematically in terms of second-order differential equations (for the positions of the vehicles) with respect to time [67–69]. Clearly, the reliability of the predictions of the OV model depends on the appropriate choice of the OV function. The simplest choice for V opt (x) is [1324, 1325] V opt (x) = vmax (x − d),
(9.63)
where d is a constant and is the Heaviside step function. According this form of V opt (x), a vehicle should stop if the corresponding distance headway is less than d; otherwise, it can accelerate so as to reach the maximum allowed velocity vmax . A somewhat more realistic choice [1031, 1325] is ⎧ ⎪ for x < xA , ⎨0 opt V (x) = f x for xA ≤ x ≤ xB . (9.64) ⎪ ⎩ vmax for xB < x. The main advantage of the forms (9.63) and (9.64) of the OV function is that exact analytical calculations, e.g., in the jammed region, are possible [1325]. Although (9.63) and (9.64) may not appear very realistic, they capture several key features of more realistic forms of OV function [68, 69], e.g., V opt (x) = tanh[x − xc ] + tanh[xc ]
(9.65)
for which analytical calculations are very difficult. For the convenience of numerical investigation, the dynamical equation (9.62) for the vehicles in the OV model has been discretized and, then, rewritten as a difference equation [1017]. The main question addressed by the OV model is the following: what is the condition for the stability of the homogeneous solution? xhn = bn + ct,
(9.66)
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300 Homogeneous flow Congested flow Simulation
Flow Q (vehicles/ 5min)
250 A C
200 150 100
B
50 0
I 0
II 20
III IV 80 40 60 Density k (vehicles/km)
V 100
120
Figure 9.3 Fundamental diagram of the OV model. The solid line shows the OV function and the dots simulation data. One can distinguish five different density regimes with respect to the stable stationary state (from [1325]).
where b = (x)av = L/N is the constant average spacing between the vehicles and c is the constant velocity. It is not difficult to argue that in the OV models, the in general, ∂V opt 2 homogeneous flow becomes unstable when ∂x x=b > τ [68, 69]. One can distinguish five different density regimes with respect to the stability of microscopic states (Fig. 9.3). At low and high densities, the homogeneous states are stable. For intermediate densities, three regimes with jammed states exist. In region III, the jammed state is stable, whereas in regions II and IV, both homogeneous and jammed states form stable structures. Beyond the formation of jams, also hysteresis effects have been observed. Thus, the OV model is able to reproduce many aspects of experimental findings. A modified Korteweg-deVries (KdV) equation has been derived from the equation (9.62) in a special regime of the parameters [785], and the relations between its kink solutions and traffic congestion have been elucidated [960]. In order to account for traffic consisting of two different types of vehicles, say, cars and trucks, Mason and Woods [916] generalized the formulation of Bando et al. [68, 69] by replacing the constant τ by τn so that v˙n (t) =
1 opt [V (t) − vn(t)], τn n
(9.67)
where τn now depends on whether the n-th vehicle is a car or a truck. Since a truck is expected to take longer to respond than a car, we should assign larger τ to trucks
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and smaller τ to cars. Some other mathematically motivated generalizations of the OV model have also been considered [503, 987, 991–993, 1016]. In [542], the OV model was calibrated using empirical car-following data. It was found that the relaxation time is rather short (τ ≈ 1.2 s), which leads to unrealistically high accelerations and overshooting of the velocity. Nevertheless the unrealistically large decelerations are not sufficient to avoid accidents in the model [1469]. Therefore, it has been suggested to extend the OV model to the generalized force model (see Section 9.4.3). Another possible solution to these problems is to take into account the driver reaction times through a delay time td , e.g., by replacing v˙n (t) in the equation of motion (9.62) by v˙n (t + td ) [66]. Davis [262, 263] has used the maximum size of a safe platoon, i.e., a platoon of vehicles that avoids collisions, as a measure of stability. He found that safe platoons require small delay time td which is much smaller than typical reaction times. As possible solution, he suggested to replace the OV function V opt (xn (t)) by V opt (xNn(t − td ) + td vn (t − td )), i.e., a kind of anticipation of future gaps. Further related studies have been performed by Wilson et al. [1085, 1469]. As mentioned earlier, drivers often receive stimulus not only from the leading vehicle but also from a few other vehicles ahead of the leading vehicle. One possible way to generalize the OV models for taking into account such multivehicle or multianticipative interactions [855] is to write the dynamical equations as
m opt xn+j − xn Sj V (9.68) − vn , v˙n = j j=1
where Sj are sensitivity coefficients. One of the commonly used explicit forms of the OV function, e.g. (9.65), can be chosen for that of the function V opt in (9.68). Extensions to multilane traffic [264, 995, 1225] and effects of ramps [105, 264] have also been studied. Open boundary conditions have been investigated by Mitarai and Nakanishi [934, 935]. They found an oscillatory solution in the linearly unstable region, which shares some features with synchronized traffic. Computer simulations of open systems were reported in [379]. Another OV model that reproduces certain aspects of synchronized traffic has been proposed in [504]. The parameters in the OV function of this model depend on the local traffic situation. The full velocity difference model (see Section 9.4.3) can also be considered a generalization of the OV model. It has been used to describe the transition from free flow to synchronized flow [675]. Berg et al. [104] have performed a continuum limit of the OV model. In [515, 530], it has been argued that the Payne model introduced in Section 9.2.1 can be considered to be a macroscopic approximation of the OV model. Some ideas of the OV model have been utilized by Mahnke et al. [897, 899–901] in their master equation approach to the study of jam dynamics. This is discussed in Section 9.6.1.
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9.4.3. Generalized Force Models The generalized force model introduced in [542] is motivated by the social-force concept [864], which has also been applied to model pedestrian dynamics (see Section 11.5.2). It assumes that the amount and direction of a behavioral change, which is typically an acceleration in traffic dynamics, is determined by generalized forces (behavioral or social forces), which reflect the motivation of an individual in response to the state of its environment. In general, these forces do not fulfill Newton’s laws8 like actio = reactio. In the case of highway traffic, the main motivation for each driver n is to reach a desired velocity v (0), which is given by the typical relaxation term (v (0) − vn )/τ , and keeping a safe distance from other vehicles. For the latter, mainly the distance dn to the preceding car n + 1 is relevant. The general structure of the equations of motion is then given by v˙n =
v (0) − vn + f (xn , vn ; xn+1 , vn+1 ), τ
(9.69)
where f is a repulsive interaction force. The choice f (opt) (dn ) =
1 opt V (dn ) − v (0) τ
(9.70)
corresponds to the OV model. To guarantee that drivers brake early and strongly enough for large velocity differences vn = vn+1 − vn apart from f (opt) , an additional term f (brake) (vn , vn+1, dn ) = −λ(vn , dn )vn (−vn )
(9.71)
has to be taken into account so that the interaction force has the form f = f (opt) + f (brake) .
(9.72)
The Heaviside function in (9.71) guarantees that the term is only effective if the velocity vn+1 of the preceding vehicle is smaller than vn . λ(vn , dn ) is a sensitivity function, which should be chosen such that f (brake) grows with increasing velocity difference |vn | and decreasing distance dn and vanishes for dn → ∞. In [542], the sensitivity function 1 dn − d(vn ) λ(vn , dn ) = exp − τ R
(9.73)
8 There are further problems, like the validity of the superposition principle which is nevertheless usually assumed in all
models.
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was used, which is motivated by the assumption that drivers try to keep a velocitydependent safe distance d(vn ) = dn(0) + Tvn .
(9.74)
Here, d (0) is the minimal vehicle distance and T is a safe time headway (typically of the order of the reaction time). The braking time τ should be smaller than τ since deceleration capabilities are larger than acceleration capabilities. R can be interpreted as the range of the braking interaction. The full model can then be written in the form of a generalized OV model, dvn V ∗ (dn , vn , vn ) − vn(t) = dt τ∗
(9.75)
with V ∗ (dn , vn , vn ) =
τ ∗ opt τ∗ V (dn ) + (vn )vn+1 τ τ
(9.76)
and
τ = τ exp (dn − d(vn ))/R
and
1 1 (vn ) = + . ∗ τ τ τ
In [542], it was also suggested to replace the standard OV function by V opt (dn ) = v (0) 1 − e−(dn −d(vn ))/R ,
(9.77)
(9.78)
where R is the range of the acceleration interaction. A calibration using empirical carfollowing data then yielded the optimal parameter values, which are shown in Table 9.1. A different form for f (brake) has been proposed in [673] called the full velocity difference model. It uses a braking term of the form f (brake) (vn , vn+1 , dn ) = λ(vn , dn )vn .
(9.79)
This term takes both positive and negative velocity differences into account and guarantees that a driver will brake also when the preceding car is much faster and the headway is smaller than the safe distance [683]. Table 9.1 Optimal parameter values [542] for the generalized force model based on a calibration with empirical follow-the-leader data Parameter Value
v (0)
τ
d (0)
T
τ
R
R
16.98 m/s 2.45 s 1.38 m 0.74 s 0.77 s 5.59 m 98.78 m
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Different variants of the full velocity difference model have been discussed, e.g., by introducing an asymmetry between positive and negative velocity differences [453], an additional acceleration difference term [1520, 1521], or a form of anticipation by including the velocity difference vn+1 [430]. In [683, 1088], continuum analogs of the full velocity difference model have been derived. A force model that tries to capture the tendency of vehicles to keep speed and distance has been proposed in [1518]. The first effect is taken into account by an acceleration term v , a(v) = a0 1 − v0
(9.80)
where v0 corresponds, e.g., to a speed limit of 15 m/s and a0 is the initial acceleration of the order of 2 m/s2 . The second effect is modeled in analogy to molecule dynamics by a Lennard-Jones potential dopt 2 dopt 4 k + , U (d) = − d d d
(9.81)
where d is the headway and dopt = d0 + k0 v 2
(9.82)
is the optimal distance the drivers want to keep. Typical parameter values suggested in [1518] are d0 = 9 m, k0 = 0.1 s2 /m. In terms of the effective density ρ ∗ = 1/d, the equation of motion for each vehicle is then given by
v ∗ ∗ ∗ 4 ∗ 2 . (9.83) a(ρ , v) = kρ −(ρ dopt ) + (ρ dopt ) + a0 1 − v0 A closer analysis shows that problems might occur when the preceding car has velocity vprec = 0. In this case, equation (9.83) is no longer applicable.
9.4.4. Intelligent Driver Model In the intelligent driver model (IDM) [1382, 1387], each driver n adjusts speed vn depending on the velocity difference vn = vn+1 − vn and headway dn = xn+1 − xn − according to
vn v˙n = a 1 − v0
δ
d∗ − n dn
2 .
(9.84)
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Here, a is the maximum acceleration, v0 is the desired velocity, d ∗ is the minimum desired headway, and δ is an acceleration exponent. The right-hand side interpolates between a free acceleration af = a(1 − (v/v0)δ ) on a free road and braking maneuvers with deceleration −a(d ∗ /d)2 when the headway becomes too small. The desired headway depends dynamically on the velocity vn and relative velocity vn according to vn vn dn∗ (vn , vn ) = d0 + Tvn − √ , 2 ab
(9.85)
where d0 is the typical headway in a jam, Tn is a safe time headway, and b is a comfortable deceleration. The third term becomes relevant only in nonstationary traffic. It implements an accident-free (“intelligent”) driving strategy, which implies in almost all situations braking decelerations less than the comfortable value bn . In emergency situations, the decelerations can become larger to make the model collision-free [1387], at least for single-lane traffic. Note that the IDM does not take into account the reaction time of the drivers, which would lead to an effective reduction of the time headway. The IDM has a homogeneous flow solution with headway d0 + Tvh dh = 1 − (vh /v0 )δ
(9.86)
corresponding to the global density ρ = 1/( + dh ). The flux in this homogeneous state is Jh = ρvh . Simple expressions for vh only result in certain limiting cases [1387]. Table 9.2 shows typical parameter values for the IDM. The acceleration a corresponds to 38 s needed to accelerate from 0 to 100 km/h. The maximum deceleration is approximately 4.5 m/s2 , i.e., g/2 and much larger than b. A proper calibration of the IDM with empirical car-following data based on trajectory sets is described in [744]. The fitted parameters are found to have realistic values. Table 9.2 Typical parameter values for the IDM as proposed in [1387] Parameter
Typical value
Desired velocity v0 Safe time headway T Maximal acceleration a Desired deceleration b Acceleration exponent δ Headway in jam d0 Car length l = 1/ρmax
120 km/h 1.6 s 0.73 m/s2 1.67 m/s2 4 2m 5m
In [1382], slightly different values for a, b, and T have been used.
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The IDM describes most macroscopic aspects of the spatiotemporal dynamics, especially the various types of congested traffic [1388] including the scattered flow-density data in the synchronized regime [1384, 1387], the behavior in systems with inhomogeneities like ramps [524, 1387], and hysteresis phenomena [259]. The IDM with memory (IDMM) [1384] allows also to take into account memory effects in the adaption of drivers to surrounding traffic. However, like most car-following models, it produces unrealistic dynamics and crashes if realistic reaction times (of the order of 1 s) are introduced [1389], although it is crash-free for vanishing reaction times [1382, 1387]. In order to overcome the problems described above, the IDM has been extended to the human driver model (HDM) in [1389]. It is a meta-model that incorporates additional aspects into basic car-following models of the type v˙n = a(dn , vn , vn ) .
(9.87)
For a more realistic description (i) finite reaction times, (ii) estimation errors (for the headway dn and the relative velocity vn ), (iii) spatial anticipation (interactions with preceding vehicles), and (iv) temporal anticipation (of future headways and velocities) have to be included. Basically this implies that the input parameters in (9.87) have to be replaced by effective or estimated ones. The HDM includes not only the IDM, but also OV models (Section 9.4.2), the velocity difference model (Section 9.4.3), bounded rational driver models [886, 887] and the Gipps model (Section 9.5.1). Lane-changes have been implemented based on the MOBIL (minimizing overall braking induced by lane changes) concept [745].
9.4.5. Kerner–Klenov Model The Kerner-Klenov (KK) model [731, 732] was introduced to reproduce the basic features of Kerner’s three-phase traffic theory. The Kerner–Klenov–Wolf model discussed in Section 8.3.2 is the CA variant of the KK model. The basic rules of vehicle motion for a vehicle which is at time t located at x(t) and has velocity v(t) and (net) headway d(t) = xprec (t) − x(t) (xprec is the position of the preceding car) are given by [727] v(t + τ ) = max [0, min {vmax , vdes (t), vsafe (t)}] ,
(9.88)
x(t + τ ) = x(t) + v(t + τ )τ ,
(9.89)
where τ is the discrete time step and vmax the maximal velocity in free flow. The desired speed v(t) + (t) for d(t) ≤ D(t) vdes (t) = (9.90) v(t) + a(t)τ for d(t) > D(t)
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depends on the headway d(t) and the synchronization distance D(t) = D(v(t), vprec(t)) where vprec (t) is the velocity of the preceding car. D(t) will be specified later. (t) is given by (t) = max [−b(t)τ , min{a(t)τ , v(t)}]
(9.91)
with the velocity difference v(t) = vprec(t) − v(t) to the preceding car. The acceleration and deceleration functions a(t) ≥ 0 and b(t) ≥ 0, respectively, restrict speed changes during a time step. They contain a stochastic component to simulate driver time delays in acceleration and deceleration. The safe speed vsafe (t) is given by 1 vsafe (t) = min v (s) (t), d(t) + vanti τ
(9.92)
where (s) vanti = max 0, min vprec(t) − aτ , vprec(t) − aτ , dprec(t)/τ
(9.93)
is an anticipated velocity of the preceding vehicle at the next time step and v (s) (t) is the solution of the Gipps equation for a safe velocity (Section 9.5.1). The precise rules are rather complex and involve a large number of parameters (see chapter 16.3 of Kerner’s book [727]). They are designed such that in the deterministic limit, the stationary states cover a two-dimensional region in the flow-density plane (Fig. 9.4). This is realized through the speed adaption effect in synchronized flow where all vehicles move at the same constant speed and headway. Two related but somewhat simpler deterministic models that capture some essential ideas of the KK model have been suggested in [733]. The acceleration time delay (ATD) model emphasizes the importance of delays in acceleration and deceleration processes. In the speed adaption model (SAM) [733, 1431], speed adaption occurs in synchronized flow depending on the driving conditions. Both models yield congested traffic patterns that are consistent with empirical results.
9.4.6. Inertial Car-Following Model The model proposed by Tomer et al. [1205, 1371] assumes that car acceleration is affected by four factors, namely, (1) keeping a safe time gap, (2) early braking if the preceding car is much slower, (3) obeying speed limits, and (4) random noise. These factors are captured by the acceleration function of vehicle n, x0n
2 (−vn ) (9.94) − − k (vn − vmax + η), an = A 1 − xn 2(xn − D)
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F
J
U
L
ρ
Figure 9.4 Fundamental diagram of the KK model in the deterministic limit. The boundaries of the allowed region are associated with free flow (F), the synchronization gap (L), and the safe gap (U).
where xn = xn+1 − xn is the headway of vehicle n and vn = vn+1 − vn is the velocity difference. Furthermore, the safety distance x0n = vn T + D has been introduced, which is determined by the safe time gap T and the minimal distance D between cars. Finally,
(x) is the Heaviside function (with (0) = 0), A and k are sensitivity constants, vmax is the maximal velocity, and η is a noise term. The first term in (9.94) becomes important for small velocity differences vn and leads to braking (accelerating) if xn < x0n (xn > x0n ). The second term is relevant if the preceding car is approached very fast (vn vn+1 ) and corresponds to the deceleration, which is necessary to reduce the velocity difference vn to 0 as the minimal headway D is approached. The third term is dissipative and represents a repulsive force, which reduces the velocity if vmax is exceeded. The equation of motion is then given by x˙ n = vn and v˙n = an with an as defined in (9.94). Assuming periodic boundary conditions (xN +1 = x1 + N /ρ, vN +1 = v1) and neglecting the noise term η, a solution corresponding to homogeneous flow can be found: ⎧ A(1 − Dρ) + kvmax ⎪ ⎪ for ρ ≤ ρc ⎨ aρT + k (0) (9.95) vn = ⎪ 1 − Dρ ⎪ ⎩ for ρ ≥ ρc ρT with ρc =
1 D+Tvmax
so that xn(0) =
n−1 + vn(0)t. ρ
(9.96)
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In computer simulations, one finds in an intermediate density regime ρ1 < ρ < ρ2 flows which are considerably lower and velocity fluctuations occur. For small values of A, the upper critical density can become larger than the maximal density ρmax = 1/D. Therefore, one can distinguish three different flow regimes: (1) free flow for ρ < ρ1 , (2) nonhomogeneous congested traffic (NHC) for ρ1 < ρ < ρ2 , and (3) homogeneous congested traffic ρ > ρ2 . In the NHC regime, the presence of humps (dense regions) is observed, which can move forward or backward. In the stationary state, the humps are equidistant, and the NHC state is similar to the recurring humps state of [844] and the empirical observations in [742]. The NHC is not unique since, depending on the initial conditions, different wavelengths of the recurring humps and corresponding flows can be realized. In [1371], this is interpreted as indication for the existence of many different attractive limit cycles. Some of these cycles are more sensitive to noise than others. The values of ρ1 and ρ2 can be estimated by an analytical calculation [1371]: ρ1 =
1 , D + Tvmax
ρ2 =
2 . AT 2
(9.97)
Comparison with simulation data show that ρ2 ≈ ρ2 , but that ρ1 is considerably smaller than ρ1 . In the regime ρ1 < ρ < ρ1 , both the NHC and the homogeneous flow solution appear to be stable as indicated by hysteresis loops in the density-flow plane.
9.5. COUPLED-MAP MODELS In the car-following models, space is assumed to be a continuum, and time is represented by a continuous variable t. Besides, velocity and acceleration of the individual vehicles are also real variables. However, most often, for numerical manipulations of the differential equations of the car-following models, one needs to discretize the continuous variables with appropriately chosen grids. In contrast, in the coupled-map approach [708], one starts with a discrete time variable. The dynamical equations for the individual vehicles are formulated as discrete dynamical maps that relate the state variables at time t with those at time t + 1, although position, velocity, and acceleration are not restricted to discrete integer values. The unit of time in this scheme (i.e., one time step) may be interpreted as the reaction time of the individual drivers as the velocity of a vehicle at the time step t depends on the traffic conditions at the preceding time step t − 1. The general form of the dynamical maps in the coupled-map models can be expressed as follows: vn (t + 1) = Mapn [vn (t), vdes , xn (t)],
(9.98)
xn (t + 1) = vn (t) + xn (t),
(9.99)
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where vdes is a desired velocity. In general, the dynamical map Map[vn (t), vdes , xn (t)] takes into account the velocity vn (t) and the distance headway xn (t) of the n-th vehicle at time t for deciding the velocity vn (t + 1) at time t + 1. The effects of the interactions among the vehicles enter into the dynamical updating rules (9.98), (9.99) only through the distance headway xn . The model of Kerner and Klenov discussed in Section 9.4.5 can also be viewed as coupled map model with maps given by (9.88) and (9.89).
9.5.1. Gipps Model Starting point of the Gipps model [436] is the consideration of braking distances in the case of finite deceleration capability. From this, a collision-free update scheme for the vehicle velocities is derived. A velocity is considered to be safe if the driver can bring the car under any circumstances to a complete stop before colliding with the preceding vehicle. Therefore, we consider a situation where the vehicle moves at velocity vn and has an initial headway dn = xn+1 − xn − to the preceding car moving with velocity vn+1 . Here, xn and xn+ are the initial positions of the cars, and is the vehicle length. Furthermore, we denote the braking distance by D(v), which is distance required to come to a stop when driving at velocity v. Taking into the reaction τ of the drivers, the safety criterion requires D(vn ) + vnτ ≤ D(vn+1 ) + dn .
(9.100)
Assuming that the maximal deceleration of the vehicle is given b < 0, one has9 D(v) = −
v2 . 2b
(9.101)
In principle, one now has to invert (9.100) to determine the safe velocity vn . Gipps also introduces a further safety margin by replacing the true reaction time τ by a safety reaction time τ + θ . The inclusion of the parameter θ leads to earlier braking of the drivers who then gradually reduce the deceleration toward the end of the braking process. A second constraint considered in the Gipps model concerns the acceleration. It is assumed that the desired speed vmax is not exceeded. Furthermore, a relation between the acceleration and the velocity is derived by fitting empirical car-following data. These constraints give an inequality, which limits vn (t + τ ) from above [436]. The new velocity vn (t + τ ) at time t + τ is then given by the minimum of the two constraints derived from considering the acceleration and braking capabilities. The case where the velocity is limited by the acceleration bound corresponds to free flow, and 9 One could also use other forms for D(v), which, e.g., take into account aspects like comfort [790].
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the case where it is limited by the safety bound corresponds to congested traffic. If the drivers underestimate the braking capabilities of the preceding car, the Gipps model can describe overreactions of drivers and thus the existence of instabilities.
9.5.2. Krauss Model (SK Model) Krauss et al. [790–793] introduced a whole class of stochastic models by considering necessary conditions for the collision-free motion of vehicles (see also [1028] for details). The models are continuous in space and discrete in time and are related to the Gipps model discussed in Section 9.5.1. The vehicles are characterized by a maximum velocity vmax , their acceleration and deceleration capabilities a(v) and b(v), respectively, and their length , which will be taken to be = 1 in the following. Then, the update rules for the velocity v and the space coordinate x of each vehicle in the SK model are as follows: SK1: Desired velocity:
vdes = min [vmax , v + a(v), vsafe]
SK2: Randomization:
v = max [0, rand(vdes − a, vdes)]
SK3: Vehicle movement:
x → x + v.
Here, rand(v1 , v2) denotes a random number uniformly distributed in the interval [v1, v2 ). vsafe is a velocity, which guarantees collision-free motion of the vehicles. It is given explicitly by [790, 1028] vsafe = vp + b(v)
g − vp , vp + b(vp)
(9.102)
where vp is the velocity of the preceding vehicle located at xp and g = xp − x − is the distance headway. In the simplest case, the acceleration and deceleration capabilities do not depend on the velocity, i.e., a(v) = a = const and b(v) = b = const. The behavior of the model can be classified in three different families. These are sketched schematically in Fig. 9.5 and can be characterized as follows: • Class I: High acceleration Here, no spontaneous jamming exists. For a → vmax and b 1, the behavior is similar to that of a CA model without velocity memory introduced in [152], which is closely related to the Kasteleyn model [713] of statistical physics. It can also be interpreted as five-vertex model [154]. Another model belonging to this class is the Fukui–Ishibashi model (see Section 8.4.1). • Class II: High acceleration – low deceleration The outflow from a jam is identical to the maximal possible flow. The jamming transition is not a true phase transition, but rather a crossover. The limit b → ∞, a = 1
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1.6 1.4 1.2 I
a
1 0.8
II
0.6 0.4 0.2
III 0
0.5
1
1.5
2
2.5
b
Figure 9.5 Schematic sketch of the three different model families in the SK model. a and b are the acceleration and deceleration capabilities, respectively (after [790, 791]).
corresponds to a continuum version of the Nagel–Schreckenberg (NaSch) cellular automaton model. • Class III: Low acceleration – low deceleration These models exhibit phase separation and metastability. The jamming transition is of first order. The outflow from a jam is not maximal. For a vmax and b vmax , the model is closely related to the Gipps model [436] discussed in Section 9.5.1. Other models belonging to this class are the Kerner–Konhäuser model (Section 9.2.1), the OV model (Section 9.4.2), and the models with slow-to-start rules (Section 8.1). On a macroscopic level, classes I, II, and III can be distinguished by the ordering of the densities ρf and ρc , where ρf is the density of the outflow from a jam and ρc is the density where homogeneous flow becomes unstable [790, 791]. For ρc > ρf , the outflow from a jam is stable, and the system phase separates into free-flow and jammed regions. Furthermore, metastable states can be found. This is the type of the behavior found in class III. For ρc < ρf , on the other hand, the outflow from a jam is unstable, and no metastable states or phase separation can be found. This is the typical behavior of classes I and II. These classes can further be distinguished since in class I, one does not find any structure formation, like spontaneous jamming, in contrast to class II. Namazi et al. [1035] have studied the SK model with open boundary conditions and found good agreement with the predictions of the extremal principle introduced in Section 4.5.2. Anticipation effects have been studied in [344]. It leads to larger flows, short temporal headways, and a stabilization of flow in dense traffic. This is explained by an alternating structure of headways, which become strongly anticorrelated: a short
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headway is followed by a long one. However, these anticorrelations have not been found in the empirical data so that additional mechanism must be at work besides anticipation. A related model has been studied before by Migowsky et al. [932, 1451]. In this model, vehicles are also characterized by a maximum velocity vmax and a bounded acceleration capability (−bmax < x¨ < amax ) which determines the safety distance ds necessary to avoid accidents. The investigations in [932, 1451] focussed on the effect of so-called driving strategies. These strategies are characterized by a vector ( fv , fa , fs ), where fv , fa , and fs are the fraction of the vehicle’s maximal velocity, acceleration, and safety distance (n) actually used (i.e., the maximum velocity is given by vmax = fv(n) vmax ), respectively. This can lead to the possibility of accidents and allows to study the number of crashes as a function of the driving strategies.
9.5.3. Yukawa-Kikuchi Model Yukawa and Kikuchi [1508–1510] have studied coupled-map models based on the map v0 − v(t) v(t + 1) = F(v(t)) := γ v(t) + βtanh + γ
(9.103)
for the uninfluenced motion of a single vehicle. v0 is the preferred velocity of the vehicle, and β, γ , δ, and are parameters. For γ close to 1, the map becomes chaotic, but acceleration and deceleration are approximately constant far from v0. Their magnitude is determined by the parameter β. controls the difference of the acceleration and deceleration capabilities. Although the model is deterministic, fluctuations in the velocity are introduced through deterministic chaos. These fluctuations around v0 are determined by the parameter δ. If there is more than one vehicle on the road, one needs an additional deceleration mechanism to avoid collisions. This can be achieved by introducing a deceleration map. Assuming that deceleration is dominated by the headway, two models have been studied in [1508]. In model A, the deceleration map describes a sudden braking process. If the front-bumper to front-bumper distance xn to the next vehicle ahead is less than the current velocity vn (t) of the following vehicle, then the velocity is reduced to xn − l where l is the length of the vehicles. The corresponding map is B(xn (t)) = xn (t) − l. Model B has a more complex deceleration map: vn (t + 1) = G(xn (t), vn (t)) :=
F(vn (t)) − vn(t) [xn (t) − l − vn(t)] + vn (t) (α − 1)vn (t)
(9.104)
for vn (t) ≤ xn (t) − l ≤ αvn (t). The parameter α determines the range within which the deceleration map G(x, v) is used. For headways less than αvn (t), the map G is used instead of F(v). Note that G(x, v = x − l) = x − l and G(x, v = (x − l)/α) =
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F(v = (x − l)/α), i.e., G interpolates between the free-motion map F and the sudden braking map of model A. The full velocity map of model B is thus given by ⎧ F ⎪ for αvn (t) ≤ xn (t), ⎨F(vn (t), vn ) Mapn (vn (t), xn (t)) = G(xn (t), vn (t)) for vn (t) ≤ xn (t) ≤ αvn (t)), (9.105) ⎪ ⎩ for xn (t) ≤ vn (t)). B(xn (t)) Local measurements of the flow for a system of vehicles with different preferred velocities vnF produce a fundamental diagram of inverse-λ shape [1510]. Here, the nonuniqueness of the flow has a simple explanation. Due to the different vnF , platoons form behind the slowest vehicles. Whenever such a platoon passes the measurement region, a flow value on the lower branch is recorded. Otherwise, the flow corresponds to the upper branch. Measurements of the power spectral density of temporal density fluctuations, i.e., the Fourier transform of the time series of local densities, show a 1/f α -behavior with α ≈ 1.8 in the free-flow regime. Because of the deterministic dynamics, the system evolves into a state with power-law fluctuations. In [1509, 1510], it has been suggested that the origin of the 1/f α -fluctuations is the power-law distribution (∝ 1/(x)3.0 ) of the headways x, since these are related to density waves. The occurrence of jams destroys long-time correlations since vehicles loose their memory of current fluctuations when they are forced to stop in a jam [1509]. Therefore, no 1/f α -behavior can be observed in the jammed regime. In [1339], a coupled-map model based on OV functions has been introduced by discretizing the time variable of the OV model (Section 9.4.2). This allows to study systems with open boundaries and multilane systems. Furthermore, a multiplicative random noise can be imposed in the velocity update so that the velocity map is given by v(t + 1) = [v(t) + α (Vopt (x) − v(t))] (1 + fnoise ξ),
(9.106)
where ξ ∈ [−1/2, 1/2] is a uniform random variable and fnoise is the noise level.
9.5.4. Nagel-Herrmann Model Nagel and Herrmann (NH) [1024] have introduced a coupled-map model, which is related to the continuum limit of the NaSch model. A generalization has later been presented in [1227]. Vehicles are characterized by a maximal velocity vmax and a safety distance α. The velocity map for the NH model is given by for vn (t) > xn (t) − α, max(xn (t) − δ, 0) (9.107) vn (t + 1) = min(vn (t) + a, vmax) for vn (t) < xn (t) − β.
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In the velocity update step, vehicles that have a headway x smaller than the safety distance α decelerate. The headway distance after deceleration is determined by the parameter δ. Vehicles that have a large enough headway, on the other hand, accelerate. The acceleration coefficient a is determined by a = amax max(1, xn (t)/γ ). Since the dynamics of the model is deterministic, the behavior depends strongly on fluctuations of the initial state [1227]. For equidistant starting positions of the vehicles, the fundamental diagram consists of two linear branches with maximum flow f (ρ ∗ ) = vmax ρ ∗ at density ρ ∗ = 1/(vmax + β). For homogeneous starting positions, the system is free flowing up to a critical density ρcrit . Beyond this, density free-flowing and congested areas coexist.
9.6. OTHER APPROACHES 9.6.1. Probabilistic Traffic Flow Theory Mahnke et al. [897, 899, 900] have developed a stochastic approach based on the description of jam dynamics, i.e., the formation and dissolution of traffic congestions. It allows to make systematic use of analogies with first-order phase transitions and nucleation phenomena in physical systems like supersaturated vapor. One important concept is the mean first passage time, i.e., the average time a system needs to reach a given state for the first time if started from some other state. In traffic flow, several quantities of this type appear, e.g., the lifetime of metastable states or the resolution time of a jam. The first step is to consider the dynamics of a single vehicle cluster or jam [896, 901]. Under the assumption that, at a time, only one vehicle can go into or come out of a jam, the master equation governing the probability distribution P(n; t) of the jam sizes n is given by dP(n; t) = W+(n − 1)P(n − 1; t) + W−(n + 1)P(n + 1; t) dt − [W+(n) + W−(n)]P(n; t),
(9.108)
where W+ and W− , are the growth (n → n + 1) and decay (n → n − 1) transition rates, respectively. It has been argued [896, 901] that W− (n) = 1/τ is constant since a vehicle would require a constant average waiting time τ of the first car to escape from a jam. An Ansatz for W+ (n) can be obtained from the OV approach (Section 9.4.2). Assuming that a vehicle changes its velocity from V opt (xf ) in free flow to V opt (xc ) in a jam and approaches the cluster as soon as the distance to the last car in the jam is less
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than xc , one obtains the Ansatz W+ (n) =
V opt (xf (n)) − V opt (xc ) . xf (n) − xc
(9.109)
Next, xf and xc have to be specified as functions of the cluster length n. Since empirically the jam density is independent of the length of a jam, it is reasonable to assume that xc (n) = xc is constant. This determines the total length Lc = nxc of the jam cluster which in turn allows to calculate the length L f = L − L c of the freeflow region. From this, xc can be obtained although subtleties for n = 1 have to be taken into account [899, 900]. From the calculation of the mean cluster size n , one can identify two critical densities. At ρ1 , the spontaneous formation of a large cluster starts, and for ρ > ρ2 , unstable car clusters with the smallest possible value xfmin can exist [900]. The fundamental diagram can also be obtained from the local density ρ(x) and velocity v(x) in the stationary state. For the OV function, V opt (x) = vmax
(x)2 D 2 + (x)2
it is given by [900] ⎧ 2 ⎪ ⎨ vmax ρ(1 − ρ) for 0 ≤ ρ ≤ ρ1 and ρ2 < ρc , J (ρ) = (Dρ)2 + (1 − ρ)2 ⎪ ⎩1 − ρ − ρ τ V opt (x ) − x for ρ ≤ ρ ≤ ρ , c c 1 2
(9.110)
(9.111)
where ρc = n/L c is the density in the cluster. The first nonlinear part corresponds to homogeneous flow, whereas the second linear part represents the nonhomogeneous congested states. Using the analogy with physical systems, three different regimes can be distinguished: (1) free flow for ρ < ρ1 as a gaseous phase, (2) a liquid phase of heavy traffic for ρ > ρ2 , and (3) a transition regime with both free and congested vehicles for ρ1 < ρ < ρ2 . This approach can further be used to study the phase transitions, especially the metastability, in more detail and to develop the theory further into a thermodynamical approach to traffic flow [898, 900]. The single cluster approach does not take into the merging or splitting of jams. Therefore, an extension to the multicluster situation has been proposed in [718, 897, 899]. Another extension is a stochastic perturbation parameter, which describes the probability for a free car to decelerate and form a jam without any obvious reason. Indeed, this parameter plays a similar role as the randomization in the NaSch model since there is no longer a sharp phase transition at ρ1 if p > 0.
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The approach has also been developed into a description of traffic breakdown as nucleation phenomenon [820, 821, 899]. It includes the empirically observed sequence “free flow → synchronized flow → jam,” which is characterized by additional order parameters, e.g., for the multilane correlations [884, 885]. The effects of ramps can also been taken into account [820]. In a related approach, a description the zero-range process (ZRP) (Section 3.3) has been used to describe phase separation in traffic systems [719]. It is assumed that the escape rate depends on the cluster size n according to w(n) = w∞
b 1+ σ n
(9.112)
for n ≥ 2. The value w(1) is specified separately since it is related to the free motion of cars. The monotonically decreasing form of w(n) can be interpreted as slow-to-start rule: due to the loss of attention, a driver restarts more slowly the longer he or she has been standing in a jam. On the other hand, the choice is motivated by the criterion for phase separation in the ZRP (Section 3.5.3): for the case where either σ > 1 or σ = 1 and b = 2, the homogeneous phase is stable for all densities. In contrast, if either σ < 1 or σ = 1 and b > 2, a condensation transition occurs at a critical density ρc . The fundamental diagram has been calculated in [719, 900]. It is given by J (ρ) = (1 − ρ) w where the mean stationary transition rate w depends on the density. In the regime of phase separation, i.e., for ρ > ρc , one has w = w∞ , and this give a linear branch. Below ρc , w has to be determined from an implicit equation leading to a nonlinear relation between flow and density. It is also possible to determine the metastable branch that looks different from the familiar inverse-λ form (Fig. 6.9) since it has also parts with negative slope and extends to higher densities [719, 900].
9.6.2. Cell Transmission Model The cell transmission model has been introduced by Daganzo [253, 254] as discretization of the LWR model. As in CA models, the road is divided into cells, but now the cell size l is not determined by the length of a car. Instead, for a given time step t, it is identified with the distance a car travels under free-flow conditions (which might be different for different sections of road). Thus, cells are much larger compared to CA models, and no strict exclusion principle has to be imposed. Instead each cell j can be occupied by at most Nj = lρjam vehicles where ρjam is the maximal jam density. Within a cell, no order of the cars needs to be specified. A second constant is the input capacity Qj , the maximum number of vehicles that can move into cell j within one time step. In a more general model, Nj and Qj are allowed to vary in time, e.g., to model transient traffic incidents. Denoting the number
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of cars in cell j at time t by nj (t), the dynamics of the cell transmission model is then given by the master equation (setting t = 1) nj (t + 1) = nj (t) + yj (t) − yj+1(t) .
(9.113)
The current yj (t) from cell j − 1 to j is the minimum of the number of cars nj−1 (t) in cell j − 1, the number of “vacancies” Nj (t) − nj (t) in cell j, and the capacity Qj (t), i.e. yj (t) = min nj−1 (t), Nj (t) − nj (t), Qj (t) .
(9.114)
In order to obtain a closed recursion, boundary conditions have to be specified. Typically the output cell has infinite size (Nout = ∞) and a suitable capacity Qout (t). In order to control the inflow, a pair of cells are introduced. The source cell “00” always contains an infinite number of vehicles (n00 (t) = ∞), which can move into an infinite gate cell “0” (N0 (t) = ∞). The inflow capacity Q0 (t) of the gate cell can than be specified such that the desired input flow at time t + 1 is realized. These equations can be understood as discretizations of the LWR model [253] with fundamental diagram trapezoid form (Fig. 9.6) JCTM(ρ) = min vmax ρ, v back (ρjam − ρ), Jmax ,
(9.115)
J
Jmax
ρΑ
ρmax
ρB
ρ
Figure 9.6 Fundamental diagram of the cell transmission model. The slope of the two branches are given by vmax and −v back , respectively.
Vehicular Traffic IV: Non-CA Approaches
where vmax is the free-flow speed, v back is the speed at which disturbances propagate backward, and Jmax ≤ vmax ρjam /2 is the maximum flow. The LWR model corresponds to the case v back = vmax .
9.6.3. Queueing Models The cell transmission model can be seen as queueing model. Queueing models are often used to study traffic on networks. Links corresponds to roads or road sections and nodes to intersections, crossings with traffic lights, etc. Dynamics inside each node is usually not taken into account. Each node is specified by a certain service rate, and vehicles move to the next queue (usually corresponding to the next node) as soon as they have been “served.” Cars that enter a queue have to wait at least for a time which corresponds to the free-flow travel time along the link before they are allowed to leave. The waiting time is further affected by the capacity (maximum throughput), the storage capacity of the link, and the queuing discipline, i.e., the order in which the cars leave a queue. The most natural choice for the latter is the first-in-firstout discipline. Examples for queueing models can be found in [342, 343, 506, 513, 650, 1411, 1472].
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CHAPTER TEN
Transport on Networks Contents 10.1. Networks and Transport
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10.2. BML Model of City Traffic 10.2.1. Phase Transition 10.2.2. Generalizations and Extensions of the BML Model 10.2.3. More Realistic Description of Streets and Junctions
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10.3. Chowdhury–Schadschneider Model 10.3.1. Crossroads with Signals 10.3.2. ChSch Model 10.3.3. Traffic Signal Optimization
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10.4. Highway and City Networks 10.4.1. Online Simulation of Traffic Networks 10.4.2. Network Analysis 10.4.3. Braess Paradox
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10.5. Computer Networks and Internet Traffic
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Traffic has a strong impact on economy and environment [156]. One of the ultimate goals of traffic research is traffic control or even traffic forecasting. This requires an understanding of traffic dynamics on networks [1259]. In the following, we will consider various aspects of network traffic, focussing on urban and highway networks.
10.1. NETWORKS AND TRANSPORT So far, we have considered vehicular traffic on a single lane, possibly with some inhomogeneities due to ramps or open boundary conditions. In reality, however, traffic usually happens on networks of connected roads or tracks with intersections, etc. The same is true for other transport systems as pedestrian dynamics and especially biological transport, which will be discussed later. The network structure of these systems brings about new and fascinating aspects, which we briefly discuss in the following. Considering the dynamics, in principle, one has to distinguish two aspects: • Dynamics on networks: This is the natural extension of the problems discussed so far where it is assumed that the structure of the network basically remains unchanged
Stochastic Transport in Complex Systems DOI: 10.1016/B978-0-444-52853-7.00010-5
Copyright © 2011, Elsevier BV. All rights reserved.
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on the timescale relevant for the transport processes. Vehicular traffic is the prime example since the road network changes on a timescale of years. • Dynamics of networks: The opposite limit is that where the network itself changes fast, e.g., on a similar timescale as the transport processes itself. This happens often in biological systems like intracellular transport. Another example is the Internet where routers and the connections between them changes on a timescale of hours. According to their structure, different types of networks can be distinguished. One classification is based on the properties of the node degree distribution function P(k), e.g., random networks (P(k) Poissonian), exponential networks (P(k) ∝ exp(−k/k0 )), scale-free networks (P(k) ∝ k−α ), and small-world networks. For further general results on networks, we refer to several comprehensive reviews, e.g., [10, 143, 318, 319, 377]. Typically road networks are random networks. They have a typical degree of the nodes (intersections) of approximately 4. Higher degrees occur very rarely. However, other transport networks, especially in biological systems, can have a different structure. To give an example for the size of typical traffic networks, we consider the highway network of North Rhine-Westphalia, the most populated state of Germany with 18 million inhabitants. The network has a total length of about 2.250 km and includes 67 highway intersections and 830 on- and off-ramps. The whole Autobahn network of Germany has a length of 12.000 km, which corresponds to about 60.0000 single-lane kilometers (2 directions, on average 2.5 lanes per direction). We will later see that even such large networks can be simulated faster than real time, which in principles makes traffic forecasting possible. Traffic networks can be represented in different ways. The natural way is to identify streets with links and intersections with nodes. However, one can also study traffic networks by considering the dual lattice. In the dual lattice [1420], streets correspond to nodes1 and intersections to links. In the dual representation, a vehicle stays on a node until it changes to a different road (e.g., by turning), which implies a jump to a different node. Obviously, this has the disadvantage of losing any information about the length of the roads. Therefore, one needs to find different ways of restoring this information [592].
10.2. BML MODEL OF CITY TRAFFIC In the Biham-Middleton-Levine (BML) model [115], each of the sites of a square lattice represents the crossing of a east–west street and a north–south street. All the streets parallel to the xˆ -direction of a Cartesian coordinate system are assumed to allow only single-lane east-bound traffic, while all those parallel to the yˆ -direction allow only single-lane north-bound traffic. Let us represent the east-bound (north-bound) vehicles 1 More precisely, road segments are grouped into axial lines, which correspond to cells.
Transport on Networks
Figure 10.1 Example for a configuration in the BML model of city traffic.
by an arrow pointing toward east (north); see Fig. 10.1. In the initial state of the system, vehicles are randomly distributed among the streets. The states of east-bound vehicles are updated in parallel at every odd discrete time step, whereas those of the north-bound vehicles are updated in parallel at every even discrete time step following a rule which is a simple extension of the totally asymmetric simple exclusion process (TASEP): a vehicle moves forward by one lattice spacing if and only if the site in front is empty; otherwise the vehicle does not move at that time step. Thus, the BML model is also a driven lattice gas model where each of the sites can be in one of three possible states: either empty or occupied by an arrow ↑ or →. Note that for parallel update rules, the BML model is fully deterministic, and therefore, it may also be regarded as a deterministic CA. The randomness arises in this model only from the random initial conditions [413]. Suppose that N→ and N↑ are the numbers of the east-bound and north-bound vehicles, respectively, in the initial state of the system. If periodic boundary conditions are imposed in all directions, the number of vehicles in every street is conserved since turning of vehicles is not allowed. In a finite L × L system, the densities of the east-bound and north-bound vehicles are given by ρ→ = N→ /L 2 and ρ↑ = N↑ /L 2 , respectively, while the global density of the vehicles is ρ = ρ→ + ρ↑ .
10.2.1. Phase Transition Computer simulations of the BML model with periodic boundary conditions indicate that a first-order phase transition takes place at a density ρc , where the average velocity of
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the vehicles vanishes discontinuously signaling complete jamming; this jamming arises from the mutual blocking (“gridlocking”) of the flows of east-bound and north-bound traffic at various different crossings (see [1334, 1335] for the corresponding results of the BML model with open boundary conditions). At concentrations just above ρc , in the jammed phase, all the vehicles together form a single cluster which is stretched along the diagonal connecting the south-west to the northeast of the system. In other words, the lowest-density jammed configurations consist of a single diagonal band where the → and ↑ occupy nearest-neighbor sites on the band in a zigzag manner. With further increase of density, more and more vehicles get attached to the band in the form of off-diagonal branches, and the infinite cluster of the jammed vehicles looks more and more random. Thus, in general, a typical infinite cluster of the jammed vehicles consists of a backbone and dangling vehicles which are the analogs of the backbone and the dangling ends of the infinite percolation clusters in the usual site/bond percolation [1309]. However, in contrast to the infinite percolation cluster in the usual random site/bond percolation, the infinite spanning cluster of vehicles in the BML model emerges from the self-organization of the system. Nevertheless, concepts borrowed from percolation theory have been used to characterize the structure of the infinite cluster of jammed vehicles in the BML model at ρ > ρc [477, 1336, 1337]. The distribution of the waiting times of the vehicles at the signals (i.e., at the lattice sites) has also been investigated through computer simulations [957, √ 975]. A simple mean-field estimate gives ρ = ρc = 6 − 32 0.343 for the symmetric case ρ→ = ρ↑ = ρ which, in spite of the approximations made, is surprisingly close to the corresponding numerical estimate obtained from computer simulation [115]. Moreover, this estimate is also consistent with the more rigorous result that ρc is strictly less than 1/2 [192, 194]. Ishibashi and Fukui [412, 629] claimed that complete jamming can occur in the BML model only for ρ = 1. However, a plausible flaw in their arguments was pointed out by Chau et al. [192]. In [1289], it has been argued that ρc ∝ L −0.14, i.e., ρc = 0 in the thermodynamic limit. D’Souza [322, 323, 876] has found that there is no sharp phase transition from the free flow to the jammed state. Instead an intermediate phase occurs where jams and freely flowing regions coexist. The properties of this intermediate regime strongly depend on the geometry of the system, e.g., the aspect ratio. A microscopic mean-field theory for the BML model has been developed in [944]. An analytical theory in terms of Pauli operators allows to implement the exclusion principle in an elegant way and has been analyzed within a mean-field approximation [717]. The BML model in three dimensions, although not relevant for vehicular traffic, has also been studied [193].
10.2.2. Generalizations and Extensions of the BML Model The BML model has been generalized and extended to take into account several realistic features of traffic in cities.
Transport on Networks
One natural extension is to distribute the vehicles asymmetrically among the eastbound and north-bound streets [971], i.e., ρ→ = ρ↑ . For convenience, let us write ρ↑ = ρfa and ρ→ = ρ(1 − fa ), where fa is the fraction of the vehicles moving toward north. Obviously, no gridlocking occurs for fa = 0 and fa = 1, i.e., ρc = 1 for both fa = 0 and fa = 1. Moreover, ρc decreases with decreasing asymmetry in the distribution of the vehicles; ρc is the smallest for fa = 1/2, i.e., symmetric distribution of the vehicles. The phase diagram in the ρ − fa plane and ρc ( fa ) has been determined in [971]. Another natural extension is to allow turning of vehicles. In [248, 913], each vehicle n is assigning a preferred direction of motion, Wn (x, y), which depends on its current location (x, y). It jumps to the next site toward east with probability Wn (x, y), while 1 − Wn (x, y) is the corresponding probability that it hops to the next site toward north. Thus, vehicles can take a turn, but this process is stochastic. In the simplest case, N /2 vehicles are assigned Wn (x, y) = γ, while the remaining N /2 vehicles are assigned Wn (x, y) = 1 − γ where 0 ≤ γ ≤ 1/2; this implies that N /2 vehicles move preferentially eastward, whereas the remaining N /2 vehicles move preferentially northward. In the limit γ = 0, no vehicle can turn, and we recover the original BML model with deterministic update rules. The most dramatic effect of the stochastic turning is that the discontinuous jump v of the average velocity v decreases with increasing γ, and eventually, the first-order jamming transition ends at a critical point where v just vanishes. In real traffic, turning is not a stochastic process independently of the other vehicles. A vehicle is likely to turn if its forward movement is blocked by other vehicles ahead of it in the same street. This has been considered in [980] where an east-bound (north-bound) vehicle turns north (east) with probability pturn if blocked by another vehicle in front of it. Computer simulations of this model show that ρc (pturn ) increases with increasing pturn . In a slightly different model [982], on being blocked by a vehicle in front, an eastbound (north-bound) vehicle hops with probability pja to the next east-bound (northbound) street toward north (east). In another extension, different maximum velocities were considered for east- and north-bound vehicles [418] where fast vehicles move according to the rules of the Fukui–Ishibashi model [414]. The BML allows also to take into account the effects of overpasses (or two-level crossings), i.e., sites that can accommodate two vehicles simultaneously [973]. The overpasses weaken the gridlocking in the BML model, and a mean-field approach [1440] predicts that ρc increases with increasing fo . Beyond a critical value of fo , the jammed phase disappears altogether and ρc = 1. These results remain true for asymmetric distributions of the vehicles among the east-bound and north-bound streets [1440]. In [226], an extension to study the effects of faulty traffic lights has been proposed. At these faulty traffic lights, both the north- and the east-bound vehicles are allowed to hop onto the empty crossing, irrespective of whether the corresponding time step of updating is odd or even. If an east-bound vehicle and a north-bound vehicle simultaneously
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attempt to enter the same crossing, then one of them is selected randomly. An increasing density of faulty traffic lights increases the effect of gridlocking thereby decreasing ρc . Various authors have incorporated the effects of static hindrances or roadblocks (e.g., vehicles crashed in traffic accident), i.e., stagnant points [474, 972, 978] where a vehicle has to stay for Tp time steps before attempting to move out of it, or stagnant streets [974], where the local density ρs of the vehicles is initially higher than that in the other streets. In contrast to static roadblocks, a stagnant street or road offers a time-dependent hindrance to the vehicles moving in the perpendicular streets. The time-dependent phenomenon of spreading of the jam from the blockage site during the approach of the system to its jammed steady-state configurations has also been investigated [411, 972, 978]. The special situation where only one north-bound street exists that cuts all the L equispaced mutually parallel east-bound streets of length L has been studied in [1018]. In [1375], a variant of the BML model, which allows to study the effects of a different synchronization scheme of the traffic lights, has been proposed. This green-wave synchronization is often used along main streets in cities to allow continuous flow. The green-wave model of [1375] replaces the parallel updating scheme of the BML model by an updating scheme which is partly backward-ordered sequential (see Section 2.1). At odd time steps, an east-bound vehicle moves by one lattice spacing if the target site was empty at the end of the previous time step or has become empty in the current time step (this is possible because of the backward-ordered sequential updating at every time step). Similarly the positions of the north-bound vehicles are updated at every even time step. The main difference between the BML model and the green-wave model (see Fig. 10.2) is that in the green-wave model, vehicles move as “convoys” (a cluster of vehicles with no empty cell between them) thereby mimicking the effects of greenwave synchronization of the traffic lights in real traffic. The jamming transition in the green-wave model has been investigated by a combination of a mean-field argument and numerical input from computer simulations [1375].
10.2.3. More Realistic Description of Streets and Junctions At first sight, the BML model may appear very unrealistic because the vehicles seem to hop from one crossing to the next. However, it may not appear so unrealistic if each unit of discrete time interval in the BML model is interpreted as the time for which the traffic lights remain green (or red) before switching red (or green) simultaneously in a synchronized manner, and over that timescale, each vehicle, which faces a green signal, gets an opportunity to move from j-th crossing to the ( j + 1)-th (or, more generally [418], to the ( j + r)-th where r > 1). In the original version of the BML model, the vehicles are located on the lattice sites, which are identified as the crossings. Brunnet and Goncalves [160] considered a modified version where, instead, the vehicles are located on the bonds and, therefore,
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BML model
Green-wave model
Time
0
2
4
5
6
Figure 10.2 Comparison of the update procedure in the BML model (left) and the green-wave model (right) (from [1375]).
never block the flow of vehicles in the transverse direction. Consequently, in this version of the CA model of city, traffic jams of only finite sizes can form, and these jams have finite lifetime after which they disappear while new jams may appear elsewhere in the system; an infinitely long-lived jam spanning the entire system is possible only in the trivial limit ρ = 1. In contrast, Horiguchi and Sakakibara [589] generalized the BML model by replacing each of the bonds connecting the nearest-neighbor lattice sites by a bond decorated with an extra lattice site in between. In [590], a generalization to s extra lattice sites between crossings has been presented. However, the vehicles are still allowed to hop forward by only one lattice spacing. Moreover, by generalizing the rules in [248, 913], probabilistic turning of the vehicles was allowed. The model exhibits a transition from the flowing phase to a completely jammed phase. The streets in the original BML model were assumed to allow only one-way traffic. This restriction was relaxed in a more realistic model proposed by Freund and Pöschel [403], which allows two-way traffic on all the streets. Thus, each east–west (north– south) street is implicitly assumed to consist of two lanes: one of which allows eastbound (north-bound) traffic, while the other allows west-bound (south-bound) traffic. Moreover, each site is assumed to represent a crossing of a east–west street and a north– south street where four numbers associated with the site denote the number of vehicles coming from the four nearest-neighbor crossings (i.e., from north, south, east, and west) and queued up at the crossing under consideration. So, in this extended version of the
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BML model, each site can accommodate at most 4Q particles if each of the 4Q of vehicles associated with it is allowed to grow to a maximum length Q. The finite space of the streets in between successive crossings does not appear explicitly in the extension of the BML model suggested by Freund and Pöschel [403] although it is more realistic than the BML model because it implicitly takes into account the possibility of formation of queues by vehicles approaching one crossing from another. Chopard et al. [204] have developed a more realistic CA model of city traffic where the stretches of the streets in between successive crossings appear explicitly. In this model also, each of the streets consists of two lanes that allow oppositely directed traffic. The rule for implementing the motion of the vehicles at the crossing is formulated assuming a rotary to be located at each crossing. Depending on the details of the rules to be followed by the vehicles at the rotary, the system can exhibit a variety of phenomena. For example, the flow can be metastable at all densities if each of the vehicles on the rotary is required to stop till the destination cell becomes available for occupation [204]. Moreover, the bottleneck created by the vehicles on the rotaries at the junctions can lead to a plateau in the fundamental diagram which is analogous to that caused by a static hindrances on a highway [651, 652, 1397, 1398].
10.3. CHOWDHURY–SCHADSCHNEIDER MODEL The BML model focusses on the occurrence of a jamming transition and makes several simplifying assumptions that are clearly unrealistic for urban traffic flow. Before we introduce a more realistic extension of this model, the Chowdhury–Schadschneider (ChSch) model, we briefly review results for the simplest possible urban network consisting of just two crossing streets.
10.3.1. Crossroads with Signals In many situations, the behavior of traffic and transport networks is controlled by the nodes, not by the links. Therefore, it is useful to study first a single node which in the case of an urban street network corresponds to an intersection of two streets. Indeed, it has been shown in [159] for the ChSch model that this can be the basis for a good phenomenological description. In [628, 977], the phase diagram of a system consisting of one east-bound and one north-bound street with one crossing has been investigated. The dynamics of the vehicles on each street is given by the deterministic TASEP (CA184). Already this system without signals shows a rich phase diagram [628, 631, 632]. Variations of this model have been studied, e.g., with Nagel–Schreckenberg (NaSch) dynamics [392], yielding [391, 1452] or open boundary conditions [324]. Fouladvand et al.
Transport on Networks
[88, 398, 400, 401] have investigated various scenarios including different signalization schemes for intersections with signals. Using the asymmetric simple exclusion process (ASEP) with deterministic bulk dynamics, the phase diagram for various types of roundabouts and rotaries has been derived. An overview of these results has been given by Huang [604]. More realistic models for roundabouts have been studied, e.g., in [386, 399, 1449].
10.3.2. ChSch Model A more detailed “fine-grained” description of city traffic than that provided by the BML model should take into account both the finite distances between crossings and the competition between the timescales set by the signal periods and the vehicle dynamics. Such a model has been proposed by Chowdhury and Schadschneider (ChSch model) who combined the dynamics of the BML model with that of the NaSch model [218]. First, one decorates each bond [589, 1018] with D − 1 (D > 1) sites to represent D − 1 cells in between each pair of successive crossings [204, 1018]. Each segment of the streets in between successive crossings is modeled by the NaSch model [204, 1296] to take into account the interactions among the vehicles moving along the same street. Moreover, one should flip the color of the signal periodically at regular interval of T time steps, where during each unit of the discrete time interval, every vehicle facing green signal should get an opportunity to move forward from one cell to the next. Such a CA model of traffic in cities has, indeed, been proposed in [218, 1238] where the rules of updating have been formulated in such a way that (1) a vehicle approaching a crossing can keep moving, even when the signal is red, until it reaches a site immediately in front of which there is either a halting vehicle or a crossing; and (2) no gridlocking would occur in the absence of random braking. Let us model the network of the streets as a N × N square lattice. For simplicity, let us assume that the streets parallel to xˆ and yˆ axes allow only single-lane east-bound and north-bound traffic, respectively, as in the original formulation of the BML model. Next, we install a signal at every site of this N × N square lattice, where each of the sites represents a crossing of two mutually perpendicular streets. We assume that the separation between any two successive crossings on every street consists of D cells so that the total number of cells on every street is L = N × D. Each of these cells can be either empty or occupied by at most one single vehicle at a time. Unlike the BML model [115], which corresponds to D = 1, and the model of Horiguchi and Sakakibara [589], which corresponds to D = 2, D (< L) in this model is to be treated as a parameter that introduces a new length scale into the problem. Figure 10.3 shows an illustration of the network and the relevant quantities. The signals are synchronized in such a way that all the signals remain green for the east-bound vehicles (and, simultaneously, red for the north-bound vehicles) for a time
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dn Sn
Figure 10.3 Snapshot of the model topology. In this case, the number of intersections in the quadratic network is set to N × N = 16. The length of the streets between two intersections is chosen to D − 1 = 4. Vehicles move from west to east on the horizontal streets or from south to north on the vertical ones. The magnification shows a segment of a west–east street. The traffic lights are synchronized, and therefore, all vehicles moving from south to north have to wait until they switch to “green light.”
interval T , and then, simultaneously, all the signals turn red for the east-bound vehicles (and green for the north-bound vehicles) for the next T time steps. Clearly, the parameter T introduces a new timescale into the problem. Thus, in contrast to the BML model, the forward movement of the individual vehicles over smaller distances during shorter time intervals is described explicitly in this “unified” model. If turning of the vehicles is not allowed, as in the original BML model, the total number of vehicles on each street is determined by the initial condition and does not change with time because of the periodic boundary conditions. Following the prescription of the NaSch model, we allow the speed v of each vehicle to take one of the vmax + 1 integer values v = 0, 1, . . . , vmax . Suppose that vn is the speed of the n-th vehicle at time
Transport on Networks
t while moving either toward east or toward north. At each discrete time step t → t + 1, the arrangement of N vehicles is updated in parallel according to the following rules: CS1: Acceleration: vn → min(vn + 1, vmax). CS2: Deceleration (due to other vehicles or signal): Suppose that dn is the gap in between the n-th vehicle and the vehicle in front of it, and sn is the distance between the same n-th vehicle and the closest signal in front of it (see Fig. 10.3). Case I: the signal is red for the n-th vehicle under consideration: vn → min(vn , dn − 1, sn − 1). Case II: the signal is green for the n-th vehicle under consideration: Suppose that τ is the number of the remaining time steps before the signal turns red. Now, there are two possibilities in this case: (i) When dn ≤ sn , then vn → min(vn , dn − 1). The motivation for this choice comes from the fact that when dn ≤ sn , the hindrance effect comes from the leading vehicle. (ii) When dn > sn , then vn → min(vn , dn − 1) if min(vn , dn − 1) × τ > sn ; else vn → min(vn , sn − 1). The motivation for this choice comes from the fact that when dn > sn , the speed of the n-th vehicle at the next time step depends on whether or not the vehicle can cross the crossing in front before the signal for it turns red. CS3: Randomization: vn → max(vn − 1, 0) with probability p (0 ≤ p ≤ 1); p, the random deceleration probability, is identical for all the vehicles and does not change during the updating. CS4: Vehicle movement: For the east-bound vehicles, xn → xn + vn , while for the north-bound vehicles, yn → yn + vn . The rule in case II of step CS2 can be simplified without changing the overall behavior of the model [1238]: Case II: If the signal turns to red in the next time step (τ = 1): vn → min(vn , sn − 1, dn − 1) else vn → min(vn , dn − 1). These rules are not merely a combination of the rules proposed by Biham et al. [115] and those introduced by Nagel and Schreckenberg [1027], but also involve some modifications. For example, unlike all earlier BML-type models, a vehicle approaching a crossing can keep moving, even when the signal is red, until it reaches a site immediately in front of which there is either a halting or a crossing vehicle. Moreover,
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if p = 0, every east-bound (north-bound) vehicle can adjust speed in the deceleration stage so as not to block the north-bound (east-bound) traffic when the signal is red for the east-bound (north-bound) vehicles. Initially, we put N→ and N↑ vehicles at random positions on the east-bound and north-bound streets, respectively. We update the positions and velocities of the vehicles in parallel following the rules mentioned above. After the initial transients die down, at every time step t, we compute the average velocities vx(t) and vy(t) of the east-bound and north-bound vehicles, respectively. A phase transition from the free-flowing dynamical phase to the completely jammed phase takes place in this model at a vehicle density ρc (D). The dependence on the dynamical parameters p, vmax , and T is not clear yet [1238]. The data obtained so far from computer simulations do not conclusively rule out the possibility that the transition density only depends on the structure of the underlying lattice, similar to the percolation transition [1309], and is independent of the dynamical parameters. The intrinsic stochasticity of the dynamics, which triggers the onset of jamming, is similar to that in the NaSch model, while the phenomena of complete jamming through self-organization and the final jammed configurations (see Fig. 10.4) are similar to those in the BML model. The variations of vx and vy with time and with ρ, D, T , and p in the flowing phase are certainly more realistic than that in the BML model [218].
Figure 10.4 A typical jammed configuration of the vehicles (N = 5, D = 8). The east-bound and north-bound vehicles are represented by the symbols → and ↑, respectively.
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The ChSch model has been formulated intentionally to keep it as simple as possible and at the same time to capture some of the interesting features of the NaSch model and the BML model. This model can be generalized (1) to allow traffic flow both ways on each street which may consist of more than one lane, (2) to make more realistic rules for the right-of-the-way at the crossings and turning of the vehicles, (3) to implement different types of synchronization or staggering of traffic lights [1296] (including green-wave), etc. Various simplified models have also been considered, e.g., limiting cases of the ChSch model with only a single street, deterministic dynamics and vmax = 1 [1369, 1370, 1414], or the dynamics of two cars at a single intersection [1452]. A continuum version of ChSch model has been derived in [926] by combining the lattice Boltzmann approach with the BML model. This leads to a mesoscopic description of urban traffic networks.
10.3.3. Traffic Signal Optimization The ChSch model can be used to study the effect of different signal optimization strategies. In [73, 75, 159], different strategies have been compared: • Synchronized strategy: All traffic lights switch synchronously to green (red) for the east (north) bound vehicles and vice versa. • green-wave strategy: Adjacent traffic lights switch with a defined offset Toff . • Random strategy: Adjacent traffic lights switch with a random offset Toff (i, j). These are global strategies where the signal period and the offsets are fixed. 5 4 3 2 1 0
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The performance of the different strategies can be estimated by using the velocity and flow (Figs. 10.5 and 10.6) as an indicator.2 The typical dependence between the time periods of the traffic lights and the mean flow in the system is shown in Fig. 10.6a for synchronized traffic lights. For low densities, one finds a strongly oscillating curve with maxima and minima at regular distances whereby the optimal traffic states are determined by the travel times between the intersections. The traffic light cycle time corresponding to the maximum system flow is equal to Tmax = D/2vfree where vfree = vmax − p is the velocity of free-flowing vehicles. Similar oscillations can be found even at very high densities. Surprisingly the random strategy can lead to remarkably higher flows than the synchronized strategy. Furthermore, the strong oscillations are suppressed by the randomness in the switching (Fig. 10.6). An explanation for the improvement at high densities is the fact that some parts of the network are completely jammed, while in other parts, cars can move freely. This additional gain due to the inhomogeneous distribution of vehicles indicates that an autonomous traffic light control based on local decisions could be more effective than the analyzed global schemes. The optimal green-wave strategy is to adjust the time delay such that the first vehicle passing an intersection will arrive at the next one exactly when it switches to green. The corresponding optimal delay time for low densities is given by Toff = D/vfree . By definition, no “green-wave” can be established at high densities, but a suitable offset in the switching between successive traffic lights can lead anyhow to an improved flow. Obviously, the dynamics for high densities is governed by the motion of large jams that 0.3
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Figure 10.6 (a) The mean flow strongly oscillates in the case of synchronized traffic lights. (b) A random shift in the switching leads to a more flexible strategy, e.g., without oscillations. The mean flow is remarkably higher in comparison to synchronized traffic lights. 2 There are other possible choices for the indicator, e.g., average trip times, the percentage of stopped cars, total waiting
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move oppositely to the driving direction. The optimal system state would be reached if a jam moves backward from one intersection to the one before and blocks it while the traffic light is red anyway so that afterwards moving vehicles (outflow of the jam) can take advantage of the green phase as much as possible. For high densities, the optimal delay time is equal to Toff = D/vjam , where vjam is the velocity of backwards moving jams. The effects of random perturbations in the car velocity have been studied in [1414] where a possible relation with emergent phenomena, resonance, and critical behavior was pointed out. Figure 10.7 shows the performance of the global strategies for optimized cycle times T [75]. The system optimum is given by the free flow of the underlying NaSch model for low densities (no interaction with crossings) and also by the high density flow of the NaSch model for high densities (interaction among vehicles is dominant). In the intermediate region, a plateau is formed corresponding to half of the maximum flow of the NaSch model since the network has two equitable directions. All three are close to the system optimum, especially for low and high densities. However, the best results are obtained for the green-wave strategy. Figure 10.7 also shows that the optimal cycle time depends on the density. Since the traffic demand and thus the density can vary strongly during the day, this optimal values are not always be achieved for fixed cycle times. Here, adaptive strategies can be more flexible. The random strategy turns out to be useful if a control strategy is required, which is not very sensitive to the adjustment of cycle times [159]. Besides the global strategies described above, various adaptive strategies are possible where the parameters depend on the actual traffic situation. Examples [75] are (1) switching based on queue length: the traffic signal switches if the queue length in front of a red light exceeds a certain length; (2) switching based on waiting time: the traffic light switches to red if the green phase is not used by a vehicle for a certain time; and (3) switching in 0.4
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analogy to a neural network: the traffic light switches like an integrate-and-fire neuron, i.e., the number of passing vehicles is integrated and determines the cycle time (potential) of the traffic light; after the switching process (“fire”), the potential is reset to zero again. The performance of these adaptive strategies in the simplest scenarios is similar to that of the global strategies. The main advantage, however, is their flexibility in more realistic situations with inhomogeneous densities [75]. The effect of signal synchronization has also been investigated in [606]. Other strategies have been studied for the ChSch model in [867]. Nagatani [1004–1012] has performed extensive studies based on different model approaches. Autonomous strategies based on local decisions were proposed in [382]. Strategies for the optimization of traffic light control based on the self-organization of the traffic lights have been considered, e.g., in [237, 432, 433, 834]. For a better understanding of urban traffic and signal optimization also, the fundamental diagram is an important quantity. For urban traffic, it has been studied, e.g., in [258, 431, 514].
10.4. HIGHWAY AND CITY NETWORKS 10.4.1. Online Simulation of Traffic Networks A large fraction of the available resources are spent by the governments, particularly in the industrialized developed countries, to construct more highways and other infrastructural facilities related to transportation. The car-following models, the coupled-map lattice models, and the CA models have been used for computer simulation with a hope to utilize the results for online traffic control. For planning and design of the transportation network [1021, 1023], e.g., in a metropolitan area [203, 352, 1022, 1185], one needs much more than just microsimulation of how vehicles move on an idealized linear or square lattice under a specified set of vehicle–vehicle and road–vehicle interactions. For such a simulation, to begin with, one needs to specify the roads (including the number of lanes, ramps, bottlenecks, etc.) and their intersections. Then, times and places of the activities, e.g., working, shopping, etc., of individual drivers are planned. Microsimulations are carried out for all possible different routes to execute these plans; the results give information on the efficiency of the different routes, and this information is utilized in the designing of the transportation network [1023]. Some socioeconomic questions and questions on the environmental impacts of the planned transportation infrastructure also need to be addressed during such planning and design. For a thorough discussion of these aspects, we refer to the reviews [538, 1021, 1023]. The next step after applications in transport planning is traffic forecasting. But before a forecast can be made, one has to know the current state of the network. This sounds simpler than it actually is! Usually the detectors are not distributed evenly over the whole
Transport on Networks
network. Therefore, for certain parts, one has very accurate information about the traffic situation, whereas for other parts, almost nothing is known. Further complications arise through the sources and sinks in form of on- and off-ramps. In order to obtain information about the state of the whole network, it is necessary to combine local traffic counts with the network structure. Data from inductive loops are used as input for flow simulations of the network. This provides a sort of interpolation for the parts of the network without detectors. An advantage of such a microscopic approach, which is called online simulation [352, 1436], is that all aspects that have to be considered for the interpolation, like network structures and traffic lights, are directly incorporated into the simulation dynamics. The information is then made available on the Internet (www.autobahn.nrw.de) in the form of a road-usage map (see Fig. 10.8). This strategy can be applied not only to highways but also to urban networks and has been implemented, e.g., for the city of Duisburg [1436]. Related projects, sometimes even more ambitious, have been carried out for other urban [1022, 1185] and highway models [1187] (see also [1023] for a more complete review). The traffic forecasts based on online simulations are already quite reliable. However, for further improvement, one does not need better traffic models, but rather a better understanding of the human reaction to this forecast! Here, one has to take into account that human drivers might change their plans if they consult a traffic forecasting system, e.g., because the occurrence of jams on their planned route is predicted. Thus, they might take a different route, travel at a different time, use public transport, or do not
Figure 10.8 Screenshot of www.autobahn.nrw.de showing the current state of a part of the highway network of North Rhine-Westphalia.
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travel at all. This implies that, very much in contrast to the weather forecast, the public announcement of the forecast might hurt its validity! Therefore, activity patterns and route choice behavior have to be understood in a much more detailed way [1259]. Computer simulations [265, 315, 848, 1434, 1435] and decision experiments [539, 1279] have been performed to see how drivers learn from their daily experience with different route choices or how they react to traffic information. For more details, we refer to the book [1259] and the review [538].
10.4.2. Network Analysis Often it is impossible to increase the performance of a traffic network simply by building new roads. To optimize an existing network, e.g., by traffic control methods like ramp metering, one first has to identify and characterize the sections of the network which limit the performance, i.e., the bottlenecks. This program has been carried out in [778, 1242] for the highway network of North Rhine-Westphalia, the most populated state of Germany. In order to allow an evaluation of the traffic load, the probability to find a jam is calculated. Traffic volumes larger than the highway capacity can be related directly to a large jam probability, and thus, bottlenecks of the highway network can clearly be identified. As expected, the jam probability shows temporal variations. During the weekend, there are less jam than during the rest of the week. From Monday to Thursday, one can clearly identify two rush-hour peaks in the morning and the afternoon. On Fridays, the afternoon peak is more pronounced. Through a statistical analysis of empirical single-vehicle data at different locations, the main bottlenecks that lead to congestions could be identified [778, 1242]. Correlations between different bottlenecks are observed, which allow to determine the dominating bottleneck that induces jams at other locations. These are predominantly on- and off-ramps rather than topological peculiarities of the highway network. It is the large in- or out-flow at ramps that perturbs the stream of vehicles on the main highway. It is expected that ramp metering systems are able to counteract this destabilization of the flow and reduce the formation of jams [619, 1207], especially in the presence of synchronized traffic. In this way, a restricted flow on the ramps may lead to a significant increase of the capacity of the main highways. Of special interest for a capacity optimization is the existence of “hot spots” in the network: the highest jam probabilities are spatially and temporally well localized. These are the parts of the network, where the flow has to be optimized. Presently these sections self-organize in a congested state, leading to flows that are far below their capacity. The situation can be improved a lot by controlling the number of entering cars and by optimizing the traffic stream at the on- and off-ramps at this section. The small number of bottlenecks that are present in the network shows that it is possible to improve the capacity with a reasonable technical effort. Of particular
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interest are sections where the main input is from other, less crowded, highways. In these cases, a restricted input does not lead to a collapse of the urban traffic.
10.4.3. Braess Paradox In order to avoid or at least reduce congestion, why not just build new roads? First, in many densely populated areas, this is simply not possible. Second, additional roads do not necessarily lead to an improvement! This surprising result is known as Braess paradox [145, 233, 1100, 1311]. A simple example is shown in Fig. 10.9. Suppose that six drivers want to move from A to D at the same time. They have the choice between two different routes ABD and ACD. The travel time on each road segment depends on the number f of cars using it. Explicit expressions are given in Fig. 10.9. For example, cars move from node A, representing a residence area, to node D, representing the business area. They can decide between two different routes, either ABD or ACD. For a total of six vehicles, the optimum is achieved if both routes are chosen by three vehicles. The total travel 6
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Figure 10.9 Illustration of Braess’ paradox (after [1311]). The travel times on a given street depend on the number f = 0, 1, ... , 6 (encircled numbers) of cars using this street. In the situation (a), the optimum corresponds to fABD = fACD = 3. With the additional street BC as in (b), the situation where all three routes are chosen by two drivers ( fABD = fACD = fABCD = 2) corresponds to a user optimum.
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time for each car is then Ta = 83. This corresponds to a Nash equilibrium, i.e., no driver can decrease the travel time by changing the route unilaterally. In that case, travel times will increase. Thus, there exists no better choice for individual drivers, i.e., the solution corresponds to a user optimum. Now assume that an additional street is build that connects B and C. This new road is fast with travel time function f + 10. If each of the now possible route choices ABD, ABCD, and ACD is taken by two drivers, the new travel time is Tb = 92 for everybody, i.e., larger than without the new street! However, it is minimal in the sense that if just one driver takes a different route, his or her travel time will increase to T˜ b = 93 or 103. Thus, there exists no better choice for individual drivers, i.e., the solution corresponds to a user optimum or Nash equilibrium. Of course, the drivers could ignore the new road, but this would require a collective effort and coordination. This surprising result shows that the system optimum can be quite different from the user optimum. The fact that the lack of coordination can lead to socially suboptimal solutions has been called price of anarchy [1297, 1502]. Its effects in transportation networks have been studied by Youn et al. [1502]. They found that it can lead to a considerable waste of travel time and that, counterintuitively, blocking of streets can improve the traffic conditions. The Braess paradox is not a true paradox. It results from a situation where user optimum and system optimum are different, e.g., as in Fig. 10.9b, whereas they agree in Fig. 10.9a. Although the choice of travel times in the present example is somewhat artificial, Braess paradox is of some relevance for real traffic (and probably other!) networks. It has also its correspondence in real physical systems, like coupled springs [234, 1100].
10.5. COMPUTER NETWORKS AND INTERNET TRAFFIC In recent years, the Internet has become the most popular medium for information transfer in the world. Terms like “e-mail” and “e-commerce” are nowadays well known to almost everybody. Because of the enormous increase of Internet users and a still growing demand, the network already reaches its maximum capacity at some times. Almost every user has been annoyed by decreasing transfer rates and increasing waiting times caused by congestions in the Internet. The heterogeneity of the network, e.g., due to different transport protocols and operating systems, and its enormous expansion in the last years make it necessary to understand the basic properties of data transport in the Internet for planning new connections and optimizing the usage of the existing resources. Inspired by the success of the methods of statistical mechanics outside the traditional domain of physics, its tools have also been applied to analyze fundamental properties of information traffic on the international network of computers (Internet)
Transport on Networks
[245, 1076, 1349, 1350, 1413]. Messages in the form of information packets are continuously being emitted from the hundreds of millions of host computers and transported to their destination computers through this network. Each of these packets is relayed through the so-called routers on its way. The routers can deal with the packets one by one. Each router has a finite buffer where the arriving packets get queued up and forwarded one by one from the head of the remaining queue to their respective next destinations. Since packets run with the velocity of light through the cables, information congestion does not take place inside the cables. It is the routers that give rise to the information congestion on the Internet. Measuring the fluctuations in the round-trip time (RTT) taken by a message on the Internet (using the ping command of the UNIX operating system), 1/f -like power spectrum has been observed [245, 1349, 1350]. Especially the influence of routers (network nodes) with low transfer rates, which are considered to be the reason for the congestions, and the collective behavior of routers are main targets of the investigations. Real data measurements like those for the ping statistics [245, 1346, 1348, 1349] or the load of a single router [854, 1466] on various kinds of networks and their analysis are the basis for a better understanding of Internet traffic. Moreover, there are investigations by Huberman and Lukose [620] on the social aspects of the Internet and the “human factor” in the system. Empirical results for the load of single routers show a self-similar behavior of Internet traffic which Willinger et al. [1466] explained as a superposition of ON/OFF sources with heavy tailed distributions of the duration lengths of the ON/OFF periods. Another method to characterize a nonequilibrium system like an Internet connection is the survey of ping time series, first presented by Csabai [245] and later by Takayasu et al. [1349, 1350]. Here, the travel times of data packets from a source to a destination host and back to the source host, the so-called RTT, are measured. The analysis of the respective power spectra shows characteristic statistics for different “traffic” states. One can distinguish a free flow and a jammed phase separated by a transition regime. On the basis of these measurements, various models were introduced to reproduce the characteristic stochastic properties. Huisinga et al. [625] have used a generalization of the ASEP for a microscopic description of the transport of data packets along a fixed path in the Internet. In this generalization, each site corresponds to a router and the particles to data packets. In contrast to the ASEP, each site n can accommodate Bn particles to take into account that each router has a finite capacity (Fig. 10.10). Data packets move with router specific probability pn to the next router if there is enough space. Computer simulations have revealed the appearance of a free flow and a jammed phase separated by a (critical) transition regime. The analysis of travel times shows the typical power spectra of real Internet traffic in the two regimes, i.e., white noise for free flow and 1/f 1/2 for the jammed system. In the transition regime between these two phases, the model shows a characteristic 1/f -noise.
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Takayasu et al. [1346, 1348, 1392] proposed a simple model based on the contact process [567] to explain 1/f -noise in the travel times of data packets and to reproduce the distribution of the congestion duration length of routers. The model of Yuan et al. [1505] is based on a reinterpretation of the well-known cellular automaton approach for vehicular traffic. Data transport is realized by changing headways between “moving routers.” This method does not give any access to the travel times of data packets. In [1076], a two-dimensional model has been suggested. Measurements of the travel times indicate the existence of a phase transition into a jammed phase. The influence of the structure of the network, namely the branching number, has been investigated in [1413] for a simple stochastic model on a Cayley tree. In the square lattice model of a computer network developed by Ohira and Sawatari [1076], information packets are generated at the sites on the boundary at a rate λ with the corresponding destination addresses chosen randomly from among the boundary sites. The packets can form queues of unlimited length at the inner nodes, which act as routers of the network. At every time step, the packets from the heads of the queues at the routers are forwarded to the tail ends of the queues at the next router. Both deterministic and probabilistic strategies have been considered for selecting the next router to which the individual packets are to be forwarded. On reaching their individual destinations, the packets die. The average number of time steps between the birth and death of a packet is referred to as the average lifetime of a packet. Computing the average lifetime as a function of the birth rate λ of the packets, Ohira and Sawatari [1076] observed a transition from a low-congestion phase to a high-congestion phase at a non-zero finite value λ∗ .
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Based on these computer models, many different strategies to improve network traffic, especially networks for data transport like the Internet, have been proposed. The strategies either are dynamical, like rerouting, or involve structural changes of the network. Currently, three different theories for the observed self-similar properties of Internet traffic exist: •
• •
Self-similar traffic has its origin in the behavior of the users. Theoretical models describe the Internet as a large number of ON/OFF sources, which reflect the flows, with identical duration distribution. The origin of the self-similarity is at the transport layer, e.g., the TCP congestion control algorithms. Self-similar traffic is interpreted as indication of critical phenomena in data traffic near a transition point from free flow to congested traffic.
For a deeper discussion and an overview over other aspects of Internet traffic, we refer to the review [1299].
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CHAPTER ELEVEN
Pedestrian Dynamics Contents 11.1. Introduction
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11.2. Empirical Observations and Collective Phenomena 11.2.1. Individual Properties 11.2.2. Observables 11.2.3. Fundamental Diagram 11.2.4. Flows at Bottlenecks 11.2.5. Collective Phenomena
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11.3. Cellular Automata Models 11.3.1. Fukui–Ishibashi Model 11.3.2. Blue–Adler Model 11.3.3. Gipps–Marksjös Model
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11.4. Floor Field CA 11.4.1. General Principle 11.4.2. Update Rules 11.4.3. Construction of the Static Floor Field 11.4.4. Conflicts and Friction 11.4.5. Other Generalizations and Interactions 11.4.6. Moving Beyond Nearest Neighbors: vmax > 1 11.4.7. Collective Effects 11.4.8. Evacuation Simulations
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11.5. Other Models 11.5.1. Fluid-Dynamic and Gas-Kinetic Models 11.5.2. Social-Force Models 11.5.3. Lattice Gas Models 11.5.4. Optimal Velocity Model 11.5.5. Active Walker Models
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In this chapter, we want to give an introduction into the modeling of pedestrian dynamics. It is somewhat different from most of the other systems discussed in this book since the motion is genuinely two dimensional. Pedestrians appear to follow certain conventions (like staying right), but there are no formal regulations such as speed limits and passing rules. This makes the topic more challenging theoretically and gives rise to new phenomena related to the spatiotemporal evolution and organization. We also provide a brief glimpse into different applications in the area of evacuations, safety analysis, etc.
Stochastic Transport in Complex Systems DOI: 10.1016/B978-0-444-52853-7.00011-7
Copyright © 2011, Elsevier BV. All rights reserved.
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In the following, we will use the terminologies “particle,” “pedestrian,” and “agent” in a rather loose way. In a strict sense, pedestrian should refer to a person, whereas a particle is a representation of a pedestrian in a model. An agent, on the other hand, is an “intelligent” particle, which is not only mobile but also capable of autonomous action, i.e., an agent makes decisions, communicates, has goals, etc. [65, 181].
11.1. INTRODUCTION For a physicist, modeling pedestrian dynamics is much more challenging than highway traffic. One reason is its generically two-dimensional nature. Vehicular traffic is mostly one-dimensional or quasi-one-dimensional through the existence of lanes that allow only unidirectional motion. Therefore, one usually deals with a well-defined ordering of vehicles (at least in the same lane), which is rarely changed through overtaking manoeuvres. Interactions with nearest neighbors (i.e., the next car ahead) are most important, which simplifies modeling considerably. In pedestrian dynamics, however, the situation is more complex, and interactions with other individuals (which might cross the path of walking) in a certain range have to be taken into account. The most important interactions are not necessarily with the nearest individuals, but might also depend on the relative direction of the velocities. For example, interactions with a person who is still 5 m away, but moving on a potential “collision course,” can be stronger than those with a person moving at your side at 50 cm distance. This complex nature of the mutual interactions between pedestrians gives rise to several self-organization phenomena, which are not observed in vehicular traffic. Therefore, only very few models exist, which can reproduce the empirically observed behavior accurately. A comprehensive theory of pedestrian dynamics has to take into account three different levels of behavior [249, 581] which we have discussed in Section 5.1. At the strategic level, pedestrians plan their activities (shopping, going to work, meeting friends). Then, on the tactical level, short-term decisions are made, e.g., choosing the precise route taking into account obstacles and density of pedestrians. Finally, the operational level describes the actual walking behavior of pedestrians, including the interactions with others and collision avoidance. Since the processes at the strategic and tactical level are exogenous and require input from other disciplines, they will not be considered in the following. However, some of the more sophisticated models are designed in such a way that would easily allow to incorporate such information. Among biological systems, human transportation networks are certainly the most advanced. Surprisingly the natural form of human motility, namely, walking, bares a strong resemblance to the behavior found in ant colonies (see Section 12.4).
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11.2. EMPIRICAL OBSERVATIONS AND COLLECTIVE PHENOMENA For the motion of pedestrians, much less empirical data exist than for highway traffic. One obvious reason is the fact that so far automatic measurements are much more difficult. Also, if one restricts oneself to pure observation, usually one deals with very inhomogeneous situations. People move in different directions, for different purposes, etc. Currently, the best way to obtain quantitative data on pedestrian motion is the analysis of video footage, either of “natural” situations or of specially designed experiments [137, 585, 685, 686, 882, 1281, 1283]. However, the automated analysis is still in its infancy, and therefore, the analysis has to be performed “by hand” most of the time. The software tool PeTrack [136, 137] is able to generate automatically very precise data for individual trajectories from videos. However, this still requires laboratory conditions.
11.2.1. Individual Properties Pedestrians usually have a preferred walking speed, although they may speed up or slow down if necessary. In contrast to vehicular traffic, they can accelerate and brake almost instantaneously. This walking speed is not constant, but depends on the travel purpose and other external factors. Weidmann [162, 1453] has collected and analyzed empirical data (Table 11.1). According to his findings, pedestrians traveling for business have the largest walking speed (1.61 m/s), while in leisure situations, it is lowest (1.10 m/s). In commuting, the average velocity is 1.49 m/s, and during shopping, it is 1.16 m/s. Overall the velocities in a crowd are almost Gaussian distributed with an average of 1.34 m/s (4.83 km/h) and standard deviation of 0.26 m/s. Gender differences exist, with men being approximately 10% faster than women (1.41 m/s versus 1.27 m/s). The influence of the characteristics of the walking infrastructure also plays a role. In [252], it was found that the speed in unidirectional flows (1.54 m/s) is larger than in bidirectional one (1.41 m/s) or crossing streams (1.35 m/s). Other factors that determine the desired velocity are the age (with velocity decreasing with age) or, better, the fitness level, and the slope of the path (uphill motion is slower than downhill or on even ground). Staircases are special since age has an increased influence on the walking speed, which Table 11.1 Typical average velocities of pedestrians from empirical observations [1453] Group/Purpose
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also depends rather strongly on many other factors, like length and width of the stairs and step height. This leads to surprising observations, e.g., that especially younger people sometimes accelerate when walking upstairs compared to walking on the horizontal ground [806]. Also temperature and time of the day has a clear influence with people moving faster in colder conditions and in the morning. Cultural differences can also have a strong impact on the typical walking speeds. The study [1471] has revealed that people in Singapore walk three times faster than people in Blantyre (Malawi) (1.71 m/s versus 0.57 m/s). However, for most large cities, the average speed lies between 1.3 and 1.6 m/s. However, most empirical studies have been taken with some care, since the circumstances might vary slightly (and are often not well documented). Since we have seen that already the average walking speed depends on several different parameters, this can be of relevance. Other characteristic parameters are the average step length and the frequency of stepping. Typical results are approximately 0.65 m for the step length and a frequency of 2 Hz. However, in general, the step length will depend on the walking speed [1284].
11.2.2. Observables Similar to vehicular traffic, the main characteristic quantities for the description of pedestrian streams are flow and density. However, in pedestrian dynamics, additional complications arise due to the two-dimensional nature of the flows. Also, there is no strict lane-discipline, which even in one-dimensional flows (e.g., in narrow bottlenecks) leads to modifications. The flow J of a pedestrian stream is defined as the number of pedestrians crossing a fixed location of a facility per unit of time. The most natural approach determines the times ti at which pedestrians have passed a fixed measurement location. The flow is then calculated from the time gaps ti = ti+1 − ti between two consecutive pedestrians i and i + 1: N 1 1 (ti+1 − ti ) . (11.1) J= where ti = ti N i=1 Another method to measure the flow is suggested by the analogy with fluid dynamics. The flow through a facility of width b is related to the average density ρ and the average speed v of the pedestrian stream, where the specific
J = ρvb = Js b ,
(11.2)
Js = ρv
(11.3)
flow1
1 In strictly one-dimensional motion, often a line density (dimension: 1/length) is used. Then, the flow is given by J = ρv.
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gives the flow per unit width. This relation is the hydrodynamic relation that we have already encountered in vehicular traffic (see Section 6.4). The quantities involved can be obtained by determining the entrance and exit times for a test section of known length and width, e.g., from video recordings or time-lapse photography. This allows to calculate the velocity of each pedestrian. The associated density ρ=
N A
(11.4)
is measured by counting the number of pedestrians N within the selected area A at the time when the moving pedestrian was at the center of the section. In [406], the pedestrian area module, defined as the reciprocal of the density, has been introduced as another way of quantifying the pedestrian load of facilities. Predtechenskii and Milinskii [1138] propose the (dimensionless) density j fj , (11.5) ρ˜ = A which is the ratio of the sum of the projection area fj of the bodies and the total area of the pedestrian stream A. Since the projection area fj depends strongly on the type of person, the densities for different pedestrian streams consisting of the same number of persons and the same stream area can be quite different. Another alternative density definition [531, 685] is based on averaging over a circular region of radius R, f (rj (t) − r ) , (11.6) ρ(r , t) = j
where rj (t) are the positions of the pedestrians j in the neighborhood of r and f (...) is a Gaussian, distance-dependent weight function exp(−|rj (t) − r |2 /R 2 ). Besides technical problems, e.g., due to camera distortion and perspective, the measurements suffer from several conceptual problems. This includes the choice of observation area, test population, and the definition of the density of objects with nonzero extent. For technical reasons, most measurements also combine a velocity or flow averaged over time with some instantaneous density (e.g., obtained as spatial average). This is another factor why measurements in similar settings can differ strongly. 11.2.2.1. Influence of the Measurement Method The strong influence of the different measurement methods can be demonstrated for a particularly simple system, the movement of pedestrians along a line with closed boundary conditions. The situation is similar to that of vehicular traffic, which allows us to adopt the discussion in [727, 857] to the case of pedestrian streams. Two principal approaches to measure observables like flow, velocity, and density exist (Fig. 11.1).
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Δx
x
Figure 11.1 Illustration of different measurement methods to determine the fundamental diagram. Local measurements at a cross section with position x averaged over a time interval t have to be distinguished from measurements at a certain time averaged over space x.
The first method is based on local measurements of the observable O at a certain location x, averaging then over a time interval t. Such averages will be denoted by Ot . Measurements at a certain location allow a direct determination of the flow J and the velocity v: J t =
N 1 = t ti t
and
vt =
N 1 vi . N i=1
(11.7)
The flow is given as the number of persons N passing a specified cross section at x per unit time. Usually, it is taken as a scalar quantity since only the flow normal to the cross section is considered. To relate the flow with a velocity, one measures the individual velocities vi at location x and calculates the mean value of the velocity vt of the N pedestrians. In principle, it is possible to determine the velocities vi and crossing times ti of each pedestrian and to calculate the time gaps ti = ti+1 − ti defining the flow as the inverse of the mean value of time gaps over the time interval t. The second method averages the observable O over space x at a specific time tk , which gives Ox. By introducing an observation area with extent bx, the density ρ and the velocity v can be determined directly: N ρx = b x
N
and
1 vi . vx = N i=1
(11.8)
This method is often used in combination with time-lapse photos. Due to cost reasons, often only the velocity of single pedestrians and the mean value of the velocity during the entrance and exit times were considered [1037, 1080]. The hydrodynamic relation J = ρvb allows to relate the two methods and to change between different representations of the fundamental diagram. Using the definition of
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the observables introduced above together with the distance ˜x = t vt , one obtains J=
N b˜x N = = ρ˜ b vx t b˜x t
with
ρ˜ =
N . b˜x
(11.9)
As in vehicular traffic, the mean values vx and vt do not necessarily correspond. This is illustrated by the situation in Fig. 11.1 where the upper lane consists of faster pedestrians than the lower lane. Averaging over x does not consider the last pedestrian in the lane above, while averaging over t at x for appropriate t does. Thus, densities calculated by ρ˜ = J t /vt can differ from direct measurements via ρx . The strong deviations that can result from the differences between the two approaches have been exemplified for a simple scenario, the single-file movement in a periodic system [1280, 1282]. Using highly accurate measurements based on trajectories determined automatically from video recordings [137], it was found that the deviations become most relevant at high densities when jam waves are present.
11.2.3. Fundamental Diagram As in highway traffic (see Section 6.4), the fundamental diagram is an important quantity to characterize the collective properties of a group of pedestrians and their mutual interactions. However, in principle, one now has to distinguish different scenarios. For instance, in the presence of counterflow, one could argue that interactions with oncoming persons are more relevant than the interactions with those walking in the same direction. Therefore, a priori, one has to distinguish unidirectional from bidirectional fundamental diagrams. Currently, there is no consensus whether there is a notable difference between the two scenarios (see the discussion in [1241]). Nevertheless the fundamental diagram contains useful information and allows to check and calibrate model approaches. Although several attempts to measure the fundamental diagram of pedestrian flow have been made, a lot of points are still controversial. So it is still not clear what the maximal density is which can be observed in pedestrian streams. Estimates range from about2 4 P/m2 up to more than 12 P/m2 (Fig. 11.2). Note that in strictly one-dimensional situations, sometimes a line density is used, which is measured in P/m. Figure 11.2 shows various fundamental diagrams used in planning guidelines and measurements of two selected empirical studies representing the overall range of the data. Certain characteristics are shared by most fundamental diagrams that have been obtained empirically, e.g., a linear increases of the flow at low densities, a single maximum of the flow, and an asymmetry toward smaller densities. These features are similar to the characteristics of fundamental diagrams of highway traffic (compare Section 6.4). 2 In the following, we use the abbreviation “P” for “persons” in units.
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2.5 SFPE WM PM Older Helbing et al.
2
Js (m/s)−1
1.5
1
0.5
0
0
2
4
6
8
10
ρ (m−2)
Figure 11.2 Fundamental diagrams for pedestrian movement in planar facilities. The lines refer to specifications according to planing guidelines (SFPE Handbook [1038], Predtechenskii and Milinskii (PM) [1138], Weidmann (WM) [1453]). Data points give the range of experimental measurements (Older [1080] and Helbing et al. [531]).
On the other hand, the comparison in Fig. 11.2 reveals that the curves disagree considerably. In particular, the maximal flow giving the capacity Jc ranges from 1.2 to 1.8 m/s, the density ρc where this flow is reached ranges from 1.75 to 7 m−2 , and, most notably, the density ρmax where the velocity approaches zero due to overcrowding ranges from 3.8 to 10 m−2 . Several explanations for these deviations have been suggested, including cultural and population differences [531], differences between unidirectional and multidirectional flow [1037, 1154], short-ranged fluctuations [1154], influence of psychological factors given by the incentive of the movement [1138], or the type of traffic (commuters, shoppers) [1075]. The most elaborate fundamental diagram has been given by Weidmann [1453] who collected 25 data sets. He gave an approximate analytical form v(ρ) = v0 [1 − exp (−ρ0(1/ρ − 1/ρmax ))] ,
(11.10)
where v0 = 1.34 m/s is the average walking speed, ρmax = 5.4 P/m2 is the maximal observed density, and ρ0 = 1.913 P/m2 . An examination of the data included
Pedestrian Dynamics
in his analysis shows that most measurements with densities larger than ρ = 1.8 m−2 are performed on multidirectional streams. Weidmann neglected differences between unidirectional and multidirectional flow in accordance with Fruin, who states in his often-cited book [406] that the fundamental diagrams of multidirectional and unidirectional flow differ only slightly. This disagrees with results of Navin and Wheeler [1037] who found a reduction of the flow depending on directional imbalances. Here, lane formation in bidirectional flow has to be considered. Bidirectional pedestrian flow includes unordered streams, as well as lane-separated and thus quasi-unidirectional streams in opposite directions. Another explanation is given by Helbing et al. [531] who argue that cultural and population differences are responsible for the deviations between Weidmann and their data. In contrast to this interpretation, the data of Hankin and Wright [493] based on measurements in the London subway (UK) are in good agreement with the data of Mori and Tsukaguchi [946] measured in the central business district of Osaka ( Japan), both on strictly unidirectional streams. Johansson [684] has argued that the discrepancies can be understood by considering the net-time headway (or time-to-collision), which results from finite reaction times. This headway is found to be approximately constant over various empirical data sets. Based on this observation, he comes to the conclusion that this is an unconscious response, which is the key mechanism behind the dynamics of pedestrian streams. The effect of cultural differences has also been studied under laboratory conditions [189]. It was found that the speed of Indian test persons is less dependent on density than that of German test persons. This might be related to differences in the self-organization behavior, which was observed to be more ordered in the German test group than in the Indian group which were less concerned about personal space. This brief discussion clearly shows that up until now there is no consensus on the precise form of the fundamental diagram and even on the origin of the observed discrepancies. This is not only unsatisfactory with regard to applications especially in safety planning, but also makes the validation and calibration of models rather difficult.
11.2.4. Flows at Bottlenecks One of the most important practical questions is how the capacity of the bottleneck increases with increasing width. Although the first studies have been performed almost 100 years ago [309, 390], the issue is up to now still controversially. This is surprising since a proper understanding of bottleneck flows, which limit the overall performance, e.g., in evacuations, is essential for the design and dimensioning of bottleneck facilities. Beyond a certain bottleneck width, the formation of lanes inside the bottleneck is observed (Fig. 11.3). At first sight, this suggests a stepwise increase of capacity. For independent lanes, where pedestrians in one lane are not influenced by those in others, capacity increases only if an additional lane can be formed. The lanes can be
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y x
Figure 11.3 Sketch of the zipper effect with continuously increasing lane distances in x-direction. The distance in walking distance (y-direction) decreases with increasing lateral distance (x-direction). Density and velocities are the same in all case, but the flow increases continuously with the width.
overlapping, i.e., their width is smaller than the effective width of the pedestrians [582, 583]. However, the study of Seyfried et al. [1281] found that the lane distance increases continuously (Fig. 11.3). A closer analysis shows [1281, 1282] that results from other studies [804, 955, 958, 967] are compatible with this finding, although slightly different experimental setups have been used. The continuous increase is connected with a weak dependence of the density and velocity inside the bottleneck on its width. Thus, in reference to J = ρvb, the flow does not necessarily depend on the number of lanes as illustrated in Fig. 11.3. Figure 11.4 shows another surprising result: the data differ considerably in the values of the bottleneck capacity. In particular, the flow values in [958, 967] are much higher than those in empirical fundamental diagrams. In fact, they are more as twice as large as the maximal flows observed in planar facilities or periodic systems of constant width (Section 11.2.3). This seems to exclude an explanation based on purely finite-size effects. A comparison of the different experimental setups shows that the exact geometry of the bottleneck is of only minor influence on the flow, while a high initial density in front of the bottleneck can increase the resulting flow values. This leads to another interesting question, namely, how the bottleneck flow is connected to the fundamental diagram. According to the general results for driven diffusive systems described in Section 4.5.2, open boundary conditions only select between the different states of the undisturbed system instead of creating completely different ones. Therefore, it is surprising that the measured maximal flow at bottlenecks can exceed the maximum of the empirical fundamental diagram. Here, further studies of the dependence of the flow on the length, width, and shape of the bottleneck [250, 251, 517, 870] are necessary. These questions are related to the common jamming criterion. Generally, it is assumed that a jam occurs if the incoming flow exceeds the capacity of the bottleneck.
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4.5 Kretz Mueller Muir Nagai Seyfried
4 3.5
J (1/s)
3
1.9 b
2.5 2 1.5 1 0.5
0.4
0.6
0.8
1
1.2 b (m)
1.4
1.6
1.8
Figure 11.4 Influence of the width of the bottleneck on the flow. The data are from different experiments [804, 955, 958, 967, 1281] performed under laboratory conditions where the test persons have been advised to move normally.
In this case, one expects the flow through the bottleneck to continue with the capacity (or lower values). The data presented in [1281] show a more complicated picture. While the density in front of the bottleneck amounts to ρ ≈ 5.0(±1) m−2 , the density inside the bottleneck tunes around ρ ≈ 1.8 m−2 . 11.2.4.1. Influence of Motivation Muir et al. [955] have studied the combined influence of bottleneck width and motivation on the flow. More precisely their empirical study considers egress from an aircraft where passengers either show cooperative or competitive behavior. Considering the egress time T from the aircraft, two regimes were found, depending on the width w of the exit. These regimes are separated by a critical width wc such that Tcomp > Tcoop Tcomp < Tcoop
for for
w < wc , w > wc ,
(11.11)
where Tcomp and Tcoop are the egress times in the competitive and cooperative mode, respectively. Thus, competition is beneficial only for wide exits, but it is harmful for narrow one. Empirically one finds wc ≈ 70 cm [955]. This can be related to results of [958, 1138] where it has been found that funnellike geometries support the formation of arches and thus blockages. This is similar to
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the phenomenon of arching encountered in granular materials [925, 1475]. Here, arches (sometimes called bridges) are self-supporting structures, also known from architecture, which form through the contact of the grains under the influence of friction. This can lead to cooperative dynamics where arches are deformed rather than destroyed. Arching is relevant in many granular materials, and in typical packings, more than 70% of particles belong to bridges. It leads to jamming in the outflow from containers, chutes, etc.
11.2.5. Collective Phenomena One of the reasons why the investigation of pedestrian dynamics is attractive for physicists is the large variety of interesting collective effects and self-organization phenomena that can be observed. Here, we only give a brief overview and refer to [512, 1241, 1260] for a more comprehensive discussion. 11.2.5.1. Jamming At large densities, various kinds of jamming phenomena occur, e.g., when many people try to leave a large room at the same time and the flow is limited by a door (see Fig. 11.5) or narrowing. Therefore, the occurrence of this kind of jamming phenomenon does not depend strongly on the microscopic dynamics of the particles. Rather it is a consequence of an exclusion principle. The space occupied by one particle is not available for others. This clogging effect is typical for a bottleneck situation. It is important for practical applications, especially evacuation simulations. Other types of jamming occur in the case of counterflow where two groups of pedestrians mutually block each other. This happens typically at high densities and when it is not possible to turn around and move back, e.g., when the flow of people is large.
(a)
(b)
(c)
Figure 11.5 Clogging near a bottleneck (door). Shown are typical stages of an evacuation simulation: (a) initial stage, (b) middle stage, and (c) final stage with only a few particles left.
Pedestrian Dynamics
Figure 11.6 Schematical depiction of lane formation in counterflow in a narrow corridor.
11.2.5.2. Lane Formation In counterflow, i.e., two groups of people moving in opposite directions, a kind of spontaneous symmetry breaking occurs (see Fig. 11.6). The motion of the pedestrians can self-organize in such a way that (dynamically varying) lanes are formed where people move in just one direction [535]. In this way, strong interactions with oncoming pedestrians are reduced, and a higher walking speed is possible. The occurrence of lane formation does not require a preference of moving on one side; it also occurs in situations when there is no left or right preference. However, empirically such a preference appears to exist as, e.g., in central Europe, people prefer to move on the right side. Usually, the preferred side corresponds to the traffic rules in each country, but there are exceptions to this rule. Although this preference is not essential for the phenomenon itself, it has an influence on the kind of lanes formed and their order. The number of lanes usually is not constant and might change in time, even if there are relatively small changes in density. It can also vary considerably with the total width of the flow area. The number of lanes in opposite directions is not always identical. This implies a sort of spontaneous symmetry breaking. Unfortunately not many quantitative studies of lane formation have been performed so far. Yamori [1491] has introduced a band index to quantify the extent of lane formation (Fig. 11.7). It is basically the ratio of pedestrians in lanes to their total number. High band indices are usually only observed for large numbers of pedestrians (above 100). The number of lanes as a function of the width has been studied in [1037], and the number of clusters as a function of the density has been studied in [583]. Further results can be found in [1357]. Experimental results have been reported, e.g., in [583, 805] where two groups with varying relative sizes had to pass each other in a corridor of width b = 2 m. One surprising result is that the total flow in a counterflow situation can be larger than the sum of two comparable unidirectional flows. Similar theoretical findings were reported in [59] although the relation with the experiments is not clear. It is worth mentioning that lane formation has also been observed, both theoretically and experimentally, in physical systems. The studies have been performed for colloidal
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(a)
(b)
Figure 11.7 Typically in bidirectional pedestrian motion, lanes (a) and clusters (b) are formed. These can be distinguished by the the band index [1491], which is basically the ratio of pedestrians in lanes to their total number. This index is close to 1 in (a) (from [768]).
mixtures where colloidal particles (with typical size of the order of 1 μm) of opposite charge were driven by an external electrical field [183, 184, 280, 331, 1181–1183, 1484]. Lane formation was experimentally verified in [856]. It is characterized as a first-order nonequilibrium phase transition with order parameter N 1 φj φ= N j=1
(11.12)
where φi for each particle i = 1, . . ., N is set to be 1, when the lateral distance rl = |xi − xj | to all particles of the other type is larger than same suitable length scale, e.g., rl > 2√1 ρ . Similar investigations have been performed on binary granular mixtures [340] and, on board of the International Space Station (ISS), on binary complex plasmas [1327]. 11.2.5.3. Flow Oscillations In counterflow at bottlenecks, e.g., doors, one can observe oscillatory changes of the direction of motion. Once a pedestrian is able to pass the bottleneck, it becomes easier for others to follow him or her in the same direction until somebody is able to pass (e.g., through a fluctuation) the bottleneck in the opposite direction (see Fig. 11.8). 11.2.5.4. Patterns in Intersecting Flows Intersecting flows can produce various collective patterns, like lane formation in the case of counterflow. Similar patterns occur when the flows intersect in a different way, e.g., the formation of diagonal stripes at crossings [517, 583]. In these stripes, clusters
Pedestrian Dynamics
Figure 11.8 Flow oscillations at a narrow bottleneck with counterflow. The flow direction through the bottleneck changes irregularly.
of pedestrians move in the same direction and with the same speed.3 Another typical example for collective patterns at crossings are short-lived roundabouts, which make the motion more efficient [517, 536]. Even if these are connected with small detours, the formation of these patterns can be favorable since they allow for a “smoother” motion. 11.2.5.5. Emergency Situations, “panic” In emergency situations, various collective phenomena have been reported that have sometimes misleadingly been attributed to panic behavior. However, there is strong evidence that panic behavior has played no role in most cases. Although a precise accepted definition of panic is missing, but usually certain aspects are associated with this concept [720]. Typically “panic” occurs in situations where people compete for scarce or dwindling resources (e.g., safe space or access to an exit), which leads to selfish, asocial, or even completely irrational behavior and contagion that affects large groups. A closer investigation of many disasters related to emergency situations have revealed that most of the above characteristics have played almost no role in the tragic events and most of the time have not been observed at all (see, e.g., [228, 692, 1190]). Therefore, the term “panic” should be avoided, crowd disaster being a more appropriate characterization. Although empirical data on crowd disasters exist, e.g., in the form of reports from survivors or even video footage, it is almost impossible to derive quantitative results from them. Models that aim at describing such scenarios make predictions for certain counterintuitive phenomena that should occur. In the faster-is-slower effect [518, 520], a higher desired velocity leads to a slower movement of a large crowd. In the freezingby-heating effect [519], increasing the fluctuations can lead to a more ordered state. For a thorough discussion, we refer to [518, 520] and references therein. However, it is still unclear in how far these effects occur in real emergency situations since it is difficult to perform “realistic” experiments. 3 Intersecting streams of pedestrian and vehicular traffic have been investigated, e.g., in [529, 662].
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There are, however, several experimental studies of evacuation scenarios. Helbing et al. [528] studied evacuations from a classroom focussing on the influence of the initial positions of the evacuees. They found a rather broad distribution of evacuation times. The experiments in [633, 634] showed that disoriented persons show a very different behavior than under normal conditions with good visibility. In [1517], the influence of the premovement time, i.e., the time between the alarm and the beginning of the movement of the persons, has been studied for evacuations from classrooms. A more complex scenario (office building) has been considered in [868]. Nagai et al. [966, 967] have performed experiments with walkers and crawlers (people moving on all fours) in counterflow and evacuating from a corridor. In [383], the exit choice behavior was investigated. Usually, the nearest exit is preferred, even if other exits are relatively close. However, at large densities, flow rates through such exits become comparable indicating a self-organized optimization. Evacuation experiments have also been performed with animals to study the influence of “panic.” In these studies, mice [1208, 1209] and ants [21, 22] have been exposed to external hazard (water and repelling liquids, respectively). These experiments have found a clear tendency toward herding, i.e., a preference of one of the available exits. 11.2.5.6. Crowd Turbulence At extreme densities, strong forces occur in large crowds. These forces can quickly change direction. As a consequence, individuals are pushed around and perform random, intended irregular displacements in all possible directions. This phenomenon has been termed crowd turbulence [531, 532, 1503]. Such situations can lead to tragedies as persons who stumble and fall are often not able to get on their feet again. Similar to fluids, crowd turbulence results from a sequence of instabilities in the flow patterns. It is associated with a sharply peaked probability density function of the velocity increments, e.g., vx(r, t + τ ) − vx(r, t), if the time shift τ is small enough [531]. The analogy with turbulence in fluids is limited since no large eddies have been observed. It has been suggested to use this as an indicator for critical situations in large crowds. To this end, a crowd pressure P(r, t) = ρ(r, t)Varr,t (v) has been introduced where Varr,t (v) is the local velocity variance [531]. The transition to the turbulent state is then supposed to happen if the crowd pressure exceeds a certain value. 11.2.5.7. Trail Formation In parks, on snow, and other nonpaved areas, often the formation of human trails can be observed (Fig. 11.9). Although these trails are usually shortcuts, typically their structure is too complex to be explained just by the influence of the terrain and the desire to take the shortest path. Sometimes, trail networks are formed that consist of different trails [447]. Also it seems that a criterion for the formation of a shortcut is not just the absolute length difference, but rather the relative length difference that has to be larger than a certain threshold.
Pedestrian Dynamics
Figure 11.9 Typical trail formed as a shortcut.
An obvious application of a theory of trail formation is landscape planing where the unintentional formation of trails could be reduced or even prevented by a proper design of the walkways.
11.3. CELLULAR AUTOMATA MODELS While there are many different cellular automata (CA) models for highway traffic, only a few models for pedestrian dynamics have been developed. A natural space discretization for a model of pedestrian dynamics can be derived from the fact that maximal densities of about 6 P/m2 can be observed in dense crowds (except in very extreme situations). Then, a pedestrian occupies an area of 40 × 40 cm2 [1453], which will be identified with the cell size. Each cell can then either be empty or be occupied by exactly one particle (pedestrian). This reflects that the interactions between them are repulsive for short distances, i.e., one likes to keep a minimal distance to others in order to avoid bumping into them. For special situations, it might be desirable to use finer discretizations, e.g., such that each pedestrian occupies four cells instead of one, but we restrict to the simplest case which is sufficient for the applications studied here. There are two definitions of nearest neighbors that are commonly used, namely the von Neumann and the Moore neighborhoods. The von Neumann neighborhood of a given cell consists of all cells that share an edge with it. The Moore neighborhood, on the other hand, comprises all cells that share a corner with the central one. On a square
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lattice, the von Neumann neighborhood consist of four cells. The Moore neighborhood additionally includes the cells in diagonal directions and thus consists of eight cells.4 Throughout the literature, the terminology is not used consistently. Sometimes, these models are also called lattice gas models [1013] (see, however, Section 11.5.3), especially when particles do not reside inside the cells but on the vertices of the lattice. In multiagent systems (MAS) [65, 181], the particles are provided with some intelligence. Agents are autonomous entities that can make decisions and carry out actions depending on the environment and interactions with other agents. From a physics point of view, MAS often can be considered CA models with particles that have internal degrees of freedom. Most CA models for pedestrian dynamics proposed so far are rather simple [118, 417, 769, 959, 1013] and can be considered generalizations of the BML model for city traffic [115] (see Section 10.2) or two-dimensional variants of the asymmetric simple exclusion process (ASEP). Particles can move to one of the neighboring cells based on certain transition probabilities. These are determined by three factors: (1) the desired direction of motion, i.e., to find the shortest connection, (2) interactions with other pedestrians, and (3) interactions with the infrastructure (walls, doors, etc.).
11.3.1. Fukui–Ishibashi Model One of the first CA models for pedestrian dynamics has been proposed by Fukui and Ishibashi [416, 417]. They have studied bidirectional motion in a long corridor, which is represented as a square lattice of length L and width W < L with periodic boundary conditions. “Eastbound” pedestrians are allowed to move at odd time steps, whereas “westbound” pedestrians move at even time steps. The dynamical rules are symmetric with respect to the preferred direction of motion, and in the following, we will only describe the rules for westbound pedestrians. During the update step, each pedestrian tries to move forward to the next cell ahead. If this is occupied by another westbound particle, he or she will stay at his or her present position. However, if it is occupied by a eastbound particle, a lane change will be attempted. Two different rules for lane changing have been proposed [416, 417]. In the side stepping model, a von Neumann neighborhood is considered, and pedestrians try to move to an empty neighbor cell either to the north or to the south. If both are empty, one site is chosen randomly. However, forward moving pedestrians are given priority over lane changing pedestrians. The diagonal stepping model uses a Moore neighborhood, and so motion in diagonal direction is allowed (Fig. 11.10). If diagonal motion is not possible, side steps are attempted as in the side stepping model. 4 Sometimes the central cell is also included in the definition so that the von Neumann neighborhood consists of five
and the Moore neighborhood consists of nine cells.
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2
2
1
1
1
2
Figure 11.10 Allowed transitions at an odd time step in the pedestrian model of Fukui and Ishibashi. Eastbound (westbound) pedestrians are represented by open (full) circles. Allowed transitions are indicated by arrows. Numbers refer to the priority in the diagonal stepping model. In the side stepping model, only transitions indicated by “2” are allowed.
For the side stepping model with one westbound pedestrian moving in a stream of eastbound pedestrians of density ρ, two second order phase transitions at densities ρc and 1 − ρc are found where ρc =
1−W . 2W
(11.13)
For 0 ≤ ρ ≤ ρc , the eastbound flow grows linearly, for ρc ≤ ρ ≤ 1 − ρc , it remains constant, and finally, for ρc ≤ ρ ≤ 1, it decreases linearly. This behavior is well-known from systems with a static defect site (see Section 4.6.1). For the diagonal stepping model, no plateau regime is observed. Instead the flow is linearly increasing up to density 1 − ρc beyond which it becomes monotonically decreasing. Again the transition at 1 − ρc is of second order. In [416], the case of an equal number of eastbound and westbound pedestrian has been considered. Here, the numerical results indicate a first-order transition to a completely blocked state with vanishing average velocity at a critical density ρc (W ) 1/2. At low densities, a kind of lane formation is observed, which allows all pedestrians to move in each time step corresponding to an average velocity of 1. Close to the critical density, metastable states of finite flow exist, which are reached for special initial conditions. 11.3.1.1. Extensions of the FI Model Various groups have used variations of the Fukui–Ishibashi (FI) model to study pedestrian dynamics in special situations. Nagatani and collaborators [959, 961, 962, 1013, 1342] have especially investigated situations where a jamming transition can occur. Their basic
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p (e) y
p (a) y p(a) x
p(b) x
p (d) y p(c) x p−(d)y
p−(b)y
p−(a)y
p (fy ) p(e) x p−(g)y
Figure 11.11 Definition of the transition probabilities for a right-walker in the Nagatani model. Depending on the configuration of neighbor cells, eight different situations have to be distinguished. The corresponding transitions for left-walkers are obtained by symmetry.
model is similar to the side stepping model, but uses a random-sequential update and a different parameterization of the transition probabilities in terms of a drift parameter D. The transition probabilities px(C) now depend on the configuration C in the neighborhood of the particle. Figure 11.11 shows the eight different situations that have to be considered for a right-walker. The standard parameterization in terms of the drift D is as follows: 1 px(a) = D + (1 − D), 3 1 (b) px = D + (1 − D), 2 1 px(c) = D + (1 − D), 2 1 (d) py(d) = p−y = , 2 (g)
px(e) = py(f ) = p−y = 1, (h)
px(h) = py(h) = p−y = 0.
1 (a) py(a) = p−y = (1 − D), 3 1 (b) p−y = (1 − D), 2 1 py(c) = (1 − D), 2
(11.14) (11.15) (11.16) (11.17) (11.18) (11.19)
The corresponding transition probabilities for a left-walker can be obtained from the (C) symmetry px(C) → p−x . Note that in all case except (h), the probability of not moving vanishes. Also the probabilities for lane changing, i.e., moving in y-direction, are symmetric. In [959], also an asymmetric variant of the model has been introduced where walkers prefer lane changes to the right. This reflects the fact that in many countries, pedestrians prefer (or even are obligated) to walk on the right side. In the asymmetric
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(C)
model py(C) < p−y even in the cases where both neighbor lanes are accessible as in the configurations (a) and (d). One focus of studies based on this model was jamming transitions in various situations. For a long corridor, a first-order phase transition into a completely jammed phase is found when the density at the entrance of corridor reaches a critical values ρc , which depends on the drift D and the width W of the corridor [959]. A fully parallel variant has been studied in [1454]. In this case, one has to take care of conflicts where two or more pedestrians try to move to the same target cell. In such cases, only one of them will be chosen randomly and allowed to move. Including back stepping, i.e., motion opposite to the preferred direction, increases the critical density since pedestrians become more flexible to avoid the deadlock. It increases monotonically with increasing back step probability. Surprisingly, Maniccam [911] found that on large two-dimensional lattices (W = L) back stepping leads to a decrease of the critical density, i.e., makes congestion worse. In another variant, the effect of allowing simultaneous (exchange) motion of pedestrians standing “face-to-face” was considered [661]. From an investigation of the fundamental diagrams, it was concluded that a realistic value for the exchange probability is as large as 0.2. Other studies of counterflow can be found in [1014, 1343, 1352, 1454, 1458, 1495]. The jamming transition persists in the two-dimensional version of the model where W = L [961] with periodic boundary conditions. The jammed pattern in a two-way situation is similar to the diagonally striped pattern found in the BML model [115] for city traffic. A four-way extension of this model, with walkers moving in all four principal directions of the square lattice, also shows a first-order jamming transition, but with a different structure of the jammed states. Maniccam [911] has investigated the influence of the update scheme and back stepping in the two-dimensional case. Both have only a quantitative influence on the jamming transition although the structure of the jammed state depends on the update type. Another factor is the lattice structure. In [910], the model was investigated on a hexagonal lattice. For large drift, the critical density is larger than for the square lattice. The difference becomes smaller for smaller drifts. The jamming transition also depends on the shape of the particles. In [968], a system of slender particles has been investigated, which occupy 1 × n cells (orientation parallel to the preferred direction). Then, the critical density and the structure of the jammed state depend strongly on the size n of the particles. The critical density increases with increasing particle size in correspondence with a change of the structure of the jammed state. For longer particles, avoid oppositely moving particles and thus tend to form lines. Similar systems have been studied in [408, 680, 918]. Other situations that have been studied with variants of the basic models are different types of crossings, e.g., a four-way crossing [962] or a T-shape channel [1341].
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Bottlenecks have been studied in [1342] and merging flows in [1000]. Also various evacuation scenarios have been considered (see, e.g., [196, 528, 635, 1340, 1351]). In order to obtain more realistic fundamental diagrams and study mixtures of different pedestrian types, the basic model has also been generalized to large velocities, i.e., larger interaction ranges [682, 1457]. The dynamic parameter model [1506, 1507] only extends the vision-conscious field and combines ideas of the FI models with that of the floor field model. A variant taking into account the effect of social forces has been proposed in [1300].
11.3.2. Blue–Adler Model Blue and Adler [117, 118, 121, 122] have extended two-lane variants of the Nagel– Schreckenberg model for the description of pedestrian dynamics. The two-dimensional grid on which the pedestrians move is therefore considered to be a multilane system. The structure of the rules is similar to the basic two-lane rules suggested in [1186] (see Section 8.6). The update is performed in four steps, which are applied to all pedestrians in parallel and take into account forward movement, side stepping, and conflict mitigation. In the first step, each pedestrian chooses the lane in which (s)he will prefers to move. In the second step, the lane changes are performed. In the third step, the velocities are determined based on the available gap in the new lanes. Finally, in the fourth step, the pedestrians move forward according to the velocities determined in the previous step. In order to study the effects of inhomogeneities, the pedestrians are assigned different maximal velocities vmax . Fast walkers have vmax = 4, standard walkers have vmax = 3, and slow walkers have vmax = 2. The best agreement with empirical observations has been achieved with 5% slow and 5% fast walkers [122]. In contrast to vehicular traffic, pedestrians reach their normal walking speed almost instantaneously. In order to take this into account, not the NaSch dynamics is used, but the deterministic limit of the FI [414] or aggressive-driver models is used [828, 1228]. Dynamical formation of lanes (Section 11.2.5) can be enforced by additional rules [121]. In [117], unidirectional flows in a corridor consisting of several parallel lanes were studied. This has been generalized later to several bidirectional modes, namely separated directional flows, interspersed directional flows, and flows with lane formation [118, 119, 121], and four-directional movement, e.g., at streets corners [120]. Most of these studies focus on the qualitative and quantitative properties of fundamental diagram. In the absence of lane formation generically, a lowered speed and capacity is observed.
11.3.3. Gipps–Marksjös Model A more sophisticated discrete model has been suggested by Gipps and Marksjös [437] already in 1985. One motivation for developing a discrete model was the limited
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computer power at that time. Therefore, a discrete model, which reproduces the properties of pedestrian motion realistically, was in many respects a real improvement over the existing continuum approaches. In addition, it was one of the first microscopic models, and thus, it was possible to include different characteristics of the pedestrians and issues like route choice. Interactions between pedestrians are assumed to be repulsive anticipating the idea of social forces (see Section 11.5.2). The pedestrians move on a grid of rectangular cells of size 0.5 × 0.5 m. To each cell, a score is assigned based on its proximity to other pedestrians (and also walls and obstacles). This score represents the repulsive interactions, and the actual motion is then determined by the competition between these repulsion and the gain of approaching the destination. In the simplest version, a cell occupied by a pedestrian is given a very large score (e.g., 1000) that reflects the strict hardcore repulsion. The four neighbor cells receive a medium score (e.g., 40), and the four next nearest-neighbor cells a small score (e.g., 13). The interaction range can even be extended. The score is approximately inversely proportional to the square of the separation. Applying this procedure to all pedestrians, each cell gets assign a potential value that is the sum of the individual contributions. The pedestrian selects then the cell of its nine neighbors (Moore neighborhood), which leads to the maximum benefit. This benefit is defined as the difference between the gain of moving closer to the destination and the cost of moving closer to other pedestrians as represented by the potential. This requires a suitable chosen gain function P. It cannot simply be a function of the change in separation since this would lead to an unrealistic preference of diagonal steps. The reason for this preference is a geometric effect: the step length in diagonal direction is longer than in the four main directions. This problem can be solved by using P(σj ) = K | cosσj | cosσj ,
(11.20)
where σj is the angle by which the pedestrian deviates from a straight line to his or her destination when moving to cell j. From geometry, this angle is given by cosσj =
(ri − s) · (d − s) , |ri − s| |d − s|
(11.21)
where ri is the location of the target cell, s is the location of the pedestrian, and d is the destination. For ri = s, one sets P = 0. The constant K allows to control the relative importance of the two factors: the gain of moving in a straight line and the costs of approaching others too closely. The updating is done sequentially in order to avoid conflicts of several pedestrians trying to move to the same position. In order to model different velocities, faster pedestrians are updated more frequently. This is done by assigning them to different lists,
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according to their velocities. For five different velocities, five lists are used, which are processed in the order 5,4,3,5,2,4,5,3,4,5,1,4,5,3,2. If such a cycle corresponds to one second of real time, the speeds range from 0.5 to 2.5 m/s.
11.4. FLOOR FIELD CA In [170, 171, 755, 1232], a more sophisticated CA model has been introduced, which by now has become the standard CA approach to pedestrian dynamics. Despite its relative simplicity, it is able to reproduce the observed collective effects.
11.4.1. General Principle The model takes its inspiration for the implementation of interactions from the process of chemotaxis (see [94, 574, 575] and Section 12.4) used by some insects like ants for communication. They produce chemical substances called pheromones to mark their way and thereby guide other individuals to food sources. A similar principle underlies the formation of human trails (Section 11.2.5), but here the trace is visible. In the approach of [171, 1232], the pedestrians also create a trace. In contrast to trail formation and chemotaxis, however, this trace is only virtual although one could assume that it corresponds to some abstract representation of the paths of others in the mind of the pedestrians. Its main purpose is to transform effects of long-ranged interactions (e.g., following people walking some distance ahead) into a local interaction (with the trace). This allows for a much more efficient simulation on a computer, especially in complex geometries where it has not to be checked explicitly whether interactions are blocked by walls, etc. [1234]. This idea is implemented in a two-dimensional stochastic cellular automaton, now called floor field model for reasons that should become clear soon. Although it is usually studied on a square lattice with only nearest-neighbor movements allowed (both for Moore and von Neumann neighborhoods), this can be generalized to other situations. Typically, the motion of a pedestrian is influenced by three factors: • The desired direction of motion toward the destination, • Interactions with other pedestrians, • Interactions with the infrastructure, e.g., walls or obstacles. The floor field model incorporates these three contributions in a unified framework. The dynamics of the particles in the model is defined by transition probabilities to the allowed neighbor cells. In contrast to the previous models, these transition probabilities are not constant, but can change in time in response to changing situations. In order to make the model computationally efficient in simulations, local interactions are desirable. This is not the case for pedestrian–pedestrian interactions, which are not even restricted to nearest neighbors as (to a large extent) in vehicular dynamics.
Pedestrian Dynamics
Usually, the motion is determined by taking into account the situation in a neighborhood, trying to estimate whether avoidance reactions become necessary, etc. In force-based models, this can quickly lead to a large number of relevant interactions that have to be taken into account. This is where the idea of chemotaxis comes in. It allows to transform interactions that are “long-ranged” in space into local ones. Since the trace exists for a while, the “nonlocal” spatial interaction becomes an interaction which is “nonlocal” in time (“memory”). In the model, this is achieved by the introduction of so-called floor fields. The transition probabilities for all pedestrians depend on the strength of the floor fields in their neighborhood in such a way that transitions in the direction of larger fields are preferred. Interactions between pedestrians are repulsive for short distances. One likes to keep a minimal distance to others in order to avoid bumping into them ( personal space or private sphere). This is taken into account through hard-core repulsion, which prevents multiple occupation of the cells. An additional effectively repulsive interaction is introduced by using a parallel updating scheme for the dynamics. Similar to the case of models of ASEP type, this introduces particle-hole attraction (see Section 4.4.3). For longer distances, the interaction is often attractive. For example, when walking in a crowded area, it is usually advantageous to follow directly behind the predecessor. Large crowds may also be attractive due to curiosity, and in emergency situations, often herding behavior can be observed [520]. The long-ranged part of the interaction is implemented through floor fields, which correspond to the pheromone trace in chemotaxis. The static floor field is constant in time and represents the constant properties of the infrastructure. The dynamic floor field models the dynamic interactions between the pedestrians. It corresponds to a virtual trace, which is created by the motion of the pedestrians, and in turn influences the motion of other individuals. Furthermore, it has its own dynamics, namely through diffusion and decay, which leads to a dilution and finally the vanishing of the trace after some time. The static floor field does not change with time since it only takes into account the effects of the surroundings. Therefore, it exists even without any pedestrians present. It allows to model, e.g., preferred areas, walls, and other obstacles. The introduction of the floor fields allows for a very efficient implementation on a computer since now all interactions are local. In this way, one has translated the long-ranged spatial interaction into a local interaction with memory. Therefore, the number of interaction terms grows only linearly with the number of particles. Another advantage of local interactions can be seen in the case of complex geometries [1234]. Because of the presence of walls, not all particles within the interaction range interact with one another. Therefore, one needs an algorithm to check whether two particles “see” each other or whether the interaction is blocked by some obstacle. All this is not necessary here. Furthermore, the pedestrians do not need to be provided with some sort of “intelligence.” The floor fields are sufficient to achieve the formation of complex structures and collective effects by means of self-organization. The pedestrians behave like simple
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p−1, −1
p− 1,0
p− 1,1
p0, − 1
p0,0
p0,1
p1, −1
p1,0
p1,1
Figure 11.12 A particle, its possible directions of motion, and the corresponding transition probabilities pij for the case of a Moore neighborhood. For the von Neumann neighborhood, the diagonal probabilities p±1,±1 are zero.
“particles” in a field without explicit intelligence. Since also no detailed assumptions about the human behavior are made, this allows to keep the model simple. Nevertheless it is able to reproduce many of the basic phenomena. In contrast to vehicular traffic, the time needed for acceleration and braking is negligible in pedestrian motion. The velocity distribution of pedestrians is sharply peaked [549]. These facts naturally lead to a model where the pedestrians have a maximal velocity vmax = 1, i.e., only transitions to neighbor cells are allowed. The desired speed and direction of motion toward the destination can be taken into account by introducing another field called matrix of preference. It is given by a 3 × 3 matrix, which contains the probabilities for a move of the particle. The central element describes the probability for the particle not to move at all, and the remaining eight correspond to a move to one of the neighboring cells5 (see Fig. 11.12). The probabilities can directly be related to observable quantities, namely, the velocity and the longitudinal and transversal standard deviations (see [169, 171] for details). In some cases, it is not necessary to use the matrix of preference. Instead the same information can be implemented into the static floor field. Typically this can be done if all pedestrians have the same destination, as in the case of an evacuation. Figure 11.13 shows the static floor field used for the simulation of evacuations from a room with a single door. Its strength decreases with increasing distance from the door. Since the pedestrian prefers motion into the direction of larger fields, this is already sufficient to find the door.
11.4.2. Update Rules We now define the stochastic dynamics of the model by specifying the transition probabilities pij for a motion to a neighboring cell in direction (i, j) where i, j ∈ {−1, 0, 1} (Fig. 11.12). As explained above, the long-ranged interactions with other pedestrians 5 If motion in diagonal directions is not considered, the matrix of preference has at most five nonzero elements.
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Pedestrian Dynamics
Figure 11.13 Static floor field for the simulation of an evacuation from a large room with a single door. The door is located in the middle of the upper boundary, and the field strength is increasing with increasing intensity.
and the surrounding is encoded in two fields, the dynamic floor field D and the static floor field S, respectively. Furthermore, a matrix of preference M can be used to specify the walking direction, speed, and fluctuations for each individual. The transition probability pij in direction (i, j) is then determined by all three contributions. Explicitly it is given by pij = N ekD Dij ekS Sij Mij (1 − nij )ξij .
(11.22)
Dij and Sij are the strengths of the dynamic and static floor field at the target cell, and of preference for a motion in the direction (i, j). Mij is the matrix element of the matrix N is a normalization factor to ensure (i,j) pij = 1 where the sum is over the possible target cells. The factor 1 − nij , where nij is the occupation number of the neighbor cell in direction (i, j), takes into account that transitions to occupied cells are forbidden. Note that nij has to be interpreted as occupation number of other particles. Since it is also possible that the particle stays at its present position, one has to take n00 = 0. ξij is a geometry factor (obstacle number), which is 0 for forbidden cells (e.g., walls) and 1 else. Finally, we have introduced two coupling constants kD and kS so that we can vary the coupling strengths to each field individually. The actual values of the parameters kD and kS depend on the situation. A large kS implies that the pedestrians choose their path mainly due to the surrounding without being distracted too much by other people. A large coupling kD to the dynamic field, on the other hand, corresponds to a strong herding behavior. Here, the pedestrian tries to follow the lead of others, e.g., in case of emergency situations or insufficient knowledge
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about the surroundings. The influence of these couplings will be discussed in more detail in Section 11.4.8. As mentioned before, the dynamic floor field is created by the motion of the pedestrians and corresponds to a virtual trace. At t = 0, it is zero everywhere. Whenever a particle moves from site (x, y) to one of its neighbors (x + i, y + j), the field Dxy at the origin cell is increased by one (Dxy → Dxy + 1). Thus, Dxy has only nonnegative integer values, which can be interpreted as the number of “bosons” located at site (x, y). The dynamic floor field is not only changed by the motion of the pedestrians but also subject to diffusion and decay. This first leads to a spreading and dilution of the trace and finally to its vanishing after some time. Diffusion and decay are controlled by two parameters α ∈ [0, 1] and δ ∈ [0, 1]. In each time step of the simulation, each boson of the dynamic field D decays with the probability δ and diffuses with the probability α to one of the neighboring cells. The update rules of the full model including the interaction with the floor fields then have the following structure: 1. The dynamic floor field D is modified according to its diffusion and decay rules. 2. Using equation (11.22), for each pedestrian, the transition probabilities pij for a move to an unoccupied neighbor cell (i, j) are determined by the matrix of preference and the local dynamic and static floor fields. 3. Each pedestrian chooses a target cell based on the probabilities of the transition matrix P = ( pij ). 4. The conflicts arising by any two or more pedestrians attempting to move to the same target cell are resolved (see below, Section 11.4.4). 5. The pedestrians who are allowed to move execute their step. 6. The pedestrians alter the dynamic floor field Dxy of the cell (x, y) they occupied before the move. One detail is worth mentioning. If a particle has moved in the previous time step, the boson created then is not taken into account in the determination of the transition probability. This prevents that pedestrians get confused by their own trace. One can even go a step further and introduce inertia [171, 1060], which reduces transition probabilities that correspond to a change in the direction of motion. The rules have to be applied to all pedestrians at the same time (parallel dynamics). This introduces a timescale into the dynamics, which can roughly be identified with the reaction time treac [171]. The existence of a timescale in the dynamics of the model is essential if one wants to make quantitative predictions for real processes. In the deterministic limit, corresponding to the maximal possible walking velocity in the model, a single pedestrian (not interacting with others) moves with a velocity of one cell per time step, i.e., 40 cm per time step. Taking a typical empirical value of about 1.3 m/s (Section 11.2) for the average velocity of a pedestrian gives an estimate for the real time corresponding
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to one time step in our model of approximately 0.3 s. This is indeed of the order of the reaction time treac and thus consistent with the microscopic rules. It also agrees nicely with the time needed to reach the normal walking speed, which is about 0.5 s.
11.4.3. Construction of the Static Floor Field Let us now consider how to construct the static floor field for complex rooms of arbitrary geometry [1060]. This becomes possible by using the combination of the visibility graph and Dijkstra’s algorithm. This allows to calculate the minimum Euclidian (L 2) distance of any cell to a door with arbitrary obstacles between them. We illustrate the idea behind this method by using the configuration in Fig. 11.14 where there is an obstacle in the middle of the room. We now calculate the minimum distance between a cell P and the door O by avoiding the obstacle. If the line PO does not cross the obstacle A–H, then the length of the line gives the minimum. If, however, as given in Fig. 11.14, the line PO crosses the obstacle, one has to make a detour around it. In this case, we obtain two candidates for the minimum distance, i.e., lines PBAO and PCDHO. The shorter one finally gives the minimum distance between P and O. If there is more than one obstacle in the room, we apply the same procedure to each of them repeatedly. Here, it is important to note that all the lines pass only the obstacle’s nodes with an acute angle. It is apparent that the obtuse nodes, e.g., E and F, can never be passed by the minimum lines. We can simply check whether a node is acute or not by considering the sign of the outer product of the two attached bonds. To incorporate this idea into the computer program, one uses the concept of the visibility graph in which only the nodes that are visible to one another are bonded [266], O 10.2 A
H
A F
O 15.0
5.0
8.3
E
D
B
C P (a)
B
10.8
5.0 C
P (b)
12.0
60 40
D
26.0
12.0
80
H
G
15.0
G
100
8.4
20 0
0
20
40
60
80
100
(c)
Figure 11.14 (a) Example for the calculation of the static floor field for a room with one obstacle. The door is at O, and the obstacle is represented by lines A–H. (b) The corresponding visibility graph for this room. Each node is connected by a bond if there are no obstacles between them. The real number on each bond represents the distance between nodes. (c) Contour plot of the static floor field for a room with four obstacles and two doors. The darkness of shading is inversely proportional to the distance from the nearest door.
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i.e., “visible” means here that there are no obstacles between them. The set of nodes consists of a cell point P, a door O, and all the acute nodes in the room. To save numerical costs in the case of a room with complex obstacles, one only chooses acute nodes. In the case of Fig. 11.14a, the node set is {P, O, A, B, C, D, G, H }, and the bonds are connected between A–B, A–H, and so on (Fig. 11.14b). Each bond has its own weight, which corresponds to the Euclidian distance between them. Once the visibility graph is determined, the distance between P and O can be calculated by adding the weight of the bonds between them. There are several possible paths between P and O, and the one with minimum weight in total represents the shortest route between them. The optimization task is easily performed by using the Dijkstra method [266], which allows to determine the minimum path on a weighted graph. Performing this procedure for each cell in the room yields the static floor field for arbitrary geometries. Other methods for the determination of floor fields have also been proposed [615, 756, 801, 803]. The most straightforward approach is flood filling [801, 803]. Here, starting from the exit or destination successively, the value of each cell with nonzero potential and all its neighbors is increased by a certain amount. In this way, the potential decreases with increasing distance from the exit. In principle, this process can also be carried in each time step, taking into account the positions of other pedestrians. Another approach [756, 801, 802] uses the quickest path instead of the shortest for the calculation of the field strength.
11.4.4. Conflicts and Friction Step 4 of the update procedure of the floor field models contains the resolution of conflicts, which occur if m ≥ 2 particles choose the same destination cell in step 3. This becomes relevant in high-density situations. In order to avoid multiple occupancies of cells, only one particle is allowed to move, while the others keep their position. In [171, 755], the conflicts between pedestrians were solved in the following way: when m > 1, particles share the same target cell, one (l ∈ {1, . . . , m}) is chosen to move, while its rivals for the same target keep their position. There are two natural ways to determine the moving particle l: 1. With equal probability, i.e., each particle moves with probability m1 . 2. According to the relative probabilities with which each particle chooses target cell, its (l) (s) m i.e., the probability for the l-th particle to move is pij /N with N = s=1 pij . For many problems, the details of the conflict resolution turned out to play only a minor role. The influence of conflicts will be discussed in more detail below. In [754], the model has been extended by an additional friction parameter μ ∈ [0, 1] in order to describe clogging effects between the pedestrians more accurately. Whenever two or more pedestrians try to attempt to move to the same target cell, the movement of all involved particles is denied with the probability μ, i.e., all pedestrians remain at their
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μ
t
t+1
Figure 11.15 Refused movement due to the friction parameter μ (for m = 4).
site (see Fig. 11.15). This means that with probability 1 − μ, one of the individuals moves to the desired cell. Which particle actually moves is then determined by one of the rules for the resolution of conflicts described above. With this definition of μ and the extended update rule, it is easy to see that μ works as some kind of local pressure between the pedestrians. If μ is high, the pedestrians handicap each other trying to reach their desired target sites. This local effect can have enormous influence on macroscopic quantities like flow and evacuation time [754]. Note that the kind of friction introduced here only influences interacting particles, not the average velocity of a freely moving pedestrian. Conflicts are a consequence of the combination of discrete time dynamics and the exclusion principle. They might appear to be undesirable effects, which reduce the efficiency of the execution of simulations and should therefore be avoided by choosing a different update scheme. However, it turns out that this is not the case and that they are important for a correct description of crowd dynamics [754], especially in clogging situations encountered in large crowds, near intersections and bottlenecks. In real life, this often leads to dangerous situations and injuries during evacuations. Conflicts can be avoided by modifying the dynamics, e.g., by using a randomsequential update. But then the model dynamics has no longer a well-defined timescale, and thus, a calibration becomes impossible. Ordered-sequential updates have the disadvantage that the particles are no longer treated on equal footing and that different sequences might lead to different behavior. The shuffled update [743, 1479, 1480] solves this latter problem by choosing a different random sequence in each time step, but again the calibration is difficult. In addition, most of these schemes are not easy to justify, e.g., from psychological arguments, and appear to be rather unrealistic. Also, since the conflicts appear to be a realistic effect, the choice of other update schemes would only disguise their importance.
11.4.5. Other Generalizations and Interactions Because of its flexibility, the floor field model can be easily extended to incorporate additional interactions, psychological effects, etc. A natural extension is to allow for disorder, i.e., agents with different properties as destinations, walking speeds, etc.
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11.4.5.1. Inertia Effects Inertia effects have been introduced in [171, 1060]. In the simplest case, an additional factor exp(kI ) if direction of motion does not change (I ) (11.23) pij = 1 if direction of motion changes with a sensitivity constant kI is introduced in the calculation of the transition probabilities in (11.22). This factor suppresses changes in the direction of motion compared to the previous time step reflecting the effects of inertia. A more complex inertia factor based on centrifugal forces has been used in [809]. Inertia effects were also considered in [476]. For a more realistic description of the behavior near exits, a bottleneck parameter β has been introduced in [1493], which reduces the velocity of pedestrians and probability of backward motion at neighboring cells of the exits. Explicitly the transition probabilities pij according to Eqn (11.22) for these neighbor cells are replaced by for βpij (B) pij = βp00 + (1 − β) for
(i, j) = (0, 0) . (i, j) = (0, 0)
(11.24)
In order to describe inertia effects in the vicinity of the exits in a more detail way, Yanagisawa et al. [1492] have further introduced a turning function τ (θ ) = exp(−η|θ |) where θ ∈ [−π, π ] is the angle of deviation from the former direction of motion. The parameter η represents the strength of the inertia effects, i.e., the difficulties in turning with large angles. The transition probabilities pij in (11.22) are then replaced by (T ) pij
=
τ (θi,j )pij
τ (0)p00 + 1 − i,j τ (θi,j )pi,j
for
(i, j) = (0, 0)
for
(i, j) = (0, 0)
,
(11.25)
where θi,j is the angle between the former direction of motion and the direction of cell (i, j). 11.4.5.2. Interactions with Walls and Other Pedestrians Interactions with walls can be included explicitly by an additional wall potential [1060, 1061] (W )
pij
= exp(kW min[Dmax , dij ]),
(11.26)
where kW is a sensitivity constant, Dmax is the range of the wall potential, and di j is the minimum distance of the pedestrian from all walls.
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As an extension of the exclusion principle, a politeness factor (P)
pij = exp(−kP Np (i, j))
(11.27)
has been introduced in [809, 1239]. NP (i, j) is the number of pedestrians in the Moore neighborhood of (i, j). This factor reflects that pedestrians try to keep some distance toward other pedestrians and suppresses motion in the direction of high-density areas. Furthermore, it smoothens density changes near the boundaries of a crowd [809]. 11.4.5.3. Directed Floor Fields Henein and White [551, 553] have modified the floor field model by incorporating physical forces. In their swarm force model, an additional force floor field is defined, which represents the force experienced by an agent at its current position. In contrast to the other floor fields, it is a vector field to represent the directionality of the force. Agents apply forces in two circumstances: (1) an agent that wants to move to a blocked cell pushes the occupant of the desired cell, and (2) agents apply force to neighboring agents in order to maintain their space in the crowd. Forces can propagate through the crowd. It is also possible to include injuries by immobilizing agents when forces exceed a certain threshold. Later Henein and White also included information propagation into the model, leading to the swarm information model [552, 554]. Here, an additional information field is introduced, which allows for inter-agent communication, e.g., about blocked exits. 11.4.5.4. Spatial Resolution Another modification that appears to be necessary to reproduce empirical observations concerns the size of the cells. The cell size generically chosen corresponds to the space requirement of a single agent, i.e., 40 × 40 cm. Since an agent occupies exactly one cell, this does not allow to model overlapping lanes like those occurring in the zipper effect (see Section 11.2.4). This indicates that the cell size used in simulations should be smaller, so that, e.g., an agent occupies 2 × 2 cells [753] (see also [476, 1300, 1485]). Another important reason for finer spatial resolution is applications to safety engineering, etc. With 40 × 40 cm cells, it would, e.g., be unclear how to discretize a corridor of with 1 m. Therefore, from this practical point of view, smaller cell sizes are necessary. 11.4.5.5. Real-Encoded CA Another problem that is related to the discreteness of the underlying lattice is illustrated in Fig. 11.16. The distance to the exit along path B is much shorter than that of path A. However, using a cellular automaton with von Neumann neighborhood (with Manhattan metric), both paths have the same length. Using a Moore neighborhood instead would require to rescale the probabilities in diagonal directions to account for the longer step size.
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Exit
A
B
Exit
A B
(a)
(b)
Figure 11.16 Example of an evacuation toward the exit, with two paths of A and B. (a) Movement in real situation without grid points. (b) Movement on a discrete CA lattice. Paths A and B have now the same length.
An alternative solution to this problem is the real-coded cellular automata (RCA) [1489, 1490], which is based on the real-coded lattice gas developed for fluid simulation [502]. In the RCA, position and velocity of each pedestrian are not restricted by the underlying lattice, but are described by real numbers. Intermediate positions calculated using real velocities not restricted to lattice vectors are repositioned to lattice points by a rounding procedure. Then, the direction of motion is readjusted toward the exit. This procedure is able to generate the expected realistic path in such a situation [1490] (Fig. 11.16).
11.4.6. Moving Beyond Nearest Neighbors: vmax > 1 The comparison with the empirical results of Section 11.2.3 shows that the observed asymmetry of the fundamental diagram is not reproduced correctly. The origin of this discrepancy is the restriction to models with nearest-neighbor interactions, which do not capture essential features like the dynamic space requirement [1284, 1453] of the agents, which depends on their velocity (and thus density). Modifications of the floor field model [753, 808] take this effect into account. Here, motion is not restricted to nearest-neighbor cells. This is equivalent to a motion at different instantaneous velocities v = 0, 1, 2, . . . , vmax where v is the number of cells an agent moves. Then, vmax = 1 corresponds to the case where motion is allowed only to nearest neighbors. Different extensions of this type are possible, depending on how one treats crossing trajectories of different agents [753]. But in all cases, the fundamental diagrams become more realistic since the maximum of the flow is shifted toward smaller densities with increasing vmax (Fig. 11.17), in accordance with the empirical observations. Higher velocities also require the extension of the neighborhood of a particle which is no longer identical to the cells adjacent to the current position. A natural definition of
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vmax = 1 vmax = 2 vmax = 3 vmax = 4 vmax = 5
0.6
J
0.4
0.2
0.0 0.0
0.2
0.4
0.6
0.8
1.0
ρ
Figure 11.17 Fundamental diagrams of the floor field model for vmax = 1, ... , 5. The maximum is shifted toward smaller densities for increasing vmax .
“neighborhood” corresponds to those cells that could be reached within one time step. In this way, the introduction of higher velocities also reduces the problem of space isotropy as the neighborhoods become more isotropic for larger velocities (F.A.S.T. model [808, 809, 1332]). Other solutions to this problem have been proposed. One way is to count the number of diagonal steps and let the agent suspend from moving following certain rules which depend on the number of diagonal steps [1262]. A similar idea is to sum up √ the real distance that an agent has moved during one round: a diagonal step counts 2, and a horizontal or vertical one counts 1. An agent has to finish its round as soon as this sum is bigger than its speed [753, 768]. A third possibility – which works for arbitrary speeds – is to assign selection probabilities to each of the four lattice positions, which are adjacent to the exact final position [1489, 1490]. Naturally these probabilities are proportional to the square area between the exact final position and the lattice point, as in this case the probabilities are normalized by construction if one has a square lattice with points on all integer number combinations. However, one also could think of other methods to calculate the probability.
11.4.7. Collective Effects 11.4.7.1. Lane Formation As the most prominent example, we want to discuss lane formation out of a randomly distributed group of pedestrians. This corresponds to a spontaneous breaking of the symmetry of the particle number distribution in space. Simulations show that an even and
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Figure 11.18 Snapshot of a simulation of counterflow along a corridor. The left part shows the parameter control. The central window is the corridor, and the light and dark squares are right- and left-moving pedestrians, respectively. The right part shows the dynamic floor fields for the two species.
odd number of lanes may be formed. The latter corresponds to a spontaneous breaking of the left–right symmetry of the system. Figure 11.18 shows simulations of a rectangular corridor, which is populated by two species of pedestrians moving in opposite directions [170, 171]. Parallel to the direction of motion the existence of walls is assumed. Orthogonal to the direction of motion both periodic and open boundary conditions had been investigated. With periodic boundary conditions, the density of pedestrians is fixed for each run. The numbers of left- and right-movers are equal, and each species interacts with its own dynamic floor field. For open boundaries, we fix the rate at which pedestrians enter the system at the boundaries. The pedestrians leave the system as soon as they reach the opposite end of the corridor. Figure 11.18 shows the graphical front end running a simulation of a small periodic system. Lanes have already formed in the lower part of the corridor and can be spotted easily, both in the main window showing the cell contents and the small windows on the right showing the floor field intensity for the two species. The formation of the lanes can also clearly be seen in the velocity profile (Fig. 11.19), which has been measured at a cross section perpendicular to the direction of flow [170, 171]. In a certain density regime, the lanes are metastable. Spontaneous fluctuations can disrupt the flow in one lane causing the pedestrians to spread and interfere with other lanes. Eventually the system can run into a jam by this mechanism.
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1 0.5 0 −0.5 −1
5
0
10
15
20
25
Figure 11.19 Velocity profile of a periodic system with ρ = 0.12 exhibiting lane formation.
(a)
(b)
Figure 11.20 (a) Oscillations of the direction of flow: a group of particles of the same species break through a blockade at a door. (b) Typical flow pattern that emerges at a crossing. The shaded circle marks the area of highest disorder.
11.4.7.2. Oscillations at Bottlenecks Flows at bottlenecks have first been investigated with the floor field model in [170]. For narrow doors, the empirically observed behavior with randomly changing flow directions is found. For wider doors, oscillations on two timescales are observed for medium densities. The longer timescale characterizes oscillations between jammed configurations and bidirectional flows. The shorter timescale characterizes the behavior in the jammed state. Here, from time to time, small groups of pedestrians can break through the blockage in randomly changing directions (Fig. 11.20). In [810], the oscillations of the flow direction at a bottleneck have been investigated quantitatively for an extension of the floor field model (Section 11.4). There a situation was studied where two groups of oppositely moving pedestrians meet at a bottleneck (a door), which can be passed by only one pedestrian at a time, after starting from symmetric initial positions. In principle, several scenarios for the flow through the bottlenecks are
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possible. The simplest ones are (1) no oscillations occur, i.e., the flow direction changes only once after one group has passed through the bottleneck, (2) zipper principle where the flow direction changes after each pedestrian, and (3) uncorrelated oscillations, i.e., the statistics of the flow changes corresponds to a coin tossing experiment. It was found that neither of these three simple scenarios is realized. Instead nontrivial correlations can be observed, even in the case of vanishing dynamic floor field. If the coupling to the dynamical floor field is nonzero, the oscillations become time dependent as long as the system does not have reached the stationary state. 11.4.7.3. Patterns at Intersections The behavior of intersecting four-directional flows (without turning) has been investigated using the floor field model in [170]. The system is surrounded by walls that have doors in the middle through which the system is closed periodically. Several flow patterns arise from these boundary conditions. The most common one is shown schematically in Fig. 11.20b. In each of the two roads, lane formation is observed, but with different orientations: in one, the particles tend to walk on the right side, and in the other, they tend to walk on the left. The region of highest disorder lies in the center of the system where the two roads meet. Placing an obstacle in this area can lead to an improvement of the overall flow by reducing the mutual hindrance of pedestrians [169, 517].
11.4.8. Evacuation Simulations In the simulation of evacuation scenarios, often no matrix-of-preference is used. The information about location of exits, etc. is encoded in the static floor field. For each lattice site, S is calculated using some distance metric [755] so that the field values increase in the direction of the door (see Section 11.4.3). The value of kS , the coupling to the static field, can be viewed as a measure of the knowledge of the pedestrians about the location of the door. It determines the effective velocity in the direction of exit. Large kS implies a motion on the shortest possible path. For vanishing kS , on the other hand, the particles will perform a random walk and just find the exit by chance. So the case kS 1 is relevant for processes in dark or smokefilled rooms where people do not have full knowledge about the location of the exit. The parameter kD for the coupling to the dynamic field controls the tendency to follow the lead of others, sometimes called herding. Large values of kD imply a strong herding behavior which has been observed in the case of emergencies [520]. Three main regimes for the behavior of the particles can be distinguished [755] for the simple scenario of an evacuation from a structureless room with one exit. For strong coupling to kS and very small coupling to kD , one finds an ordered regime where particles only react to the static floor field and the behavior than is almost deterministic. The disordered regime characterized by strong coupling to kD and weak coupling to kS leads to a maximal value of the evacuation time. The behavior here corresponds to emergency
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scenarios. Between these two regimes, an optimal regime exists where the combination of interaction with the static and the dynamic floor fields minimizes the evacuation time. Since evacuations typically lead to large densities at exits and other bottlenecks, friction effects become relevant. Kirchner et al. [754] have studied their influence in simple scenarios. At low densities, increasing the friction μ usually has only a very weak effect on evacuation times. Here, conflict situations, even close to the door, are rare since almost no jamming occurs. However, for large coupling kS to the static field when all particles find the shortest way even for low densities, a jam will form at the exit. But since typically these jams are small, an increase of evacuation times is only observed for large values of μ. The behavior is different at high densities. Here, quickly a large jam at the exit forms. Increasing μ may lead to a sharp increase of the evacuation time which diverges for μ → 1, since for large μ the outflow is strongly suppressed. The pedestrians hinder one another due to strong competition for the unoccupied sites near the exit. Thus, the case of large kS together with a large value of μ describes a typical emergency situation, where an ordered outflow is inhibited due to local conflicts near bottlenecks or doors, resulting in strongly increased evacuation times. Figure 11.21 shows the influence of an increased coupling strength to the static field for fixed μ. For kS → 0, the particles perform a pure random walk, and the evacuation times are very large and almost independent of μ. In this situation, conflicts are not very important for the dynamics. In contrast, for kS → ∞, they choose the shortest way to 15000 μ = 0.0 μ = 0.6 μ = 0.9
T
10000
5000
0
0
2
4
6
8
10
kS
Figure 11.21 Dependence of evacuation times on the friction parameter μ and kS for density ρ = 0.3.
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the exit. One, therefore, expects that evacuation times decrease with increasing coupling strength to the static field since also the effective velocity in the direction of the door is increased. This is indeed true for vanishing friction μ = 0. However, for μ → 1, the number of unsolved conflicts increases due to strong jamming at the exit leading to clogging phenomena and highly increased evacuation times. For very high values, one finds a minimal evacuation time for an intermediate coupling kS ≈ 1. This means that a larger kS , which implies a larger average velocity of freely moving pedestrians, leads to larger evacuation times. This is the faster-is-slower effect described in Section 11.2.5. A similar local minimum is found in the cooperative regime as function of the coupling kD instead of kS [755]. Besides the evacuation time, also the time evolution of an evacuation, i.e., the number N (t) of persons leaving the room up to time t, yields interesting information. For large densities that lead to strong clustering at the door already at the beginning of the evacuation, N (t) shows a nearly linear increase. For μ = 0, the evacuation process is almost deterministic, and the fluctuations (due to the random initial conditions and the dynamics) are very small. With increasing μ, the number of conflicts increases, and the fluctuations become much stronger. The time evolution of one single sample exhibits an interesting dynamics, e.g., short periods of time where no person leaves the room leading to plateaus in the N (t) curves. This irregular behavior is well-known from granular flow and is typical for clogging situations [533, 925, 1475]. The plateaus are formed stochastically and cannot be observed after averaging over various samples. The variance of the evacuation time increases strongly with μ, and thus, the average evacuation time T is no longer a meaningful quantity for safety estimates. In Section 11.2.4, we have described the surprising results of Muir et al. [955] who have found a critical exit width which separates regimes where cooperative and competitive behavior lead to smaller evacuation times. Within the framework of the floor field model, competition can be described as an increased assertiveness (large kS ) and strong hindrance in conflict situations (large μ). Cooperation is represented by small kS and μ = 0 [752]. This allows reproducing the experimentally observed crossing of the two curves at a small door width [955]. Without friction (μ = 0), increasing kS alone always decreases the egress time T . The effect is therefore only obtained by increasing both kS and μ. Thus, there are two factors that determine the egress of persons and the overall evacuation time in this scenario: walking speed (controlled by the parameter kS ) and friction (controlled by μ). These parameters depend in a different way on the door width: the influence of the friction dominates for very narrow doors which leads existence of the critical width wc . This is well reproduced by simulations with the floor field model [752]. These results show that friction is indeed an essential part of the dynamics. Conflicts close to the exit are most important since they have a direct influence on the evacuation
Pedestrian Dynamics
time. Therefore, in case of competitive behavior and narrow doors, it is important to find other means in order to reduce the number of conflicts occurring at the exits. In [520], it has been proposed to place an additional column in front of the exit. Surprisingly, this can lead to a reduction of evacuation times [518, 520]. This is confirmed by simulations of the floor field model [754]. However, this effect is based on the assumption that the basic behavior (i.e., the model parameters) of the pedestrians is not changed by the presence of the column. It is questionable whether this is satisfied in real situations, e.g., since the column might be perceived as hindrance. However, recent experimental studies [517, 1492] under laboratory conditions indeed found an increase of the outflow in the presence of an obstacle due to the reduction of the number of conflicts. The flow increase is larger if the column not put at the center, but slightly shifted to one side. In [1301], the collective effects occurring in evacuation scenario have been investigated using a variant of the floor field model. It is shown that the results are very similar to those of the social-force model (see Sec. 11.5.2). Evacuation processes with obstacles have were studied in [1415]. In [224], the floor field model has been used to optimize facility designs for emergency evacuation.
11.5. OTHER MODELS 11.5.1. Fluid-Dynamic and Gas-Kinetic Models Pedestrian dynamics has some obvious similarities with fluids, e.g., the motion around obstacles appears to follow “streamlines.” Motion at intermediate densities is restricted (short-ranged correlations). Therefore, it is not surprising that, very much like for vehicular dynamics, the earliest models of pedestrian dynamics belonged to the populationbased approaches and took inspiration from hydrodynamics or gas-kinetic theory [507, 550, 617, 621–623]. Henderson [549, 550] has tried to establish an analogy of large crowds with a classical gas. From measurements of motion in different crowd fluids in a low density (“gaseous”) phase, he found a good agreement of the velocity distribution functions with a MaxwellBoltzmann distribution [549]. Motivated by this observation, he has later developed a fluid-dynamic theory of pedestrian flow [550]. Modeling the interaction between the pedestrians as a collision process where the particles exchange momenta and energy, a homogeneous crowd can be described by the well-known kinetic theory of gases. However, the interpretation of the quantities is not entirely clear, e.g., what the analogs of pressure and temperature are in the context of pedestrian motion. Temperature could be identified with the velocity variance, which is related to the distribution of desired velocities, whereas
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the pressure expresses the desire to move with a certain velocity. It is a consequence of the momentum changes in the random motion of the particles. Hughes [617, 621–623, 1393] has derived a fluid-dynamical model based on three assumptions: 1. The walking speed of a pedestrian is determined solely by the surrounding density and behavioral characteristics of other pedestrians. 2. The motion of pedestrians toward their destination is governed by a potential. Since there is no perceived advantage moving along a line of constant potential, the direction of motion is always perpendicular to the potential. 3. Pedestrians seek to minimize their estimated travel time. Deviations from that behavior are possible to avoid extremely high densities. These assumptions can be formulated in a mathematical way which, e.g., requires the introduction of a function measuring the discomfort of a crowd at a given density. The resulting equation for the time evolution of the density has certain analogies with the Lighthill–Whitham–Richards theory (Section 9.1.1). The applicability of classical hydrodynamical models is based on several conservation laws. The conservation of mass, corresponding to conservation of the total number of pedestrians, is expressed through a continuity equation of the form ∂ρ(r, t) + ∇ · J(r, t) = 0 , ∂t
(11.28)
which connects the local density ρ(r, t) with the current J(r, t). This equation can be generalized to include source and sink terms. However, the assumption of conservation of energy and momentum is not true for interactions between pedestrians, which in general do not even satisfy Newton’s Third Law (“actio = reactio”). In [507], several other differences to normal fluids were pointed out, e.g., the anisotropy of interactions or the fact that pedestrians usually have an individual preferred direction of motion. In [507], a better founded fluid-dynamical description was derived on the basis of a Boltzmann-like gas-kinetic model. The procedure is very similar to the derivation of gas-kinetic models for highway traffic (see Section 9.3). To take into account the twodimensional nature of pedestrian dynamics, different groups μ of particles are defined, (0) which correspond to different desired directions of motion (or velocities vμ ). Their densities ρμ change in time due to four different effects: (0)
1. Since the pedestrians try to reach their intended velocities vμ , the density approaches
(0)
(0) (0) (0) , t := δ vμ − vμ ,t . ρ˜μ r, vμ ρμ(0) r, vμ , vμ
(11.29)
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(0)
(0)
Here, ρ˜μ is the density of pedestrians with desired velocity vμ but arbitrary actual velocity vμ. The approach is then characterized by a relaxation time τ = mμ /γμ . 2. The interaction between pedestrians is modeled by a Stosszahlansatz as in the Boltzmann equation. Here, pair interactions between types μ and ν occur with a total rate that is proportional to the densities ρμ and ρν . 3. Pedestrians are allowed to change from type μ to ν, which, e.g., accounts for turning left or right at a crossing. 4. Additional gain and loss terms allow to model entrances and exits where pedestrian can enter or leave the system. The resulting fluid-dynamic equations derived from this gas-kinetic approach are similar to that of ordinary fluids (see Section 1.6). However, due to the different types of pedestrians, corresponding to individuals who have approximately the same desired velocity, one actually obtains a set of coupled equations describing several interacting fluids. These equations contain additional characteristic terms describing the preference of an intended velocity and the change of fluid type due to interactions in avoidance maneuvers. Since equilibrium is approached through the tendency to walk with the intended velocity, not through interactions as in ordinary fluids, the viscosity increases with density. Also temperature and pressure play a different role than in ordinary fluids. The analog of “temperature” is determined by the variance of the intended velocities. The pressure is not external but has its origin in the desire to move at a certain desired velocity. Other problems occur at low densities [507] which is the normal situation in crowds. In (fire-safety) engineering, simplified fluid-dynamic approaches have been developed since the middle of the 1950s [1366]. Some of the these could also be classified as queuing models since the central idea is to describe pedestrian dynamics as flow on a network with links of limited capacities. Knowledge of the flow J = ρvb and the technical data of the facility are then sufficient to evaluate evacuation times, etc. One essential ingredient is a systematic use of the equation of continuity, especially at bottlenecks. These methods allow to calculate evacuation times in a relatively simple way that does not require any simulations. Parameters entering in the calculations can be adapted to the situation that is studied. Often they are based on empirical results, e.g., evacuation trials. A more detailed description and further references can be found in [1191, 1241].
11.5.2. Social-Force Models The social-force model [535] is a continuum model for the description of pedestrian dynamics. As indicated by the name, the interactions between the pedestrians are implemented by using the concept of a social force or social field [864]. It is based on the idea that changes in behavior can be understood in terms of fields or forces. Applied to pedes(soc) represents the influence of the environment (other trian dynamics, the social force Fj
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pedestrians, infrastructure) and changes the velocity vj of pedestrian j. Thus, it is responsible for acceleration which justifies the interpretation as a force. The basic equation of motion for a pedestrian of mass mj is then of the general form mj
dvj (pers) (soc) (phys) + Fj + Fj . = Fj dt
(11.30)
(soc) (soc) (pers) = l = j Fjl is the total force due to the other pedestrians, and Fj denotes a Fj “personal” force which makes the pedestrians attempt to move with their own preferred (0) velocity vj . It thus acts as a driving term and is given by (pers)
Fj
=
mj (0) vj − vj , τj
(11.31)
where τj is a reaction or acceleration time. In high-density situations also, physical forces (phys) become important, e.g., friction and compression when pedestrians make contact. Fjl (soc)
comes from the territorial The most important contribution to the social force Fj effect, i.e., the personal space or private sphere. Pedestrians feel uncomfortable if they get too close to others, which effectively leads to a repulsive force between them. Similar effects are observed for the environment, e.g., people prefer not to walk too close to walls. Since social forces are difficult to determine empirically, some assumptions have to be made. Usually, an exponential form is assumed. Describing the pedestrians as disks of radius Rj and position (of the center of mass) rj , the typical structure of the force between the pedestrians is described by [520] Rjl − rjl (soc) Fjl = Aj exp (11.32) njl ξj with Rjl = Rj + Rl , the sum of the disk radii, rjl = |rj − rl |, the distance between the r −r centers of mass, njl = j r l , the normalized vector pointing form pedestrian l to j (see jl Fig. 11.22). Aj can be interpreted as strength, and ξj can be interpreted as the range of the interactions. Sometimes, an additional anisotropy factor [518] λj + (1 − λj )
1 + cosϕjl 2
(11.33)
is introduced in (11.32) that takes into account the anisotropy of the interactions between the pedestrians. Usually, the situation in front of a pedestrian is more relevant for his/her behavior than that behind. The angle ϕjl between velocity and the direction of the force
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tjl
vl Rl
vj
njl
ϕjl
Rj
Figure 11.22 Definition of the most important quantities in the social-force model. v
is defined by cosϕjl = −njl · ej with ej = |vjj | . λj as an anisotropy factor such that λj = 1 corresponds to the isotropic case. An important difference between physical and social forces is that the latter in general will not satisfy Newton’s Third Law (“actio=reactio”). This is natural since the feelings of two persons toward each other are usually not symmetric, e.g., if ξj = ξl or λj = λl . However, this does not appear to be essential [1385], since in most studies so far the forces have been assumed to satisfy the Third Law. Upon contact between two pedestrians, i.e., for rjl ≤ Rjl , the physical force F(phys) jl becomes relevant. It has two contributions, namely from pushing and friction: (push)
= k(Rjl − rjl )njl ,
(11.34)
Fjl(fric)
= κ(Rjl − rjl )tjl .
(11.35)
Fjl (push)
(fric)
Fjl represents a force that tries to prevent compression, and Fjl represents a sliding friction force in tangential direction tjl , i.e., orthogonal to njl . Sometimes, κ is chosen as a function of the relative tangential velocity of the pedestrians. The Heaviside step function takes into account that the force vanishes when the two pedestrians are not in contact. The interactions with the infrastructure, e.g., walls, have a similar structure and can also be divided into a social and a physical component. The values of the various model parameters as used in [520] are shown in Table 11.2. However, in other publications and for other situations, different parameter sets have been used. Calibrations of the model using empirical data have been reported, e.g., in [584, 586, 686, 807, 1094].
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Table 11.2 Typical parameter values for the social-force model. Parameter
Value
v(0) j k Aj ξj τj κj Rj
0.6 – 1.5 m/s 1.1–1.2 · 105 kg/s2 2 · 103 N 0.08 m 0.5 s 2.4 · 105 kg/m/s ∈ [0.25 m, 0.35 m]
The resulting equations of motion for a large crowd are very similar in structure to those describing, e.g., the behavior of granular matter. In order to solve the system, methods known from molecular-dynamics simulations are used (see, e.g., [833, 1284]). The equations have to be discretized in an appropriate way to guarantee accuracy, as well as allowing to study large systems on a sufficiently large timescale (of the order of minutes, which is in contrast to most molecular-dynamics simulations). In addition, a straightforward implementation of the equations of motion can lead to unrealistic movement of single pedestrians, e.g., negative velocities in the main moving direction. Therefore, not only different specifications of the forces have been used [520, 535, 1459], but also additional restrictions on the degrees of freedom have been used (see, e.g., [535]). Other modifications were proposed in [833, 1284, 1504]. The appeal of the social-force model comes to a large part from the analogy with Newtonian dynamics. But the additional restrictions of the degrees of freedom and interventions into the dynamics by corrective mechanisms indicate that the analogy between the dynamics of classical particles and the dynamics of pedestrians has its limitations. First, problems arise since the equations of motion describe particles with inertia. This could cause overlaps between pedestrians violating the volume exclusion or oscillations leading to velocities opposite to the direction of the intended velocity. The violation of the volume exclusion [833] leads to unrealistically high flow values [222]. Increasing the strength or range of the force to avoid violations of the exclusion principle will increase oscillations. Vice versa, a decrease of the force will reduce oscillations but increase potential overlaps between particles. Another problem arises if the forces act in the direction of motion. This additional “push” can lead to velocities larger than the desired velocity even in normal situations. Also there are problems arising due to the superposition of forces [1310]. This can already be seen from the simple example of queue formation since the gap between pedestrians waiting in line will depend on the length of the queue. Further modifications of the model dynamics have resulted from the recent attempts [584, 586, 686, 807, 1094] to calibrate the model using empirical data.
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Apart from the ad hoc introduction of interactions, the structure of the social-force model can also be derived from an extremal principle [577, 580]. It follows under the assumption that pedestrian behavior is determined by the desire to minimize a certain cost function which takes into account not only kinematic aspects and walking comfort but also deviations from a planned route. The social-force model has been used to study pedestrian dynamics in various situations. It can reproduce the main collective effects like lane formation and oscillations at bottlenecks, etc. [535]. Furthermore, it has been applied to evacuation simulations predicting phenomena like the faster-is-slower effect [518, 520] or freezing-by-heating [519]. Further studies of evacuation processes focussing on clogging behavior and the structures observed near exits have been reported in [1091–1093]. 11.5.2.1. Centrifugal Force Models A variant of the social-force model is the centrifugal force model (CFM) proposed in [1504]. The repulsive force exerted by pedestrian j on pedestrian i is assumed to depend on the relative velocity between them and decays as the inverse of their distance rij , Frep jl
= −mj Kjl
vjl2 rjl
njl ,
(11.36)
with vjl =
1 (vj − vl ) · njl + |(vj − vl ) · njl | 2
(11.37)
i.e., vjl is the projection of the relative velocity in the direction njl , and Kjl =
1 vj · njl + |vj · njl | . 2 vj
(11.38)
The coefficient Kjl reduces the action of the repulsive force to the angle of vision (180◦ ) of each pedestrian. Besides this repulsive force, which has the same form as the centrifugal force in mechanics, the driving force (11.31) is used. In addition, a collision detection technique [1504] is introduced to avoid unrealistic behavior in high-density situations. Due to the relative velocity term, the force (11.36) does not become large enough for small distances to avoid partial or even total overlapping of pedestrians. To solve these problems and improve numerical stability, Chraibi et al. [222, 223] have proposed a generalized CFM. It keeps the basic structure of the repulsive centrifugal force but takes into account additional effects so that a collision detection technique that takes over control at high densities is not required. The repulsive forces acting
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(0)
on pedestrian j are assumed to be proportional to its desired velocity vj so that faster moving pedestrians tend to feel stronger repulsive interactions. Furthermore, it incorporates the dynamic space requirement, i.e., faster pedestrians require more space due to the increasing step size [1284, 1453]. Therefore, the radius of the particles representing pedestrians has the form Rj (vj ) = Ra + γ vj with parameters Ra and γ that can in principle be determined empirically. The repulsive force is defined by Frep jl = −mj Kjl
(0)
νvj + vjl
2
rjl − Rjl (vj , vl )
njl ,
(11.39)
where the term in the denominator is the distance between the pedestrians, with Rjl (vj , vl ) = Rj (vj ) + Rl (vl ). The parameter ν controls the strength of the force and is in principle the only free adjustable parameter of the model. As mentioned earlier, the generalized CFM does not require the collision detection technique to avoid extreme overlaps of particles, which also improves the performance of the implementation and reduces numerical instabilities.
11.5.3. Lattice Gas Models In 1986, Frisch, Hasslacher, and Pomeau [405] have shown that one does not have to take into account the detailed molecular motion within fluids in order to obtain a realistic picture of (2d) fluid dynamics. Instead the fluid can be constructed from fictitious particles of identical mass that move with the same speed, differing only in their velocities. In fact, it is sufficient to restrict to only six different velocities. They proposed a lattice gas model [202, 1200, 1201] on a triangular lattice with hexagonal symmetry,6 which is similar in spirit to CA models, but the exclusion principle is relaxed. The dynamics is based on a succession of collision and propagation that can be chosen in such a way that the coarse-grained averages of this microscopic dynamics is asymptotically equivalent to the Navier–Stokes equations of incompressible fluids. In [912], a kind of mesoscopic approach inspired by these lattice gas models has been suggested as a model for pedestrian dynamics. In analogy with the description of transport phenomena in fluids (e.g., the Boltzmann equation), the dynamics is based on a succession of collision and propagation. N pedestrians are modeled as particles moving on a triangular lattice. The particles are characterized by their velocities or, more precisely, through the direction ci of the velocity, where ci (i = 1, . . ., z) is a vector pointing to one of the z neighbor sites of a lattice with coordination number z. For the hexagonal lattice, we thus have z = 6. The speed is given by |ci | = vmax in all cases. In addition, one defines the direction c0 = 0, 6 Meaning that each vertex has a hexagonal neighborhood.
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Collision
Propagation
Figure 11.23 The dynamics of lattice gas models proceeds in two steps. Pedestrians coming from neighboring sites interact in the collision step where velocities are redistributed. In the propagation step the pedestrians move to neighbor sites in the directions determined in the collision step.
which denotes a resting particle. Denoting the number of pedestrians entering site r at time t by ni (r, t), the total number of pedestrians is given by N = r zi=1 ni (r, t). The system is prepared in an initial state such that no particles with identical velocities occupy the same site. Thus, there can be at most seven pedestrians per site. The motivation for relaxing the exclusion principle is that the discretization of space on a lattice does not yield any specific scale. Furthermore, conflicts in the pedestrian motion are avoided. As in a lattice gas model [1201], the dynamics now consists of two steps (Fig. 11.23). In the propagation step, each pedestrian moves to the neighbor site in the direction of its velocity vector, which can be any of the ci (i = 0, . . ., z). In the collision step, the particles interact and new velocities (directions) are determined. In contrast to physical systems, momentum does not need to be conserved during the collision step. In [912], a specific collision step has been suggested that allows to reproduce several of the collective phenomena discussed in Section 11.2. First, it is assumed that each pedestrian has a preferred direction of motion denoted by cF , which is typically assumed to be constant. The collision rules are then defined to reflect some empirical observations. Pedestrians will try to move in their preferred direction cF , but at the same time, there is a tendency to follow the direction of flow. Each individual will thus consider only a reduced number ξ of optional directions ci around his or her favorite choice cF . Furthermore, at high densities, the crowd motion is influenced by a kind of friction that slows down the pedestrians. This is achieved by reducing the number of individuals allowed to move to neighboring sites. These considerations lead to a collision step that takes into account the favorite direction cF , the local density ρ = zi=0 ni (r, t) (the number of pedestrians at the collision site), and a quantity called mobility μ(r + ck , t) at all neighbor sites r + ck defined by 1 ni (r, t)ci . ρ i=0 z
μ=
(11.40)
The mobility can be interpreted as a normalized measure of the local flow after the collision.
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The parameters of the model are (1) critical density ρ0 that determines the minimum number of pedestrians per site which lead to a hindrance of free motion, (2) the disorder parameter ξ ∈ {0, . . ., z/2}, and (3) a parameter η ∈ [0, 1] describing the preference to move into the direction of higher mobility. The collision step is then given by the following: 1. choose the target cell r + ct among the neighboring cells r + cj that maximizes the quantity 2ηcj · μ(r + cj , t) + 2(1 − η)cj · cF ,
(11.41)
where only j = F, F ± 1, . . ., F ± ξ are considered. In order to avoid a systematic bias, F + i and F − i are considered in random order in case of an equal score. 2. move to the target cell r + ct with probability 1 if ρ ≤ ρ0 , (11.42) p= ρ0 /ρ if ρ > ρ0 otherwise stay at the current site, i.e., choose direction c0 . These rules are designed that in a densely occupied site on average ρ0 , pedestrians are allowed to move. This reflects the mesoscopic nature of the model, since this can be interpreted such that only people at the boundary of the cell area are able to move in high-density situations. The choice of the destination cell is determined by the agreement between the mobility at the target cell (first term in (11.41)) and the favorite direction (second term in (11.41)). The relative importance is determined by the parameter η. This reflects the fact that the actual motion is determined by finding a compromise between the desired direction and the direction of the local flow. The disorder parameter ξ represents the ability to focus on the favorite direction (ξ = 0). For ξ > 0, the individual can also consider neighboring directions. It has been suggested [912] that this parameter offers the opportunity to model crowd disasters: a stressed individual with a clear destination will not consider alternative directions (ξ = 0), whereas without clear destination, all directions will be explored (ξ = z/2) to find, e.g., an exit.
11.5.4. Optimal Velocity Model The optimal velocity model originally introduced for the description of highway traffic (see Section 9.4.2) can be generalized to higher dimensions [1033, 1034, 1322, 1323]. This allows to describe several types of transport processes, such as collective motion in biological systems, granular flow in a liquid, and pedestrian dynamics.
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In the natural extension of the OV model to the two-dimensional case, the equation of motion for particle i is given by ⎧ ⎫ ⎨ d2 ri dri ⎬ (11.43) (t) = a V0 + V(rij (t)) − (t) . ⎩ dt 2 dt ⎭ j
ri = (xi , yi ) is the position of particle ith and rij = rj − ri . The function V(rij ) expresses the interaction between two particles with the following form: V(rij ) = f (rij )(1 + cosϕ) nij , f (rij ) = α{tanhβ(rij − b) + c},
(11.44) (11.45)
where rij = |rj − ri |, cosϕ = (xj − xi )/rij and nij = (rj − ri )/rij . V0 is a constant vector that represents a “desired velocity” at which an isolated particle would move. The strength of the interaction depends on the distance rij between the ith and jth particles and on the angle ϕ between the directions of rj − ri and the current velocity dtd ri . Because of the term (1 + cosϕ), a particle reacts more sensitively to particles in front than to those behind. Now two cases can be distinguished, namely repulsive and attractive interactions. The former is relevant for pedestrian dynamics or particles moving in a liquid through a pipe, whereas the latter is more suitable for biological motion. Therefore, for pedestrian motion, one chooses c = −1, which implies f < 0, i.e., repulsive interactions. In the case c = 1, the force is attractive for all rij , whereas for 1 > c > −1, both repulsive and attractive interactions coexist. Thus, the parameter c controls the border of the regions with repulsive and attractive interactions. At low densities, the homogeneous flow rj = Rj + vt
(11.46)
is realized [1033], where Rj = (Xj , Yj ) is a constant vector representing a site on a triangular lattice and v is the constant velocity at which all particles move. The distance r between nearest-neighbor particles in the homogeneous solution is the same for all pairs. A detailed stability analysis [1033] leads to a classification according to the type of the unstable mode (longitudinal, transversal, mixed) and a phase diagram in the a − r-plane that consists of four different phases [1033]. The phase diagrams in terms of patterns of group formation is discussed in [1323]. A generalization to the case of additional attractive interactions can be found in [1034]. In counterflow situations lane formation is observed at low densities (large r) [1033]. These lanes are stable but become unstable at higher densities. Although still temporarily lanes are formed, the system ends in a blocking state. Since the pedestrians are represented
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by point particles, the blocking state is not completely frozen, but particles can escape through tiny spaces after some time. At very short distances (r < 1), even the temporary lanes disappear, and the blocking state emerges immediately.
11.5.5. Active Walker Models Active walker models [534, 541] are no genuine models of pedestrian dynamics since their focus is usually not on the (direct) interactions of pedestrians with each other. Instead they are used to describe the formation of human or animal trails, etc. [447]. Here, the walker leaves a trace by modifying the underground on his or her path. This modification is real in the sense that it could be measured in principle, similar to the process of chemotaxis described in Section 11.4.1. For trail formation, vegetation is destroyed by the walker. For the simulation of the formation of human trail systems,7 special models have been suggested. The basic mechanism behind the formation is that frequently used shortcuts become more attractive to others, e.g., since the vegetation is destroyed and the path becomes better visible. On the other hand, if an existing path is rarely used, vegetation can grow again so that finally the trail vanishes. To capture this kind of self-organization process, active walker models have been suggested [534, 541]. Active walkers are random walkers who can change their environment locally. In turn, these changes influence the behavior of the random walker, e.g., its transition probabilities. The model suggested in [541] is a continuum model where the markings (e.g., the destroyed vegetation) are described by a ground potential G(r, t). The natural (i.e., undisturbed) ground conditions are given by G0 (r, t). A trail will then correspond to large values of G(r, t). The markings typically have a certain lifetime T (r). Walkers create new markings at their current position rα (t) with strength Qα (rα , t). For the formation of human trails, the choice Qα (rα , t) = I (r)([1 − G(r, t)/Gmax(r)] is used, which reflects a saturation effect since the clarity of the trail is limited by a maximal value Gmax (r). The evolution of the ground potential is given by dG(r, t) 1 Qα (rα , t)δ(r − rα (t)) . = [G0(r, t) − G(r, t)] + dt T (r) α
(11.47)
The motion of the active walker α is then determined by the Langevin equations drα (t) = vα (t) , dt
(11.48)
(0)
dvα (t) vα eα (rα , vα , t) − vα(t) + 2α /τα ξ α (t) . = dt τα 7 Similar considerations also apply to the formation of ant trails.
(11.49)
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Here, vα (t) is the current velocity of walker α. τα is a relaxation time of velocity adaption and acts as a friction term. Fα (t) represents various deterministic influences, especially the preferred direction of motion or interactions with other walkers. Finally, ξ α (t) is a stochastic force of intensity α that usually is assumed to be Gaussian white noise. The first term models the adaption of current walking velocity vα to the desired (0) velocity vα eα . This includes the adaption of the walking direction and the walking speed. This process is specified by the orientation relation eα (rα , vα , t) = eα ({G(r, t)}rα , vα ). The walking direction is influenced by both the destination dα and the existing trails. The latter are described by a trail potential Vtr (rα , t) of the form (11.50) Vtr (rα , t) = d2 re|r−rα |/σ (rα ) G(r, t) , which reflects the attractiveness of walking near r. σ (rα ) can be interpreted as the range of visibility. Now, a reasonable choice for the orientation relation is eα (rα , t) =
dα − rα + ∇rα Vtr (rα , t) . |dα − rα + ∇rα Vtr (rα , t)|
(11.51)
On homogeneous ground (Vtr (rα , t) constant), the walking direction eα (rα , t) ∝ dα − rα is only determined by the destination dα . In contrast, without a destination, the pedestrian will move in the direction of the largest increase of the trail potential Vtr (rα , t). The form (11.51) of the orientation relation takes the arithmetic average of these two limiting cases. The full dynamic equations can be simplified since usually the relaxation time τα is much smaller than the timescale of trail formation given by Tk . Then, the time derivative in (11.49) can be neglected, and one obtains the equation of motion √ drα (t) = vα (t) ≈ vα(0)eα (rα , vα , t) − vα(t) + 2α τα ξ α (t) . dt
(11.52)
Through rescaling of the equations [541], two local parameters can be identified that determine the dynamics of the trail formation: κ(r) =
I (σ r)T (σ r) , σ
λ(r) =
V0 T (σ r) , σ
(0)
(11.53)
where V0 is the mean value of the desired velocities vα . In [534, 541], multiagent simulations of this model have been performed with different origins and destinations for the pedestrians who start at randomly chosen times.
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In the case of very small attractiveness of the trails, a direct way system is formed where all pedestrians take the shortest path from their origin to the destination. For the case of a large attractiveness, this is true only in the initial stage of the simulation. At later times, and for realistic parameter values, a so-called minimal detour system develops. This is a compromise between a direct way system and a minimal way system, i.e., the shortest way system that would connect all origins and destinations. These findings are in good agreement with the empirically observed behavior.
CHAPTER TWELVE
Traffic Phenomena In Biology Contents 12.1. Introduction 12.1.1. Different Types of Traffic in Biology
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12.2. TASEP for Hard Rods: Minimal Model of Transcription and Translation 12.2.1. TASEP for Hard Rods: Minimal Models of Traffic of Ribosomes and RNAPs 12.2.2. TASEP for Hard Rods with Internal States: Effects of Individual Mechano-Chemistry
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12.3. TASEP for Particles with Langmuir Kinetics: Minimal Model of Kinesin Traffic 12.3.1. TASEP-Like Generic Models of Molecular Motor Traffic 12.3.2. Traffic of Interacting Particles with “Internal States”and Langmuir Kinetics: Effects of Individual Mechano-Chemistry of KIF1A
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12.4. Traffic in Social Insect Colonies: Ant-Trails 12.4.1. Model of Single-Lane Unidirectional Ant-Traffic 12.4.2. Model of Single-Lane Bidirectional Ant-Traffic 12.4.3. Model of Two-Lane Bidirectional Ant-Traffic 12.4.4. Experimental Investigations of Ant-Traffic 12.4.5. Empirical Results for Fundamental Diagrams of Ant-Trails
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12.1. INTRODUCTION Motility is the hallmark of life. From intracellular molecular transport and crawling of amoebae to the swimming of fish and flight of birds, movement is one of life’s central attributes. All these “motile” elements generate the forces required for their movements by actively converting some other forms of energy into mechanical energy. However, in this chapter, we are interested in a special type of collective movement of these motile elements. What distinguishes a traffic-like movement from all other forms of movements is that traffic flow takes place on “tracks” or “trails” (like those for trains and street cars or like roads and highways for motor vehicles) for the movement of the motile elements. We are mainly interested in the general principles and common trends seen in the mathematical modeling of collective traffic-like movements at different levels of biological organization. We begin with an overview of the models of traffic of macromolecular motors involved in gene expression. Then, we review the intracellular traffic of motor proteins that transport molecular cargo along filamentous tracks. Finally, we look at the
Stochastic Transport in Complex Systems DOI: 10.1016/B978-0-444-52853-7.00012-9
Copyright © 2011, Elsevier BV. All rights reserved.
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collective movement of social insects, like ants and termites, on trails from the perspective of traffic science.
12.1.1. Different Types of Traffic in Biology Now we shall give a few examples of the traffic-like collective phenomena in biology to emphasize some dynamical features of the tracks, which makes biological traffic phenomena more exotic as compared with vehicular traffic. In any modern society, the most common traffic phenomenon is that of vehicular traffic. The changes in the roads and highway networks take place over periods of years (depending on the availability of funds), whereas a vehicle takes a maximum of a few hours for a single journey. Therefore, for all practical purposes, the roads can be taken to be independent of time while studying the flow of vehicular traffic. In sharp contrast, the tracks and trails, which are the biological analogs of roads, can have nontrivial dependence on time during the typical travel time of the motile elements. We give a few examples of such traffic (see Figs. 12.1 and 12.2). • Time-dependent track whose length and shape can be affected by the motile element: Microtubules, a class of filamentous proteins, serve as tracks for two superfamilies of motor proteins called kinesins and dyneins [591, 1249]. Interestingly, microtubules are known to exhibit an unusual polymerization–depolymerization dynamics even in the absence of motor proteins. Moreover, in some circumstances, the motor proteins interact with the microtubule tracks so as to influence their length as well as shape; one such situation arises during cell division (the process is called mitosis). • Time-dependent track/trail created and maintained by the motile element: A DNA helicase [1155] unwinds a double-stranded DNA and uses one of the two strands thus opened as the track for its own translocation. Ants are known to create the trails by dropping a chemical, which is generically called pheromone [574]. Because the pheromone gradually evaporates, the ants keep reinforcing the trail in order to maintain the trail networks. • Time-dependent track destroyed by the motile element: A class of enzymes, called MMP-1, degrades their tracks formed by collagen fibrils [970, 1465]. In this chapter, we present an overview of the common trends in the mathematical modeling of these traffic-like phenomena. Although the choice of the physical examples and modeling strategies are biased by our own works and experiences, we put these in a broader perspective by relating these with works of other research groups.
12.2. TASEP FOR HARD RODS: MINIMAL MODEL OF TRANSCRIPTION AND TRANSLATION Nucleic acids (DNA and RNA) are, effectively, linear polymers. The monomeric subunits of nucleic acids are nucleotides and, therefore, nucleic acid strands are also
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Synapse Synapse
Kinesin Cell body
Axon
Dynein
Dendrite
Vesicles, and so forth
Kinesin
Dynein
Microtubule
Figure 12.1 Schematic view of a nerve cell. It is divided into three parts: cell body, dendrites, and axon. The axon and dendrites are filaments that extrude from the cell body. Synaptic signals from other neurons are received by the cell body and dendrites, while signals to other neurons are transmitted by the axon. In the axon, different motor proteins move along microtubules: kinesin moves toward the end of the microtubule, and dynein generally moves toward the cell body.
referred to as polynucleotides. In contrast, a protein is a linear polymer of amino acids that are linked together by peptide bonds and, therefore, often referred to as a polypeptide. The process of polymerization of a messenger RNA (mRNA) from the corresponding DNA template is called transcription and is carried out by a molecular machine called RNA polymerase (RNAP). Triplets of nucleotides on an mRNA form one single codon. The sequence of amino acids to be selected for polymerizing a protein is decided by the corresponding sequence of codons on the mRNA template. Synthesis of a protein from the corresponding mRNA template is carried out by a ribosome, which is one of the largest and most sophisticated macromolecular machines within the cell and the process is referred to as translation (of genetic code). Transcription and translation form the most important steps in gene expression whereby proteins are synthesized according to the genetic blueprint. In this section, we study the spatio-temporal organization of RNAP and ribosomes on the respective tracks during transcription and translation of genetic code, respectively.
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12.2.1. TASEP for Hard Rods: Minimal Models of Traffic of Ribosomes and RNAPs The stretch of mRNA between a start codon and a stop codon serves as a template for the synthesis of a polypeptide. The process of translation itself can be divided into three main stages: (1) initiation, during which the ribosomal subunits assemble on the start codon on the mRNA strand, (2) elongation, during which the nascent polypeptide gets elongated by the formation of peptide bonds with new amino acids, and (3) termination, during which the process of translation gets terminated at the stop codon and the polypeptide is released by the corresponding ribosome. Often many ribosomes move simultaneously on a single mRNA strand while each synthesizes a separate copy of the same protein (Fig. 12.2). Such a collective movement of the ribosomes on a single mRNA strand has superficial similarities with vehicular traffic and is, therefore, referred to as ribosome traffic. In all the models of ribosome traffic [205, 316, 317, 831, 892, 893, 1286, 1288], the sequence of codons on a given mRNA is represented by the corresponding sequence of the equispaced sites of a regular one-dimensional lattice. A typical ribosome is much larger than the size of a single codon. Therefore, most of earlier models treat an individual ribosome as a “self-driven” hard rod by ignoring the details of molecular composition and architecture of the ribosome. Thus, in those models, the length of the ribosome is an integer (more precisely, units of the lattice constant). On the lattice, the steric interaction of the ribosomes is taken into account naturally by imposing the condition of mutual exclusion, i.e., no site of the lattice can be simultaneously covered by more than one rod.
Polypeptide
mRNA
Ribosome
Codons
Figure 12.2 The process of biopolymerization: Ribosomes attach to mRNA and read the construction plan for a polypeptide, which is stored in the genetic code formed by the sequence of codons.
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The dynamics of the system was formulated in terms of the following update rules: A hard rod, whose forward edge is located at the site i, can hop forward by one lattice spacing with the forward hopping rate qi , provided the target site is not already covered by another hard rod. Moreover, initiation and termination were assumed to take place with the corresponding rates α and β, respectively, which are not necessarily equal to any of the other rate constants qi . For the sake of simplicity of analytical calculations, one usually replaces this intrinsically inhomogeneous process by a hypothetical homogeneous one by assuming [892, 893] that qi = q, irrespective of i. In such special situations, this model reduced to the totally asymmetric simple exclusion process (TASEP) for hard rods without any defect or disorder [1273]. The rate of protein synthesis and the ribosome density profile in the model developed by MacDonald et al. [892, 893], as well as in some other closely related models have been investigated in detail. If one imposes periodic boundary conditions on this simplified version of the model, the steady-state flux of the ribosomes is given by [83] ρ(1 − ρ ) (12.1) J =q 1 − ρ( − 1) where ρ is the number density of the ribosomes; if N is the total number of ribosomes on the lattice of length L, then ρ = N /L. In the special case = 1, the expression (12.1) reduces to the well-known formula (see Section 4.1.3.1) J = qρ(1 − ρ)
(12.2)
for the steady-state flux in the TASEP (see Section 4.1.3.1). Comparing equation (12.1) with (12.2), ρ/[1 − ρ( − 1)] has often been identified as an effective particle density while the corresponding effective hole density is given by 1 − ρ. The corresponding phenomenological hydrodynamic theory [1288] has also been derived [1255] from the TASEP-like dynamics of the hard rods of size on the discrete lattice. The fundamental diagram implied by the expression (12.1) exhibits a maximum at the density ρm and the value of flux at this maximum is Jm , where 1 ρm = √ √ ( + 1)
q and Jm = √ . ( + 1)2
(12.3)
Only in the special case = 1, this fundamental diagram is symmetric about ρ = 1/2; the maximum shifts to higher density with increasing . Klumpp and Hwa [764] developed a model of RNAP traffic to explore the effects of pausing, termination, and antitermination. But, just like the TASEP-type models of translation, this model of transcription also did not explicitly capture the mechanochemical cycle of the machine.
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12.2.2. TASEP for Hard Rods with Internal States: Effects of Individual Mechano-Chemistry Strictly speaking, neither ribosome nor RNAP is a hard rod; their mechanical movement along the respective tracks are coupled to their internal mechano-chemical processes that drive the synthesis of the macromolecule (protein and mRNA, respectively). Chowdhury and coworkers [82, 83, 422, 1394, 1396] have extended the TASEP type models of ribosome traffic and RNAP traffic by assigning several “internal” states, which capture the different chemical and conformational states of an individual ribosome and RNAP, respectively, during their biochemical cycle. Let Pμ(i) be the probability of finding the left edge of the motor at site i, in the internal state μ. Basu and Chowdhury [83] wrote down the master equations for Pμ(i) for ribosome traffic under mean-field approximation imposing periodic boundary conditions. The corresponding equations for RNAP traffic were reported by Tripathi and Chowdhury [1394]. Solving these equations in the steady state and using the definition of flux, one obtains the fundamental diagram. However, the corresponding master equations for open boundary conditions could be solved only numerically even under the steady-state conditions. Analogs of time-headway and distance-headway have been defined also for molecular motor traffic. The distributions of these quantities have been calculated for cytoskeletal motor traffic [207], as well as for traffic of RNAPs and ribosomes [421, 1394, 1395]. Going beyond these usual measures of characterization of traffic, the distributions of the dwell times of ribosomes and RNAPs at the successive positions have also been calculated [422, 1396].
12.3. TASEP FOR PARTICLES WITH LANGMUIR KINETICS: MINIMAL MODEL OF KINESIN TRAFFIC Intracellular transport is carried by motor proteins that can directly convert the input chemical energy into mechanical energy required for their movement along filaments, constituting what is known as the cytoskeleton [591, 1249] (Fig. 12.1). Three superfamilies of these motors are kinesin, dynein, and myosin. Most of the kinesins, dyneins, and some families of the so-called unconventional myosins are like porters in the sense that these move over long distances along the respective filamentous tracks without getting completely detached. In this chapter, we shall focus mostly on the effects of mutual steric interactions of these motors on their collective spatio-temporal organization. Often a single filamentous track is used simultaneously by many motors and, in such circumstances, the intermotor interactions cannot be ignored. Fundamental understanding of these collective physical phenomena may also expose the causes of many diseases (e.g., Alzheimer’s disease), which arise from malfunctioning of the molecular motor transport system.
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Derenyi and collaborators [282, 283] developed a one-dimensional model of interacting motors in the context of Brownian ratchet. They modeled each motor as a rigid rod subjected to a time-dependent potential and formulated the dynamics of the system in terms of a Langevin equation in the overdamped limit; the steric interactions of the rods were incorporated through the mutual exclusion. Another model of interacting Brownian motors was also proposed [4] in the same spirit. Usually, in this type of models, the time-dependence of the potential is captured by its switching between a flat form and a sawtooth form. Note that in these models, the particles representing the motors are “field-driven,” rather than “self-driven,” where the hopping probabilities are obtained from the slope of the instantaneous local potential.
12.3.1. TASEP-Like Generic Models of Molecular Motor Traffic The fundamental diagram of the model [4], computed imposing periodic boundary conditions, is very similar to those of TASEP. This observation indicates that further simplification of the model proposed in ref. [4] is possible to develop a minimal model for interacting molecular motors. However, attachment of motors to any unoccupied site on the track and also their possible detachments before reaching the last site on the track should be included in the model for a more realistic description. Indeed, such models have been developed over the last few years [366, 404, 696, 765– 767, 878, 879, 1052, 1053, 1095, 1096, 1119]. In the TASEP with Langmuir kinetics ( TASEP-LK) [1095, 1096], which was discussed already in Section 4.6.3, the molecular motors are represented by particles, whereas the sites for the binding of the motors with the cytoskeletal tracks (e.g., microtubules) are represented by a one-dimensional discrete lattice. Just as in TASEP, the motors are allowed to hop forward, with probability q, provided the site in front is empty. However, unlike TASEP, the particles can also get “attached” to an empty lattice site, with probability A, and “detached” from an occupied site, with probability D (see Fig. 12.3) from any site except the end points. The state of the system was updated in a random-sequential manner. Parmeggiani et al. demonstrated a novel phase of this ωd
ωa
q α
β
Figure 12.3 A schematic description of the TASEP-LK. Just as in TASEP, the motors are allowed to hop forward, with probability q. In addition, the motors can also get “attached” to an empty lattice site, with probability A, and “detached” from an occupied site, with probability D from any site except the end points; the rate of attachment at the entry point on the left is α, while that at the exit point on the right is β.
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model where low- and high-density regimes, separated from each other by domain walls, coexist. They interpreted this spatial organization as traffic jam of molecular motors. A cylindrical geometry of the model system was considered by Lipowsky, Klumpp, and collaborators [765–767, 878, 879, 1052, 1053] to mimic the microtubule tracks in typical tubular neurons. The microtubule filament was assumed to form the axis of the cylinder, whereas the free space surrounding the axis was assumed to consist of Nch channels each of which was discretized in the spirit of lattice gas models. They studied concentration profiles and the current of free motors, as well as those bound to the filament by imposing a few different types of boundary conditions. This model enables one to incorporate the effects of exchange of populations between two groups, namely, motors bound to the axial filament and motors that move diffusively in the cylinder. They have also compared the results of these investigations with the corresponding results obtained in a different geometry, where the filaments spread out radially from a central point.
12.3.2. Traffic of Interacting Particles with “Internal States” and Langmuir Kinetics: Effects of Individual Mechano-Chemistry of KIF1A The models of intracellular traffic described so far are essentially extensions of the TASEP that includes Langmuir-like kinetics of adsorption and desorption of the motors. In reality, a motor protein is an enzyme, whose mechanical movement is loosely coupled with its biochemical cycle. Nishinari et al. [1063] have modeled the traffic of a particular family of kinesins, called KIF1A, which are single-headed kinesin motors [1070, 1077– 1079]. The movement of a single KIF1A motor was modeled earlier with a Brownian ratchet mechanism [700, 1177]. In contrast to the purely TASEP-type models of molecular motor traffic, which take into account only the mutual interactions of the motors, our model explicitly incorporates also the Brownian ratchet mechanism of individual KIF1A motors, including its biochemical cycle that involves adenosine triphosphate (ATP) hydrolysis. The TASEP-like models successfully explain the occurrence of shocks, i.e., domain walls separating the high-density and low-density regions. But because most of the biochemistry is captured in these models through a single effective hopping rate, it is difficult to make direct quantitative comparison with experimental data, which depend on such chemical processes. In contrast, the model proposed by Nishinari et al. [1063] incorporates the essential steps in the biochemical processes of KIF1A, as well as their mutual interactions and involves parameters that have one-to-one correspondence with experimentally controllable quantities. The biochemical processes of kinesin-type molecular motors can be described by the four-states model shown in Fig. 12.4: bare kinesin (K), kinesin bound with ATP (KT), kinesin bound with the products of hydrolysis, i.e., adenosine diphosphate (ADP) and phosphate (KDP), and, finally, kinesin bound with ADP (KD) after releasing phosphate.
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K
KT
KDP
KD
P
ATP state 1
K ADP
state 2
Figure 12.4 The biochemical and mechanical states of a single KIF1A motor. In the chemical states on the left of the dotted line, KIF1A binds to a fixed position on the MT (state 1), while in those on the right KIF1A diffuses along the MT track (state 2). At the transition from state 1 to 2, KIF1A detaches from the MT. Brownian, ratchet
α −
0
1
1
0
0
2
0
δ 1
2
1
2
γ 1,2
1
+
β 1,2 Attachment
Detachment
Figure 12.5 A three-state model for molecular motors moving along a MT. 0 denotes an empty site, 1 is K or KT, and 2 is KD. Transition from 1 to 2, which corresponds to hydrolysis, occurs within a cell, whereas movement to the forward or backward cell occurs only when motor is in state 2. At the minus and plus ends, the probabilities are different from those in the bulk.
Experiments [1070] revealed that both K and KT bind to the MT in a stereotypic manner (historically called “strongly bound state,” and here we refer to this mechanical state as “state 1”). KDP has a very short lifetime, and the release of phosphate transiently detaches kinesin from MT [1070]. Then, KD rebinds to the MT and executes Brownian motion along the track (historically called “weakly bound state,” and here referred to as “state 2”). Finally, KD releases ADP when it steps forward to the next binding site on the MT utilizing a Brownian ratchet mechanism, and thereby returns to the state K. Thus, in contrast to the earlier TASEP-like models, each of the self-driven particles, which represent the individual motors KIF1A, can be in two possible internal states labeled by the indices 1 and 2. In other words, each of the lattice sites can be in one of three possible allowed states (Fig. 12.5): empty (denoted by 0), occupied by a kinesin in state 1, or occupied by a kinesin in state 2. For the dynamical evolution of the system, one of the L sites is picked up randomly and updated according to the rules given below together with the corresponding probabilities (Fig. 12.5): Attachment : 0 → 1 with ωa dt
(12.4)
Detachment : 1 → 0 with ωd dt
(12.5)
Hydrolysis : 1 → 2 with ωh dt 2 → 1 with ωs dt Ratchet : 20 → 01 with ωf dt 20 → 02 with ωb dt Brownian motion : 02 → 20 with ωb dt
(12.6) (12.7) (12.8)
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The probabilities of detachment and attachment at the two ends of the MT may be different from those at any bulk site. We chose α and δ, instead of ωa , as the probabilities of attachment at the left and right ends, respectively. Similarly, we took γ1 and β1 , instead of ωd , as probabilities of detachments at the two ends (Fig. 12.5). Finally, γ2 and β2 , instead of ωb , are the probabilities of exit of the motors through the two ends by random Brownian movements. It is possible to relate the rate constants ωf , ωs , and ωb with the corresponding physical processes in the Brownian ratchet mechanism of a single KIF1A motor (see ref. [463] for the details). Good estimates for the parameters of the model could be extracted by analyzing the empirical data [463]. For example, ωd 0.0001 m/s is independent of the kinesin concentration. However, ωa , which depends on the kinesin concentration, could be in the range 0.0001 m/s ≤ ωa ≤ 0.01 m/s. Similarly, ωb 0.6 m/s, ωs 0.145 m/s, ωf 0.055 m/s, and 0 ≤ ωh ≤ 0.25 m/s. Let us denote the probabilities of finding a KIF1A molecule in the states 1 and 2 at the lattice site i at time t by the symbols ri and hi , respectively. In mean-field approximation, the master equations for the dynamics of motors in the bulk of the system are given by dri = ωa (1 − ri − hi ) − ωh ri − ωd ri + ωs hi + ωf hi−1 (1 − ri − hi ), dt dhi = −ωs hi + ωh ri − ωf hi (1 − ri+1 − hi+1 ) dt − ωb hi (2 − ri+1 − hi+1 − ri−1 − hi−1 ) + ωb (hi−1 + hi+1 )(1 − ri − hi ).
(12.9)
(12.10)
The corresponding equations for the boundaries are also similar. Assuming that each time step of updating corresponds to 1 ms of real time, we performed simulations up to 1 min. In the limit of low density of the motors we have computed, for example, the mean speed of the kinesins, the diffusion constant, and mean duration of the movement of a kinesin on a microtubule from simulations of our model (see Table 12.1); these are in excellent quantitative agreement with the corresponding empirical data from single molecule experiments. Table 12.1 Predicted transport properties from our model in [1063] in the low-density limit for four different ATP concentrations. τ is calculated by averaging the intervals between attachment and detachment of each KIF1A ATP(mM)
∞ 0.9 0.3375 0.15
ωh (1/ms)
v (nm/ms)
D/v (nm)
τ (s)
0.25 0.20 0.15 0.10
0.201 0.176 0.153 0.124
184.8 179.1 188.2 178.7
7.22 6.94 6.98 6.62
Traffic Phenomena In Biology
Figure 12.6 Formation of comet-like accumulation of kinesin at the end of MT. Fluorescently labeled KIF1A (lower dark spots) was introduced to MT (upper light string) at 10 pM (top), 100 pM (middle), and 1000 pM (bottom) concentrations along with 2 mM ATP. The length of the white bar is 2 μm.
The collective pattern formed by the red fluorescent-labeled kinesins in Fig. 12.6, is a typical example of the effects of mutual interactions of the motors on the same track. Using the model described earlier, we have also calculated the flux of the motors in the mean-field approximation imposing periodic boundary conditions. Although the system with periodic boundary conditions is fictitious, the results provide good estimates of the density and flux in the corresponding system with open boundary conditions. In contrast to the phase diagrams in the α − β-plane reported by earlier investigators [1095], we have drawn the phase diagram of our model in the ωa − ωh plane (see Fig. 12.7) by carrying out extensive computer simulations for realistic parameter values of the model with open boundary conditions. The phase diagram shows the strong influence of hydrolysis on the spatial distribution of the motors along the MT. For very low ωh , no kinesins can exist in state 2; the kinesins, all of which are in state 1, are distributed rather homogeneously over the entire system. In this case, the only dynamics present is due to the Langmuir kinetics. At a small, but finite, rate ωh both the density profiles ρj1 and ρj2 of kinesins in the states 1 and 2 exhibit a localized shock. Interestingly, the shocks in these two density profiles always appear at the same position. Moreover, the position of the immobile shock depends on the concentration of the motors, as well as that of ATP; the shock moves toward the minus end of the MT with the increase of the concentration of kinesin or ATP or both (Fig. 12.7). Because of limitations of space, we have excluded several biological traffic phenomena. These include, for example, bidirectional transport along microtubules where the same cargo moves along the same microtubule track using sets of opposing motors [467, 1456].
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ω h (ms–1) (ATP[mM])
x t
0.2 (0.9)
0.1 (0.15)
0.01 (0.0094) ω a (ms–1) (KIF1A[nM])
Blue: state_1 Red: state_2 0.00001 (1)
0.00005 (5)
0.001 (100)
Figure 12.7 Phase diagram of the model of [1063] in the ωh − ωa plane, with the corresponding values for ATP and KIF1A concentrations given in brackets. These quantities are controllable in experiment. The boundary rates are α = ωa , β1,2 = ωd , γ1,2 = δ = 0. The position of the immobile shock depends on both ATP and KIF1A concentrations.
12.4. TRAFFIC IN SOCIAL INSECT COLONIES: ANT-TRAILS Termites, ants, bees, and wasps are the most common social insects, although the extent of social behavior, as compared with solitary life, varies from one subspecies to another [574]. The ability of the social insect colonies to function without a leader has attracted the attention of experts from different disciplines [28, 138, 141, 336, 424, 618, 1358, 1359]. Insights gained from the modeling of the colonies of such insects are finding important applications in computer science (useful optimization and control algorithms) [891], communication engineering [139], artificial “swarm intelligence” [332], and micro-robotics [811], as well as in task partitioning, decentralized manufacturing [23–27, 1171, 1171], and management [140]. In this section, we consider only ants. When observed from a sufficiently long distance, the collective movement of ants on trails resemble the vehicular traffic observed from a low flying aircraft. Therefore, instead of studying the emergence of an ant-trail, we study here the collective movements of ants on preexisting trails from the perspective of traffic science.
Traffic Phenomena In Biology
Ants communicate with each other by dropping a chemical (generically called pheromone) on the substrate as they move forward. Although we cannot smell it, the trail pheromone sticks to the substrate long enough for the other following sniffing ants to pick up its smell and follow the trail. Couzin and Franks [242] developed an individual-based model that not only addressed the question of self-organized lane formation but also elucidated the variation of the flux of the ants with two important parameters of the model. The “internal angle” α may be interpreted as angle of local visual field or that of olfactory perception, or tactile range of the antennae of the individual ants. Moreover, each individual ant is assumed to turn away from others within these zones by, at most, an angle θa t in time t. Imposing periodic boundary conditions, Couzin and Franks [242] computed the flux of ants in their model by computer simulations. The flux was found to be a nonmonotonic function of both α and θa . At low α, ants cannot detect others ahead, whereas at high α they spend most of their time avoiding others even through collisions with others may be unlikely; both these reduce the flux considerably. Similarly, at low θa , ants cannot turn sufficiently rapidly to avoid collision, whereas at high θ , they change their direction quickly so that not many move in the same direction at any time. Thus, only in the intermediate range of values of α and θa , the ants are optimally sensitive. Therefore, the flux exhibits a maximum both as a function of α and as a function of θa . In the recent years, we have developed discrete models that are not intended to address the question of the emergence of the ant-trail [541, 1278], but focus on the traffic of ants on a trail, which has already been formed. A continuous model has been proposed in [691].
12.4.1. Model of Single-Lane Unidirectional Ant-Traffic In our model of unidirectional ant-traffic, the ants move according to a rule, which is essentially an extension of the TASEP. In addition, a second field is introduced that models the presence or absence of pheromones (see Fig. 12.8). The hopping probability of the ants is now modified by the presence of pheromones. It is larger if a pheromone is present at the destination site. Furthermore, the dynamics of the pheromones has to be specified. They are created by ants and free pheromones evaporate with probability f per unit time. Assuming periodic boundary conditions, the state of the system is updated at each time step in two stages (see Fig. 12.8). In stage I, ants are allowed to move, whereas in stage II, the pheromones are allowed to evaporate. In each stage, the stochastic dynamical rules are applied in parallel to all ants and pheromones, respectively. Stage I: Motion of ants An ant on a site cannot move if the site immediately in front of it is also occupied by another ant. However, when this site is not occupied by any other ant, the probability
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State at time t q
Q
q
Stage I
f
f
f
Stage II
State at time t + 1
Figure 12.8 Schematic representation of typical configurations of the unidirectional ant-trail model. The symbols • indicate the presence of pheromone. This figure also illustrates the update procedure. Top: Configuration at time t, i.e., before stage I of the update. The nonvanishing probabilities of forward movement of the ants are also shown explicitly. Middle: Configuration after one possible realisation of stage I. Two ants have moved compared with the top part of the figure. The open circle with dashed boundary indicates the location where pheromone will be dropped by the corresponding ant at stage II of the update scheme. Also indicated are the existing pheromones that may evaporate in stage II of the updating, together with the average rate of evaporation. Bottom: Configuration after one possible realization of stage II. Two drops of pheromones have evaporated and pheromones have been dropped/reinforced at the current locations of the ants.
of its forward movement to the ant-free site is Q or q, depending on whether or not the target site contains pheromone. Thus, q (or Q) would be the average speed of a free ant in the absence (or presence) of pheromone. To be consistent with real ant-trails, we assume q < Q, as presence of pheromone increases the average speed.
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2000
91 000 90 800
1500
90 600 1000 90 400 500
0
90 200 100 200 300 400 500 600 700 800 900 1000 (a)
90 000 350 400 450 500 550 600 650 700 750 800 850 (b)
Figure 12.9 Snapshots of the spatial configurations demonstrating coarsening of the clusters of ants in the early stages (a) and in the late stages (b) of time evolution starting from random initial condition. The position of each ant is denoted by a black dot.
Stage II: Evaporation of pheromones Trail pheromone is volatile. So, pheromone secreted by an ant will gradually decay unless reinforced by the following ants. In order to capture this process, we assume that each site occupied by an ant at the end of stage I also contains pheromone. However, pheromone in any “ant-free” site is allowed to evaporate; this evaporation is also assumed to be a random process that takes place at an average rate of f per unit time. The total amount of pheromone on the trail can fluctuate although the total number N of the ants is constant because of the periodic boundary conditions. In the two special cases f = 0 and f = 1, the stationary state of the model becomes identical to that of the TASEP with hopping probability Q and q, respectively. One interesting phenomenon observed in the simulations is coarsening. At intermediate time usually several noncompact clusters are formed (Fig. 12.9(a)). However, the velocity of a cluster depends on the distance to the next cluster ahead. Obviously, the probability that the pheromone created by the last ant of the previous cluster survives decreases with increasing distance. Therefore clusters with a small headway move faster than those with a large headway. This induces a coarsening process such that after long times only one noncompact cluster, also called loose cluster, survives (Fig. 12.9(b)). A similar behavior has been observed also in the bus route model1 [208, 1081] discussed in Section 8.8. If the system evolves from a random initial condition at t = 0, then during coarsening of the cluster, its size R(t) at time t is given by R(t) ∼ t 1/2 . 1 In the bus route model, each bus stop can accommodate at most one bus at a time; the passengers arrive at the bus stops
randomly at an average rate λ, and each bus, which normally moves from one stop to the next at an average rate Q, slows down to q, to pick up waiting passengers.
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0.25
0.6
0.2 Flux
Average speed
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0.4
0.15 0.1
0.2 0
0.05 0.2
0.4
0.6 Density (a)
0.8
1
0
0.2
0.4 0.6 Density (b)
0.8
1
Figure 12.10 Variation of the average speed (a) and flux (b) of the ants with their density on trail.
In vehicular traffic, usually, the intervehicle interactions tend to hinder each other’s motion so that the average speed of the vehicles decreases monotonically with increasing density. In contrast, in our model of unidirectional ant-traffic, the average speed of the ants varies nonmonotonically with their density over a wide range of small values of f because of the coupling of their dynamics with that of the pheromone (Fig. 12.10). This uncommon variation of the average speed gives rise to the unusual dependence of the flux on the density of the ants in our unidirectional ant-traffic model (see Fig. 12.10). Furthermore, the flux is no longer particle-hole symmetric. Analytical Results: So far, an exact solution for the stationary state of the ant-trail model (ATM) has not been achieved. However, it is possible to describe the dynamics rather well using approximate analytical theories. First, mean-field type approaches have been suggested [210, 1058]. However, homogeneous mean-field theories fail in the intermediate density regime. Here the loose cluster dominates the dynamics that cannot be described properly by the mean-field theories, which assume a uniform distribution of the ants. A better theoretical treatment2 can be formulated by realizing [827, 1058] that the model is closely related to the zero-range process (ZRP) introduced in Section 3.3. The mapping from the ATM to the ZRP proceeds in the same way as for ASEP by identifying the particles (ants) with the sites of the ZRP and the gaps between the ants with the particles of the ZRP. In the ATM representation, the hopping probability u of a particle representing an ant can be expressed as u = q(1 − g) + Qg, 2 Similar considerations have been used before for the bus route model [1082].
(12.11)
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where g is the probability that there is a surviving pheromone on the first site of a gap. Assume that the gap size is x and the average velocity of ants is v. Since g(t + 1) = (1 − f )g(t) holds at each time step, we obtain g(x) = (1 − f )x/v after iterating it by x/v times, which is the time interval of between the passage of successive ants through any arbitrary site. In this argument, we have implicitly used a mean-field approximation, i.e., that the ants move with the mean velocity v maintaining equal spacing x. Thus, in the ant-trail model, the hopping probability u is related to gaps x by [210] u(x) = q + (Q − q)(1 − f )x/v ,
(12.12)
which become the occupation-dependent hopping probabilities in the ZRP representation. For the latter, one can now use the results derived in Section 3.3. Note that because the ATM uses parallel dynamics, we need to consider also the ZRP with parallel update. Considering a system with L sites and N ants, i.e., density ρ = N /L, the average velocity v of ants is then calculated in the way described in Section 3.3 using v=
L−N
u(x)p(x)
(12.13)
x=1
since the number of particles in the ZRP representation is L − N . p(x) is the probability of finding a gap of size x, which is given by p(x) = f (x)
ZL−x−1,N −1 , ZL,N
(12.14)
where f (x) are the single-site weights given in (3.38). The partition function Z is obtained by the recurrence relation ZL,N =
L−N
ZL−x−1,N −1 h(x),
(12.15)
x=0
with Zx,1 = h(x − 1) and Zx,x = h(0), which is easily obtained by (12.14) with the normalization p(x) = 1. The fundamental diagram of the ATM can be derived by using (12.13) with changing ρ from 0 to 1. The velocity v in (12.12) can be set to v = q, which is known to be a good approximation for v [1082]. Strictly speaking, v should be determined self-consistently by (12.12) and (12.13). The good agreement with simulation data confirms that the ZRP provides an accurate description of the steady state of the ATM. The fundamental diagrams obtained from simulations indicate the possibility of a phase transition in the thermodynamic limit. Again using the results from the mapping
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to the ZRP, one can conclude that there is no phase transition in the whole density interval 0 ≤ ρ ≤ 1 as long as f > 0 [827]. The situation drastically changes in the limit f → 0, where a phase transition takes place at the critical density ρc =
Q−q . Q − q2
(12.16)
The corresponding average velocity is found as vc = q.
(12.17)
ρc can also be obtained by the intersection point of the line J = vc ρ and the ASEP current (4.29) with hopping rate Q. Note that the limits L → ∞ and f → 0 do not commute [1082]. If one takes f → 0 before L → ∞, then the flow is given by the ASEP result (4.29). This order is relevant for the case of numerical simulations. However, if f → 0 is taken after L → ∞, then the anomalous variation of the average velocity with density disappears. The ATM has also been studied with open boundary conditions [827]. Here boundary-induced phase transitions occur between a high-density, low-density, and maximal current phase. The phase diagram has a structure similar to that of the ASEP (Section 4.2.4), but now the location of the phase transitions depends on the evaporation rate f . Finally, we mention a realization of the ATM in terms of a multiple robot experiment [212, 1064]. In this experiment, the robots communicate through a virtual pheromone system in which the chemical signals are mimicked by graphics projected on the floor. The action of the robots is then determined from the color information of the graphics. The experiments indeed confirm the expected behavior, e.g., the formation of loose clusters at intermediate densities or the anomalous features of the fundamental diagram [1064].
12.4.2. Model of Single-Lane Bidirectional Ant-Traffic The single-lane model of unidirectional ant traffic, which we have discussed earlier, has been extended [826] to capture the essential features of bidirectional ant-traffic in some special situations like, for example, on hanging cables. In this model, the right-moving (left-moving) particles, represented by R(L), are never allowed to move toward left (right); these two groups of particles are the analogs of the outbound and nest-bound ants in a bidirectional traffic on the same trail. Thus, no U-turn is allowed. In addition to the TASEP-like hopping of the particles onto the neighboring vacant sites in the respective directions of motion, the R and L particles on nearest-neighbor sites and facing each other are allowed to exchange their positions, i.e.,
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K
the transition RL → LR takes place, with the probability K . This might be considered as a minimal model for the motion of ants on a hanging cable as shown in Fig. 12.11. When an outbound ant and a nest-bound ant face each other on the upper side of the cable, they slow down and, eventually, pass each other after one of them, at least temporarily, switches over to the lower side of the cable. Similar observations have been made for normal ant-trails, where ants pass each other after turning by a small angle to avoid head-on collision [164, 242]. In our model, as commonly observed in most real ant-trails, none of the ants is allowed to overtake another moving in the same direction. Let us now introduce a third species of particles, labeled by the letter P, which are intended to capture the essential features of pheromone. The P particles are deposited on the lattice by the R and L particles when the latter hop out of a site; an existing P particle at a site disappears when a R or L particle arrives at the same location. The P particles cannot hop but can evaporate, with a probability f per unit time, independently from the lattice. None of the lattice sites can accommodate more than one particle at a time. The state of the system is updated in a random-sequential manner. Because of the periodic boundary conditions, the densities of the R and the L particles are conserved. In contrast, the density of the P particles is a nonconserved variable. The distinct initial states and the corresponding final states for pairs of nearest-neighbor sites are shown in Fig. 12.12 together with the respective transition probabilities.
Figure 12.11 A snapshot of an ant-trail on a hanging cable. It can be regarded as strictly onedimensional. But, nevertheless, traffic flow in opposite directions is possible as two ants, which face each other on the upper side of the cable, can exchange their positions if one of them, at least temporarily, switches over to the lower side of the cable.
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Initial
Final
Rate
Initial
Final
Rate
RL
RL LR RP R0 0R PR R0 0R PR
1−K K (1 − f ) (1 − Q) f (1 − Q) fQ (1 − f )Q 1−q fq (1 − f )q
PR
PR 0R
1−f f
P0
P0 00
1−f f
PP
PP P0 0P 00
(1 − f )2 f(1 − f ) f(1 − f )
RP
R0
f2
Figure 12.12 Nontrivial transitions and their transition rates in the PRL model [826]. Transitions from initial states PL, 0L, and 0P are not listed. They can be obtained from those for LP, L0, and P0, respectively, by replacing R ↔ L and, then, taking the mirror image.
Suppose N+ and N− = N − N+ are the total numbers of R and L particles, respectively. For a system of length M , the corresponding densities are ρ± = N± /M with the total density ρ = ρ+ + ρ− = N /M . Of the N particles, a fraction φ = N+ /N = ρ+ /ρ are of the type R, while the remaining fraction 1 − φ are L particles. The corresponding fluxes are denoted by J± . In both the limits, φ = 1 and φ = 0, this model reduces to the model reported in refs. [210, 1058] and reviewed in the preceding section. One unusual feature of this PRL model is that the flux does not vanish in the densepacking limit ρ → 1. In fact, in the full-filling limit ρ = 1, the exact nonvanishing flux J+ = K ρ+ ρ− = J− at ρ+ + ρ− = ρ = 1 arises only from the exchange of the R and L particles, irrespective of the magnitudes of f , Q, and q. In the special case, Q = q =: qh , the hopping of the ants become independent of pheromone. This special case of the PRL model is identical to the AHR model [51] (see Section 4.7.3) with q− = 0 = κ. A simple mean-field approximation (MFA) yields the estimates J± ρ± qh (1 − ρ) + K ρ∓
(12.18)
irrespective of f , for the fluxes J± at any arbitrary ρ. We found that the results of MFA agree reasonably well with the exact values of the flux [1161] for all qh ≥ 1/2, but deviate more from the exact values for qh < 1/2, indicating the presence of stronger correlations at smaller values of qh . For the generic case q = Q, the flux in the PRL model depends on the evaporation rate f of the P particles. In Fig. 12.13, we plot the fundamental diagrams for wide ranges of values of f (in Fig. 12.13(a)) and φ (in Fig. 12.13(b)), corresponding to one set of hopping probabilities. First, note that the data in Fig. 12.13 are consistent with the physically expected value of J± (ρ = 1) = K ρ+ ρ− because in the dense packing limit, only the exchange of the oppositely moving particles contributes to the flux. Moreover,
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0.20
0.15
0.15 Flux (F+)
Flux (F+)
0.10
0.05
0.00 0.0
f = 0.001 f = 0.005 f = 0.01 f = 0.05 f = 0.10 f = 0.25
0.2
0.4 0.6 Density (a)
0.8
0.10
φ = 0.00 φ = 0.20 φ = 0.30 φ = 0.50 φ = 0.70 φ = 0.80 φ = 1.00
0.05
1.0
0.00 0.0
0.2
0.4 0.6 Density
0.8
1.0
(b)
Figure 12.13 The fundamental diagrams in the steady state of the PRL model [826] for several different values of (a) f (for φ = 0.5) and (b) φ (for f = 0.001). The other common parameters are Q = 0.75, q = 0.25, K = 0.5, and M = 1000. Nonmonotonic variation of the average speeds of the ants with their density on the trail gives rise to the unusual shape of the fundamental diagrams.
the sharp rise of the flux over a narrow range of c observed in both Fig. 12.13(a) and (b) arise from the nonmonotonic variation of the average speed with density, an effect which was also observed in our earlier model for unidirectional ant traffic [210, 1058]. In the special limits, φ = 0 and φ = 1, this model reduces to our single-lane model of unidirectional ant traffic; therefore, in these limits, over a certain regime of density (especially at small f ), the particles are expected to form “loose” (i.e., noncompact) clusters. Therefore, in the absence of encounter with oppositely moving particles, τ± , the coarsening time for the right-moving and left-moving particles would grow with system size as τ+ ∼ φ 2 M 2 and τ− ∼ (1 − φ)2 M 2 . In the PRL model under periodic boundary conditions, the oppositely moving loose clusters collide against each other periodically where the time gap τg between the successive collisions increases linearly with the system size following τg ∼ M . This scaling relation has been verified numerically (see the typical space-time diagram in Fig. 12.14). During a collision, each loose cluster “shreds” (i.e., cuts into pieces) the oppositely moving cluster; both clusters shred the other equally if φ = 1/2. However, for all φ = 1/2, the minority cluster suffers more severe shredding than that suffered by the majority cluster because each member of a cluster contributes in the shredding of the oppositely moving cluster. In small systems, the shredded clusters get opportunity for significant recoarsening before getting shredded again in the next encounter with the oppositely moving particles. But, in sufficiently large systems, shredded appearance of the clusters persists. However, we observed practically no difference in the fundamental diagrams for M = 1000 and M = 4000. The data (Fig. 12.15) corresponding to φ = 1 are consistent with the asymptotic growth law R(t) ∼ t 1/2. In sharp contrast, for φ = 0.5, R(t) saturates to a much smaller
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Figure 12.14 Space-time plot of the PRL model for Q = 0.50, q = 0.25, f = 0.005, ρ = 0.2, φ = 0.3, K = 1.0, and M = 4000. The black and gray dots represent the right-moving and left-moving ants, respectively. 103
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Figure 12.15 Average size of the cluster R in the PRL model [826] as function of time t for φ = 1.0, and φ = 0.5, both for the same total density ρ = 0.2; the other common parameters being Q = 0.75, q = 0.25, K = 0.50, f = 0.005, and M = 4000.
value (Fig. 12.15) that is consistent with highly shredded appearance of the corresponding clusters. Thus, coarsening and shredding phenomena compete against each other, and this competition determines the overall spatio-temporal pattern. Therefore, in the late stage of evolution, the system settles to a state where, because of alternate occurrence of
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shredding and coarsening, the typical size of the clusters varies periodically. Moreover, we find that, for given ρ and φ, increasing K leads to sharper speeding up of the clusters during collision so long as K is not much smaller than q. Both the phenomena of shredding and speeding during collisions of the oppositely moving loose clusters arise from the fact that, during such collisions, the dominant process is the exchange of positions, with probability K , of oppositely-moving ants that face each other.
12.4.3. Model of Two-Lane Bidirectional Ant-Traffic It is possible to extend the model of unidirectional ant-traffic to a minimal model of twolane bidirectional ant-traffic [688]. In such models of bidirectional ant-traffic, the trail consists of two lanes of sites. These two lanes need not be physically separate rigid lanes in real space. In the initial configuration, a randomly selected subset of the ants move in the clockwise direction in one lane while the others move counterclockwise in the other lane. The numbers of ants moving in the clockwise direction and counterclockwise in their respective lanes are fixed, i.e., ants are allowed neither to take U-turn. The rules governing the dropping and evaporation of pheromone in the model of bidirectional ant-traffic are identical to those in the model of unidirectional traffic. The common pheromone trail is created and reinforced by both the outbound and nest-bound ants. The probabilities of forward movement of the ants in the model of bidirectional ant-traffic are also natural extensions of the similar situations in the unidirectional traffic. When an ant (in either of the two lanes) does not face any other ant approaching it from the opposite direction, the likelihood of its forward movement onto the ant-free site immediately in front of it is Q or q, respectively, depending on whether or not it finds pheromone ahead. Finally, if an ant finds another oncoming ant just in front of it, as shown in Fig. 12.16, it moves forward onto the next site with probability K . Since K
K
Figure 12.16 A typical head-on encounter of two oppositely moving ants in the model of bidirectional ant-traffic [688]; the corresponding hopping probability is denoted by K. This process does not have any analog in the model of unidirectional ant-traffic.
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Figure 12.17 Fundamental diagrams of the model for bidirectional traffic [688] for the cases q < K < Q (a) and K < q < Q (b) for several different values of the pheromone evaporation probability f . The densities for both directions are identical and therefore only the graphs for one directions are shown. The parameters in (a) are Q = 0.75, q = 0.25, and K = 0.5. The symbols ◦, •, , , ∗, +, , 3, and correspond, respectively, to f = 0, 0.0005, 0.005, 0.05, 0.075, 0.10, 0.25, 0.5, and 1. The parameters in (b) are Q = 0.75, q = 0.50, and K = 0.25. The symbols ◦, , , , , and correspond, respectively, to f = 0, 0.0005, 0.005, 0.05, 0.5, and 1. The inset in (b) is a magnified replot of the same data, over a narrow range of density, to emphasize the fact that the unusual trend of variation of flux with density in this case is similar to that observed in the case q < K < Q (a). The lines are merely guides to the eye. In all cases, curves plotted with filled symbols exhibit nonmonotonic behavior in the speed–density relation.
ants do not segregate in perfectly well-defined lanes, head-on encounters of oppositely moving individuals occur quite often although the frequency of such encounters and the lane discipline varies from one species of ants to another. In reality, two ants approaching each other feel the hindrance, turn by a small angle to avoid head-on collision and, eventually, pass each other. At first sight, it may appear that the ants in our model follow perfect lane discipline and, hence, unrealistic. However, that is not true. The violation of lane discipline and head-on encounters of oppositely moving ants is captured, effectively, in an indirect manner by assuming K < Q. But, a left-moving (right-moving) ant cannot overtake another left-moving (right-moving) ant immediately in front of it in the same lane. It is worth mentioning that even in the limit K = Q, the traffic dynamics on the two lanes would remain coupled because the pheromone dropped by the outbound ants also influence the nest-bound ants and vice versa. Figure 12.17 shows fundamental diagrams for the two relevant cases q < K < Q and K < q < Q and different values of the evaporation probability f for equal densities on both lanes. In both cases, the unusual behavior related to a nonmonotonic variation of the average speed with density as in the unidirectional model can be observed [688]. An additional feature of the fundamental diagram in the bidirectional ant-traffic model is the occurrence of a plateau region. This plateau formation is more pronounced
Traffic Phenomena In Biology
in the case K < q < Q than for q < K < Q because they appear for all values of f . Similar plateaus have been observed earlier [651, 1397] in models related to vehicular traffic where randomly placed bottlenecks slow down the traffic in certain locations along the route. Further results on the spatio-temporal organization of the ants in this model can be found in [407, 689]. A variation of the model was studied for different boundary conditions in [612].
12.4.4. Experimental Investigations of Ant-Traffic In vehicular traffic, normally the average speed monotonically drops as the density increases leading, eventually, to traffic congestion and jamming. Are such jams possible in ant-traffic? This question was first addressed by Burd et al. [165] who used the leaf-cutting ant species Atta cephalotes. Because of the large scatter in their data, the interpretation of the date remained questionable. In [691], it was concluded from laboratory experiments that no functional relation between the average velocity (or flow) and density exists. However, in that particular case heavy bidirectional traffic was considered without taking into account the density in counterdirection. Later John et al. [690] analyzed data collected from natural unidirectional trails of the ant species Leptogenys processionalis. The data unambiguously establish that no jam forms in the traffic of this species of ants even at very high densities (see below). Related results for bidirectional trails were reported by John et al. in [689]. Our observation clearly indicates that the traffic rules in the world of ants need not be identical to those followed by the drivers in vehicular traffic. In the last few years, Dussutour, Deneubourg, and their collaborators have carried out several interesting experiments to understand the rules of ant traffic. Dussutour, Deneubourg, and Fourcassié [329] experimentally investigated the temporal reorganization of black garden ant Lasius niger in a bottleneck region in a bidirectional traffic. The ants had to pass through a narrow bridge of 3 mm width; at the two ends of the bridge wider space was available to the ants. In this experiment, Dussutour et al. discovered a “traffic rule” that essentially specifies the “right of the way”: ants arriving at the entrance of the bottleneck give way to ants coming from the opposite direction and, therefore, queue up. The queue of waiting ants becomes longer during the time it takes to clear ants approaching from the opposite side of the bottleneck region. Then the cluster of queued up ants move into the bottleneck and cross it while on the other side incoming ants queue up. Thus, alternating clusters of inbound and outbound ants cross the bridge. Although alternating clusters of inbound and outbound ants were observed also in a similar experiment with leaf-cutting ant Atta colombica, “the right of the way” was more elaborate [328].
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For black garden ant Lasius niger, right of the way is symmetric, i.e., same priority for both incoming and outgoing ants at a bottleneck. Nest-bound loaded ants do not get higher priority. But, most of the clusters of inbound leaf-cutting ants Atta colombica are headed by laden ants. The rule for right of the way is asymmetric: outbound ants always give way to loaded (i.e., cargo-carrying) inbound ants. Why is the right of the way asymmetric in Atta colombica traffic? It has been speculated that this preferential treatment of the inbound laden ants ensures high rate of food flow to the nest. Note that the unladen inbound ants (Atta colombica) have to slow down to follow the laden inbound ants ahead in the same cluster. Do they gain any advantage by not overtaking the slower laden ants? Perhaps, the unladen inbound ants also gain the right of the way by staying behind the laden inbound ants thereby minimizing head-on collisions with outbound ants which would have slowed it down anyway. How does an ant select a branch at a bifurcation? Dussutour, Fourcassié, Helbing, and Deneubourg [330] used a setup with two branches in their experiment with black garden ant Lasius niger. At low densities (less crowding), the ants use only one of the two branches although the traffic is bidirectional. At high density, both branches are used. But, flow remains bidirectional in each branch. How do the ants select a branch after reaching the junction? Suppose, an ant A is approaching the junction where it has to select either branch 1 or branch 2. If another ant B approaches the junction from the opposite side along, say, the branch 1, then upon collision with B, the ant A naturally selects the branch 2 for its own forward movement. Couzin and Franks [242] studied the spatial organization of army ants Eciton burchellis. As the density of ants on the trail increases, the ants organized themselves into three lanes, rather than two! The outbound ants took the two outer lanes, while the inbound ants carried food back to the nest along the central lane. This self-organization is, perhaps, dictated by the fact that the emerged pattern is more robust than a possible competing pattern with only two lanes. In any case, a biological advantage of the three-lane pattern is that the food reaches the nest safely along the central lane under the protection of the outbound ants in the two outer lanes. But, what mechanism underlying the interactions of the ants is responsible for such three-lane formation on the trail? Couzin and Franks [242] showed that the food-carrying ants, being more sloth, turn by a smaller angle, compared with the outbound ants, during a face-to-face collision. This feature of the interactions facilitate the formation of three lanes. Similar multilane formation also takes place in termite Longipeditermes longipes where the outbound termites in outer lanes protect the food brought in by inbound termites in inner lanes [938].
12.4.5. Empirical Results for Fundamental Diagrams of Ant-Trails As mentioned earlier, the collective movement of ants on preexisting trails has also been investigated empirically. The study by John et al. [690] focussed on the unidirectional
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trails, which allows to adopt methods and concepts from traffic engineering and compare the spatio-temporal organization with that on highways. The experimental data were collected using video recordings of natural trails (Fig. 12.18) that allowed to determine densities, velocities, and so on. Surprisingly, no overtaking was observed so that the order of the ants on the track remained unchanged. This considerably simplifies the analysis of the data. All relevant quantities can be determined from the times tA ( j) and tB ( j) at which ant j enters and leaves the measurement section [690]. Figure 12.19 shows a typical fundamental diagram obtained in this way. The most unusual feature is that, unlike vehicular traffic, there is no significant decrease of the average velocity with increasing density. Consequently, the
Figure 12.18 A snapshot of a trail section of length L used for measurements. A and B indicate the positions of the virtual detectors at which entrance and exit times are measured for each ant. Ants labeled by squares and circles move in opposite directions. 4
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Figure 12.19 Empirical fundamental diagram for a unidirectional ant-trail. The velocity depends only very weakly on the density. The inset shows the flow–density relation, which clearly shows the absence of a jammed branch. “bl” denotes the body-length of an ant, which has been used as length unit.
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flux increases approximately linearly over the entire regime ρ ∈ [0, 0.8] of the observed density. The jammed branch of the fundamental diagram, which is commonly observed in vehicular traffic and which is characterized by a monotonic decrease of flow with increasing density, is completely missing. Obviously, effects of mutual blocking, which are normally expected to become dominant at high densities are strongly suppressed in ant traffic. Other important quantities that characterize the spatial distribution of the ants on the trail are velocity and distance-headway distributions [690]. Surprisingly, the velocity distribution becomes much sharper with increasing global density, whereas the most probable velocity decreases only slightly. Also distance-headway distribution becomes much sharper with increasing density while the maximum shifts only slightly to smaller headways. At low densities, predominantly large distance-headways are found. The corresponding distribution is well described by a negative-exponential distribution, which is characteristic of the so-called random-headway state [922]. In contrast, at very-high densities, mostly very short distance-headways are found. In this regime, the log-normal distribution appears to provide the best fit to the empirical data. The absence of a jammed phase in the fundamental diagram is closely related to the spatio-temporal organization of the ants along the observed section of the trail. The dominant, and directly observable, feature is the formation of platoons, as predicted by the models discussed earlier. Ants inside a platoon move with almost identical velocities, maintaining small distance-headways. In contrast, larger distance-headways are interplatoon separations. Ants within a platoon move at a slower average velocity, whereas solitary ants can move faster if they detect a strong pheromone trace created by a preceding platoon. Moreover, since fluctuations of velocities of different platoons are larger than the intraplatoon fluctuations, the distribution becomes sharper at higher densities because the platoons merge thereby reducing their number and increasing the length of the longest one. It is worth pointing out that physical origin of the occurrence of the nearly constant average velocity of the vehicles in highway traffic is very different from the constant velocity of ants in ant traffic. In vehicular traffic, the average velocity of the vehicles remains practically unaffected by increasing density, provided the density is sufficiently low, because at those densities, the vehicles are well separated from each other and, therefore, can move practically unhindered in the free-flow state. However, in ant traffic, this constant velocity regime is a reflection of the fact that ants march together collectively forming platoons, which reduce the effective density. Thus, in spite of some superficial similarities, the characteristic features of ant traffic seem to be rather different from those of vehicular traffic. Perhaps, ant traffic is analogous to human pedestrian traffic [171, 329, 689, 1240] (see Chapter 11), as was conjectured beautifully by Hölldobler and Wilson in their classic book [574].
GUIDE TO THE LITERATURE
Although the following bibliography contains more than 1500 references it is far from being complete! In many parts of the book we have tried to give an extensive list of references for the benefit of the reader who is interested in further details or wants to explore more “exotic” parts of the field. Here we want to point out some major resources that we have found extremely useful and that partly have inspired our book and partly give slightly different views of the topic. We therefore give a short commented list of these works. Due to our background we have focussed on physics-related literature. However, there is also a large body of work in the mathematical and engineering literature. This is only partly known to most physicists (including us!), but contains a wealth of results to be (re)discovered. One problem for a physicist with the mathematics literature is often the slightly different formalism used. We recommend the books by Liggett [872, 873], Kipnis and Landim [751], and Spohn [1305] as good starting points. The classic introduction to the mathematical background and the general theoretical approaches for stochastic systems is still the book by van Kampen [1409]. The major resource for the concepts, methods, and formalisms underlying the theory of stochastic systems as outlined in Part I was the extensive review by Schütz [1273]. For the chapter on zero-range (and related) processes the review by Evans and Hanney [363] is invaluable. It contains a wealth of relevant material including more details about the topics covered here. For everything related to the matrix product Ansatz in stochastic systems the “solver’s guide” by Blythe and Evans [123] is an extremely useful and comprehensive resource. It gives a clear exposition of the approach and provides a wide variety of related material, also in areas not covered in this book. Much of the second part of the book is based on our review [216] from 2000 and subsequent updates [206, 215, 1231, 1233, 1235]. Other relevant reviews with slightly different focus are by Helbing [512], Nagatani [1001], Mahnke et al. [899] and Maerivoet and De Moor [895]. The books by Kerner [727, 730] provide not only his personal view on the current status of traffic science, but also a wealth of empirical based material. Other aspects of vehicular traffic are covered in various books and reviews [423, 426, 512, 899, 900, 1001, 1023, 1028, 1142]. Major resources for a more traffic engineering oriented perspective are the books by May [922] and Leutzbach [857]. We also recommend the proceedings to biannual conference seriesTraffic and Granular Flow [38, 419, 526, 587, 1243, 1261, 1473] and Pedestrian and Evacuation Dynamics [420, 760, 1260, 1437]. These conferences are truly interdisciplinary and the proceedings give excellent snapshots of the state-of-the-art including related fields. Stochastic Transport in Complex Systems DOI: 10.1016/B978-0-444-52853-7.00013-0
Copyright © 2011, Elsevier BV. All rights reserved.
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The Encyclopedia of Complexity and System Science [931] contains introductory and review articles on many areas which are related to the topics covered here. It ranges from theoretical and conceptual work right up to specific applications. The books by Howard [591] and Schliwa [1249] give comprehensive overviews of intracellular transport with molecular motors. Our review [219] provides an introduction to biological transport and features additional topics not covered in Chapter 9.
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INDEX
1/f noise, 240 1/f power law spectrum, 293
A ABC model, 183–184 absorbing state(s), 189, 306 acceleration noise, 241 acceleration time delay model, 369 (see also “ATD”) accident(s), 294, 303, 311, 323, 326, 332–333 accident criteria, 333 action point(s), 241 active (Brownian) particles, 9, 10 active walker models, 458–460 adaptive strategies, 397, 398 ADM model, 305 (see also “aggressive driving model”) agent(s), 210, 212, 213, 408, 424 aggressive driving model, 305 (see also “ADM model”) AHR model, 96, 184–185 algebra, 43, 47, 48, 51, 52, 136, 143, 146, 154, 155, 194, 201–204 Alzheimer’s disease, 466 anticipation, 238, 241, 294, 295, 305, 307–309, 343, 349, 368, 374 anticipated velocity, 295 ants, 462, 472–475, 483–488 ant-trail, 472–473, 486–488 ant-trail model (ATM), 474, 476–478 application time, 241 ARAP, 97, 103–104 (see also “asymmetric random average process”) arching, 418 Arnoldi algorithm, 60, 62, 63 ASAP, 191–192 (see also “asymmetric avalanche process”) ASEP, 74–75, 110, 188–189, 248, 290, 391 (see also “asymmetric simple exclusion process”) ASEP-LK, 175–177, 467 asymmetric avalanche process, 191–192 (see also “ASAP”) asymmetric random average process, 97, 103–104 (see also “ARAP”)
asymmetric simple exclusion process, 74–75, 110, 188–189, 248, 290, 391 (see also “ASEP”) ATD, 369 (see also “acceleration time delay model”) ATM, 476 (see “ant-trail model”) ATP, 468, 471 autocorrelation function, 10, 220, 240, 266 automated highway systems, 308 autonomous models, 41 autonomous strategies, 398 Aw–Rascle model, 348–349
B back stepping, 427 backbone, 386 backward-sequential update, 35, 127 balanced vehicular traffic model, 350 Ballot numbers, 199 band index, 419, 420 BBGKY hierarchy, 16–17 BCA, 55, 290 (see also “Burgers CA”) behavioral force, 358 Bethe Ansatz, 41–43, 52, 79–81, 113–115, 142–143, 148 bicycle traffic, 314 bidirectional, 322, 471, 478, 483 Biham–Middleton–Levine model, 180, 384–390 (see also “BML model”) biopolymerization, 464 BJH model, 289 BL model, 300 blanks, 221, 232 Blue–Adler model, 428 BML model, 180, 384–390, 427 (see also “Biham–Middleton–Levine model”) Boltzmann equation, 16–17, 20, 214, 351–357 boomerang effect, 236 Bose–Einstein(-like) condensation, 172, 324, 330 bottleneck(s), 164, 221, 326–328, 415–418, 443 bottleneck parameter, 438 bottleneck-induced jams, 221, 222 boundary-induced phase transition(s), 24, 158–163, 270–271
549
550
INDEX
brake light model, 295–299 (see also “BL model”) bridge model, 185 bridges, 418 Braess paradox, 401–402 BRM, 329, 330 (see also “bus-route model”) Brownian motion, 9 Brownian ratchet, 468, 469 Burgers CA, 55, 290 (see also “BCA”) Burgers equation, 18, 55, 144, 341–342, 344 bus-route model, 329, 330 (see also “BRM”)
C CA, 19, 214, 336, 423–430 cancellation mechanism, 46, 189 capacity, 216, 328, 331, 400, 414–417 capacity drop, 228 car-following model(s), 311, 357–371 car-following theories, 357 car-oriented, 38, 120, 251, 259–260, 277 (see also “COMF”) cell transmission model, 379–381 cellular automata (cellular automaton), 19, 214, 336, 423–430 (see also “CA”) centrifugal force model (CFM), 453–454 characteristics, 212, 338, 348 charge conjugation, 169, 248 (see also “particle-hole transformation”) chemotaxis, 430, 431, 458 chipping model, 105–106, 306 Chowdhury–Schadschneider model, 390–398 (see also “ChSch model”) ChSch model, 390–398 (see also “Chowdhury–Schadschneider model”) clogging, 88, 418 cluster approximation, 38–40, 117–119, 256–257 cluster size distribution, 113 coarse-graining, 19 coarsening, 94–95 codon, 177, 463, 464 coexistence line, 142, 148, 159, 198 coherent moving blocks, 326 Cole-Hopf transformation, 144, 196, 248, 341 collective effects, 441–444 collective velocity, 159, 160 collision detection, 453 collision step, 455, 456 collisions, 17, 310, 453, 454, 473
colloidal particles, 420 column, 196, 297, 447 COMF, 38, 120, 251, 259–260, 277 comfortable driving model, 295–299 (see also “BL model”) compressible fluid, 16, 18, 336, 454 conflict(s), 36, 129, 145, 427, 436–437 conformal invariance, 43 congestion, 221, 222, 402 congested states, 232, 236, 378 constant headway states, 237 continuity equation, 53, 133, 143, 343, 448 convection term, 342 corner transfer matrix, 63 correlation function, 184, 220, 261 correlation length, 23, 138, 185 counterflow, 413, 419, 442 coupled-map models, 214, 371–377 critical, 22, 32 critical dynamics, 22–23 critical exponent(s), 23, 189, 293 critical slowing down, 23, 25 crosscorrelation function, 220, 232, 240 crossing(s), 387, 390, 427 crowd disaster, 421 crowd pressure, 422 crowd turbulence, 422 cruise-control limit, 188, 291–294 cultural differences, 410, 415 current, 5, 140, 149, 205, 216, 217 (see also “flow”, “flux”) cytoskeletal motor traffic, 466 cytoskeletal tracks, 467 cytoskeleton, 466
D dangerous situations, 332, 333 DDRD, 194–195 (see also “directed diffusion of reconstituting dimers”) decimation, 61 decision time, 241 DEEP, 177 (see also “dynamical extending exclusion process”) defects, 164, 173, 327–328 defect particles, 164, 194 density, 165, 218 density matrix, 61–63
551
INDEX
density-matrix renormalization group, 61–63 (see also “DMRG”) density profile, 338 density wave, 338 desired velocity, 353, 360, 457 detailed balance, 12, 211 determinant representation, 115–116 diagonal stepping model, 424 diffusion, 74, 105, 340 diffusion constant, 113, 160, 162 diffusion equation, 55, 144, 341 Dijkstra’s algorithm, 435 direct way system, 460 directed diffusion of reconstituting dimers, 194–195 (see also “DDRD”) directed percolation, 189 discrete time update (dynamics), 33, 48–51 disorder, 83, 164, 173–174, 324–329 disorder TASPEP (DTASEP), 169, 205 distance-headway, 217, 219 DMPA, 51–52 (see also “dynamical MPA”) DMRG, 61–63 (see also “density-matrix renormalization group”) DNA, 178, 462 Doi–Peliti formalism, 53 domain wall(s), 158–160 (see also “shock”) downstream, 222, 235 drop-push model, 81, 186, 187 dual lattice, 384 Dyck paths, 142 dynamic exponent, 73, 115 dynamic floor field, 431, 434 dynamic parameter model, 428 dynamical critical exponent, 23, 146 dynamical extending exclusion process, 177 (see also “DEEP”) dynamical MPA, 51–52 (see also “DMPA”) dynein(s), 305, 462, 463, 466 Dyson gas, 237
E e-mail, 402 edge effect, 173 Edwards-Wilkinson (universality class), 146 electrical length, 218 elevator traffic, 331 elongation, 464 enantiodromy, 54
energy equation, 357 environment, 61, 398 equation of continuity, 18, 137, 336, 343 equilibrium traffic state, 337 equilibrium, thermodynamic, 4–5, 24 equivalent spacing, 228 ER model, 311 ergodic, 58 ergodicity, 31, 104 Euler form, 315 Euler representation, 20 Eulerian scaling, 53, 144 Euler–Lagrange transformation, 315–316 exclusion principle, 72, 124, 311, 439, 455 exponential networks, 384 extended particles, 177 extremal principle, 161, 270, 453
F F.A.S.T. model, 441 faster-is-slower effect, 421, 446, 453 fast-to-start rule, 288 FI model, 304, 305, 314, 316, 317, 328, 424–428 (see also “Fukui–Ishibashi model”) Fick’s law, 147 fidelity, 213 finite segment mean field theory, 173 finite size algorithm, 62 first-order model, 342 five-vertex model, 254, 324, 373 floating-car data, 216, 239 flood filling, 436 floor field(s), 430–447 floor field model, 430, 431, 436, 437, 439–441, 443, 444, 446, 447 flow, 14–19, 216, 217, 226, 230–235, 290–291, 377–379 (see also “flux”, “current”) flow-density relation, 228, 231, 233, 295, 347 fluctuation theorem(s), 13, 163 fluctuation-dissipation theorem, 5 fluid-dynamic models, 214, 347 (see also “hydrodynamic models”) flux, 216 (see also “flow”, “current”) Fokker–Planck equation, 13–14, 177 follow-the-leader model, 358–360 force floor field, 439 forward-sequential update, 35, 50, 157
552
INDEX
free-flow, 231 freezing-by-heating effect, 421 friction, 436–437 friction parameter, 437, 445 frustration effects, 241 Fukui–Ishibashi model, 304, 305, 314, 316, 317, 328, 424–428 (see also “FI model”) full velocity difference model, 349, 365 fundamental diagram, 161, 226–228, 248–255 fundamental diagram-based approach, 295
G Galilei-invariant model, 309–311 Garden of Eden states, 124, 257 (see also “GoE”) gas-kinetic based traffic model, 345 (see also “GKT model”) gas-kinetic models, 214, 351–357, 447–449 gene expression, 463 general pattern, 235 generalized CFM, 453 generalized force, 358 generalized force model, 364–366 generalized OV model, 365 generalized ZRP, 99, 100, 129 generating function, 228 Gershgorin’s theorem, 66 Ginzburg–Landau equations, 23 Gipps equation, 369 Gipps model, 372–373 Gipps–Marksjös model, 428–430 GKT model, 345 (see also “gas-kinetic based traffic model”) global strategies, 395 GoE, 124, 257 (see also “Garden of Eden states”) Green function, 115 green-wave model, 388 green-wave strategy, 395 green-wave synchronization, 388 Greenshields model, 227, 341–342 gridlock(ing), 385 gross headways, 220
H Hamiltonian, 28–31 HDM model, 368 (see also “human driver model”) head-on collision, 323 headway (see “time-headway”, “distance-headway”, “spatial headway”)
headway-distance effect models, 289 Heisenberg model, 41, 114 herding, 431, 444 high-density phase, 139–140, 150–151, 198 hole, 117, 248, 253, 260 human driver model, 368 (see also “HDM model”) hydrodynamic limit, 52–53, 143–144 hydrodynamic models, 286 (see also “fluid-dynamic models”) hydrodynamic(al) relation, 411 hysteresis, 96, 228–230, 282–291
I IDM model, 366–368 (see also “intelligent driver model”) IDMM model, 368 IPDF (see “interparticle distribution functions”) incompressible fluids, 16, 18, 454 individual-based, 7–10 induction loops, 216 inertia, 434 inertia effects, 438 inertial car-following model, 369–371 infinite size algorithm, 62 information field, 439 initiation, 434 injection-produced slowdown, 271 integrability, 41 intelligent driver model, 366–368 (see also “IDM model”) interacting subsystem approximation, 173 intermediate headway states, 237 Internet, 402–405 interparticle distribution functions, 40–41 inverse-λ form, 228 irreducible string, 194 irreversibility, 10–13 Ising measure, 190 Ising model, 22, 75, 196–197
J jam, 221, 224–226, 263–264, 345–348 jam outflow phase, 286 jam-gap distribution, 285 jammed phase, 231 jamming, 418 jamming transition, 374, 387, 388, 390, 427 jump moments, 14
553
INDEX
K
K3 model, 345 (see also “Kühne–Kerner–Konhäuser model”) Kac ring model, 16 Kawasaki dynamics, 21 Kerner–Klenov model, 348, 368 (see also “KK model”) Kerner–Klenov–Wolf model, 299–301 (see also “KKW model”) Kerner–Konhäuser model, 345 KIF1A, 468–472 kinematic wave(s), 337–340 kinesin(s), 462, 466–472 kinetic equation, 356–357 kinetic theory, 17–19 KK model, 348, 368 (see also “Kerner–Klenov model”) KKW model, 299–301 (see also “Kerner–Klenov–Wolf model”) KLS model, 21, 88, 176, 190–191, Kolmogorov consistency conditions, 40, 117, 259 Korteweg-de Vries equation (KdV equation), 362 KPZ equation, 196 Kramers–Moyal expansion, 13 Krauss model, 373–375 (see also “SK model”) Krebs–Sandow theorem, 68–70 Kühne model, 345 Kühne–Kerner–Konhäuser model, 345 (see also “K3 model”)
L Lagrange form, 316–318 Lagrange representation, 20, 213 Lanczos algorithm, 60 Landau theory, 22 lane-changing rules, 319, 320 lane formation, 419–420, 441–443 lane inversion, 321 Langevin equation, 9–10, 458, 467 Langmuir isotherm, 174 Langmuir kinetics, 85–86, 174–177, 466–472 large deviation function, 162–163 lattice gas, 20–22, 424, 454–456 lattice Boltzmann equation, 20 Lee-Yang zeroes, 142 Lennard-Jones potential, 366 level of service, 230–231 lifetime distribution, 265
lifetimes of jams, 265–266 Lighthill–Whitham–Richards model, 337–340, 344, 448 (see also “LWR model”) Lincoln-tunnel, 290 line density, 413 linear response, 8, 211 Liouville equation, 15–16 Liouville operator, 29 local breakdown effect, 346 local cluster effect, 346 local equilibrium, 52 local structure theory, 38 localization length, 160 localized cluster, 236 loose cluster(s), 475 low-density phase, 138–139, 149–150, 198 LWR model, 337–340, 342, 344, 348, 379 (see also “Lighthill–Whitham–Richards model”)
M macroscopic fluctuation theory, 163 macroscopic models, 212 Manhattan metric, 439 Markov matrix, 29 Markovian approximation, 38 Markov-process inspired model, 313 MAS, 214, 424 (see also “multi-agent systems”) massive, 32 master equation, 10–13, 28–31, 187 matrix of preference, 432 matrix-product Ansatz, 43–52, 120, 135–136 (see also “MPA”) maximal current phase, 136, 149 maximum principle, 158 mean-field approximation, 37–41, 205, 480 mean-field theory(ies), 116–117, 120–123, 132–133 mechanical restrictions model, 301–303 (see also “MR model”) mega-jam, 283 memory effects, 241 mesoscopic models, 212 metastable branch, 286, 310, 318, 379 metastable state(s), 228, 290–291 method of empty intervals, 40 method of moments, 356 MFT, 37 (see “mean-field theory”) micro-shocks, 162
554
INDEX
microscopic models, 6, 24, 211 microtubule(s), 462 minimal way systems, 460 mitosis, 462 MOBIL, 322, 368 mobility, 455 modified comfortable driving model, 298 molecular chaos, 17, 355 molecular dynamics, 8, 452 molecular motor(s), 10, 305, 467 momentum equation, 342, 357 Monte Carlo, 57, 94, 173, 184 Moore neighborhood, 423 motor proteins, 466 Motzkin paths, 142 moving bottlenecks, 236 moving routers, 404 MPA, 43–52, 120, 135–136 (see also “matrix-product Ansatz”) MR model, 301–303 (see also “mechanical restrictions model”) mRNA (messenger RNA), 463, 464 multi-agent systems (multi-agent models), 214, 424 (see also “MAS”) multichannel ASEP, 181 multiclass models, 348 multispecies models, 181–185 multianticipative interactions, 363 multivehicle interactions, 355 myosin, 466
N Nagel-Herrmann model, 376–377 (see also “NH model”) Nagel–Schreckenberg model, 117, 243–279 (see also “NaSch model”) NH model, 376–377 (see also “Nagel-Herrmann model”) NaSch model, 117, 243–279 (see also “Nagel–Schreckenberg model”) Nash equilibrium, 402 Navier–Stokes equation (theory), 14, 17–19, 342–343, 356 net-time headway, 415 network(s), 383–405 Newtonian dynamics, 7, 16, 53 Newton’s Third Law, 448, 451 noise, 213
nonequilibrium, 3–25 nucleation, 23, 24, 377, 379 nucleic acids, 462
O occupancy, 219 off-ramp, 222, 319, 328, 350, 399, 400 on-ramp, 222, 328, 329, 336, 350 one-transit walks, 142 online simulation, 398–400 operational level, 210, 408 optimal-velocity (OV) function, 238–239, 360, 363, 365, 376, 378 optimal-velocity (OV) model, 309, 312, 360, 368, 374 optimistic driving, 301 optimum ground state, 66–68 order parameter, 22, 268 ordered-sequential update, 35–36, 49, 126–127, 151–153 oscillating moving jams, 298 Ostwald ripening, 5, 24 overfeeding, 160, 287 overlaps, 38, 452, 454 overreaction(s), 245, 246, 260, 286, 296
P pairwise-balance, 12, 211 panic, 421–422 paradisical mean-field theory, 123–124 paradisical states, 124, 257 (see also “Garden of Eden states”) parallel update, 50–51, 154–156, 246 (see also “synchronous update”) parking garage problem, 178 partially asymmetric simple exclusion process, 111, 145, 148–151 (see also “PASEP”) particle-hole attraction, 253, 260 particle-hole symmetry, 117 particle-hole transformation, 117, 248 particle-hopping model, 30, 71–108 particle-oriented, 20, 37–38 particlewise disorder, 164, 171–173 PASEP, 111, 145, 148–151 (see also “partially asymmetric simple exclusion process”) Paveri-Fontana model, 353–356 Payne’s equation, 344 Payne model, 344, 346, 363
555
INDEX
pedestrian(s), 407–460 pedestrian area module, 411 perception time, 241 percolation, 189, 386, 394 permeability, 166 Perron–Frobenius theorem, 66 personal space, 431, 450 pessimistic driving, 301 phantom (traffic) jam(s), 223, 347 phase separation, 5, 24, 90, 95–96, 164, 169, 172, 184, 285, 287, 309, 328, 331, 348, 374, 379 phase transition, 22–25, 141–142, 158–163, 231–234, 268–271, 385–386, 394 pheromone(s), 430, 462, 473, 475 Phillips model, 344, 346, 348 ping-pong lane changes, 320 pinned layer, 298 plateau, 254, 328, 397, 446 platoon formation, 172, 294, 326, 376, 488 politeness factor, 322, 439 polynucleotides, 463 polypeptide, 463 population-based, 6, 11–14 porters, 466 price of anarchy, 402 Prigogine theory, 351, 352 private sphere, 431, 450 PRL model, 480, 481 probability path method, 38 product state, 69, 120 projected entangled pair states (PEPS), 68 propagation step, 455 protein, 463, 466 protein synthesis, 177, 465 public conveyence model, 331
Q quantum formalism, 28–37, 45–48 quenched disorder, 164–170, 324–329 q-deformed harmonic oscillator algebra, 203 queueing model, 381 queueing theory, 290
R ramp(s), 235–236, 328–329 (see also “on-ramp”, “off-ramp”) ramp metering, 329, 400
random networks, 384 random number generator(s), 59 random strategy, 395 random-headway state, 237 random-sequential dynamics, 59, 111–113 random-sequential update, 33, 214 random-walk, 286 random walker, 74, 293, 458 Rankine–Hugoniot equation, 159 ratchet model, 185 RCA, 440 (see also “real-coded cellular automata”) reaction time, 241 real-coded cellular automata, 440 (see also “RCA”) relaxation time(s), 32, 266–267 renormalization group, 53, 61–63 reservoir(s), 10, 22, 29, 31, 131, 178–180, 271 resolving probability, 286 restricted ASEP, 188–189 reverse-bias phase, 150 requiem, 348 ribosome, 463–465 ribosome traffic, 464 RNA, 462, 463 RNA polymerase (RNAP), 463–465 robot, 478 round-trip time (RTT), 403 roundabout(s), 391 router(s), 403, 404 RTT (round-trip time), 403 rule-184 CA, 111, 304, 314
S SAM, 369 (see also “speed adaption model”) scale-free networks, 384 scattering process, 42 second-class particle(s), 161, 173, 181–183 second-order model, 348 selection principle, 161 self-organized criticality, 89, 293, 324 self-similar traffic, 405 shock, 158, 159, 162, 340 shock dynamics, 161–162 shock velocity, 158, 159, 339 shock wave, 19, 339, 342 shock-tracking probes, 162 shredding, 481–483 shuffled update, 128–129, 437 side stepping model, 424–426
556
INDEX
similarity transformation, 54, 114, 145 single-step model, 114, 195, 248 single-vehicle data, 217, 219, 400 site-oriented, 20, 116–117, 255 sitewise disorder, 164–170 six-vertex model, 114, 167 SK model, 373–375 (see also “Krauss model”) slow-to-start rule, 188, 282–291, 296, 303, 311, 314, 374, 379 small-world networks, 384 social field, 449 social force(s), 8, 19, 214, 449–454 social insects, 462, 472 social-force model, 449–454 SOV, 312 (see also “stochastic optimal velocity model”) space mean speed, 218 spatial headway, 217, 237, 286 specific flow, 410 speed adaption effect, 369 speed adaption model, 369 (see also “SAM”) speed-density relation, 228, 484 spinodal decomposition, 5, 24, 172 spontaneous jam(s), 221, 224–226, 342 stability analysis, 359, 457 static floor field, 431–433, 435–436 stationary state, 5, 12, 31, 48, 97–101, 260–261 (see also “steady state”) steady state, 5, 12, 38, 100–101, 323, 481 (see also “stationary state”) steady state selection principle, 161 stimulus, 8, 311, 357–358 stochastic Hamiltonian, 28–31, 62–63 stochastic matrix, 31, 82 stochastic optimal velocity model, 312 (see also “SOV”) stochastic similarity, 54 stop-and-go, 224, 230, 234, 236, 240, 288, 313, 346, 353 Stosszahlansatz, 17, 449 strategic level, 210, 408 structure factor, 267–268 sublattice-parallel update, 36, 50, 127–128, 154 superblock, 61 superjamming, 266 swarm force model, 439 swarm information field, 439 swarm intelligence, 472
sweep, 36, 62 symmetry breaking, 22, 181, 268, 419 synchronization distance, 299, 369 synchronized flow pattern, 235 synchronized phase (synchronized traffic), 231–235, 238, 294, 298, 307, 349, 363 synchronized strategy, 395 synchronous update, 36, 214 (see also “parallel update”) system optimum, 397, 402
T tactical level, 210, 408 tagged particle, 113 TASEP, 22, 24, 51, 74, 75, 110, 111, 116, 124, 126, 127, 129, 146, 154, 173, 246, 313, 385, 462–469, 478 (see also “totally asymmetric simple exclusion process”) TASEP-LK, 467 tensor product, 46, 66–68 termination, 464, 465 termite(s), 462, 472, 486 territorial effect, 450 three-phase traffic theory, 231–233, 301, 307, 368 time-headway, 217, 219, 220, 237, 238, 263, 264, 303, 307, 466 time-oriented CA model, 306–307 (see also “TOCA”) time-to-collision, 296, 415 time mean speed, 218 TMRG, 63–65 (see also “transfer matrix DMRG”) TOCA, 306–307 (see also “time-oriented CA model”) tollbooths, 328 totally asymmetric simple exclusion process, 22, 24, 51, 74, 75, 110, 111, 116, 124, 126, 127, 129, 146, 154, 173, 246, 313, 385, 462–469, 478 (see also “TASEP”) traffic forecasting, 383, 384, 398, 399 traffic lights, 179, 290, 388, 392, 395–398 traffic pattern, 230, 234, 235, 329 traffic phase, 230–234, 294 traffic pressure, 343, 345, 348, 357 traffic state(s), 220, 230, 232, 234–236 traffic volume, 216, 400 trail(s), 458, 462, 472–473, 486–488 trail formation, 422–423, 430, 458, 459 trail network(s), 422, 462
557
INDEX
trail system, 458 transcription, 462–463 transfer matrix, 34–36, 63, 114, 120, 190 transfer matrix DMRG, 63–65 (see also “TMRG”) transition probability, 11, 125, 433, 434 translation, 463, 464 transmission probability, 166 transport term, 342 Trotter-Suzuki decomposition, 63, 64 Trotter-Suzuki mapping, 63 Trotter dimension, 64 trucks, 236, 323, 326, 362 TT model, 288–289, 315 (see also “TT slow-to-start model”) TT slow-to-start model, 317 turning function, 438
velocity equation, 343–345, 349, 350 vertex-state models, 68 visibility graph, 435, 436 volume exclusion, 74, 452 von Neumann neighborhood, 423, 424, 432, 439
W walking speed, 409–410, 448, 459 wall potential, 438 wave, 338 wide (moving) jams, 223, 232–236, 286, 298, 346–347 wrong-way travel, 348, 349
X
U
XXZ chain, 52, 142 XXZ model, 114, 145
ultradiscrete method, 54–57, 313 ultradiscrete OV model, 313 upstream, 221, 223, 232 user optimum, 402
Y
V
Z
variance equation, 344 VDR model, 283–288, 294, 300 vehicular chaos, 355, 356 velocity-dependent braking, 305–306 velocity-dependent randomization, 283–288, 294, 300 (see also “VDR model”)
Zamolodchikov algebra, 52 zero-range process, 53, 75, 83–86, 88–90, 123, 312, 379, 476 (see also “ZRP”) zipper effect, 416, 439 ZRP, 53, 75, 83–86, 88–90, 123, 312, 379, 476 (see also “zero-range process”)
Yukawa-Kikuchi model, 375–376
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