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SUPERBRANES AND SUPEREMBEDDINGS
Dmitri SOROKIN Humboldt-UniversitaK t zu Berlin, Mathematisch-Naturwissenshaftliche Fakultat, Institut fuK r Physik, Invalidenstrasse 110, D-10115 Berlin, Germany INFN, Sezione Di Padova, via F. Marzolo, 8, 35131 Padova, Italy
AMSTERDAM } LAUSANNE } NEW YORK } OXFORD } SHANNON } TOKYO
Physics Reports 329 (2000) 1}101
Superbranes and superembeddings Dmitri Sorokin Humboldt-Universita( t zu Berlin, Mathematisch-Naturwissenshaftliche Fakultat, Institut f u( r Physik, Invalidenstrasse 110, D-10115 Berlin, Germany INFN, Sezione Di Padova, via F. Marzolo, 8, 35131 Padova, Italy Received August 1999; editor: J. Bagger Contents 1. Introduction 2. Doubly supersymmetric particles 2.1. The bosonic particle 2.2. Spinning particles 2.3. Superparticles, i-symmetry 2.4. Spinning superparticles 2.5. Worldline super"eld actions for standard superparticles. The superembedding condition 2.6. Coupling to supersymmetric Maxwell "elds 2.7. Superembedding into curved superspaces 3. Superstrings 3.1. Green}Schwarz formulation 3.2. Doubly supersymmetric N"1 superstrings 3.3. Coupling to supergravity background 3.4. Heterotic fermions 3.5. Type II superstrings
4 11 11 12 14 17
18 35 37 39 39 42 52 53 55
4. Basic elements of (super)surface theory 4.1. Bosonic embedding 4.2. Superembedding 4.3. i-symmetry from the point of view of superembedding 4.4. N"1, D"10 superparticles and superstrings from the general prespective of superembedding 5. M-theory branes 5.1. The supermembrane 5.2. The super-5-brane 6. Other developments and applications 6.1. The generalized action principle 6.2. D-branes, L-branes and branes ending on branes 6.3. Non-linear realizations and superembeddings 7. Concluding remarks and outlook Acknowledgements References
56 56 62 68
70 72 74 80 89 89 91 92 94 95 95
Abstract We review a geometrical approach to the description of the dynamics of superparticles, superstrings and, in general, of super-p-branes, Dirichlet branes and the M5-brane, which is based on a generalization of the
Alexander von Humboldt Fellow. On leave from Kharkov Institute of Physics and Technology, Kharkov, 310108, Ukraine. E-mail address:
[email protected];
[email protected] (D. Sorokin) 0370-1573/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 1 0 4 - 0
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elements of surface theory to the description of the embedding of supersurfaces into target superspaces. Being manifestly supersymmetric in both, the superworldvolume of the brane and the target superspace, this approach uni"es the Neveu}Schwarz}Ramond and the Green}Schwarz formulation and provides the fermionic i-symmetry of the Green}Schwarz-type superbrane actions with a clear geometrical meaning of standard worldvolume local supersymmetry. The dynamics of superbranes is encoded in a generic superembedding condition. Depending on the superbrane and the target-space dimension, the superembedding condition produces either only o!-shell constraints (as in the case of N"1 superparticles and N"1 superstrings), or also results in the full set of the superbrane equations of motion (as, for example, in the case of the M-theory branes). In the "rst case worldvolume superspace actions for the superbranes can be constructed, while in the second case only component or generalized super"eld actions are known. We describe the properties of the doubly supersymmetric brane actions and show how they are related to the standard Green}Schwarz formulation. In the second part of the article basic geometrical grounds of the (super)embedding approach are considered and applied to the description of the M2-brane and the M5-brane. Various applications of the superembedding approach are reviewed. 2000 Elsevier Science B.V. All rights reserved. PACS: 11.15.!q; 11.17.#y Keywords: Superstrings; Supergravity; Branes; Surface theory
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1. Introduction String theory to much more extent than any other theory requires for its description various important developments of classical and quantum "eld theory, group theory, geometry and topology. To comprehend di!erent properties of string theory and to uncover underlying relations between them one should use for the analysis di!erent (often intertwined) mathematical tools and methods. One of such basic geometrical tools is surface theory which describes the embedding of surfaces into higher-dimensional manifolds. The application of surface theory to the description of string theory is quite natural, since the string is a one-dimensional relativistic object which sweeps a two-dimensional surface (worldsheet) when it propagates in a (target) space}time. The dynamics of the string completely determines the geometrical properties of the worldsheet describing its embedding into the target space, and vice versa, specifying geometrical properties of the embedding of a surface into a target space one can, in principle, get the full information about the details of the dynamics of a string whose worldsheet is associated with this surface. For instance, a metric g (m) (m, n"0, 1) of a string worldsheet locally parametrized by coordiKL nates mK"(q, p) is an induced metric which is related to a D-dimensional target-space metric g (X) KL (m, n"0, 1,2, D!1) through the condition g (m)"R XK g (X(m))R XL , (1.1) KL K KL L which is the simplest example of the embedding condition. Note that with taking into account worldsheet reparametrization symmetry the induced metric condition (1.1) amounts to Virasoro constraints on string dynamics (see [1}3] for details on superstring theory). The classical trajectories of the string are surfaces of a minimal (or more generally, extremal) area. This follows from the Nambu}Goto string action
S"!¹ dm(!det g (X(m)) , KL
(1.2)
whose geometrical meaning is to be the total area of the worldsheet of a string with the tension ¹. The variation of the action (1.2) vanishes for minimal area surfaces, which produces the string equations of motion. Therefore, from the geometrical point of view the string equations describe minimal (area) embedding of the worldsheet into target space. A direct application of the geometrical methods of surface theory in string theory was initiated in [4,5] and revealed a connection of the string equations of motion with two-dimensional (exactly solvable) non-linear equations, such as the sin-Gordon and Liouville equation. Though, of course, all string formulations imply that string worldsheet is a surface embedded into a target space}time, the geometrical embedding approach explores this in the most direct way. In what follows the underlined indices will correspond to target-space objects (coordinates, "elds, etc.), while not underlined indices will correspond to the worldsurface. This will allow us to escape, to some extent, the proliferation of indices.
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It deals with such objects as induced vielbeins of the surface, extrinsic curvature and torsion of the surface, and reduces the string equations to the system of the Codazzi, Gauss and Ricci equations completely determining the embedding of the surface (see [7] for the details on surface theory and [6] on its applications to strings). The embedding approach was also used in connection with the problem of formulating a consistent quantum string theory in non-critical space}time dimensions and has been developed in a number of papers (see [6,8] and references therein). In addition to particles and strings a variety of other extended relativistic objects have been discovered in D"10 superstring theories and D"11 supergravity. This includes a D"11 membrane [9], a D"11 "ve-brane [10] and Dirichlet branes [11,12]. Collectively all these extended objects are called superbranes or super-p-branes, where p denotes the number of spatial dimensions of a given brane. The existence of the superbranes re#ects and causes important duality chains which connect D"11 supergravity with "ve basic D"10 superstring theories and string theories among themselves (see [13,3] for a review on dualities). This gives rise to a conjecture that D"11 supergravity and the D"10 string theories can be associated with di!erent vacua of a single underling quantum theory called U-theory [14]. This theory is also often called M-theory [16], but following Sen [14] and Schwarz [15] we shall reserve the latter name for a D"11 sector of U-theory whose low energy e!ective "eld theory limit is D"11 supergravity, and which also contains a membrane (M2-brane) and a "ve-brane (M5-brane) as part of its physical spectrum. Since the superbranes are surfaces in target superspaces it is natural to apply for studying their properties the same geometrical methods as for strings, i.e. to describe the propagation of superbranes by specifying the embedding of brane worldvolumes in target superspaces. For instance, a bosonic part of a super-p-brane worldvolume action is a (p#1)-dimensional analog of the Nambu}Goto action (1.2). It produces bosonic equations of motion which are equivalent to minimal (volume) embedding of the corresponding worldvolume into space}time. As we have already mentioned, at the classical level the use of geometrical embedding methods allow one to "nd a connection between equations of motion of bosonic branes and integrable [17], and in the case of strings, with exactly solvable non-linear equations [4}6,8,18}20]. Analogous relations were also found for superstrings [21,22]. As a result, one can relate, for example, (super)strings to exactly solvable Wess}Zumino}Novikov}Witten (WZNW) models [23,24,22]. (for a review on WZNW models and related topics see [25,26] and references therein). This provides us with useful information about the details of brane dynamics. In the case of supersymmetric extended objects the use of the embedding approach has turned out to be even more fruitful, since it has allowed one to clarify the geometrical origin of local symmetries of superbrane worldvolumes [27}45], to relate di!erent formulations of superbrane dynamics [46}49], and to get equations of motion for those superbranes for which the construction of worldvolume actions encountered problems, such as D(irichlet) branes [50] and the M5-brane [51]. It has also allowed one to make a progress towards a solution of the problem of covariant quantization of superstrings [53,33,48,54,140,55]. Let us also mention that supergravity can also be described as a theory of supersurfaces [56]. To apply the embedding approach to the description of superbranes one should "rst generalize the method itself and "nd supersymmetric analogs of bosonic embedding conditions such as the induced metric condition (1.1) and the minimal embedding of the superbrane worldvolume which
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would be equivalent to the superbrane equations of motion [45]. A reasonable generalization is to consider superbrane worldvolumes as supersurfaces locally parametrized by (p#1) bosonic coordinates mK and n fermionic coordinates gI (k"1,2, n) embedded (in a speci"c way) into target superspaces locally parametrized by D bosonic coordinates XK and 2 " Grassmann spinor coordinates HI (where brackets denote the integer part of "). Thus we get a superembedding. The assumption that such a superembedding should underlie the worldvolume dynamics of superbranes is prompted by a well known fact that the D"10 superstring theories can be formulated in two di!erent ways. In the Neveu}Schwarz}Ramond (or spinning string) formulation [57] superstring propagation is described by a supersurface, possessing n"1 local worldsheet supersymmetry, embedded into bosonic space}time. Space}time supersymmetry appears in this model only upon quantization as a symmetry of quantum string physical states singled out by the Gliozzi}Scherk}Olive projection [59]. On the other hand in the Green}Schwarz formulation [58] the superstring worldsheet is a bosonic surface embedded into a target superspace. This formulation is manifestly space}time supersymmetric and, in addition, possesses a local worldsheet fermionic symmetry called i-symmetry. The number of independent i-symmetry transformations is half the number of target superspace Grassmann coordinates. i-symmetry was "rst observed in the case of superparticles [60,61] and is inherent to all superbranes in the Green}Schwarz formulation [1,9,62}64]. It plays an important role ensuring that superbranes form stable, so called, Bogomol'nyi}Prasad} Sommerfeld con"gurations whose presence in a superspace background preserves half of the space}time supersymmetries of a background vacuum. For D"10 superstrings this means, in particular, that their quantization results in consistent quantum supersymmetric theories. At the same time local fermionic i-symmetry causes problems with performing the covariant Hamiltonian analysis and quantization of superbrane theories. This is due to the fact that the "rst-class constraints corresponding to i-symmetry form an in"nite reducible set, and in a conventional formulation of superparticles and superstrings it turned out impossible to single out an irreducible set of the fermionic "rst-class constraints in a Lorentz covariant way. (Note however that it is possible to do this in the case of D-branes [65].) It is therefore desirable to replace i-symmetry with something more natural and simple. To do this one can notice that i-symmetry transformations resemble supersymmetry transformations. Space}time supersymmetry transformations of XK and HI with a supersymmetry parameter eI are dHI "eI ,
dXK "iHM CK dH ,
(1.3)
while i-symmetry transformations of XK and HI on the superbrane worldvolume have the following form dXK (m)"!iHM CK dH, (CM )"1 (1.4) dHI (m)"(1#CM )I iJ (m), J where (1#CM ) is a projection operator constructed from the CK matrices. The projector is speci"c for each type of the superbranes, and its presence implies that the number of independent i-symmetry transformations is twice less than the number of the space}time supersymmetries (1.3). Notice the di!erence in the sign of the XK -variations in (1.3) and (1.4). This di!erence implies that if ordinary supersymmetry is associated with left boosts in the target superspace, the i-symmetry is associated with right boosts.
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It is instructive to understand how the presence of the branes breaks symmetries of target superspace vacua. For this consider an in"nite #at membrane in a D"11 Minkowski superspace. (i) D"11 Lorenz symmetry SO(1, 10) is broken down to d"3 Lorentz symmetry SO(1, 2) of the worldvolume times SO(8) rotations in directions orthogonal to the membrane; (ii) D"11 Poincare invariance is broken down to a translational symmetry along the membrane worldvolume (generated by momenta P , m"0, 1, 2), K (iii) therefore, unbroken supersymmetries are those generated by supercharges whose anticommutators close on the unbroken translations along the membrane worldvolume. There are 16 such generators Q (a"1,2, 16) among 32. ? +Q, Q,"2iP CK . K 16 target-superspace Grassmann coordinates (let us call them g?) which are shifted by the action of the 16 unbroken supercharges can be regarded as ones ¶llel' to the membrane. It is important to stress that it is the unbroken supersymmetry transformations of g? which can be compensated by appropriate worldvolume i-symmetry transformations, and, hence, these 16 Grassmann coordinates can be associated with fermionic directions which extend the bosonic worldvolume of the membrane to a superspace embedded into the target superspace. Other 16 target-space Grassmann coordinates (let us call them h?Y) correspond to the spontaneously broken supersymmetries. They are Goldstone fermion physical modes of the brane. From the above analogy we see that, since i-symmetry is a local worldvolume fermionic symmetry, it would be natural to endow it with the direct meaning of being a manifestation of conventional n-extended local supersymmetry of the superbrane worldvolume. This would allow one to get rid of &unpleasant' features of i-symmetry by dealing directly with well understood properties of conventional linearly realized supersymmetry. Such an interpretation of i-symmetry was proposed in [27]. For this one should construct a formulation of superbrane dynamics which would be manifestly supersymmetric on both the worldvolume and the target superspace. Such a doubly supersymmetric formulation would then unify the NSR and GS formulations. Note that the NSR formulation of super-branes with p'1 is unknown and the doubly supersymmetric formulation may provide one with an idea what &spinning' branes might look like. To have the worldvolume and target space supersymmetry manifest the doubly supersymmetric formulation should be constructed as a super"eld theory on both the worldvolume superspace and the target superspace, the former being (super)embedded into the latter. We thus again arrive at the superembedding approach. The idea to use doubly supersymmetric models for a combined treatment of the NSR and the GS superstrings "rst appeared in [66]. It has then been applied to the construction of an interesting class of so called spinning superparticle [67] and spinning superstring [68] models. In general, however, these models describe objects with twice larger number of physical states than, for example, corresponding NSR spinning strings or GS superstrings have themselves. They also do not resolve the problem of i-symmetry, since the latter is still present in the spinning superbranes as an independent symmetry in addition to double supersymmetry. A main reason why spinning superbranes have &redundant' symmetries and physical states is that embedding of their superworldvolumes into target superspaces is too general. To describe
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conventional (i.e. not spinning) superbranes in the superembedding approach (with i-symmetry replaced by local worldvolume supersymmetry) one should "nd an appropriate superembedding condition which would amount to conditions required for the description of the dynamics of the &standard' superbranes. The basic superembedding condition was "rst found in [27] for superparticles and then, as a result of the development of the superembedding approach [27}51], was proved to be generic to all known types of superbranes. From the geometrical point of view this condition is quite natural and simple. Consider the pullback onto the superworldvolume (with n"2 " \ Grassmann directions) of the vector component E? (Z) of a target space supervielbein E+ (Z)"(E? , E? ) (where Z+ "(XK , HI ), and the indices from the beginning of the alphabets correspond to the (co)tangent space of the target superspace which has 2n"2 " Grassmann directions). The pullback of E? is a onesuperform on the superworldvolume and, therefore, can be expended in a superworldvolume supervielbein basis e (m, g)"(e?, e?) (1.5) E? (Z(m, g))"e?E? #e?E? . ? ? The superembedding condition reads that the pullback components of the vector supervielbein E? (Z) along the Grassmann directions of the superworldvolume are zero (1.6) E? (Z(m, g))"0 . ? The geometrical meaning of this condition is that at any point on the superworldvolume the worldvolume tangent space Grassmann directions form a subspace of the Grassmann tangent space of the target superspace. In a certain sense Eq. (1.6) is analogous to the induced metric condition (1.1) (actually, Eq. (1.6) implies a superspace generalization of (1.1)). As we shall see below, the dynamical meaning of the superembedding condition (1.6) is that it produces Virasoro-like constraints on the dynamics of the superbranes, and in many cases its integrability conditions lead to superbrane equations of motion. It also provides a link between i-symmetry of the standard GS formulation and local worldvolume supersymmetry of the doubly supersymmetric formulation of superbranes [27], as well as a classical relation between the NSR and GS formulation in the case of superparticles [46,47] and superstrings [48,49]. In the cases when the superembedding condition does not produce the superbrane equations of motion one can construct worldvolume super"eld actions for corresponding superbranes. Such actions have been constructed for N"1, D"3, 4, 6, 10 massless superparticles [27,31,32,36,38,39], N"1, D"3, 4, 6, 10 superstrings [28,29,33,34,37,40], N"2, D"2, 3, 4, 6 massless and massive superparticles [35,38,42,69], an N"2, D"3 superstring [41,42] and an N"1, D"4 supermembrane [70]. In the case where worldvolume superembedding corresponds to the type II D"10 superstrings and D-branes, the D"11 supermembrane and the super-"ve-brane the superembedding condition produces superbrane equations of motion [42,45,50,51]. It is remarkable that in such cases the
The capital N stands for the number of the spinorial supercharges of target-space supersymmetry.
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superembedding condition contains full information about the physical contents of supersymmetric theories on the worldvolumes of the superbranes. For instance, in the case of the D-branes one derives from (1.6) that the D-branes carry on their worldvolumes vector gauge "elds of the Dirac}Born}Infeld-type, and the M5-brane carries a self-dual second-rank tensor gauge "eld. The superembedding approach allowed one to get the full set of non-linear equations of motion of D-branes [50] and the M5-brane [51,52] before the Green}Schwarz-type worldvolume actions for these objects were constructed in [62}64]. When the superembedding condition produces superbrane equations of motion one cannot construct worldvolume super"eld actions for these branes. The problem is the same as in the case of extended supersymmetric "eld theories and supergravities when constraints imposed on super"elds put the theory on the mass shell. In these cases one should either consider component actions (such as Green}Schwarz-type actions), or apply a generalized action principle [72] based on a groupmanifold approach to the description of supersymmetric theories [73]. In the latter case one gets an action which produces the super"eld superembedding condition as an equation of motion, though the action itself is not a fully #edged super"eld action in the sense that the integral is taken only over a bosonic submanifold of the superworldvolume. In some cases the superembedding condition is not enough even for the o!-shell descriptions of superbranes [50]. This happens, for instance, with D6- and D8-branes of the type IIA superstring theory [74] or with space-"lling branes, such as the D9-brane of the type IIB superstring [75]. In these cases an additional superworldvolume condition is required to constrain super"elds describing gauge "elds propagating in the worldvolume of the branes. As another important and profound feature of the superembedding approach it is worth mentioning that it provides a natural link between space}time and twistor [76,77] description of relativistic systems. Actually, in the "rst series of papers on the doubly supersymmetric description of superbranes the approach was called twistor-like, and only quite recently it has acquired the present name of the geometrical [45,78] or simply the superembedding [79] approach. In the superembedding approach auxiliary commuting spinors appear as superpartners of the target superspace Grassmann coordinates and have properties of twistors [76,77,80,27] and/or Lorentz harmonics [87}94,53]. The use of twistor-like variables to formulate the theory of supersymmetric relativistic objects has the following deep grounds. When a superbrane propagates in a nontrivial gauge super"eld or supergravity background i-symmetry requires background super"elds to satisfy super"eld constraints, which (in most of the cases) are equivalent to background super"eld equations of motion. For instance, the i-symmetry of a massless superparticle propagating in a super-Yang}Mills background requires geometrical integrability of the background along light-like lines of the superparticle trajectories, which implies constraints on background super"elds. In the case of N"3, 4, D"4 or N"1, D"10 super-Yang}Mills background, for example, this is equivalent to the SYM "eld equations [95,96]. Analogous results were obtained for supergravity backgrounds
A gauge "xed action for a space-"lling D3-brane was "rst constructed in [71] as a result of studying partial breaking of N"2, D"4 supersymmetry.
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[95,97]. And it was understood that the correspondence between geometrical integrability along massless superparticle orbits and super-Yang}Mills and supergravity equations is based on a twistor interpretation of these theories. In this connection twistor formulations of superparticles and superstrings have been studied in a number of papers [30,38,54,80}86,97}100]. As we shall see, the superembedding condition (1.6) contains a Cartan}Penrose relation between vectors and commuting spinors. And the Cartan}Penrose relation turns out to be the twistor-like solution of the Virasoro-type constraints which govern the dynamics of any brane. Therefore, through the superembedding condition one establishes the relationship between various space}time, twistor-like and harmonic formulations of superparticles, superstrings and superbranes. In this review we shall consider basic features of the superembedding approach. We shall show how one arrives at the superembedding condition by constructing an appropriate worldline super"eld action for a massless superparticle. We shall demonstrate how local worldvolume supersymmetry transformations reduce to i-symmetry transformations upon the elimination of the auxiliary components of worldvolume super"elds which made local supersymmetry manifest. For those superbranes for which the superembedding condition does not put the theory on the mass shell we shall present doubly supersymmetric (worldvolume super"eld) actions which produce the superembedding condition, and will discuss their symmetry properties. We shall perform the analysis of the superembedding condition in cases when it puts the theory on-shell producing superbrane equations of motion. In these cases, such as the D"11 M2-brane and the M5-brane, we shall demonstrate how superbrane equations of motion arise as a requirement of the integrability of the superembedding condition, and will discuss the problem of constructing doubly supersymmetric actions for such branes. Since the (super)embedding approach has a rather wide range of applications, it is not possible in one review to cover in detail all points where this approach has been found to be useful. Some of them we shall just sketch referring the reader to corresponding original literature. For instance, we leave aside the relation of the method to integrable models, and an important and interesting problem of covariant quantization of Green}Schwarz superstrings, which itself requires a separate review article. The details on solving this problem with methods related to the superembedding approach the reader may "nd in [54,33,48,140,53,55]. The article is organized as follows. It can be conventionally divided into two parts. The "rst part consists of Sections 2 and 3, and Sections 4 and 5 constitute the second part. In Section 2 we introduce the basic ideas of the superembedding approach with a simple example of superparticles. This instructive example should facilitate the reader's understanding of all the features, ingredients and techniques of the approach, which will be applied (in exactly the same way) in subsequent Sections to the description of more complicated models of superstrings and superbranes. In Section 3 we discuss the superembedding formulation of N"1, D"3, 4, 6, 10 superstrings. As in the case of N"1, D"3, 4, 6, 10 superparticles, the superembedding condition does not produce dynamical equations of motion of the N"1 superstrings, and one can construct worldsheet super"eld actions, from which the dynamical equations are derived. An interesting feature of these actions is that the string tension appears there through an auxiliary dynamical variable [40]. This is a realization of the idea of a dynamical generation of the brane tension, various aspects of which
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have been discussed in Refs. [101}103]. We then describe worldsheet super"eld actions for N"1 superstrings in curved target superspaces and show that the latter should obey supergravity constraints. Finally, we introduce into the doubly supersymmetric construction chiral (heterotic) fermions which extend the N"1, D"10 closed superstring to a D"10 heterotic string. In Section 4 we present basic geometrical ideas which underlie the theory of bosonic surfaces and then extend them to the description of superembeddings. In Section 5 we apply general tools of superembeddings to the description of classical dynamics of the supermembrane and the super-5-brane of M-theory. In Section 6 we brie#y review other applications of the superembedding approach.
2. Doubly supersymmetric particles 2.1. The bosonic particle We start by recalling the form of actions for a massless bosonic particle which we will then subject to various kinds of supersymmetrization. This will allow us to describe the dynamics of particles having spin degrees of freedom. A well known quadratic action for the massless bosonic particle propagating in D-dimensional Minkowski space is
S" dq ¸(x, x , e)" dq
1 x K x L g , KL 2e(q)
(2.1)
where g is the diagonal Minkowski metric (!,#,2,#), q is the time variable parametrizing KL a particle worldline xK (q) (m"0, 12, D!1) in the target space}time, x K ,R xK and e(q) is an O auxiliary "eld which can be regarded as a (nondynamical) gravitational "eld on the worldline of the particle. The latter ensures the invariance of the action (2.1) under worldline reparametrizations (di!eomorphisms) q "f (q), the worldline "elds xK (q) and e(q) transforming as scalars and a onedimensional covariant vector, respectively, dq 1 x K (q )"xK (q), e (q )" e(q)" e(q) . dq fQ (q)
(2.2)
The consequence of the reparametrization invariance is the constraint on the dynamics of the particle 1 x K x "0 , K e or pK p "0 , (2.3) K which is obtained by varying (2.1) with respect to e(q). In Eq. (2.3) p "d¸/dx K "(1/e)x K is the K canonical momentum of the particle. The constraint (2.3) implies that the particle is massless. Using
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the canonical momentum variable p (q) we can rewrite action (2.1) in the "rst-order form K e S" dq p x K ! p pK . K 2 K
(2.4)
We will now generalize both versions of the massless particle action to describe various supersymmetric particles. 2.2. Spinning particles Let us assume that the trajectory of a particle in a bosonic space}time is not a line but a supersurface parametrized by one bosonic variable q and one fermionic variable g (which can be regarded as a one-dimensional worldline spinor). The functions XK (q, g) describing the embedding of this supersurface into D-dimensional bosonic target space become worldline super"elds XK (q, g)"xK (q)#igsK (q) ,
(2.5)
where, as above, xK (q) are the coordinates of the particle in the target space and sK are their Grassmann-odd superpartners. The one-dimensional graviton e(q) becomes a member of a (supergravity) super"eld E(q, g)"e(q)#2igt(q)
(2.6)
and thus acquires a gravitino "eld t(q) as its Grassmann-odd superpartner. The particle momentum p enters a super"eld K (2.7) P (q, g)"p (q)#igo (q) . K K K To get an action for a spinning superparticle one can supersymmetrize both action (2.1) or its "rst-order form (2.4). Since the generalization of the latter is more sophisticated [104] we shall not describe it below. Action (2.1) is generalized to a superworldline integral as follows [105]:
i 1 S"! dq dg DXK R X "! dq dg DXK D(DX ) , O K K 2E 2E
(2.8)
where R R 1 D" #ig , D" +D, D,"iR O Rg Rq 2
(2.9)
is a Grassmann covariant derivative. Action (2.8) is invariant under local worldline superreparametrizations which include bosonic worldline reparametrizations and local supersymmetry
In (2.5) and below capital letters stand for worldvolume super"elds, while corresponding small letters denote leading components of these super"elds which are associated with conventional dynamical variables of the superbranes.
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transformations. The in"nitesimal superreparametrizations are q !q"dq"K(q, g)!gDK , g !g"dg"! DK , (2.10) D !D"dD"!KQ D , where K(q, g)"a(q)#iga(q) is the superreparametrization parameter which contains the worldline bosonic reparametrization parameter a(q) and the local supersymmetry parameter a(q). Under (2.10) the super"eld XK (q, g) (2.5) transforms as a scalar XYK (q , g )"XK (q, g), its components having the following variation properties (dXK ,XYK (q, g)!XK (q, g)) i dxK "!a(q)x K ! a(q)sK , 2 dsK "!a(q)sK !a (q)sK !a(q)x K . The super"eld P (q, g) also transforms as a scalar P (q , g )"P (q, g) K K K i dp "!a(q)p ! a(q)o , K K K 2
(2.11)
do "!a(q)o !a (q)o !a(q)p . K K K K Finally, the transformation properties of the super"eld E(q, g) are
(2.12)
i dE"!R (KE)# DKDE , O 2
(2.13)
from which one derives the variation of the worldline graviton e(q) and the gravitino t(q) de"!R (ae)!ia(q)t , O (2.14) dt"!a(q)tQ !a t!a(q)e !a(q)e . Let us note that the geometry of the superworldline associated with the superreparametrizations de"ned in Eq. (2.10) is conformally #at. This means that worldline supervielbeins e (q, g)"(eO, eE) describing such a geometry di!er from #at supervielbeins e (q, g)"(eO , eE ) by a conformal factor (the subscript 0 indicates the #at basis). Namely, eO"E(q, g)eO "E(q, g)(dq#ig dg) , eE"E (q, g) dg!ieO DE . (2.15) In the case of the conformally #at worldline supergeometry (and a corresponding worldsheet supergeometry of superstrings) it is convenient to use #at supercovariant derivatives (2.10) and supervielbeins rather than curved ones and just take into account conformal factors where they are required for the covariance. (Note that, as in the #at case, such a form of the supervielbein satis"es the worldline torsion constraint ¹O,deO"!ieEeE.)
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Integrating (2.8) over g (using the Grassmann integration rules dg"0 and dg g"1) we get the component action
S" dq
1 i (x K x #isK s )! ts x K . K K 2e e K
(2.16)
In addition to the &masslessness' constraint (2.3) action (2.16) yields the fermionic constraint which comes from the variation of the last term in (2.16) with respect to t(q) sK p "0 , K where now the canonical momentum is
1 i p " x ! ts K e K e K
.
(2.17)
(2.18)
(Note that since s are Grassmann-odd, s sK ,0.) K K Upon quantization the constraint (2.17) becomes the Dirac equation imposed on a "rstquantized spinorial wave function. Therefore, action (2.3) or (2.16) describes a relativistic particle with spin 1/2, and the Grassmann vector sK describes the spinning degrees of freedom of the particle [106,105]. To get an action for relativistic particles of an arbitrary spin n/2 (where n is a natural number) one should consider the worldline of a particle to be a supersurface parametrized by n fermionic variables gO (q"1,2, n) [107,108]. 2.3. Superparticles, i-symmetry Let us now do an opposite thing. Instead of considering the embedding of a worldline supersurface into a bosonic D-dimensional space}time consider an embedding of a bosonic worldline into a #at target superspace parametrized by bosonic coordinates xK (m" 0, 1,2, D!1) and Grassmann spinor coordinates hI (k"1,2, 2 " ). Depending on the dimen sion D the spinors h can be chosen to be Dirac, Majorana or Majorana}Weyl ones. The global supersymmetry transformations of the target superspace coordinates are dhI "eI , dxK "ihM CK dh ,
(2.19)
and we would like to construct an action for a particle which would be invariant under these transformations. For the construction of such an action one uses supercovariant one-forms E "(E? , E? ) which form an orthogonal supervielbein basis in the #at target superspace E? "(dxK !i dhM CK h)d? , E? "dhI d? . (2.20) K I The target superspace geometry is #at, i.e. it has zero curvature, but non-zero torsion (the external di!erential acts from the right) T? "dE? "!i dhM C? dh .
(2.21)
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[When superspaces are #at we shall usually not make a distinction between tangent-space &#at' indices (from the beginning of the alphabets) and coordinate indices (from the middle of the alphabets), since in this case the two types of indices are simply related by the unit matrix, as in the de"nition of the supervielbeins (2.20).] The pullback of supervielbeins (2.20) onto the particle worldline with the image Z+ (q)"(xK (q), hI (q)) is (2.22) E? (Z(q))"dq E? (Z(q))" dq (R x? !iR hM C? h) , O O O E? (Z(q))"dq R h? (q) . O The superparticle action is obtained by replacing x K in (2.1) or (2.4) with the vector component E? of the supervielbein pullback (2.22). Thus we get [109}111] O 1 E? E@g , (2.23) S" dq 2e(q) O OM ?@
or in the "rst-order form
e S" dq p E? ! p p? . ? O 2 ?
(2.24)
The superparticle equations of motion derived from (2.23) or (2.24) are R
1 E? "R p? "0, (E? C )? R h@M "(p? C )? R h@M "0 , O e(q) O O ? @M O O ? @M O
(2.25)
E? "R x? !iR hM C? h"e(q)p . O ? O O By construction actions (2.23) and (2.24) are invariant under the target-space supersymmetry transformations (2.19), and under the worldline reparametrization of xK , e(q) (2.2), p (q )"p (q) K K and h (q )"h(q), which is responsible for the mass-shell condition (2.3). In addition, these actions possess a hidden worldline fermionic i-symmetry [60,61]. The i-transformations of the worldline "elds are d hI "i(p CK )I iJ (q) , K J G d xK "!ihM CK d h N i E? ,dZ+ E? "0 , G G + G (2.26) d e"4hMQ i , G d p "0 , G K where iI (q) is a Grassmann spinor parameter of the i-transformations. One can notice that not all iI (q) contribute to the i-transformations, and that, in fact, transformations (2.26) are in"nitely reducible. Consider a particular choice of iI (q) when iI (q)"i(p CK )I i J (q) . K J
(2.27)
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Then, due to the de"ning relations for the Dirac matrices +CK , CL ,"CK CL #CK CL "2gKL ,
(2.28)
the transformation of h takes the form (2.29) d hI (q)"!p pK i I . K G This transformation vanishes on the mass shell p"0 (2.3). Therefore, the parameters of form (2.27) do not eliminate any on-shell gauge degrees of freedom of the superparticle. If in (2.27) we choose i "p CK i we will see that on mass shell (2.3) parameter (2.27) turns to K zero. We can continue such a substitution an in"nite number of times and "nd that at any stage there are i-parameters for which the i-transformations are trivial on the mass shell. The reason for this (in"nite) reducibility is that the spinorial matrix p CK has the rank which on mass shell (2.3) is K half the maximum rank, i.e. half the dimension 2 " of the spinor. This means that among the 2 " components of iI only half of the components are independent and e!ectively contribute to the i-symmetry transformations. This also concerns fermionic constraints on the superparticle dynamics whose appearance is the consequence of i-symmetry. In the Dirac terminology [112] these constraints belong to the "rst class, and in the canonical Hamiltonian formulation they are regarded as the generators of the local i-symmetry. The constraints in question are n (q)(p CK )I "0 , K J I where
(2.30)
d¸ n " "ihM (p CK )J J K I I dhQ I is the momentum canonically conjugate to the Grassmann coordinate h. Since the expression for n does not contain time derivatives it is a fermionic spinorial constraint (2.31) D "n !ihM (p CK )J "0 . I J K I I Under the canonical Poisson brackets [p , xL ]"dL , +n , hJ ,"dJ the constraints (2.31) have the K I K I anticommutation properties of the super-Poincare algebra (2.32) +DM I , D ,"!2i(p CK )I . J K J As we have seen, the matrix on the right-hand side of (2.32) is degenerate and has the rank one half of the maximum rank. Thus half the constraints (2.32) are of the second class and another half are of the "rst class, the anticommutator of the latter being (weakly) zero. Without introducing auxiliary "elds it turns out to be impossible to split the fermionic spinor constraints (2.31) into irreducible Lorentz covariant sets of "rst and second class constraints because the spinor representation of the Lorentz group is the fundamental one and cannot be decomposed into any other Lorentz group representations.
The brackets [ ,] and + ,, stand for commutation and anticommutation relations, respectively.
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The maximum we can do in the present situation is to get an in"nite reducible covariant set of the "rst class fermionic constraints (2.30) multiplying (2.31) by p CK , or to break manifest Lorentz K covariance. If we try to covariantly quantize the superparticle model, the in"nite reducibility of i-symmetry and of the corresponding constraints will require the introduction of an in"nite set of ghosts for ghosts [113] and, in addition, we should also manage with the second class constraints contained in (2.31). All this makes the problem of the covariant quantization of superparticles and superstrings a di$cult one. To be able to split the fermionic constraints (2.31) into irreducible sets of "rst and second class constraints it has been proposed to enlarge the space of superparticle (and superstring) variables with auxiliary bosonic spinor variables (twistors [77,80}84,27], or Lorentz harmonics [87}94,53]) and to use these variables to carry out a covariant split of the constraints. Below we shall see that these auxiliary variables naturally appear in the doubly supersymmetric formulations of the super-p-branes. 2.4. Spinning superparticles Let us now make the next step and to construct particle models which would be invariant under both the worldline (2.10) and the target-space (2.19) supersymmetry transformations. The worldline of such a particle is a supersurface z+"(q, g) embedded into the target superspace Z+ "(XK , HI ). I.e. the particle trajectory is Z+ (z+)"Z+ (q, g)"(XK (q, g),HI (q, g)) ,
(2.33)
where the worldvolume super"eld XK (q, g) is the same as in (2.5) and HI (q, g)"hI (q)#gjI (q) .
(2.34)
We see that the Grassmann spinor coordinate h(q) of the particle acquires a commuting spinor superpartner j(q). Under worldline superreparametrizations (2.10) H(q, g) transforms as a scalar super"eld dhI "!a(q)hQ I !a(q)jI , 1 i djI "!a(q)jQ I ! a (q)jI ! a(q)hQ I . 2 2
(2.35)
The pullback onto the superworldline of the superinvariant forms E "(E? , E? ) (2.20) takes the form E? (Z(z+))"dz+E? (Z(z))"dz+ R H? (z)"eO R H? #eE DH? , + + O (2.36) E? (Z(z))"dz+E? (Z(z))"eO (R X? !iR HM C? H)#eE (DX? !iDHM C? H) , O + O where we have expanded the target superforms in the #at supervielbein basis (2.15) of the superworldline. Note that the worldline spinor components of the superforms (2.36) are basic ones in the sense that the &q'-components can be constructed from the former by applying to the &g'-components the
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covariant derivative D. We thus have R H"!iD(DH) , O (2.37) E? "R X? !iR HM C? H"!iD(DX? !iDHM C? H)!DHM C? DH . O O O Therefore, it is su$cient to use the basic &g'-components to construct doubly supersymmetric actions. It turns out that depending on which action (2.8), (2.23) or (2.24) is generalized to acquire the second supersymmetry we get di!erent doubly supersymmetric particle models. The second supersymmetrization of actions (2.8) and (2.23) results in a same doubly supersymmetric action which describes the dynamics of so called spinning superparticles [67,114]. To get this action we should, for example, simply replace DXK in (2.10) with the basic &g'-component of the target-space vector supervielbein pullback (2.36) E? "DX? !iDHM C? H . E The resulting action is [67,114]
S"! dq dg
1 E? DE . 2E E E?
(2.38)
(2.39)
We shall not discuss the properties of the particle model described by the action (2.39). The reader may "nd the details in Refs. [67,114,31]. We only mention that the "rst-quantized spectrum of this particle is the direct product of the spectra of the corresponding spinning particle (2.8) and superparticle (2.23). And this is the reason for the name &spinning superparticle'. For instance, in D"4 the "rst-quantized states of the spinning particle (2.8) have spin . The quantization of the N"1, D"4 massless superparticle results in states described by a chiral supermultiplet (0, ) which contains one complex scalar and one Weyl spinor. The direct product of these spectra "ts into an N"1, D"4 chiral scalar (0, ) and vector (, 1) supermultiplet. Therefore, the "rst-quantized states of the D"4 spinning superparticle have two spins 0, two spins and spin 1. This spin content is the same as one described by an N"2, D"4 vector supermultiplet. The latter arises as a result of the quantization of an N"2, D"4 superparticle [115]. The equivalence of the two models was demonstrated in [114,31]. We now turn to the construction of a doubly supersymmetric action which would describe a particle model with the same physical content as the standard superparticle model based on actions (2.23) and (2.24), the role of the local worldline supersymmetry being to substitute the i-symmetry. 2.5. Worldline superxeld actions for standard superparticles. The superembedding condition We have seen that the number n of the independent i-symmetry transformations is half the number of components of the Grassmann spinor coordinates of the superparticle. Therefore, if a superparticle propagates in superspace with D bosonic and 2 " fermionic directions, one should construct an n"2 " \ worldline supersymmetric action for being able to completely replace n i-symmetries with n-extended worldline supersymmetry. We shall see that such actions can be
D. Sorokin / Physics Reports 329 (2000) 1}101
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constructed for N"1, D"3, 4, 6 and 10 massless superparticles. A particular property of these critical bosonic dimensions is that there exists a twistor-like (Cartan}Penrose) representation of a light-like vector in terms of a commuting spinor. Note that standard classical Green}Schwarz superstrings also exist only in these dimensions [58,1], which is implicitly related to the same fact. Superparticles (including massive ones as [34,35]) and superstring models in other space}time dimensions can be obtained from these basic models by a dimensional reduction [38,69]. Let us "rst consider the simplest case of an N"1, D"3 superparticle. 2.5.1. N"1, D"3 superparticle and twistors In this case the Majorana spinor has two components and i-symmetry has only one independent parameter. Our goal is to identify the independent i-symmetry transformations of (2.26) with the local worldline supersymmetry transformations (2.10). To this end we write down a worldline supersymmetric version of the "rst-order action (2.24)
L S" dq dg P E? ! P P? , ? E 2 ?
(2.40)
where the worldline even super"eld P and the odd super"eld E? are de"ned in Eqs. (2.7) and (2.38), ? E respectively, and the Lagrange multiplier super"eld L(q, g) is Grassmann odd for the second term of (2.40) to have the right Grassmann parity. By construction action (2.40) is invariant under the worldline superreparametrizations (2.10) and target-space supersymmetry transformations (2.19), and it is also invariant under fermionic transformations (2.26), where all variables are replaced by corresponding super"elds, and d/dq is replaced by the supercovariant derivative D. At the "rst glance it seems that we have not got rid of the i-symmetry since it appeared again at the worldline super"eld level. However, as we shall see, second term of (2.40) is unnecessary [38]. Its role would be to produce a super"eld generalization of the mass shell condition (2.3), but, it turns out that Eq. (2.3) follows already from the "rst term of the action (2.40). We therefore skip the second term. Then the resulting action [27]
S"!i dq dg P E? "!i dq dg P [DX? !iDHM C? H] ? E ?
(2.41)
does not have any redundant i-symmetry, and, as we shall now demonstrate, is equivalent to the standard superparticle action (2.24) in D"3. As we shall see in Section 2.5.4 in N"1, D"4, 6 and 10 target superspaces the n"1 worldsheet super"eld action (2.41) also describes standard superparticles, since then in addition to n"1 worldsheet supersymmetry it has D!3 hidden local fermionic i-symmetries. Note that action (2.41) is of a &topological' Chern}Simons nature [32] since it is invariant under the local worldline superreparametrizations but does not contain the superworldline metric. Performing g-integration in (2.41) we get the following action for the components of the super"elds (2.5), (2.7) and (2.34)
S" dq p (x ? !ihMQ C? h!jM C? j)#i dq o (s? !jM C? h) . ? ?
(2.42)
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The second term in (2.42) reads that o and s? are auxiliary "elds satisfying algebraic equations ? o "0, s? "jM C? h . (2.43) ? The equations of motion of x? and h? are, respectively, R p "0, (p C? )? R h@M "0 . (2.44) O ? ? @M O They are the same as Eqs. (2.25) yielded by action (2.24). We should now show that the mass shell condition (2.3) also follows from action (2.42). This can be derived in two ways. The variation of (2.42) with respect to p gives ? ,x ? !ihMQ C? h"jM C? j , E? " (2.45) O E from which it follows that g E? E@M " "0 . ?@ O O E This is because the square of the r.h.s. of (2.45) is identically zero
(2.46)
(jM C? j)(jM C j),0 ? due to a property of the C-matrices in D"3
(2.47)
(CK )? (CC ) #(CK )? (CC ) #(CK )? (CC ) "0 , (2.48) K A BM K BM @M K BM @M @M A BM where C "e is the charge conjugation matrix which can be used to raise and lower the spinor ?@ ? @M indices. M Identities (2.47) and (2.48) can be easily checked in the Majorana representation of the Dirac matrices where the Majorana spinor is real, and
0 1 0 1 1 0 , C " , C " . C"C " !1 0 1 0 0 !1
(2.49)
Identity (2.48) also holds in D"4, 6 and 10 space}time dimensions, and Eq. (2.47) is valid in these dimensions for the Majorana (or Weyl), simplectic S;(2) Majorana}Weyl [117,120] and the Majorana}Weyl spinors, respectively. As a consequence of these identities any light-like vector in these dimensions can be represented as the bilinear combination of commuting spinors (2.45). Such a representation is called the Cartan}Penrose (or twistor) representation of the light-like vector. Note that its existence in the special dimensions D"3, 4, 6 and 10 is related to the existence of four division algebras associated with real, complex, quaternionic and octonionic numbers, respectively, (see [116}119] as nice reviews for physicists) which form an algebraic basis for corresponding twistor constructions [76,81]. In application to the superembedding formulation of superparticles and superstrings these structures were exploited, for example, in Refs. [80,81,27,29,82,83,121]. Let us now derive the Cartan}Penrose relation as a solution of the equation of motion of j (p C? )? j@M "0 . @M ? If the matrix p C? is non-degenerate Eqs. (2.44) and (2.50) are satis"ed only when ? R h"0, j"0 , O
(2.50)
(2.51)
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which (in virtue of (2.45)) leads to R xK "0 . O Such a solution describes a particle which is frozen in a single point of super space}time (it even does not evolve along the time direction x). Since this solution is physically trivial it can be discarded by requiring that the components of j do not turn to zero simultaneously. This requirement is in agreement with the de"nition of twistor variables [76] which are commuting spinor variables parametrizing a projective space. Then Eq. (2.50) has non-trivial solutions if the matrix p C? is degenerate, the general solution of (2.50) being ? 1 (2.52) p " jM C j, N p p? "0 , ? ? e(q) ? or, because of the C-matrix identity (2.48), 2 p (CC? ) " j j . ? @M ? e(q) ? @M
(2.53)
Thus we have again arrived at the Cartan}Penrose representation of the light-like vector. Comparing (2.52) with (2.45) we see that 1 1 " (x ? !ihMQ C? h) p? " E? " e(q) O E e(q)
(2.54)
has the meaning of the superparticle canonical momentum as in the standard formulation of Section 2.3, and the multiplier e(q) can be identi"ed with the worldline gravitation "eld. One can notice that the Cartan}Penrose relation (2.52) establishes a one-to-one correspondence between the two-component commuting spinor j? and the D"3 light-like vector p? which also has two independent components. [The graviton e(q) is completely auxiliary and can be gauge "xed to a constant by the worldline reparametrizations (2.2).] Therefore, either p? or j? can be considered as independent dynamical variables in the phase space, and the number of physical degrees of freedom of the doubly supersymmetric particle is the same as that of the standard superparticle. If we choose p? as an independent variable then independent equations of motion (2.44), (2.52) and (2.54) of the dynamical variables p? , x? and h? derived from action (2.42) coincide with the massless superparticle equations (2.25) and (2.3) yielded by action (2.24). The local worldline supersymmetry can be used to eliminate one (pure gauge) degree of freedom of h? . All this testi"es to the classical equivalence of the two actions. The only thing which remains to establish is the relationship of the worldline supersymmetry and i-symmetry. For this consider the worldline supersymmetry transformations (2.11), (2.12)
Note that in the standard formulation of relativistic particles this requirement is analogous to excluding the point e(q)"0 from the solutions of the particle equations of motion (2.25).
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and (2.35) of hI , jI , x? and p? , when the relations (2.43) for the auxiliary "elds o and s? hold, ? 1 i dhI "! jI a(q), djI " hQ I a(q) , (2.55) 2 2 i dx? " hM C? ja(q)"!ihM C? dh , 2
(2.56)
dp? "0 .
(2.57)
Without losing generality we can always replace the supersymmetry parameter a(q) with the following expression: 4i a(q)"! jM i? , e ?
(2.58)
where i (q) is an arbitrary Grassmann spinor. If we now substitute Eq. (2.58) into (2.55) and (2.56) ? and take into account (2.53) we shall see that the supersymmetry transformations become the i-symmetry transformations (2.26) for x? and hI . It remains to derive the i-variation of the worldvolume "eld e(q) (2.26). To this end we make use of relation (2.52), where e(q) appears in the formulation under consideration, and observe that when o? "0, the particle momentum is invariant (2.57) under the local supersymmetry transformations (2.12). For consistency we must require that the r.h.s. of the Cartan}Penrose representation (2.52) of p? is also invariant under the supersymmetry transformations. To show this we should "nd the supersymmetry variation of e(q) which cancels the variation of j (2.55). At this point we should also take into account that h satis"es the equation of motion (2.44). This requirement is justi"ed by the fact that the worldline supersymmetry transformations relate the bosonic kinematic equation (2.50), and its solution (2.52), (2.53), to the fermionic dynamical equation (2.44). In view of (2.50) and (2.53) the general solution of (2.44) is hQ I "jI tI (q) ,
(2.59)
where tI (q) is a Grassmann-odd worldline function. Then the required worldline supersymmetry variation of e(q) is de(q)"!ie(q)a(q)tI (q) .
(2.60)
Substituting (2.58) in (2.60) we get the i-symmetry variation (2.26) of e(q). Therefore, fermionic i-symmetry is nothing but the worldline supersymmetry transformations of the superparticle dynamical variables when the auxiliary "eld components of the corresponding worldline super"elds are eliminated. The independent i-parameter (or the supersymmetry parameter) is the (Lorentz-invariant) projection of the spinorial i-symmetry parameter onto the commuting spinor j (2.58). The (irreducible) "rst class fermionic constraint which generates n"1 worldline supersymmetry (and/or irreducible i-symmetry) is j? D "0, +jD, jD,"!2ip jM CK j"0 , ? K
(2.61)
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where D was de"ned in (2.31). This completes the proof of the classical equivalence of the doubly ? supersymmetric model based on action (2.41) and standard superparticle (2.23), (2.24) in N"1, D"3 superspace. Before going further it is worth mentioning the relation of the doubly supersymmetric action (2.41) and (2.42) with twistor formulations of (super)particles [77,80}84,31,86]. We have already observed similarity between the commuting spinor variables j? and twistors which manifests itself through the Cartan}Penrose relation between j? and p (2.52). K p in (2.42) with If we consider j? and not p as independent dynamical variables we can replace K K j? . Then hiding e(q) by rede"ning j? (e\ j? Pj? ) and dropping out the term with the fermionic auxiliary "elds we get
S" dq jM C j(x ? !ihQM C? h) . ?
(2.62)
This action is one on which the supertwistor formulation [77,80}82] of superparticles in space}times of dimension D"3, 4, 6 and 10 is based. Various versions of the supertwistor formulation have been developed in a number of papers [80}86,30,100]. The doubly supersymmetric formulation provides us with a natural link between standard (2.23) and the twistor description of superparticles. 2.5.2. On-shell relationship between superparticles and spinning particles We have seen that in the doubly supersymmetric formulation of superparticles the Grassmann superpartners sK of the particle coordinates xK are treated as auxiliary variables expressed in terms of h? and j? (Eq. (2.43)). Note that Eq. (2.43) can be regarded [46] as an odd counterpart of the Cartan}Penrose twistor relation (2.45), in fact, they are worldline supersymmetry partners. On the contrary, in the theory of spinning particles (Section 2.2) sK is a dynamical (spin) variable obeying the equations
i sK p "sK x ! ts "0 , K K e K
(2.63)
1 e sK " tx K # sK , e 2e
(2.64)
which are yielded by the spinning particle action (2.16). We shall now demonstrate that equations (2.63) and (2.64) are equivalent to the superparticle equations (2.44). Remember that the solution of the fermionic equation in (2.44) is Eq. (2.59). Substituting (2.59) into the expression for the superparticle momentum (2.54) and taking into account (2.43), i.e. that s? "jM C? h, we have 1 1 p? " (x ? !ihMQ C? h)" (x ? !itI s? ) . e(q) e(q)
(2.65)
We see that in this form the superparticle momentum coincides with the spinning particle momentum (2.18) if we take tI "(1/e)t, both momenta satisfying the same equation of motion p ? "0. Contracting (2.65) with s? "jM C? h and taking into account the Cartan}Penrose relation (2.52) for p we "nd that Eq. (2.63) is identically satis"ed. ?
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We now check that the solution sK "hM CK j is compatible with equation of motion (2.64). For this take the time derivative of sK . Then, in view of (2.59), (2.52), (2.65) and that we put tI "(1/e)t, we get
1 1 1 sK " tjM CK j#hM CK jQ " t x K ! tsK #hM CK jQ . e e e
(2.66)
jQ is found to be e jQ " j 2e
(2.67)
by solving
1 p K "R jM CK j "0 . O e Hence, the last term in (2.66) is equal to (e /2e)sK , and Eq. (2.66) coincides with Eq. (2.64). We have shown that the dynamical variables of the N"1, D"3 superparticle and the n"1 spinning particle, and their equations of motion are related to each other by the Cartan}Penrose (twistor-transform) expressions (2.43) and (2.45) 1 pK " jM CK j, sK "hM CK j . e Therefore, the doubly supersymmetric action (2.41) can be regarded as a &master' action for both types of the particles. It provides us with superparticle or spinning particle equations of motion depending on whether the Grassmann vector sK or the spinor hI is chosen as an independent dynamical variable. The two particles are thus classically equivalent, though quantum mechanically (as is well known) these are systems with di!erent spectra of states. The on-shell relationship between N"1 superparticles and n"1 spinning particles [46,47], and between N"1 superstrings and n"1 spinning strings [48,49] holds also in space}time dimensions D"4, 6 and 10. To demonstrate this relationship one should gauge "x all but one local worldsheet supersymmetries (or i-symmetries) of the target space supersymmetric objects. The remaining local worldsheet supersymmetry is then identi"ed with n"1 worldsheet supersymmetry of the corresponding spinning objects. The commuting twistor-like spinors relate the dynamical variables of the two types of models. It is well known that the equivalence between D"10 superstrings and D"10 spinning strings extends to the quantum level so that the two models describe one and the same quantum string theory, provided the NSR spinning strings have been GSO-projected. The &twistor' transform demonstrated above allows one to utilize useful properties of both the Green}Schwarz and the NSR formulation to study the problem of covariant quantization of superstring theory [54,33,48,140,55]. Another example of quantum equivalent systems (which we have mentioned in Section 2.4) is the equivalence between the free N"1, n"1 spinning superparticle and the N"2 superparticle in D"4 [114].
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2.5.3. The superembedding condition We now return to the consideration of action (2.41) and repeat its analysis directly at the worldline super"eld level with the purpose of getting the principal condition of the superembedding of the particle superworldline into target superspace and to analyze its properties. The superembedding condition is derived by varying action (2.41) with respect to the super"eld P (q, g), and it reads that the superparticle moves in target superspace in such a way that the ? pullback of the target-space supervielbein vector component along the Grassmann direction of the superworldline is zero E? "DX? !iDHM C? H"0 . E The consequence of Eq. (2.68) is obtained by hitting (2.68) with the derivative D
(2.68)
R X? !iR HM C? H!DHM C? DH"0 . (2.69) O O Eq. (2.69) implies that the component of E? along the time direction of the worldline is expressed in terms of the spinor components E? "dH? of the target space supervielbein (2.36) along the Grassmann direction of the worldline E? (q, g),R X? !iR HM C? H"DHM C? DH . (2.70) O O O This is the super"eld version of the Cartan}Penrose relation (2.45) which states that E? (q, g) is O light-like and, hence, the superparticle is massless. Condition (2.70) is the only consequence of Eq. (2.68) which one has in the case under consideration. We see that the superembedding condition (2.68) itself does not lead to the dynamical equations of motion of the superparticle (2.44). The super"eld form of the dynamical equations is obtained from action (2.41) by varying it with respect to X? and H? DP "0, P C? DH"0 . (2.71) ? ? Therefore, in the case of the N"1, D"3 superparticle the superembedding condition (2.44) &prescribes' the superparticle to be massless but does not completely de"ne its classical dynamics. The classical equations of motion of the superparticle should be derived separately, for example, from the action principle as above. It turns out that the same situation holds for superparticles in N"1, D"4, 6 and 10 target superspaces. 2.5.4. N"1, D"4, 6, 10 superparticles The i-symmetry of the N"1, D"4, 6, 10 superparticles has n"D!2"2, 4, 8 independent parameters, respectively. Hence, to replace i-symmetry with local worldline supersymmetry one should consider the embedding into N"1, D"4, 6, 10 target superspace of a superworldline M parametrized by the bosonic time variable q and n"D!2 fermionic variables gO L (q"1,2, D!2). Note that, in general, one can consider doubly supersymmetric models in D"4, 6 and 10 with the number n of worldvolume supersymmetries being less than D!2. Such models also describe standard superbranes but with only part of the i-symmetries represented as manifest n-extended worldline supersymmetry, the other D!2!n i-symmetries remaining hidden. In the case of
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superparticles and superstrings N"1 doubly supersymmetric formulations with n(D!2 are used for establishing the classical relationship between spinning and superobjects [46}49] as well as for superstring quantization [54,140]. For instance, the n"1 worldline super"eld action (2.41) is also a consistent action for standard N"1, D"4, 6, 10 superparticles, since it does possess additional fermionic symmetries which complement the n"1 worldline supersymmetry to n"D!2 i-symmetries. In the D"4 case the additional i-symmetry transformations are d HI "K(q, g)(CDH)I , C"iCCCC , G (2.72) d XK "!iHM CK dH, dP "0 , ? G where K(q, g)"i(q)#gb(q) is an n"1 super"eld parameter independent of the n"1 supersymmetry parameter K(q, g) (2.10). The bosonic component b(q) of K(q, g) allows one to gauge away one of the four real components of jI . The remaining three components correspond to three independent components of the D"4 lightlike vector jM CK j associated with the superparticle momentum. The appearance in (2.72) of the matrix C re#ects (in the language of the Majorana spinors) the presence of the complex structure inherent to the Weyl representation of the D"4 spinors. (The symmetry with the parameter b(q) then becomes local ;(1) symmetry acting on Weyl spinors j.) This allows one to promote the second fermionic symmetry (2.72) to the second manifest worldsheet supersymmetry by constructing an n"2 super"eld action in N"1, D"4 superspace as an alternative to (2.41) (see [27,36,38]). In D"6 action (2.41) should possess 4 fermionic symmetries one of which is n"1 worldline supersymmetry (2.10). To show how the other three look we should introduce the notion of the S;(2) simplectic Majorana}Weyl spinors (see [117,120] for details). The spinor index a"1,2, 8 splits into the S;(2) index i"1, 2 and the index a"1,2, 4 of a fundamental representation of S;H(4)&SO(1, 5). By de"nition the S;(2) simplectic Majorana}Weyl spinor satis"es a pseudoreality condition : hM ? "B? h@He , h?G " G @ HG where the matrix B is de"ned by the conditions
(2.73)
BcK B\"(cK )H, BHB"!1 and * denotes complex conjugation. The 4;4 matrices cK replace the D"6 Dirac matrices and are analogs of 2;2 p-matrices in D"4. They are antisymmetric cK "!cK . ?@ ?@ Then 3 hidden local fermionic symmetries of the action (2.41) in D"6 are given by
(2.74)
dH?G"K (q)(p')G DH?H , (2.75) ' H where (p')G (I"1, 2, 3) are the S;(2) Pauli matrices. X? and P are transformed as in (2.72). ? H The commuting spinor j has now 8 real components. 5 of these components correspond to "ve ?G independent components of a D"6 lightlike vector (the particle momentum), and another three
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27
are to be pure gauge degrees of freedom. Indeed, they are eliminated by local S;(2) transformations dj?G"b (q)(p')G j?H (2.76) ' H which are part of (2.75), where b (q)"DK " . ' ' E The n"4 super"eld generalization of (2.41), which involves only one twistor-like spinor j?G and makes symmetries (2.75) manifest, was constructed in [36] by the use of the harmonic superspace technique [87]. In the D"10 case the situation becomes much more complicated since to furnish all 8 fermionic symmetries of the action (2.41) in an irreducible form one should deal with the non-associative octonionic structure [29]. [A corresponding n"8 octonionic super"eld action which would directly produce (2.42) is unknown.] However, it is possible to write down a reducible set of i-transformations which complement the n"1 local supersymmetry (2.82) to eight independent fermionic symmetries of the action (2.41) in the D"10 case [30,48] d H? "(DHM C DH)(C? K)? !2DH? (DHM K) . ? G d XK "!iHM CK d H . G G The transformations (2.77) are reducible since the parameters K? (q, g) in the form
(2.77)
K"(C? K )DH, K "KI (DHM C DH) ? ? ? do not contribute to (2.77). It can be shown that the number of independent components in K? is 7. The fermionic superparameter K? (q, g) contains bosonic parameters which allow one to eliminate 7 of the 16 components of j? , so that the remaining 9 components again correspond to a lightlike D"10 vector. It is worth mentioning that the appearance in (2.72)}(2.77) of extra bosonic gauge transformations which reduce the number of independent components of jI is related to the well known fact that in D"3, 4, 6 and 10 the commuting spinors (&twistors') with n"2(D!2)"2, 4, 8 and 16 components parametrize, respectively, S, S, S and S spheres. (In each case one spinor component can be "xed to be a constant by scale transformations, which corresponds to worldline reparametrizations when these spinors appear in particle models.) These spheres are Hopf "brations ("ber bundles) which are associated with the division algebras R, C, H and O of the real, complex, quaternionic and octonionic numbers. The bases of these "ber bundles are, respectively, the spheres S, S, S and S, and the "bers are the group manifolds Z, S";(1) and S"S;(2), and a sphere S (which is not a group manifold). The base spheres correspond to and are parametrized (up to a scaling factor) by light-like vectors (massless particle momenta) in D"3, 4, 6 and 10, respectively, and the "bers correspond to pure gauge degrees of freedom associated with additional gauge symmetries of the superparticle models discussed above. We have thus shown that the n"1 super"eld action (2.41) is non-manifestly i-symmetric in D"3, 4, 6 and 10 space}time dimensions and, hence, describes N"1 superparticles. The generalization of the action (2.41) to a d"2 worldsheet will describe N"1 D"3, 4, 6, 10 tensionless superstrings. As we have already mentioned there are di!erent ways of constructing appropriate (classically equivalent) manifestly n"D!2 worldline supersymmetric actions for N"1 superparticles
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[27,36] and N"1 superstrings [28,33,34,37] in D"4 and 6 target superspaces which make use of complex and quaternionic (or harmonic) analyticity structure inherent to these superspaces. However, the simplest way (and the only known one for N"1, D"10 superparticles [58] and superstrings [33,40]) is to write down a straightforward generalization of the action (2.41). For this we should de"ne an appropriate geometry of an n"D!2 worldline superspace M . L As in the case of n"1 worldline superspace (2.10), (2.15) the supergeometry of M can always L be chosen to be superconformally #at. I.e. the worldline supervielbeins have the form eO"E(q, g)eO "E(q, g)(dq#igO dgO) , eO"E (q, g) dgO!ieO DOE , q"1,2, n (2.78) (indices p, q, r, s2 from the end of the Latin alphabet will be always reserved for a representation of an internal group transformations of gO). And one can again work with #at supervielbeins eO and eO and #at supercovariant derivatives R D " #ig R , +D , D ,"2id R . (2.79) O RgO O O O P OP O For the worldline superdi!eomorphisms z +"z +(z,) of the M coordinates z+"(q, gO) to L preserve the conformal structure on M they must be restricted to satisfy the constraint L D q !ig PD g P"0 , (2.80) O O which implies that the odd supercovariant derivatives transform homogeneously under the restricted superdi!eomorphisms D "D g PD . (2.81) O O P The in"nitesimal form of the superreparametrizations (restricted superdi!eomorphisms) (2.80) is determined (as in Eq. (2.10)) by a single unconstrained superparameter K(q, gO) dq"K(q, g)!gOD K , O i dgO"! DOK , 2
(2.82)
i 1 dD "! KQ D # [D , D ]KDP . O 4 O P O 2 In addition to the bosonic reparametrization parameter a(q)"K" and the worldline supersymE metry parameter a (q)"!iD K" the superfunction K(q, g) contains parameters of local SO(n) O O E rotations b (q)"[D , D ]K" . The fermionic coordinates gO transform under an n-dimensional OP O P E representation of SO(n). The SO(n) indices are raised and lowered by the unit matrix and hence there is no distinction between them. Let us embed the superworldline M into N"1, D"4, 6, 10 target superspace. The image of L M is described by the worldline super"elds Z+ (z+)"(XK (q, gO), HI (q, gO)) which now have many L
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29
more components than in the n"1 case XK (q, gO)"xK (q)#igOsK (q)#2 , O HI (q, gO)"hI (q)#gOjI (q)#2 , O
(2.83) (2.84)
where dots denote terms of higher orders in gO. The pullback onto M of the target-space L supervielbeins (2.20) is E? (Z(z+))"dz+ E? (Z(z))"dz+ R H? (z)"eO R H? #eO D H? , O O + + (2.85) E? (Z(z))"dz+ E? (Z(z))"eO (R X? !iR HM C? H)#eO (D X? !iD HM C? H) O O O + O We postulate that the dynamics of massless superparticles in dimensions D"4, 6, 10 is governed by the superembedding condition similar to Eq. (2.68) (2.86) E? (Z(z))"D X? !iD HM C? H"0 . O O O Let us analyze the consequences of (2.86). We shall see that (2.86) contains the mass-shell condition of massless superparticle dynamics, and that only the leading components of the super"elds (2.83) and (2.84) are independent, while all other components are auxiliary and expressed in terms of the leading components and their derivatives [33,42]. Taking the covariant derivative D of (2.86), symmetrizing the result with respect to the indices P r and q and taking into account Eq. (2.79) we get (2.87) d (R X? !iR HM C? H)"D HM C? D H , O O P OP O where the r.h.s of (2.87) is automatically symmetric in q and r because of symmetry properties of the C-matrices in D"3, 4, 6 and 10. From Eq. (2.87) it follows that 1 E? (z+)"R X? !iR HM C? H" D HM C? D H , O O O O D!2 O
(2.88)
1 D HM C? D H" d D HM C? D H . O P Q D!2 OP Q
(2.89)
and
Condition (2.88) is analogous to (2.70), and (2.89) is an additional one. It, in particular, restricts the in number of independent components of the n"D!2 commuting spinors jI "D HI " O O E super"eld (2.84) in such a way that E? determined in (2.88) is lightlike. To check this one should O simply convince oneself that the square of the r.h.s. of (2.88) is identically zero due to the properties (2.48) of the C-matrices and by virtue of relation (2.89). Hence, the superparticle with the superworldline embedding speci"ed by condition (2.86) is massless. We shall now show that the superembedding condition (2.86) completely determines the higher components of the super"elds (2.83) and (2.84) in terms of their leading components xK and hI up to bosonic local worldline reparametrizations and local SO(D!2) rotations (2.82).
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To this end one should prove that because of Eq. (2.89) the number of independent components of the bosonic matrix D HI is equal (up to the SO(D!2) rotations) to the number of the O independent components of the light-like vector E? (2.88). O The vector E? has (up to q-reparametrizations) n"D!2 independent components and parO ametrizes an SL-sphere. Indeed, the condition E? E@M g "0 implies that O O ?@ G"\ (E"\)# (EG )"(E ),(R X !iR HM C H) , (2.90) O O O O O G where i stands for D!2 transverse spatial directions and 0 denotes the time component of the vector. The r.h.s. of (2.90) can be put to 1 by gauge "xing, in an appropriate way, the worldline bosonic a(q)-reparametrizations contained in (2.82) of X (2.11). Thus the remaining D!1 spatial components of the light-like velocity E? , among which D!2 are independent, parametrize an O SL-sphere. The matrix D HI has 2(D!2) components, since the spinor representations which we choose in O D"3, 4, 6 and 10 have dimension 2(D!2). (D!2)(D!3)#1 components of this matrix can be eliminated by the use of SO(D!2) rotations and a(q)-reparametrizations (2.82), since D carries the O SO(D!2) index and D HI transforms homogeneously under "nite worldline reparametrizations O and SO(D!2) rotations D HI (z)"D g PD H I (z ) . (2.91) O O P One can derive Eq. (2.91) from the "nite extension (2.80) of the worldline superreparametrizations (2.82). Note that D g P satis"es the same di!erential conditions as D HI , i.e. O O 2i R +D , D ,g P! d g P"0 , (2.92) Q O D!2 QO Rq and algebraic constraints 1 D g PD g P#D g PD g P" d D g PD g P , O Q Q O R D!2 OQ R
(2.93)
which resemble the integrability condition (2.89). Conditions (2.93) are, in fact, the integrability conditions of constraint (2.80) for the worldline superreparametrizations to preserve the conformally #at geometrical structure of the superworldline. This guarantees that (D!2)(D!3)#1 components of D HI can be gauged away by approO priate SO(D!2)-rotations and a(q)-reparametrizations. We thus remain with 2(D!2)! (D!2)(D!3)!1" (D!2)(3D!5)!1 components. From (2.89) it follows that Eq. (2.88) holds for each value of the index q separately (without the factor n"D!2 on the r.h.s.). This means that (D!1)(D!2) components of D HI are expressed O in terms of D!1 components of the lightlike vector. Thus the number of components of the matrix D HI still remaining undetermined is (D!2)(3D!5)!1!(D!1)(D!2)" O (D!1)(D!2)!1. We now show, following the nice reasoning of Refs. [39], that these remaining components are restricted to be zero by the algebraic equations (2.89).
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Let us consider a particular form of the C-matrices, which corresponds to a choice of light-cone coordinates in D-dimensional target space. This breaks manifest Lorentz symmetry SO(1, D!1) of the target space down to its subgroup SO(1, 1);SO(D!2). (We have already made a light-cone splitting of the components of a light-like vector when we analyzed the geometrical meaning of the masslessness condition (2.90).) Further on, the case of space}time dimension D"6 should be handled separately, since there are no Majorana spinors in this dimension and we deal instead with S;(2) simplectic Majorana}Weyl spinors (2.73). The choice of an appropriate realization of the D"6 C-matrices di!ers from that in D"3, 4 and 10 space}time dimensions. Anyway, the D"6 analysis can be performed along exactly the same lines and gives the same results as the analysis of the D"4 and D"10 case below, and we leave it for the reader as an exercise. In the dimensions D"3, 4, and 10 in the Majorana representation the matrices (CC? ) are ? @M symmetric and can be chosen as follows:
CC"
1 0 0 1
,
CCG"
0
cG
(cG)2
0
, CC"\"
1
0
0 !1
,
(2.94)
where cG are spinor matrices in a (D!2)-dimensional space (D"3, 4, 10) associated with the OOY space transverse to the particle trajectory. The indices i"1,2, D!2, q"1,2, D!2 and q "1,2, D!2 are indices, respectively, of the vector and two (generally non-equivalent) spinor representations of the group SO(D!2) of transformations acting in this space. For SO(D!2) we use the same indices q, r,2 as for the worldline internal group SO(n) (n"D!2), since in an appropriate gauge both groups are identi"ed, as we shall see in a minute. When D"4 the transverse space is two dimensional. The group of transverse rotations is SO(2), and it has one 2-dimensional spinor representation. In the case of D"10 the 16;16 matrices (2.94) which act on the 16-component real Majorana}Weyl spinors are analogs of D"4 p-matrices rather than the Dirac matrices which are 32;32 matrices in D"10 (see, for example, [45] for details on D"10 and D"11 C-matrices). The transverse space is now 8-dimensional, and SO(8) has one vector and two di!erent spinor representations, all three being related by a famous triality property, which shows up in properties of octonions [118,119]. In the basis (2.94) the matrix EI "D HI splits into two (D!2);(D!2) matrices O O EI "D HI "(EP , EPY) , (2.95) O O O O and Eqs. (2.89) take the form 1 ER ER " d ER ER O P D!2 OP Q Q
(2.96)
1 d ER cG ERY EQ cG EQY#EQ cG EQY" O QQY P P QQY O D!2 OP Q RRY Q
(2.97)
1 ERYERY" d ERYERY . O P D!2 OP Q Q
(2.98)
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To proceed with the analysis of Eqs. (2.96)}(2.98) we should "rst require that the matrix EI has the O maximum rank D!2. This requirement is of the same nature as one which we introduced in the case of the N"1, D"3 superparticle (Section 2.5.1) when, to discard an unphysical &frozen' particle solution, we assumed that all components of the commuting spinor jI cannot be equal to zero simultaneously. When the rectangular matrix EI is split as in (2.95) the requirement for it to have the O maximum rank is equivalent to the requirement that the determinant of either EP or EPY is O O nonzero. From (2.96) and (2.98) we see that EP and EPY satisfy (up to a normalization) the properties of O O the orthogonal matrices SO(D!2). Hence, if, for example, det EP O0, we can use the worldline O transformations (2.91) to gauge "x E? to be the unit matrix O EP "D HP"dP . (2.99) O O O The meaning of this gauge condition is twofold. It identi"es the SO(D!2) group of superworldline transformations with the subgroup SO(D!2) of the target space Lorentz group SO(1, D!1), and it identi"es half of the target space Grassmann coordinates HI "(HP, HPY) with the Grassmann coordinates of the superworldline HP"gOdP N HP" "hP(q)"0 . (2.100) O E Thus, half of the target space supersymmetries are identi"ed with worldsurface supersymmetry. It is this half of the supersymmetries of the target space vacuum which remain unbroken in the presence of superbranes. The condition (2.99), (2.100) is the superworldsheet counterpart of the light-cone condition (C#C"\)h"0 often used in the standard formulation of superparticles and superstrings to gauge "x the i-symmetry [1]. In gauge (2.99) condition (2.96) is identically satis"ed and Eq. (2.97) takes the form 1 d cG EOY . cG EOY#cG EOY" OOY P POY O D!2 OP QOY Q
(2.101)
It can be easily checked that the general solution of (2.101) in the dimensions D!2"2 and 8 is E "cG < (z) , (2.102) OOY OOY G where < (z) is an SO(D!2)-vector super"eld. Then Eq. (2.98) is identically satis"ed. G Substituting the expressions (2.99) and (2.102) for E? into (2.88) we "nd that the vector O
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the spinor matrix E? "D H? in terms of the components of E? "R X? !iR HM C? H. Then from the O O O O O form of (2.86), (2.102) and (2.103) it follows that (up to the worldline superreparametrizations) all higher components of the super"elds X? (z) and H? (z) are expressed through their leading components which are dynamical variables in the superparticle model. This is a remarkable property of the superembedding condition given the large number of the components which constitute the worldline super"elds when, for instance, D"10 and the number of superworldline Grassmann coordinates is n"8. It implicitly follows from the solution (2.102) and (2.103) of the integrability conditions (2.88) and (2.89) that the superembedding condition does not contain the dynamical equations of motion (2.44) of the physical particle variables xK (q) and hI (q), i.e. R xK and R hI remain unrestricted. Hence O O again, as in the case of the N"1, D"3 superparticle, to completely determine the classical dynamics of the superparticle one should construct an action from which the dynamical equations of motion are derived. The straightforward generalization of the action (2.41) which produces the superembedding condition is
S"!i dq d"\g PO E? "!i dq d"\g PO [D X? !iD HM C? H] . ? O ? O O
(2.104)
A di!erence between Eqs. (2.104) and (2.41) is that now PO (q, g) is a Grassmann-odd super"eld for ? the action (2.104) to be Grassmann-even. An appropriate variation of PO ensures the invariance of ? Eq. (2.104) under the worldline superreparametrizations. The component of PO associated with the particle momentum is ? p? "eO 2O"\ D 2D P? " , (2.105) O O"\ O"\ E which can be seen from a corresponding term in the component action obtained by integrating (2.104) over g. In (2.105) eO 2O"\ is the totally antisymmetric unit tensor. The superembedding condition is obtained by varying action (2.104) with respect to PO . And the ? variations of (2.104) with respect to X? (z) and H? (z) result in the super"eld equations D PO "0, PO C? D H"0 , (2.106) O ? ? O which (as one can check) contain the dynamical equations (2.44) for the superparticle variables xK (q) and hI (q). To complete the proof that action (2.104) describes standard massless superparticles we should convince ourselves that the super"eld PO (z) does not contain any extra independent dynamical ? variables (which would otherwise mean that the model has redundant degrees of freedom). To show this one should notice that the number of independent components of PO (z) substan? tially reduces because of an additional local invariance of the action (2.104) with respect to the
We have seen that the independent components of the matrix E? parametrize the sphere S"\. In Section 4.4 we shall O demonstrate that this sphere can be realized as a compact subspace of a 2(D!2)-dimensional coset space SO(1, D!1)/SO(1,1);SO(D!2), and that the components of E? are associated with Lorentz-harmonic variables in O a spinor representation of SO(1, D!1).
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following transformations of PO [39]: ? (2.107) dPO "D (KOPQC D H) , P ? Q ? where the tensorial parameter KOPQ(z) is totally symmetric and traceless with respect to each pair of ? the indices q, r, s. Then, by analyzing Eqs. (2.106), it can be shown that the components of the super"eld PO (z) can ? either be gauged away by independent local transformations contained in (2.107) or are expressed in terms of components of D H? . The reader may "nd the details of this analysis in Refs. [39,40]. O In particular, we again get the superparticle equations of motion (2.44) and the Cartan}Penrose representation of the superparticle momentum (2.105) 1 1 1 d p? " D HM C? D H" " jM C? j N jM C? j " d jM C? j . OP P E e(q) O P O P D!2 OP Q Q e(q) O
(2.108)
When the auxiliary "elds are eliminated the worldline supersymmetry transformations of the remaining "elds take the form dh"!aO(q)j , dj "!ia hQ , O O O dx? "!ihM C? dh, dp "0 . ? Taking
(2.109) (2.110)
2(D!2)i a (q)"! jM i(q) O O e and using relation (2.108) which implies 1 1 (p? C )? " jM C? j (C )? " j? jM ? @M O ? @M e(q)(D!2) O e(q) O O@M
(2.111)
we again recover the i-symmetry transformations (2.26). Note that in general the expansion of the symmetric matrix j? j@M in the basis of C-matrices has additional terms, for instance, in D"10 O O 1 1 (jM C? 2? j )(CI 2 )? @M , (2.112) j? j@M " (jM C? j )CI ? @M # O ? O ? ? O O 16 O 16 ) 5! O where CI ? @M are the same as in Eq. (2.94) but have the vector index down and the spinor indices up. ? (Actually, CI ? @M are inverse of (CC? ) .) ? @M ? In our case the terms with the antisymmetrized product of the C-matrices vanish (2.113) jM C? 2? j "0 O O because of Condition (2.108) on the commuting spinors, which can be proved using the results of the analysis of Eqs. (2.89) and (2.95)}(2.102). The same happens in the dimensions D"4 and 6. This completes the demonstration that the worldline super"eld action (2.104) based on the superembedding condition (2.86) describes massless superparticles propagating in #at N"1, D"3, 4, 6 and 10 target superspaces.
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2.6. Coupling to supersymmetric Maxwell xelds So far we have dealt with free superparticle models. We will now consider how the interaction of superparticles with Abelian gauge "elds is described in the superembedding approach [39]. When supersymmetric gauge theory is formulated in superspace the gauge superpotential A (x, h)"(A , A ) is subject to constraints (see, for instance, [123}126] and references therein). K I + This is done to eliminate redundant "elds (especially those of higher spins) in the component expansion of the super"eld A (x, h). The constraints are imposed on the "eld strength of A (x, h) + + which is a target space two-form (2.114) F"dZ,M dZ+ F " dZ,M dZ+ (R A !(!)+,R A ) . +, + ,M ,M + F is Grassmann antisymmetric, i.e. it is antisymmetric when one or both indices are bosonic (of +, parity 0) and symmetric when both indices are fermionic (of parity 1). (In Eq. (2.114) and below the wedge product of external di!erentials and forms is implied. The external di!erential is assumed to act on the di!erential forms from the right, i.e. d(PQ)"P dQ#(!1)O dPQ for a p-form P and a q-form Q). We shall also use the expansion of the forms in the supervielbein basis such as (2.20) F"E M E F , where
(2.115)
F "(!) M >,M E+ E,M F (2.116) M +, and E+ is the supervielbein matrix inverse to E "dZ+ E . + The constraints are imposed on components of the "eld strength. The essential constraint being generic to all super-Yang}Mills theories in any space}time dimension is (2.117) F "0 . ? @M Constraints on other components of F are obtained from the consistency of the Bianchi identities dF"0 with constraint (2.117). We thus get that F "= (C )@M , (2.118) @M ? ? ? ? where = (Z) is a spinorial super"eld. In the case of N"1, D"10 super-Yang}Mills theory the @M consistency of the Bianchi identities in the presence of constraint (2.117) also implies the equations of motion for the "elds of the SYM supermultiplet, i.e. the constraint (2.117) puts D"10 SYM on the mass shell [126,95]. In [95] it was observed that i-symmetry of the action of the superparticle coupled to the super-Maxwell "eld A requires that the Maxwell "eld is integrable along the lightline trajectories + of the superparticle, i.e. that its "eld strength is zero along the superparticle trajectories. This takes place if the gauge "eld obeys constraint (2.117). Thus i-symmetry demands that the gauge "eld superbackground is constrained. In the superembedding formulation light-like integrability means that the pullback of the "eld strength F onto the superworldline satisfying the superembedding condition (2.86) vanishes [39].
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The pullback is F "E E M F , (2.119) where E "(E M , E M ) are de"ned in (2.85). Taking into account (2.86) and (2.87) we have O O F "E? E@M F , (2.120) O P ? @M OP 1 F "E? E@M F # EM C? E E@M F . (2.121) OO P O ? @M O O ? @M D!2 P We observe that components (2.120) and (2.121) vanish, and, hence, F "0 N F "R Z,M R Z+ (R A !(!)+, R A )"0 (2.122) +, + , + ,M ,M + if the super-Maxwell "eld satis"es constraints (2.117) and (2.118). On the other hand, using the properties of E M discussed in the previous subsection it can be shown that F "0 implies the super-Maxwell constraints. Condition (2.122) can also be regarded as an additional superembedding condition when the gauge "eld superbackground in target superspaces is non-trivial. We would like to get this condition (2.122) from a superparticle action. So we extend the free superparticle action (2.104) by an appropriate Maxwell coupling term. The extended action is
S"!i dq dLg PO E? # dq dLg P+(R Z+ A !R U) , ? O + + +
(2.123)
where n"D!2, P+(Z) is a Lagrange multiplier and U(z) is an auxiliary superworldline "eld. Action (2.123) is invariant under Abelian gauge transformations dA"du(Z) provided U(z) transforms under the pullback of u(Z) as a Goldstone "eld dU(z)"u(Z(z)). Condition (2.122) arises as the integrability condition of the equation of motion dS "R Z+ A !R U"0 . + + + dP+
(2.124)
The variation of action (2.123) with respect to U yields the equation R P+"0 , + whose general solution is 1 P+"R KI *+# e 2 gO 2gOL d+e , * O n! O OL
(2.125)
(2.126)
where e is a constant and KI *+(z) is an arbitrary Grassmann antisymmetric super"eld. The "eld KI *+(z) is a pure gauge if action (2.123) is invariant under the following transformations of P+ dP+"R K*+ , * where K*+(z)"!(!)*+K+*(z).
(2.127)
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The variation of the action (2.123) under (2.127) is
dS"! dq dLg K F "! dq dLg K (E? E@M F #2E? E@M F #E? E@M F ) . ? @M ?@ ?@
(2.128)
The last term in (2.128) contains E? and hence can be compensated by an appropriate variation of O the Lagrange multiplier PO in (2.123), while the "rst two terms must vanish if (2.127) is a symmetry ? of the action. This implies the super-Maxwell constraints (2.117) and (2.118). Using the symmetry (2.127) one reduces the solution for P+ to the last term in (2.126). Then substituting this term back into the action (2.123) we get the standard minimal coupling Maxwell term
S "e dq ZQ + A , + +
(2.129)
where e plays the role of the electric charge. Thus in the superembedding formulation the electric charge arises as an integration constant of an auxiliary superworldline "eld. Below we shall see that the string tension appears in the superembedding approach in a similar way. 2.7. Superembedding into curved superspaces Consider now the propagation of superparticles in curved N"1 target superspaces of bosonic dimension D"3, 4, 6 and 10. Then the super Poincare transformations of Z+M "(XK , HI ) are extended to the target-space superdi!eomorphisms Z + "Z + (Z) .
(2.130)
The #at supervielbeins E (2.20) are generalized to curved supervielbeins (2.131) E "dZ+ E (Z) , + and the whose leading components correspond to the target space graviton e? (x)"E? " K K F . gravitino t? (x)"E? " I I F Parallel transport of (spin)-tensors in curved target superspace is determined by a connection one-form X "dZ+ X taking values in the algebra of the tangent space group which is the + M M Lorentz group SO(1, D!1). The geometry of curved superspace is characterized by its torsion
E ,(dE #E M X ),¹ "E!M E M ¹ , ! M and curvature
(2.132)
R "dX #X!M X , !M M M M where "d#X is the covariant external di!erential in the curved target superspace.
(2.133)
This is analogous to how gauge coupling constants appear in Kaluza}Klein theories upon integrating out extra compact directions.
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The superembedding condition (2.86) is now imposed on the superworldline pullback of E E (Z(z))"eO(z)R Z+ E #eO(z)D Z+ E ,e E + + O O and takes the form
(2.134)
E? (Z(z)),D Z+M E? "0 . (2.135) O O + The worldline super"eld action which produces (2.135) has the same form as Eq. (2.104) but with E? replaced by E? O O
S"!i dq d"\g PO E? . ? O
(2.136)
This action is manifestly invariant under the target-space superdi!eomorphisms and superworldline restricted superdi!eomorphisms (2.82). For consistency it must also be invariant under local variations of PO which generalize the transformations (2.107) ? dPO "(d? D #X? )(KOPQC E ) , (2.137) @M ? Q @M P P@M where E stands for E? "D Z+ E? , and remember that KOPQ is totally symmetric and traceless with Q Q Q + respect to the each pair of indices q, r, s. The variation of (2.136) under (2.137) is
i dS" dq d"\g(KOPQC E )(E? E@M ¹@M #2E M E? ¹@M ) , @M Q O P ? @M O P ? 2
(2.138)
where ¹? are components of the target superspace torsion (2.132). The last term of (2.138) can be canceled by the following variation of PO : ? dPO "(!) (KOPQC E )E M ¹@M , ? @M Q P ? and the "rst term of (2.138) must vanish, which requires the constraint on the target-space torsion (2.139) ¹? "!2i(CC? ) . ? @M ? @M This is the basic constraint of the super"eld formulation of supergravity in any dimension [123,124,127}129]. Note that the #at superspace torsion (2.21) is compatible with this constraint. To conclude Sections 2.6 and 2.7 we remark that in the superembedding formulation the constraints on gauge "eld and target space superbackgrounds are not a requirement of local worldsheet fermionic symmetry (as in the standard approach) but of additional local symmetries of the action under variations of the Lagrange multipliers which ensure the superembedding conditions. In this regard it is worth mentioning that the case of the N"1, D"3 superparticle is a special one. The n"1 worldline supersymmetry action in D"3 supergravity and super-Maxwell background can be constructed in such a way that Maxwell coupling is described by the standard minimal coupling term
S "!i dq dg[P DZ+ E? #eDZ+ A ] . ? + , " +
(2.140)
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In contrast to the standard i-symmetry formulation (2.23), in the form (2.140) the N"1, D"3 superparticle action does not have any local symmetries which would require the target-space and gauge "eld background to satisfy the constraints (2.117) and (2.139). In this case the constraints should be imposed &by hand' at the level of equations of motion.
3. Superstrings We now turn to the consideration of superstrings in the superembedding approach, but "rst let us recall standard forms of the superstring action. 3.1. Green}Schwarz formulation In this formulation (see [58,1] for details) superstring dynamics is described by the embedding into a target superspace of a bosonic two-dimensional worldsheet parametrized by coordinates mK"(q, p). The requirement of target-space supersymmetry of the superstring action restricts the possible dimensions of the target superspace in which classical superstrings can propagate to D"2, 3, 4, 6 and 10 bosonic dimensions and 2N(D!2) real fermionic spinor dimensions (where N"1, 2 counts the number of spinors). To avoid dealing with boundary terms, in what follows we restrict ourselves to the discussion of closed superstrings. The intrinsic geometry of the worldsheet is described by the zweibein e?(m)"dmK e? (m) (3.1) K where a"0, 1 are SO(1, 1) vector indices of the tangent space. Below we shall also often use the light-cone basis e>>(m)"e#e, e\\(m)"e!e ,
(3.2)
where (##,!!) stand for SO(1, 1) light-cone vector components. We reserve a single (#,!) for light-cone components of worldsheet spinors. In the light-cone basis the d"2 Minkowski metric takes the form g
>> \\
"g "0 , "g "!, g >> >> \\ \\ \\ >>
(3.3)
so that e "!e>>, e "!e\\ . >> \\ The worldsheet metric is
(3.4)
g (m)"e? (m)e (m)"!(e\\e>>#e\\e>>) , L K KL K ?L K L
(3.5)
det g ,g"!(det e? ), det e? "eKLe\\e>> K L KL K K
(3.6)
and
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(e"!e"1, e"e"0). The pullback of the target-space supercovariant forms (2.20) onto the worldsheet is E? "dmK E? "dmK(R xK !iR hM 'CK h')d? "e?(D xK !iD hM 'CK h')d? , K ? ? K K K K E? '"dmK E? '"dmK R hI 'd? "e?D hI 'd? , K K ? I I where
(3.7) (3.8)
D ,eK(m)R , (3.9) ? ? K and eK(m) is the inverse of e? (m) (i.e. eK(m)e? (m)"dL ). The index I"1, 2 indicates that the number of ? K ? L K h-coordinates can be 1 or 2 depending on whether we deal with N"1 or N"2 target superspace. The N"2 Green}Schwarz superstring action is
¹ S"! dm(!ggKLE? E@M g #¹ B(m) , K L ?@ 2
(3.10)
Where ¹ is the string tension and B(m)"dmK dmL R Z,M R Z+ B (Z) is the worldsheet pullback K L + , of the target superspace two-form (3.11) B"idx? (dhM C h!dhM C h)#dhM C h dhM C h . ? ? ? ? The physical meaning of B is that, for instance, in D"10 it describes a #at limit of the Neveu}Schwarz two-form gauge "eld in the super"eld formulation of D"10 supergravity. As we see from action (3.10), the strings couple minimally to B, and the string tension is associated with the B-charge. The B-term in (3.10) is called the Wess}Zumino term since it is an integral of an external di!erential form which does not contain the worldsheet metric and is invariant under the target space supersymmetry transformations (2.19) only up to a total derivative (and only in D"3, 4, 6 and 10 where the Dirac matrices satisfy Eq. (2.48)). The de"nite relative coe$cient between the two terms of action (3.10) is required by i-symmetry. The i-symmetry transformations of the superstring dynamical variables are dh'"2iC i'K(m), C ,E? C K ? K K dxK "!ihM 'CK dh'NdZ+M E? "0 , + d((!ggKL)"!16(!g(PKJiLR h#PKJiLR h) , \ J > J where PKL are projectors ! 1 1 PKL" gKL$ eKL , ! 2 (!g
PKLg PJKY"PKKY, PKLg PJKY"0 , ! LJ ! ! ! LJ 8
For simplicity we again work in #at target superspace.
(3.12)
(3.13)
D. Sorokin / Physics Reports 329 (2000) 1}101
and the parameters i'? (m) (I"1, 2) are (anti)-self-dual worldsheet vectors K iK"PKLi, iK"PKLi . \ L > L In the light-cone basis (3.2) the components of i' are
41
(3.14)
i "eK i , i "eK i ; i "0, i "0 . (3.15) \\ \\ K >> >> K >> \\ The dynamical equations of motion of the superstring are obtained by varying (3.10) with respect to hI (m) and xK (m) C PKLR h"0 , K \ L C PKLR h"0 , K > L R [(!g(gKLR xK #2iPKLR hM CK h#2iPKLR hM CK h)]"0 . L \ L > L K Varying action (3.10) with respect to the worldsheet metric we get
(3.16)
g (gJNE? E@M g )"E? E@M g ,G . KL J N ?@ K L ?@ KL Rewritten in the light-cone basis Eqs. (3.17) take the form of the Virasoro constraints
(3.17)
"0, E? E "0 . (3.18) E? E \\ \\? >> >>? From Eqs. (3.17) it also follows that the intrinsic worldsheet metric g can be identi"ed with the KL worldsheet metric G "E? E@M g induced by embedding into the target superspace K L ?@ KL 2 1 g " G . (3.19) (!g KL (!G KL The two metrics can be made equal by gauge "xing the invariance of action (3.10) under the Weyl rescaling of g (g P (m)g ). KL KL KL Using (3.19) we can eliminate the metric g from the superstring action and rewrite it in the KL Nambu}Goto form
S"!¹ dm(!det G #¹ B(m) . KL
(3.20)
It is instructive to note that in this form of the superstring action it is more natural to replace the (anti)-self-dual worldsheet vector parameters i'? (m) of the i-symmetry transformations (3.12) with K worldsheet scalar parameters i'? (m),2i(C i'K)? , K which satisfy the condition i'? "(1#CM )'? i(@M , (@M
(3.21)
(3.22)
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where (1#CM ) is the spinor projection matrix (1#CM )(1#CM )"(1#CM ) , 1 eK K C C K, (CM )"1, ¹rCM "0 , CM " K K 2(!det G KL 1 0 K'(" , 0 !1
(3.23)
and, as in (3.12), C "E? C de"nes the worldsheet pullback of the target space Dirac matrices. K ? K Eq. (3.22) implies that, again as in the case of the superparticles, the number of independent i-transformations is half the number of components of h and is equal to N(D!2). Property (3.22) of parameters (3.21) can be checked using the (anti)-self-duality of i' (3.14) where K in projectors (3.13) the metric g is replaced by G . KL KL Following our strategy we would like to replace the i-symmetry transformations with local supersymmetry transformations. In the case of the N"2 superstrings the worldsheet supersymmetry transformations should be parametrized by n"D!2 worldsheet chiral and n"D!2 antichiral Majorana}Weyl fermionic parameters, which can be denoted as (n, n)-supersymmetry. In the case of the N"1 superstrings (when one of the h-coordinates in (3.10) is put to zero) the worldsheet supersymmetry should be of a heterotic type (n, 0) and have n"D!2 chiral or antichiral parameters. The reason why in the N"1 case the worldsheet supersymmetry associated with i-symmetry should have the heterotic (chiral) structure is that in this case there is only one i-symmetry parameter which is either the anti-self-dual (antichiral) or self-dual (chiral) worldsheet vector (see Eqs. (3.13)}(3.15)). To realize local supersymmetry on the worldsheet we should extend the latter to a supersurface M parametrized by two bosonic coordinates mK"(q, p) and N(D!2) real fermionic coordi_ L L nates g?O"(g\O, g>OY) (where a"!,# are the spinor (chirality) indices in the light-cone basis and q, q "1,2, n are indices of internal SO(D!2) group transformations which can be independent for g\ and g>). Thus real g!O satisfy the d"2 Weyl condition g!O"$ccg!O"$cg!O ,
(3.24)
where c? (a"0, 1) and c are d"2 Dirac matrices which can be chosen to have the form of (2.49). In the next subsection we consider the superembedding of a superworldsheet which describes N"1 superstrings. 3.2. Doubly supersymmetric N"1 superstrings To describe the embedding of a supersurface M with d"2 bosonic coordinates mK"(q, p) L and n"D!2 real fermionic coordinates g\O into N"1, D"3, 4, 6, 10 target superspace we
Note that the i-parameters in the form (3.21), (3.22) can also be used directly in (3.12). This would lead to a more complicated form of a rede"ned variation of the intrinsic metric g . See [9] for details on this form of i-transformations KL in the case of a supermembrane.
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should "rst specify the geometrical properties of M appropriate for the description of the L superstrings. As in the case of the superparticles, we would like to make life as simple as possible and to deal with an M geometry which would be almost #at and, at the same time, preserved by restricted L worldsheet superdi!eomorphisms containing unrestricted d"2 local reparametrizations dmK"aK(q, p) and local supersymmetry transformations dg\O"a(q, p). In d"2 superspaces this can always be achieved starting from the most general form of the supervielbeins, imposing suitable (supergravity) constraints on the superworldsheet torsion and restricting the number of independent supervielbein components by partially gauge "xing the general worldsheet superdi!eomorphisms [122]. A suitable choice of M geometry is as follows. L The constraints on the torsion ¹ "e e ¹ ,De (z) (where the covariant external di!eren! tial D"d d#u (z) contains a tangent space connection) are ¹\\ "!id , ¹>>"0 . \O \P OP The worldsheet supervielbeins e (z) are e?(z+)"(dmK!i dg\O eK (z))e? (z) , \O K e\O(z+)"dg\O , and the supercovariant derivatives D (which form the tangent basis dual to (3.26)) are 1 D "eK (z)R , D " D eK (z)R , K >> >> K \\ n \O \O
(3.25)
(3.26)
(3.27)
D "R #ieK (z)R , (3.28) \O \O \O K where eK (z) and (1/n)D eK (z) form the vector}vector components of the inverse supervielbein >> \O \O in the light-cone basis
1 (eK , eK )" D eK , eK . \\ >> n \O \O >>
(3.29)
So the supervielbein matrix is
e? 0 K e " + !e? eK d\P d\O K \P I I and its inverse is
(3.30)
eK 0 . (3.31) e+" ? eK dI \O \O Note that eK can be transformed to the component dK of the unit matrix by appropriate >> >> tangent space transformations e K"¸? (z)eK. Therefore, eK , and hence e>>, are completely >> >> ? >> K auxiliary. This allows one to consider an even simpler form [40] of supervielbeins and covariant
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derivatives (3.26)}(3.31). However, we prefer to keep eK arbitrary since together with eK >> \\ (at g\O"0) they are identi"ed with the components of the worldsheet zweibein (3.1)}(3.5). In accordance with torsion constraints (3.25) the supercovariant derivatives (3.27), (3.28) are required to satisfy the &#at' superalgebra +D , D ,"2id D , +D , D ,"0 . \O \P OP \\ \O \\ This imposes the constraint on the form of the supervielbein components eK (z) \O 2 D eK #D eK " d D eK . \O \P \P \O n OP \Q \Q
(3.32)
(3.33)
The form of (3.27) and (3.28) is preserved (up to local tangent space transformations D "¸ D , D "¸\ND which include Weyl rescaling) by restricted superdi!eomorphisms ? ? \O \O \N whose in"nitesimal form is 1 i dmK"KK! eK D K\\, dg\"! D K\\ , O 2 \O \O 2 \O
(3.34)
i dD " (D , D K\\)D , \P \O 2 \O \P i deK "!iD KK# (D K\\)D eK , \O \P \O \O 2 \P
(3.35)
where KK(m, g) and K\\(m, g) are independent worldsheet superfunction parameters. K\\(m, g) is a self-dual vector in the same sense as the ones in Eqs. (3.15). In the Wess}Zumino gauge D " "R one can use local transformations with parameters \O E \O D D 2D KK" ,2, D KK" \O \O
\OL\
E \O E to put all components of eK (z) to zero except for \O 1 D eK " "eK (m) , \\ n \O \O E
(3.36)
which is a light-cone component of the inverse zweibein (3.2). In this gauge constraint (3.33) is identically satis"ed. If we then impose on the worldsheet zweibein the conformal gauge eK(q, p)"dK, covariant derivatives (3.27) and (3.28) become #at, and the superdi!eomorphisms ? ? (3.34) reduce to the chiral (heterotic) superconformal (n, 0) transformations i i dm\\,d(q!p)"K\\(m\\, g\O)! g\OD K\\, dg\"! D K\\ ; \O O 2 \O 2 dm>>,d(q#p)"K>>(m>>) .
(3.37)
We are now in a position to write down the superembedding condition for the N"1, D"3, 4, 6, 10 superstring. It again prescribes that in the basis (3.26) the Grassmann component of the superworldsheet pullback of the target space supervielbein E? (Z) (2.20) is zero E? (Z(z))"D X? !iD HM C? H"0 . \O \O \O
(3.38)
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The only di!erence between Eq. (3.38) and the superembedding condition for the N"1 superparticles is that now D contains the zweibein components eK (z). \O \O The analysis of the superembedding condition (3.38) is carried out in exactly the same way as the superparticle case (Section 2.5.3) and results in the same conclusions [33,40]: (i) All higher components in the expansion of the super"elds XK (m, g) and HI (m, g) are expressed in terms of the leading components xK (m) and hI (m) which are dynamical variables of the superstrings. (ii) The dynamical equations of motion (3.16) of xK (m) and hI (m) do not follow from the superembedding condition. (iii) One of the Virasoro constraints (3.18) of the N"1 superstrings appears as a consequence of (3.38) which produces the Cartan}Penrose relation (3.39) d E? ,d (D X? !iD HM C? H)"D HM C? D H \\ \O \P OP \\ OP \\ "0 . (3.40) N E? E \\ \\? (iv) Because of relation (3.39), the i-symmetry (3.12) of the N"1 Green}Schwarz formulation (h"0) is identi"ed with local worldsheet supersymmetry. (v) The superstring dynamical equations and the second Virasoro condition should be derived from a worldsheet super"eld action in N"1, D"3, 4, 6, 10 target superspace. We start to construct this action by writing down the term which produces the superembedding condition (3.38)
S "!i dm dLg P\OE? "!i dm dLg P\O[D X? !iD HM C? H] , ? \O ? \O \O
(3.41)
where n"D!2. To this action one should, in principle, add a Lagrange multiplier term which takes into account the superworldsheet geometry constraint (3.33), but since this constraint is identically satis"ed in the Wess}Zumino gauge (3.36) such a term is purely auxiliary and does not contribute to the component superstring action [40]. We therefore skip this term. As in the superparticle case action (3.41) is invariant under the local transformations of the Lagrange multiplier (3.42) dP\O(z)"(D #iR eK )(KOPQ C D H) , \P K \P >> ? \Q ? where the parameter KOPQ (z) is totally symmetric and traceless with respect to the each pair of the >>? SO(n) indices q, r, s. Again, as in the superparticle case, this symmetry allows one to reduce the number of independent components of the super"eld P\O(z), and to express the remaining ones in terms of compo? of motion nents of HI (z) by solving the equations P\OC? D H"0 . (3.43) (D #iR eK )P\O"0, ? \O \O K \O ? Action (3.41), however, does not describe the fully #edged N"1, D"3, 4, 6, 10 superstrings [137]. As we shall see in the next subsection, it describes so called null (or tensionless) superstrings,
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extended objects which are characterized by having zero tension and a degenerate worldsheet metric [130]. The reason is that we have not yet incorporated a Wess}Zumino-like term into the worldsheet super"eld action, which will generate the string tension dynamically. 3.2.1. Null superstrings Null strings [130] (and in general null (super)branes or tensionless branes) have attracted certain attention from various points of view [131,101,132}135,102,136]. As far as their properties are concerned the null strings are closer to the massless particles than to the strings. Actually, they describe a continuum of massless particles moving along a degenerate light-like surface [131,135]. Moreover the null strings do not need any critical dimension of space}time to live in [132,133] and the null superstrings [131,134,135], in contrast to the ordinary superstrings, do not require a Wess}Zumino term to be i-symmetric. It has been assumed [101,135,136] that the null strings and branes may be regarded as a high energy limit of the ordinary string thus providing a way for describing strings beyond the Planck's scale. We now show that action (3.41) describes null superstrings in N"1, D"3, 4, 6 and 10 superspaces [137]. For simplicity consider the case when the superworldsheet in (3.41) has only one fermionic direction (i.e. n"1). This case directly corresponds to an N"1, D"3 superstring but action (3.41) with n"1 describes N"1, D"4, 6 and 10 superstrings as well, though in these higher dimensions only one of D!2 i-symmetries is manifest. The form of the hidden i symmetries is exactly the same as in the case of superparticles (2.72)}(2.77). Let us integrate (3.41) over the worldsheet fermionic variable and solve for algebraic equations of motion of auxiliary variables analogous to Eqs. (2.43). Then the action reduces to
S " dm p\\[eK (m)(R x? !iR hM C? h)!jM C? j ] , ? \\ K K \ \
(3.44)
where , jI "D HI (z)" , eK (m)"D eK (z)" . p\\(m)"P\\(z)" ? E E \\ \ \ E \ \ ? As in the superparticle case, from action (3.44) we derive the Cartan}Penrose relations p\\"e\(m)jM C j ? \ ? \ 1 E? (m)"eK (m)(R x? !iR hM C? h)"jM C? j " p\\ , K \ \ e\ ? \\ \\ K
(3.45)
where e\(m) is a proportionality coe$cient with four &upper minus' SO(1, 1) indices. Eq. (3.45) implies that E? satis"es the Virasoro constraint (3.40). In addition variation of (3.44) with respect \\ to eK gives \\ p\\(R x? !iR hM C? h)"0 , (3.46) ? K K which in view of (3.45) implies eL E? E "0 . \\ L K?
(3.47)
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From Eq. (3.47) it follows that (if eL is non-zero, which we assume) the induced worldsheet metric \\ G "E? E (3.48) KL K L? is degenerate det G "0 . KL This is a characteristic feature of the null strings and branes. Making use of relations (3.45) and introducing instead of eK the worldvolume vector \\
S " dm
(3.49)
One gets exactly the same form (3.49) of the null superstring action from the action (3.41) with n"D!2, though, the procedure becomes a bit more complicated because of the much greater number of auxiliary "elds entering the initial super"eld action. Thus the doubly supersymmetric action (3.41) describes N"1, D"3, 4, 6, 10 null superstrings. Note that in contrast to the Green}Schwarz action the null superstring action (3.49) does not contain the Wess}Zumino term. However it is still i-symmetric (which is the relic of the linearly realized worldsheet supersymmetry of the action (3.41)). The i-symmetry transformations are d h?"i
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(2.19). The appropriate two-form is F"e\\e>>E? E@M g #B#dA . (3.50) \\ >> ?@ It includes the worldsheet pullback of the Wess}Zumino form (3.11) (where h"0 because we deal with N"1), i.e. B"i dx? dhM C h , (3.51) ? and e>>"dmK e>>(m) and e\\"dmK e\\(m) are the light-cone components of the zweibein K K (3.1), (3.2). Since under the supersymmetry transformations B transforms as a total derivative
dB"id
i dx? ! dhM C h hM C e ? ? 3
the vector "eld A (m) must also vary as K i dA"!i dx? ! dhM C? h hM C e ? 3
to cancel the variation of B in (3.50). The vector "eld A (m) should not be a new propagating worldsheet "eld since our aim is to K describe the ordinary superstrings which do not carry such "elds. Hence we must require that on the mass shell A (m) is expressed in terms of superstring dynamical variables. This is achieved by K assuming that the two-form (3.50) vanishes on the mass shell, i.e. dA"!e\\e>>E? E@M g !B . (3.52) \\ >> ?@ Eq. (3.52) implies that the "eld strength of A (m) is not independent and hence does not describe K new redundant physical degrees of freedom. From Eq. (3.52) it also follows that its r.h.s. is a closed form, which is true for any two-form on a d"2 bosonic manifold. To get Eq. (3.52) as an equation of motion we add to action (3.44) the term
S " dm (m)eKL[e\\e>>E? E@M g #B #R A ] , K L KL K L 2 \\ >> ?@
(3.53)
where (m) is a Lagrange multiplier whose variation gives rise to Eq. (3.52). The extended action (3.44)#(3.53) S"S #S 2 is invariant under new local transformations found in [33,40] dp\\? "K\\ (m)eKLe>>e\\(E? #jM C? j ) (m) >> K L \ \\ \
(3.54)
(3.55) deL "K\\ (m)eL , deL "0 . >> >> \\ \\ If (m) is non-zero this symmetry allows one to eliminate p\\ on the mass shell when the Eqs. (3.45) ? hold.
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Let us now see how the string tension appears, and the Green}Schwarz action is recovered. Varying (3.53) with respect to A we get K R "0 N "¹ , K
(3.56)
where ¹ is an integration constant. If ¹ is non-zero, and we substitute (3.56) into action (3.53), the term eKLR A becomes a total derivative and can be skipped. As a result the action S (3.53) reduces K L 2 to the N"1 Green}Schwarz action (3.10) (with h"0). The "rst term of the Green}Schwarz action is nothing but the "rst term in (3.53), which can be checked using expressions (3.2)}(3.6). We see that ¹ is the string tension. Moreover, when (and hence ¹) is non-zero the local transformations (3.55) allow one to put p\\ satisfying (3.45) to zero. Thus the term S (3.44) drops out of the extended action (3.54) and the ? latter coincides with the N"1 Green}Schwarz action
S "¹ dm eKL[e\\e>>E? E@M g #B] . K L KL \\ >> ?@ ,
(3.57)
On the contrary, when ¹"0 the second term S (3.53) disappears from (3.54) and it reduces to the 2 null superstring action (3.44). Therefore, Eq. (3.54) is the generic action which is non-singular in the tensionless limit and describes superstrings with all values of the tension. Our next step is to lift the action S on to the worldsheet superspace. 2 So now we consider the two-form (3.50) given on the superworldsheet M , i.e. all its ingredients L depend on z+"(mK, g\O). 3.2.3. Weil triviality An interesting observation concerning properties of the two-superform (3.50) on M was made L in [33,140]. The two-superform F is closed on M when the superembedding condition (3.38) is satis"ed. L The closure of a two-form on M is a non-trivial property because of the presence of n L Grassmann directions. Let us check this property, but "rst rewrite (3.50) in a more appropriate form #B#dA . F"E? e>>E >>?
(3.58)
By de"nition of the pullback E? "e\\E? #e>>E? #e\OE? , \\ >> \O
(3.59)
we see that if the superembedding condition E? "0 (3.38) holds, the form (3.58) coincides \O with (3.50). Then, in view of the de"nition of B (3.51), )#dB dF"d(E? e>>E >>? "!(dE? )e>>E
>>?
#E? d(e>>E )#iE? dHM C dH . >>? ?
(3.60)
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Because of the target-space torsion constraint (2.21) and expansion (3.59), the "rst term of (3.60) can be rewritten as "!i dHM C dH E? #i dHM C dH(e\\E? #e\OE? ) . (3.61) !i dHM C dH e>>E >>? ? ? \\ \O ? If the superembedding condition (3.38) and its consequences (3.39) and (3.40) are satis"ed, the last two terms of (3.61) vanish, and the "rst term cancels the last term in (3.60). By virtue of the worldsheet torsion constraints (3.25) and the superembedding conditions (3.38)}(3.40) the second term in (3.60) also vanishes )"e\\E? e>>e\O E E? d(e>>E >>? \\ \O >>? &e\\E? e>>e\OD HM C D H"0 . (3.62) \\ >> ? \O The transition to the expression after the similarity sign & has been made by getting the expression for e>>e\O E as a component of the tangent space torsion constraint (2.21), \O >>? dE? "!i dH C? dH, in the worldsheet supervielbein basis and taking into account (3.25) and (3.38)}(3.40). We thus get dF"U\OE? #U\\+OP,D E? "0 , (3.63) \P \O ? \O ? where U are some three-superforms. The variation of F with respect to the M superdi!eomor L phisms z +"z+#dz+(m, g) is dF"d(i F)#i dF . (3.64) B B Now take the integral of F over the two-dimensional slice of M such that g\P"0, i.e. take the L integral over the ordinary worldsheet S "¹ 2
F .
(3.65)
M
Eq. (3.65) is nothing but the action (3.53) where (m) is replaced with ¹ (the term with A (m) is K a total derivative and can be skipped). The variation of (3.65) with respect to the worldsheet superdi!eomorphisms (3.64) is (up to a total derivative) dS "¹ 2
M
i dF . B
(3.66)
Since dF (3.63) is &proportional' to the superembedding condition, the variation of S can be 2 canceled by an appropriate variation of the Lagrange multiplier of the action (3.41) if we take the sum of the two actions. Thus, the action
S"!i dm dLg P\OE? #¹ ? \O
F
(3.67)
M
possesses local n"D!2 worldsheet supersymmetry though the second term is not an integral over the whole superworldsheet. As we have learned in the previous subsection the component version of this action describes N"1, D"3, 4, 6, 10 superstrings.
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The doubly supersymmetric action in the form (3.67) was proposed in [33]. The property of F to be a closed superform up to the superembedding condition and the corresponding nonmanifest superdi!eomorphism invariance of action (3.67) was called Weil triviality, a property which had been found to be useful for studying chiral anomalies in super Yang}Mills theories [141]. Actually, action (3.65) is a particular example of the generalized actions of the group manifold (rheonomic) approach to the description of supersymmetric "eld theories [73] and superbranes [197]. 3.2.4. Worldsheet superxeld action To make the action S (3.65) manifestly supersymmetric on the superworldsheet one constructs 2 a Lagrange multiplier term [40] which produces the on-shell condition F"0 .
(3.68)
The superstring action which includes such a term is
S"S #S "!i dm dL g P\OE? # dm dL g P+,F , 2 ? \O +,
(3.69)
where now
S " dm dL g P+,F " d m dL g P+,[E? e>> E #B #R A , ] . 2 + ,, >>? +, +, + ,
(3.70)
[ ,, denotes graded antisymmetrization of the superworldsheet indices (i.e. if one or both of the indices M, N are bosonic they are antisymmetrized, and if both of them are fermionic they are symmetrized). In addition to all symmetries discussed above action (3.69) is invariant under the local transformations of the Lagrange multiplier P+, dP +,,"R K *+,,(z) . * Indeed, the variation of the S term (3.70) with respect to (3.71) is 2
dS " dm dLg K *+,,(dF) , 2 *+,
(3.71)
(3.72)
where dF has the form of Eq. (3.63), and, hence, variation (3.72) is canceled by an appropriate variation of the Lagrange multiplier P\O in (3.69). Varying the action (3.70) with respect to A we + ? get R P +,,"0 , + whose general solution is 1 P +,,"R KI *+,,(z)# e 2 g\O ....g\OL d + d,, ¹ , * \\ >> n! O OL
(3.73)
(3.74)
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where ¹ is a constant. The "rst term in (3.74) can be put to zero by gauge "xing the local transformations (3.71). Then, substituting (3.74) (with KI "0) back into action (3.69) we see that upon the g-integration it reduces to (3.67). We have thus demonstrated that the worldsheet super"eld action (3.69) describes the N"1, D"3, 4, 6, 10 superstrings. Note that in the case when the superworldsheet has only one fermionic direction, i.e. when n"1, there exists another form of the super"eld action S [33], which involves only one Grassmann 2 component B of the M pullback of the two-form B (3.51). In this form the full action is \ >>
, S"S #S "!i dm dg P\\E? #¹ dm dg sdet(e)B \ ? \ \ >> \ 2
(3.75)
where sdet(e) is the superdeterminant of the supervielbein matrix e (3.30). + In Section 3.2 (Eqs. (3.38)}(3.40)) we demonstrated that the superembedding condition gives rise only to one of the superstring Virasoro constraints. The second constraint, namely E? E "0, >> >>? arises as a result of the variation of the full action (in any of its forms, (3.67), (3.69) or (3.75)) with respect to the supervielbein component eK . A completely &twistorized' version of the doubly \\ supersymmetric action for the N"1 superstrings, where both Virasoro constraints have a twistor origin, was constructed in [142]. 3.3. Coupling to supergravity background The generalization of superstring action (3.69) to describe a superstring propagating in curved target superspace introduced in Section 2.7 is almost straightforward [33,40]. In (3.69) and (3.70) one should replace the #at supervielbeins with curved ones E? PE? , consider B(Z) as \O \O a two-form gauge super"eld, whose leading component B (X) is the Neveu}Schwarz gauge KL potential entering the supergravity multiplet, and introduce a dilaton super"eld U(Z) coupling by rede"ning F(Z) as follows: E #B #R A , . F "eUE? e>> +, + , + ,, >>? +, The superstring action takes the form
S"S #S "!i dm dLg P\OE? # dm dLg P+,F . 2 ? \O +,
(3.76)
(3.77)
By construction (3.77) is superdi!eomorphism invariant on the superworldsheet and in the target superspace, and it must also be invariant under variation (3.71) of the Lagrange multiplier P+,. This means that the variation (3.72) of action (3.77) must be compensated by an appropriate variation of the Lagrange multiplier P\O. For this, dF must have the form of the l.h.s. of ? Eq. (3.63), i.e. dF"U\OE? #U\\+OP,D E? . (3.78) ? \O \P \O ? Performing the direct computation of the external di!erential of (3.76) one "nds that it has the form of Eq. (3.78) if the superbackground satis"es the supergravity torsion constraint (2.139) ¹? "!2i(CC? ) , ? @M ? @M
(3.79)
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and the components of the "eld strength H"dB of the two-from gauge super"eld are constrained as follows: (3.80) H "2ieU8(CC ) , H "0 ? ? @M ? @M A ? @M ? (where U(Z) is the dilaton "eld). Thus, as in the case of the Green}Schwarz formulation, the consistency of coupling the doubly supersymmetric strings to the supergravity background demands that the superbackground obeys supergravity constraints (3.79) and (3.80). In D"10 these constraints put N"1 supergravity on the mass shell [128,97]. 3.4. Heterotic fermions So far we have dealt with classical N"1 superstrings in space}time of dimensions D"3, 4, 6 and 10. Quantum consistency of superstring theory singles out 10-dimensional space}time. In addition, to be anomaly free a closed N"1, D"10 superstring should be heterotic [143], i.e. it should contain extra (heterotic) matter on its worldvolume which upon quantization produces an N"1, D"10 super-Yang}Mills "eld A (Z) taking its values in the adjoint representation of the + gauge group SO(32) or E ;E . Such worldsheet matter "elds can be 32 chiral fermions t\A(m), where &!' is the chiral spinor index of the d"2 Lorentz group SO(1, 1) and A"1,2, 32 is the index of the 32-dimensional vector representation of the target-space gauge group SO(32). Therefore, the chiral fermions can be minimally coupled to the SO(32) gauge super"eld A AB (Z). + Note that t\A(m) cannot be minimally coupled to a gauge "eld of the E ;E group whose lowest dimensional representation is 496"2;248. The E ;E gauge "elds appear in heterotic string theory in a much more subtle way (see, for instance, [1]) and we shall not describe them here. Our aim now is to introduce the heterotic fermions into the superembedding approach. In the Green}Schwarz formulation the heterotic fermion term, which is added to the N"1, D"10 superstring action (3.57), is
i S " ¹ dm t\eK (m)(R !R Z+ A )t\ . \\ K K + R 2
(3.81)
Varying (3.81) with respect to t\ we get the equation of motion which has the form of a chirality condition on t\ eK (R !R Z+ A )t\"0 . + \\ K K In the conformal gauge eK (m)"dK and when A "0. Eq. (3.82) reduces to \\ \\ + R t\A"0 \\ which implies that the heterotic fermions are worldsheet left-movers t\A"t\A(m>>)"t\A(q#p) .
(3.82)
(3.83)
(3.84)
For comparison let us note that in the conformal gauge the equations of motion of h(m) deduced from the N"1 Green}Schwarz action (3.57) imply that h(m) are right-movers, i.e. h? "h? (m\\)"h? (q!p) .
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Under the i-symmetry transformations (3.12)}(3.15) the heterotic fermions transform as follows: d t\"AM (E? C )? i@M t\ . (3.85) G ? \\ ? @M >> Therefore, they are inert with respect to the i-transformations when the SYM background is switched o! (note that when N"1, from (3.12)}(3.15) it follows that d eK "0). G \\ We would like to extend the heterotic fermion action (3.81) and the chirality condition (3.82) to the superworldsheet formulation of the heterotic string whose N"1, D"10 target-superspace part is described by the action (3.69). The generalization of the chirality condition (3.82) is straightforward. We consider the heterotic fermions as worldsheet super"elds W\A(mK, g\O) and assume that on the mass shell they satisfy the equation of motion [D !D Z+ A ]W\"0 , (3.86) \O \O + where the covariant derivative D was determined in Eq. (3.28). It can be easily checked that in the \O Wess}Zumino gauge (3.36), Eq. (3.86) reduces to Eq. (3.82), and hence, in the conformal gauge and in the absence of the SYM background, the heterotic fermions W\ are chiral (3.84) and do not transform under the &left-moving' conformal supersymmetry transformations (3.37), which is in agreement with their i-symmetry properties (3.85). Note that, as usual, i-symmetry, or (in the worldsheet super"eld formulation) the integrability of (3.86) requires the SYM constraints (2.117) and (2.118) on the background gauge "eld A . + Now the problem is to get Eq. (3.86) from a worldsheet n-extended superspace action. When n"1, action (3.81) admits the straightforward generalization
¹ dm dg W\(D !D Z+ A )W\ . S "! \ \ \ + R 2
(3.87)
When n"2 there is a generalization of (3.87) in terms of complex chiral n"2 super"elds [33]. For the worldsheet supersymmetry formulation with n'2 di!erent forms of the super"eld action of chiral fermions were proposed [144}146]. The most elegant formulation, which we shall sketch for n"8, was proposed by Howe [145]. Let us split the 32-dimensional index A of t\A(z) into an index i"1,2, 8 and an index A "1, 2, 3, 4, and identify the index i with the index of the vector representation of the group of internal isomorphisms SO(8) of the n"8 worldsheet supersymmetry (remember that g\O transform under a spinor representation of SO(8)). The index A labels the vector representation of a target superspace gauge group SO(4) and A AYBY (Z) are now SO(4)-gauge "elds which the heterotic + fermions can couple to. Note that the possible minimal coupling of the heterotic fermions is now reduced from SO(32) gauge coupling to SO(4) coupling. This is a shortcoming of the present formulation which, however, seems to be akin to the general problem of directly coupling E ;E gauge "elds mentioned at the beginning of this subsection. Assume that ow the mass shell the super"eld t\ AY(z) satis"es the constraint (3.88)
K t\G,(dG D !u\G !dG D Z+ A )t\H"cG POY, + OOY \O H \O \O \H H \O where u\G is an SO(1, 1);SO(8) superworldsheet connection, the SO(8) gamma}matrices have \O \H already been introduced in (2.94), and POYAY(z) is a super"eld in the spinor representation of SO(8) &dual' to that of g\O.
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On the mass shell POYAY(z) should vanish and then Eq. (3.88) reduces to the chirality condition analogous to (3.86) POY" K
\O
t\G"0 ,
(3.89)
thus, providing us with the equations of motion of the heterotic fermions. It seems instructive to note that constraint (3.88) resembles the superembedding condition (3.38), and, indeed, (3.38) reduces to the exactly the same form when the local worldsheet supersymmetry is gauge "xed as in Eqs. (2.99) and (2.100). To see this, take the transverse part of (3.38) which, in view of Eqs. (2.94)}(2.100), is D XG!ig\NcG D HOY !icG HOY "0 , \O NOY \O \ OOY \ i"1,2, 8; q"1,2, 8; q "1,2, 8 .
(3.90)
The second term of (3.90) can be &absorbed' by rede"ning the transverse coordinate XG"XK G!ig\NcG HOY , and we "nally get NOY \ D XK G"2icG HOY , \O OOY \
(3.91)
whose form is the same as that of the constraint (3.88). Constraint (3.88) can be explicitly solved [145], the solution being in terms of an unconstrained super"eld <\AY(z) (with eight SO(1,1) OY minuses) t\G"cG D <\ , OOY \O OY
(3.92)
where D is a di!erential operator of a 7th power in K whose explicit form the reader can "nd in \O \O Ref. [145]. The action which gives rise to Eq. (3.89) is
S " dm dg sdet(e)<\POY , \ OY R
(3.93)
where under POY one should imply Eq. (3.88) with t\G being replaced by its &prepotential' (3.92). This completes the construction of the manifest doubly supersymmetric formulation of the N"1, D"10 heterotic string, and we now brie#y discuss the case of type II superstrings. 3.5. Type II superstrings As we have already mentioned at the end of Section 3.1, to replace the i-symmetry of the N"2 superstring action in the Green}Schwarz formulation (3.10) with local worldsheet supersymmetry one should deal with an (n, n) worldsheet superspace parametrized by bosonic coordinates mK (m"0, 1) and 2n"2(D!2) worldsheet spinor coordinates g?O (a"!,#; q"1,2, D!2). The basic condition specifying the superembedding of such a supersurface into an N"2 target superspace parametrized by supercoordinates Z+ "(XK , H'I ) (I"1, 2) is E? "D Z+M E? (Z(z))"0 , ?O ?O +
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where (as above) D is a spinorial covariant derivative on the worldsheet supersurface and E? are ?O + components of the target-space supervielbein. The content and consequences of this superembedding condition were analyzed in Refs. [42,45]. It was shown that in the case of an N"2, D"3 superstring the superembedding condition does not contain dynamical equations of motion of the superstring coordinates and, hence, a doubly supersymmetric action for the N"2, D"3 superstring can be constructed in the form analogous to those of the N"1 superstrings [33,40]. In the case of IIA and IIB superstrings in tendimensional space}time the superembedding condition contains the dynamical equations, which was demonstrated in detail in Refs. [42,45]. This is similar to the appearance of the equations of motion in the constraints of higher dimensional super-Yang}Mills and supergravity theories, as we have mentioned above. In such cases standard super"eld actions cannot be constructed, since, for example, if one tried to introduce the superembedding condition into an action with a Lagrange multiplier, Lagrange multiplier components would contain propagating (redundant) degrees of freedom which would be provided with kinetic terms by the dynamical equations contained in the superembedding condition. Though the standard super"eld actions cannot be constructed in the cases when the superembedding condition puts a superbrane theory on the mass shell, in these cases the superembedding condition contains exhaustive information about the classical dynamics of the superbranes. This is very useful when one deals with new brane objects for which a detailed worldvolume theory has not yet been developed, as it was, for instance with D-branes, and especially with the M-theory "ve-brane [50,51]. We now leave the superstrings and, upon the discussion of general geometrical grounds of (super)embedding, turn to the consideration of M-theory branes, the D"11 supermembrane and the super-5-brane, whose classical properties are completely determined by the superembedding condition.
4. Basic elements of (super)surface theory 4.1. Bosonic embedding Consider a (p#1)-dimensional bosonic surface M embedded into a D-dimensional curved N> bosonic target space M . M " in the cotangent space of M A local orthogonal basis is given by a vielbein one-form N> e?(mK)"dmK e? (m) , K
(4.1)
which transforms under the vector representation of SO(1, p). The matrix e? (m) and its inverse eK(m) K ? relate the local orthogonal basis on M to the local coordinate basis dmK. They have the N>
Mathematically profound presentation of surface theory the reader may "nd in [7], and its extension to superembeddings in [45,79].
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following properties: e? (m)eK@(m)"g?@, g?@"(!,#,2,#) , K e? eL g "g (m) , (4.2) K K ?@ KL where g (m) is a metric on M . We assume that the bosonic surface has zero torsion, i.e. its KL N> connection u? (m)"dmK u? satis"es the condition @ K@ ¹?"de?#e@u? "0 . (4.3) @ Similarly, in the target space M we introduce a local orthogonal cotangent basis given by M " a vielbein one-form E? (xK )"dxK E? (x) , (4.4) K which transforms under the vector representation of SO(1, D!1) and whose components satisfy the conditions E? EK@"g?@, g?@"(!,#,2,#) , K (4.5) E? E@M g?@"g (x) , KL K L where g (x) is a metric on M which is assumed to be torsion free as well KL M " ¹? "dE? #E@M X? "0 . (4.6) @M To embed the surface M into the target space M means to relate the intrinsic geometry of N> M "how the local M frame E? behaves on the M to the geometry of M . To this end we consider N> " M M " surface, i.e. consider the pullback of E? on to M N> E? (x(m))"dmK R xK E? "e?E? , (4.7) K K ? where (4.8) E? (x(m)),eKR xK E? . ? ? K K Since the orthogonal frame E? is determined up to the local SO(1, D!1) transformations, rotating it by an appropriate SO(1, D!1) matrix u? (m), it is always possible to adapt its pullback on @ M in such a way that (up to a rescaling Mfactor) N> (4.9) E?,E@M u? "e? , @M (4.10) EG,E@M uG "e@E@M uG "0 , @M @ @M where we have split the upper index of the SO(1, D!1) matrix u? into the index a"0, 1,2, p (of @M the M directions parallel to M ) and the index i"1,2, D!p!1 (of the M directions " N> " M transverse to M ). These indices are, respectively, the indices of the subgroupsM SO(1, p) and N> SO(D!p!1) of SO(1, D!1) u? "(u? , uG ) . @M @M @M
(4.11)
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Note that, by de"nition, the SO(1, D!1) matrix u? satis"es the orthogonality conditions @M u? u@M gAB"g?@, uA uBM g "g , (4.12) ?@ A BM ? @M AB which are preserved by the independent left and right SO(1, D!1) ;SO(1, D!1) transforma* 0 tions of u? @M u( ? "(O )BM uA (O\)? . (4.13) * @M BM 0 A @M The meaning of Eqs. (4.9) and (4.10) is the following. When the surface M is embedded into N> M the vielbein frame E? of M can always be chosen in such a way that the pullback onto M " of p#1 of its components M "coincides with a given local frame e?(m) on M M , and the other N> N> D!p!1 components of E? are orthogonal to the surface. With such a choice one identi"es a subgroup SO(1, p) of the target space group SO(1, D!1) acting on u? on the right (see Eq. (4.13)) @ 0 with the group SO(1, p) of rotations in the (co)tangent space of M M . N> Eqs. (4.9) and (4.10) are invariant under the transformations of SO(1, p);SO(D!p!1) of the group SO(1, D!1) (4.13). Thus, SO(1, D!1) is broken down to SO(1, p);SO(D!p!1), 0 0 where SO(D!p!1) acts in the subspace of M transverse to the surface M . Therefore, the " N> M SO(1, D!1) matrix (4.11) which brings the pullback of E? to the form (4.9) and (4.10) is determined up to the SO(1, p);SO(D!p!1) rotations. This means that its elements, which are called (Lorentz-vector) harmonics [87,88,91], parametrize a coset manifold SO(1, D!1) . SO(1, p);SO(D!p!1) We should stress that the left group SO(1, D!1) of (4.13), which is associated with the local * Lorentz transformations of the target-space vielbein (4.4), remains unbroken. This ensures the Lorentz-covariant description of the embedding. Eqs. (4.9) and (4.10) can be regarded as embedding conditions. Their structure re#ects the fact that the embedded surface M breaks the local Lorentz symmetry SO(1, D!1) of the target N> 0 space M down to SO(1, p);SO(D!p!1) acting in the directions parallel and transversal to M " (4.9) de"nes the vielbein e? on M M . Eq. as the vielbein induced by embedding. Using (4.5), N> N> (4.9) and (4.10) one can easily check that E? de"ned in (4.8) are orthonormal ? E? E "g , (4.14) ?@ ? @? and, hence, the M metric (4.2) is the induced metric N> g (m)"R xK R xL g (x(m)) . (4.15) KL K L KL Because of the orthonormality property (4.14) the components of E? can be identi"ed with the ? u? components of the SO(1, D!1) harmonic matrix inverse to (4.11) (as usual, the Lorentz indices ? are raised and lowered by the Minkowski metrics) E? (x(m))"u? (m),g?@u@ g . @M @? ? ? Notice that in the case of strings Eq. (4.14) is equivalent to the Virasoro constraints.
(4.16)
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Remember that by virtue of (4.12) the matrix inverse to the orthogonal matrix u? is @M (4.17) (u\)? "g?AuBM g ,(u? , u? ) . @M A B@ ? G To simplify notation we shall always skip the superscript &!1' of the components of the inverse SO(1, D!1) matrices when their lower index is decomposed into the SO(1, p);SO(D!p!1) indices. This should not lead to a confusion if we keep in mind that only the upper index of the &direct' SO(1, D!1) matrices can be subject to SO(1, p);SO(D!p!1) splitting, as in Eq. (4.11). We shall now show that the spin connection u? of the surface M is related to the pullback of @ N> the spin connection X? of M by local SO(1, D!1) transformations of the latter. " @M M condition Consider the integrability of Eq. (4.7) by taking the M external covariant di!erential "
"d#X of its left- and right-hand side. Then, in view of (4.6),M (4.3) and (4.16), we get e@u? u@M "e@ E@M ,e@(du@M #u? X@M ) . (4.18) @ ? @ @ @ ? Multiplying Eq. (4.18) by u? and taking into account (4.12) and (4.16) we arrive at the expression @M (4.19) u? "( u@M )u? ,du@M u? #u? X@M u? , @ @M @ @M @ ? @M @ where du@M u? are SO(1, p)-valued components of the SO(1, D!1) Cartan form @ @M (4.20) X@M ,d(u\)A u@M "!(u\)A du@M ? ? A ? A ((u\)A was de"ned in (4.17)). X@M identically satis"es the Maurer}Cartan (zero-curvature) ? ? equations (4.21) dX@M #XA X@M "0 . ? ? A The integrability of the Maurer}Cartan equations is the basis for relating (super)branes to integrable systems [4}6,8,17}22]. As a generalization of (4.20) we also introduce a &covariantized Cartan' form (which is, of course, nothing but the Lorentz}transformed spin connection X on M ) M " (4.22) XK @M , (u\)A u@M "d(u\)A u@M #(u\)A XBM u@M . ? ? A ? A ? A BM Written in terms of the connection XK , the curvature two-form on M M " R@M ,dX@M #XA X@M (4.23) ? ? ? A takes the form (u\)A RBM u@M "dXK @M #XK A XK @M . ? A BM ? ? A
(4.24)
Remember that the spin connections are antisymmetric with respect to the Lorentz-group indices. Note also that with such a choice of u? the covariant di!erential ( K u@M )u? "(du@M #u? X@M !uA u@M )u? , which contains both connections, is @ @ @M @ @ ? @ A @M identically zero ( K u@M )u? ,0. @ @M
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From (4.19), (4.22) and (4.24) it follows that the curvature R@ of the surface M is ? N> R@ ,du? #u?u@"uA RBM u@ !XK G XK @ , ? G ? @ A A ? A BM where i"1,2, D!p!1 are the transverse indices (see Eq. (4.11)) and
(4.25)
XK G "( u@M )uG ,du@M uG #uA XBM uG , XK ?"!d g?@XK H . (4.26) ? G GH @ ? @M ? @M ? A BM Note that though only the target-space covariant derivative enters the expression (4.26) for XK G , it is ? also covariant with respect to the local SO(1, p) transformations on the surface, since the term with the surface spin connection is identically zero because of the orthogonality properties of the harmonics u@ u? uG ,0 . ? @ ? In surface theory Eq. (4.25), which expresses the surface curvature in terms of the target-space connection and curvature, is called the Gauss equation. The other two determining equations of surface theory contained in (4.24) are the Codazzi and Ricci equations (see [7,6] for details). We shall present the explicit form of these equations for superembeddings in the next subsection. Note that when the target space is #at R@M "0 and X@M can be gauge "xed to zero, the induced spin ? ? connection u@ on M coincides with the SO(1, p) components of the Cartan form X ? N> (4.27) u@ "du@M u@ ? @M ? and R "XG XG in #at M . ?@ ? @ M " Let us now derive the restriction on components of XK G (4.26) which follows from the integrability ? condition (4.18). To this end we multiply (4.18) by uG , then the left-hand side of (4.18) becomes zero @M and we get e?XK G "e?e@XK G "0 N XK G "XK G , (4.28) ? @ ? @ ? ? @ which implies that XK G is symmetric in the SO(1, p) indices a and b. This symmetric matrix is called ? @ the second fundamental form of the surface, and its trace hG"g?@XK G characterizes average ? @ extrinsic curvatures of the surface in the target space [7,6]. To summarize, conditions (4.9) and (4.10) whose integrability leads to Eqs. (4.19), (4.24) (containing (4.25)), and (4.28) completely determine the geometrical properties of embedding a general (pseudo) Riemann surface M into a (pseudo) Riemann manifold M in terms of the vielbein, N> M " words Eqs. (4.9), (4.10), connection and curvature forms of M pulled back on to M . In other N> M " geometry on the embedded (4.19) and (4.28) determine the induced surface. From the physical point of view the induced geometry of the surface means that, though the surface is a curved manifold and hence contains a (p#1)-dimensional gravity, this gravity is not &fully #edged' in the sense that the graviton on M is described by the induced metric (4.15) and, N> hence, is a composite "eld built from the worldsurface scalars xK (m). An interpretation of General Relativity as induced gravity in the worldvolume of a brane has been discussed in [147].
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We would like the embedded surface M to be associated with the worldvolume of a p-brane. N> The worldvolume which describes the classical motion of the p-brane in the target space is of a minimal volume. This is an additional condition of embedding M which is equivalent to the N> p-brane equations of motion. In the geometrical language the minimal embedding of M is speci"ed by the requirement N> that the trace of the second fundamental form XK G determined in (4.26) and (4.28) is zero ? @ g?@XK G "0 , (4.29) ? @ which implies that the average extrinsic curvatures of M in M are zero. N> M " of motion which one gets, Let us now show that Eq. (4.29) is equivalent to the p-brane equations for example, from the Nambu}Goto action (1.2). Using de"nition (4.26) of XK G and relations (4.16), ? @ (4.2), (4.5) and (4.15) we can rewrite Eq. (4.29) as follows: g?@XK G "g?@( E@M )uG "g?@ (eKR xK E? )E EL@uG ? @ K @M K ?L ? @ ? @ @M "g?@eL R (eKR xK g (x))EL@uG !g?@eKR xK g ( EL@)uG ? L @ K KL @M ? K KL ? @M 1 " R ((!ggKLR xK g (x))!gKLR xK R xJM g CN EL@uG "0 , L KL K L KN JL @M (!g K
(4.30)
where "eK(R #R xK X ), and CN (x) is the D-dimensional Christo!el symbol ? ? K K K JL CN (x),( EN )E@M "(R EN !XA EN )E@M . (4.31) JM @M L JM @M J@ A L JL In the square brackets of the last line of (4.30) one can recognize the standard equations of motion of a bosonic p-brane propagating in a D-dimensional target space. In (4.30) they are projected on to the D!p!1 directions transverse to the p-brane. This projection re#ects the fact that, because of the (p#1)-dimensional worldvolume reparametrization invariance, among the D equations obtained from the Nambu}Goto action (1.2) only D!p!1 transverse equations are independent, and these are explicitly singled out by the geometrical embedding condition (4.29), (4.30), while the other p#1 equations (¶llel' to the brane) are identically satis"ed. To see this let us in the last line of (4.30) substitute uG with uA , which corresponds to projecting the equations of motion of xK (m) @M @M along the brane worldvolume. Then &moving' in the direction opposite to that which we followed developing Eq. (4.30) we get
1 R (egKLR xK g (x))!gKLR xK R xJM g CN EL@uA L KL K L KN JL @M e K
"g?@[( E? )uA !eKuA ]"g?@[( u? )uA !eKuA ],0 , ? @ ? ? K@ ? @ ? ? K@ which is identically zero by virtue of the embedding condition (4.19) for the M spin connection. N> Thus, in the embedding approach the bosonic p-brane equations of motion are described by the minimal embedding condition (4.29) on components of the M pullback (4.22) of the target space N> connection form XK @M . ? We shall now generalize the results of this subsection to the description of superembeddings [45,50].
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4.2. Superembedding Consider a supermanifold M locally parametrized by supercoordinates Z+ "(XK , HI ) " L M is #at the index k is associated with an irreducible (m"0,2, D!1; k"1,2, 2n). When M " L spinor representation of SO(1, D!1) M and, in addition, can also include an internal index corresponding to N-extended supersymmetry, as in the case of the N"2, D"10 superstrings (Section 3). We assume that the supergeometry on M is such that it describes a D-dimensional superM " L gravity (see Sections 2.7 and 3.3). This implies that the structure group of the tangent superspace of is SO(1, D!1) (or more precisely Spin(1, D!1)) and the supervielbein components M M " L E (Z)"dZ+ E "(E? , E? ) (4.32) + satisfy the torsion constraint (3.79). We are interested in embedding into the target superspace M a supersurface M locally N> L M " L parametrized by supercoordinates z+"(mK, gI). With such an embedding we would like to describe the dynamics of superbranes whose n-dimensional i-symmetry is replaced by more fundamental worldvolume supersymmetry. So, we take a supersurface with the number of fermionic directions gI which is half the number of the target superspace fermionic directions. We should stress that the consideration of general properties of superembeddings carried out in this subsection is schematic, as far as spinors and C-matrices are concerned, since it is pretty hard to describe with one and the same set of exact formulae di!erent types of spinor representations (Majorana, Majorana}Weyl, simplectic}Majorana, etc.) which are used in each given case of superembedding depending on D and p. A local cotangent frame on M is given by the supervielbein N> L e (z)"dz+e "(e?, e?O) (4.33) + whose components are transformed under vector and spinor representations of the structure group SO(1, p);SO(D!p!1) of M (remember that D is the bosonic dimension of the target N> L superspace). The index a"0, 1,2, p is the index of the vector representation of SO(1, p), while a and q are the indices of irreducible spinor representations of SO(1, p) and SO(D!p!1), respectively, such that dim Spin(1, p);dim Spin(D!p!1)"n. These requirements on the structure group of M and its spinor representations put restrictions on the dimensions of N> L supersurfaces which can be embedded into a given target superspace. The analysis of possible superembeddings of this kind shows that they correspond to all known superbranes (super-pbranes, D-branes and M-branes) and their dimensional reductions (possibly accompanied by a dualization). This results in a brane scan [50] similar to the standard one [149].
One can study the embedding of a more general class of supersurfaces with the dimension d of the odd (fermionic) subspace being less than n. As we have seen with the examples of superparticles (Section 2.5.4) and superstrings (Section 3.2.1), such embeddings may also describe superbranes which preserve half of the target-space supersymmetries, but there may exist superembeddings which describe BPS states preserving a lower number of target-space supersymmetries. Let us also mention that the embedding of supersurfaces with d 'n, in general, seems to be too restrictive to describe physically interesting models (see, however, [148]).
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At the moment we do not make any assumption concerning the supergeometry of M , since, N> L as we shall see, it is determined by the superembedding condition. Consider the pullback on to M of the M supervielbein E N> L M " L E? "e e+R Z+ E? ,e?E? #e?OE? , (4.34) + + ? ?O (4.35) E? "e e+R Z+ E? ,e?E? #e?OE? , + + ? ?O where e+(z) is the supervielbein matrix inverse to e (z). + As in the bosonic case we postulate that the superembedding is such that with an appropriate choice of the local frame e? on M and by the use of local SO(1, D!1) rotations (4.11)}(4.13) N> L one can direct the vector component pullback (4.34) of E? on the supersurface such that E?,E@M u? (z)"e?(z) , (4.36) @M (4.37) EG,E@M uG (z)"e@E@M uG "0 . @M @ @M Eqs. (4.34), (4.36) and (4.37) imply that e?(z) is the induced supervielbein on M since it N> L coincides with the pullback of E? . We also see that (4.38) E? (Z(z))"u? (z) . ? ? But what is most important, in view of (4.34)}(4.37), we get the basic superembedding condition (4.39) E? ,e+ R Z+ E? "0 . ?O ?O + + Thus, the superembedding condition (4.39) is a natural consequence of the embedding carried out in such a way that the bosonic cotangent subspace of M is a subspace of the bosonic cotangent N> L . Alternatively, Eq. (4.39) implies that the fermionic tangent subspace of M is subspace of M " L N> L M a subspace of the fermionic tangent subspace of M . M " L The embedding conditions (4.36), (4.37) and (4.39) imply that (as in the bosonic case (4.15)) the supersurface metric g (z),e? e@ g "R Z+ E? R Z,M E@M g "E? E (4.40) KL K L ?@ K + L ,M ?@ K L? is the induced metric. In addition to the vector components the supervielbein (4.32) contains the spinor components (4.35). When the vector components are transformed by the SO(1, D!1) matrix u? , the spinor components must be simultaneously transformed by a matrix v? of the spinor @M @ representation of SO(1, D!1). The matrices u? and v? are related to each other byM the standard @ @M D-dimensional C-matrix de"ning conditions M (4.41) C@M u? ,v? C? v@M . A BM @M A ? @M BM Eq. (4.41) implies that u? are composed of v? , which have D(D!1)/2 independent components @M @M corresponding to D(D!1)/2 independent generators of Spin(1, D!1). The choice of the C-matrices depends on the dimensions D and p, but for the embedding under consideration they can always be schematically represented by the following SO(1, p); SO(D!p!1) invariant set of matrices (compare with Eq. (2.94)). In the directions parallel to
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the supersurface
c? dP C? " ?@ O ? @M 0
0
, a"0, 1,2, p , c??@dPY OY and in the directions normal to the supersurface
(4.42)
0 d? cG @ OOY , i"1,2, D!p!1 , CG " (4.43) ? @M d@ (cG)OYO 0 ? where c? are d"p#1 c-matrices, and the matrices cG obey the SO(D!p!1) Cli!ord algebra, ?@ OOY the indices q and q corresponding to (in general non-equivalent) spinor representations of SO(D!p!1). Note that in (4.42) and (4.43) the SO(1, D!1) spinor indices a, b split into the M the SO(1, p); SO(1, p);SO(D!p!1) spinor indices (aq) of the M odd subspace and N> L SO(D!p!1) spinor indices a " : (aq ) &transverse' to the supersurface. The spinor components of the supervielbein pullback (4.35) transformed with v? (z),(v?O, v?Y) ? ? @M take the form
(4.44)
. (4.45) E@M v? "(E@M v?O, E@M v?Y) where v?Y,v @M @M @M @M @M ?OY In the same way as we have done with the vector basis (4.36), we can always make a choice of a spinor basis e?O on M such that N> L E@M v?O"e?O , (4.46) @M which, in view of (4.35), implies E@M v?O"0 , ? @M E@M v?O"d? dO , @ P @P @M and, hence,
(4.47)
E@M v?Y,e?Oh?Y #e?E@M v?Y , @M ?O ? @M where
(4.49)
h?Y (z),h (z) , ?O ?O @OY is an SO(1, p);SO(D!p!1)-valued matrix on M . N> L From Eqs. (4.47)}(4.50) we derive that
(4.50)
E@M "v@M #h?Y v@M , ?O ?O ?O ?Y where by v@M and v? we have de"ned the components of the matrix ?O ?Y (v\)? ,(v? , v? ) @M ?O ?Y
(4.51)
(4.48)
(4.52)
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inverse to v@M ,(v?O, v?Y). The inverse matrices (4.17) and (4.52) are related by the equation similar to ? ? ? (4.41) C@M (u\)? ,(v\)? C? (v\)@M . A BM @M A ? @M BM
(4.53)
Note that the embedding conditions (4.36), (4.37), (4.46) and (4.49) are de"ned up to the local SO(1, p);SO(D!p!1) rotations which reduce the number of independent components of u? (z) @M and v@M (z) from D(D!1)/2 down to D(D!1)/2!(p#1)p/2!(D!p!1)(D!p!2)/2" ? (p#1)(D!p!1). Hence, as in the bosonic case, the matrices u? and v@M (z) are, respectively, @M ? Lorentz}vector and Lorentz-spinor harmonics which parametrize the coset space [92,93,45] SO(1, D!1) . SO(1, p);SO(D!p!1) As we shall see in Section 5 the physical meaning of the supersurface "eld h?Y (z) is that it appears ?O in (4.49) due to the presence of gauge "elds on the worldvolume of superbranes, such as Dirac} Born}Infeld vector "elds of the D-branes and the self-dual antisymmetric gauge "eld of the M-5-brane. h?Y (z) is expressed in terms of the "eld strengths of these "elds [50,51,197]. ?O For the ordinary super-p-branes (where there is no worldvolume matter other than scalar and spinor "elds associated with the transverse oscillations of the brane in the target superspace) h?Y (z) ?O is either auxiliary (as in the case of an N"1, D"4 supermembrane [70]) or zero [45]. We shall demonstrate this for the N"1 superparticles and superstrings in the next subsection, and for the D"11 supermembrane in Section 5.1. We now consider the consequences of the superembedding conditions (4.36)}(4.39), (4.46)}(4.51) for the properties of the induced supergeometry of M . As in the bosonic case (Eq. (4.18)), we N> L external di!erential "d#X of the vector supervielbein pullback (4.34). Then, in take the M M " L(4.39) and the target-space supergravity torsion constraint view of (4.38), ¹? , E? "dE? #E@M X? "!iE? C? E@M , @M ? @M
(4.54)
where all other components of the two-form ¹? are (conventionally) put to zero, we get !iE? C? E@M "¹?u@M !e@(u? u@M ! u@M ) . ? @M ? @ ? @
(4.55)
Remember that ¹?"e e ¹? "de?#e@u? is the supersurface torsion and E@M "u@M (4.38). @ ? ? From (4.55) it follows that (up to an appropriate rede"nition of ¹?) the M spin connection N> L u? can always be identi"ed with u@M u? , as in the bosonic case (4.19) @ @M @ u? " u@M u? ,(du@M #u? X@M )u? ,XK ? . @ @ @M @ @ ? @M @
(4.56)
Provided that the basic torsion constraint ¹? "!2iC? is imposed, all other torsion constraints are derived by ? @M ? @M solving for the torsion Bianchi identities. In addition, by rede"ning the supervielbeins and superconnection one can always put all components of ¹? except for ¹? to zero (see [151}154,128,127,129] for a detailed analysis of supergravity ? @M constraints in various dimensions).
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Then (4.55) reduces to
and
¹?,e e ¹? "!iE? C? E@M u? ,¹? u? ? @M ? ?
(4.57)
(4.58) e? E@M uG "e? u@M uG ,e?XK G "!iE? C? E@M uG . ? ? ? @M ? @M ? @M Eqs. (4.57) and (4.58) are the integrability conditions of the superembedding conditions (4.37), (4.39), (4.46) and (4.56). In particular, Eq. (4.57) implies that the superembedding conditions (4.39), (4.36) and (4.56) require the supersurface torsion to satisfy constraints, whose form depends on the concrete superembedding under consideration [45] (see Section 5). Thus, if target space supergeometry is constrained to be of the supergravity type, the superembedding induces supergeometry on the supersurface such that it corresponds to a worldvolume (induced) supergravity. The supermultiplet of the induced supergravity includes the graviton and gravitino which are composed from components of a worldvolume matter supermultiplet. The matter supermultiplet consists of the D!p!1 scalar modes xG(m) and the n/2 spinor modes h?Y of the brane transverse #uctuations, and it may also contain worldvolume gauge "elds (as in the case of the D-branes and the M5-brane). Let us now brie#y discuss the consequences of the integrability equation (4.58). In each given case of superembedding, Eq. (4.58) allows one to "nd out whether or not the superembedding condition (4.39) contains the superbrane equations of motion. From Eq. (4.58), in view of (4.41), (4.43), (4.46) and (4.49), it follows that, as in the bosonic case (4.28), the vector component of the superform XK G "e XK G is symmetric with respect to the SO(1, p) ? ? vector indices e?e@XK G "0, N XK G "XK G . (4.59) @ ? @ ? ? @ By analogy with the bosonic embedding we can identify XK G ,e+R Z+ E? uG with components + ? ? ? @ ? + of the second fundamental form of the supersurface M de"ned as N> L (4.60) K!Y "e+R Z+ ( E #u! E )E!Y + + ! + where E!Y,(uG , v?Y!v?Oh?Y ), E E!Y,0, a "( ) ? ? ? ?O @PY In many cases of embedding M into M the integrability conditions of (4.39) require " L N> L M additional restrictions on the components of XK G , such as a (possibly inhomogeneous) analog of ? the tracelessness of its vector part g?@XK G "0, which is equivalent to the bosonic "eld equations @ ? (4.30), and the condition on the spinor part of the following type [94,45] (c?)@?XK G "0 . (4.61) ?O ? As we shall see in the next section, Eq. (4.61) amounts to the fermionic "eld equations, and, hence, is a superembedding counterpart of the bosonic minimal embedding condition (4.29). Therefore, when integrability requires (4.61) or its inhomogeneous generalization due to the structure of the supergravity background, the superembedding condition (4.39) (or equivalently
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(4.36)) contains the equations of motion of the corresponding superbrane. This is in contrast to the bosonic embedding where the minimal embedding condition (4.29), being equivalent to the dynamical equations for the bosonic brane, is always imposed as an additional condition. Note that Eq. (4.61) is a generic minimal superembedding condition, and it can be also used (as an addition condition) to "nd the superbrane equations of motion in the cases where the basic superembedding condition (4.39) is o!-shell. To "nd out whether the superembedding condition (4.39) puts a given superbrane theory on the mass shell one should examine each case separately. In the next section we shall make such an analysis for the M-theory branes. To conclude the description of the induced supergeometry of M we should identify the N> L SO(D!p!1) connection AP "dz+AP (z)"AGH(c )P acting on the spinor components e?O of the GH O O +O M supervielbein (4.33). Note that the Spin(1, p) connection u@(z) is given by the standard N> L ? relation 1 1 u@(z)" u?@(c )@" XK ?@(c )@ . ? ?@ ? ?@ ? 4 4
(4.62)
external di!erential of Eq. (4.35) and multiply the To "nd the form of AP let us take the M O M " L we get result by v@P. In view of Eqs. (4.47)}(4.51) ? ¹? v@P"¹@P#e?( E? )v@P#e?Oh?Y ( v? )v@P#e?O[( v? )v@P!dP u@!d@ AP ] , (4.63) ? ?O ?Y ? O ? ? O ? ? ?O ? where ¹@P,de@P#e?Pu@#e@OAP is the spinor component of the M torsion form. ? O N> L Using the harmonic relations (4.41) and (4.53) one can show that 1 1 ( v? )v@P" dP XK ?@(c )@ # d@XK GH(c )P , ?@ ? 4 ? GH O ?O ? 4 O
(4.64)
where XK ?@ and XK GH are components of the target-space connection form de"ned in (4.22) and (4.56). In Eqs. (4.56) and (4.62) we have identi"ed u?@ with XK ?@. Then from Eqs. (4.63) and (4.64) it follows that we can identify the SO(D!p!1) connection form on M with XK GH, i.e. N> L (4.65) AGH"XK GH, AP "XK GH(c )P , GH O O and Eq. (4.63) reduces to the expression de"ning ¹@P as the induced torsion of M N> L ¹@P"¹? v@P!e?( E? )v@P!e?Oh?Y ( v? )v@P . (4.66) ? ?O ?Y ? ? ? The curvature forms associated with the induced connections u@ and AH are obtained by taking the ? G corresponding components of the pullback (4.24) of the target space curvature. R@ has the form ? similar to the Gauss equation (4.25) R@ ,du@ #uA u@"uA RBM u@ !XK G XK @ , ? ? ? A ? G ? A BM
(4.67)
Our choice of the supersurface connections (4.62) and (4.65) is such that [(d#X#u#A)v@M ]v@P,0 (see also the ?O @ footnote to Eq. (4.19)). This di!ers from the choice of Refs. [51,52] where the form of the connectionsM has been "xed in such a way that the supersurface torsion components ¹A and ¹AQ vanish. @ ?O @P
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and RH is determined by the so-called Ricci equation G RH,dAH#AIAH "uA RBM uH !XK ?XK H . (4.68) G G G I G ? G A BM The last equation contained in (4.24) is the Codazzi equation which speci"es the form of the supersurface covariant di!erential of XK G ? (4.69) DXK G ,dXK G #u@ XK G #XK H AG "uA RBM uG . ? A BM ? ? ? @ ? H From Eq. (4.69) we see that in the #at target (super)space (R"0) the (super)embedding is such that the external covariant di!erential of XK G on the (super)surface vanishes. ? To summarize, Eqs. (4.36), (4.37), (4.46), (4.56) and (4.65)}(4.69) completely determine the induced geometrical properties of the embedded supersurface in terms of the pullbacks of the targetsuperspace supervielbein and superconnection adapted to the supersurface by the use of the Lorentz harmonics. Note that, analyzing the general properties of the superembedding, we have from the beginning imposed the constraints (4.54) on target-superspace geometry, which we "nd more instructive. But instead of introducing the target-superspace constraints, we might from the beginning specify the supergeometry on the supersurface. Then the constraints on the geometry of target superspace would arise as consequences of the integrability of the superembedding condition. These two initial options of choosing the superspace constraints for studying the superembeddings are equivalent, and depending on the problem considered it is convenient to use one option or another. This equivalence follows from the general property of the basic superembedding condition (4.39). If we only impose this condition and do not assume any constraints on the supersurface and target}superspace torsion, the spinor components of the torsion ¹? and the pullback of ¹? are related by the following identity ¹? E? "E? E? ¹? , (4.70) ?O @P ? @M ?O @P ? which can be easily derived from (4.34) by taking the covariant external di!erential of both of its sides and taking into account (4.39). We shall now demonstrate how in the generic case to extract i-symmetry transformations from general local superdi!eomorphisms of the embedded supersurface [27,45,72,51,52,70]. 4.3. i-symmetry from the point of view of superembedding As we have already mentioned (see Eqs. (1.4), (2.26) and (3.12) as examples) for all known superbranes in the Green}Schwarz-type formulation the i-symmetry transformations of the superbrane coordinates Z+ (m) in target superspace have the following generic form: d Z+ E? "(1#CM )? i@M , G + @M d Z+ E? "0 , G + where (1#CM )? is a spinor projection matrix speci"c to each type of superbranes. @M
(4.71) (4.72)
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At the same time, in the superembedding approach Z+ (z+) (z+"(mK, gI)) are worldvolume super"elds transformed as scalars under the M superdi!eomorphisms z +"z M(z), their N> L in"nitesimal variations being dZ+ "dz+R Z+ "dz+e (e, R Z+ ) . + + , Multiplying (4.73) by E and splitting the index A on the vector and spinor indices (a, a) + dZ+ E? "dz+e (e, R Z+ E? ),dz+e E? , + , + +M + dZ+ E? "dz+e (e, R Z+ E? ),dz+e E? . + , + + + Because of the superembedding condition (4.39), Eq. (4.75) reduces to
(4.73) we get (4.74) (4.75)
dZ+ E? "dz+e? E? . (4.76) + ? + Comparing (4.76) with (4.72) we see that to get i-symmetry transformations from the worldvolume superdi!eomorphisms we should consider such variations of z+ for which the Grassmanneven superdi!eomorphisms are zero (4.77) dz+e? "0"dZ+ E? . + + Then only the Grassmann-odd part dz+e?O of the M superdi!eomorphisms contributes to the + N> L variation (4.74), and we get (4.78) dZ+ E? "dz+e?O E? . + ?O +M Since dz+e?O are arbitrary variations we can (without losing generality) replace them with + (4.79) dz+e?O ,i@M (z)v?O . @M + Eq. (4.78) takes the form (4.80) dZ+ E? "i@M (z)(v?OE? ) . @M ?O +M If we now recall the superembedding conditions (4.48), (4.51) (E? "v? #h?Y v? ) we shall notice ?O ?O ?O ?Y that v?OE? "P? (z) is a projector (PA P? "P? ). This projector can be identi"ed (possibly, up to @M A @ @M ?O @ M (4.71), i.e. linear transformations) with (1#CM ) M from (1#CM )? "v?OE? . (4.81) @M ?O @M We conclude that the leading (gI"0) components of the in"nitesimal superdi!eomorphisms (4.80) and (4.77) yield the i-symmetry transformations (4.71) and (4.73). Here it is also the place to note that in the superembedding formulation of the superbranes we have not introduced Wess}Zumino terms as independent objects. The reason is that in superbrane actions the Wess}Zumino term is introduced to ensure the non-manifest i-symmetry of the worldvolume action, while in this formulation the construction is manifestly worldvolume superdi!eomorphism invariant. In the superembedding approach the Wess}Zumino term shows up in the structure of the bosonic equations of motion, which we will see with the examples of the M-theory branes.
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4.4. N"1, D"10 superparticles and superstrings from the general perspective of superembedding Let us now connect the general properties of the superembeddings considered in the previous subsection with the doubly supersymmetric description of N"1, D"10 superparticles and superstrings given in Sections 2 and 3. Superembeddings which describe superparticles have been discussed in [150], and the N"1, 2, D"10 superstrings have been studied in [45]. In these cases the target superspace is M (for simplicity, we take it to be #at), and the M embedded supersurface is M (where d"1 for the superparticles and d"2 for the superstrings). B The target-space structure group SO(1, 9) is broken by the presence of the superparticle or the superstring down to SO(1, 1);SO(8). The explicit realization of the C-matrices (4.42) and (4.43) is given by Eq. (2.94), with the matrices CC and CC, or CC!!"C(C$C) corresponding to the SO(1, 1) subgroup of SO(1, 9). The indices a"(!!,##) and a"(!,#) are, the lightcone indices of the vector and the spinor representation of SO(1, 1), respectively. In the case of the superparticle the worldline coordinate q is identi"ed with the coordinate m\\ of the superworldsheet mK"(m\\, m>>). And recall that the odd coordinates g\O of M are one}component B worldline or worldsheet (chiral) spinors. The SO(1, 1);SO(8) splitting of the Lorentz-harmonic matrices (4.11), (4.44) is u? "(u\\, u>>, uG ) @M @M @M @M v? (z)"(v\O, v>OY) . @M @M @M The harmonics parametrize the coset space
(4.82) (4.83)
SO(1, 9) . SO(1, 1);SO(8) In the #at target superspace projection (4.45) of the spinor supervielbein E? "dH? is dH@M v? "(dH@M v\O, dH@M v>OY) . (4.84) @M @M @M In (4.84) we can recognize the Lorentz-covariant version of the splitting of D HI discussed in O Eqs. (2.95)}(2.99), when we made the analysis of the superembedding condition for the N"1 superparticle. To get (2.95) from (4.84) we should simply take v? "d? which implies u? "d? . This, of course, @ @M @M @M breaks both the right and the left SO(1, 9) group M associated with the Lorentz harmonics (see Eq. (4.13) and below) and, hence, breaks the manifest target}superspace covariant description of the superembedding. Then, for instance, Eq. (4.48) reduces to Eq. (2.99). Furthermore, comparing (4.49) with (2.5.47) and (2.103) (where q,!!) we "nd that h
"cG EG , (4.85) \O \OY OOY \\ where EG are &transverse' components of the vector supervielbein pullback E? " \\ \\ R X? !iR HM C? H (2.103). \\ \\ Now notice that by taking u? "d? (and v? "d? ) we do not (in general) have the embedding @M @ @ @ condition (4.37) (i.e. EG O0) for an M adaptedM localM frame such that \\ (4.86) E@M uG "0 . \\ @M
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To get (4.86) we must perform an appropriate SO(1,9) transformation of E? , as has been \\ explained in the previous subsections. Once this has been done, the tensor h (4.85) vanishes. \O \OY Thus, in the case of the N"1 superparticles and superstrings, when the pullback of the target-space supervielbein E is adapted (by the Lorentz harmonics) to the superworldline or the superworldsheet such that Eqs. (4.36), (4.37) and (4.46) are satis"ed, the h-component of E@M is zero \O and (in the #at target superspace) we have (4.87) E@M ,D H@M "v@M . \O \O \O From (4.87) it follows that its leading (g\O"0) component, which is the commuting spinor variable j@M , coincides with components of the inverse Lorentz-spinor harmonics \O . (4.88) j@M (m)"v@M " \O \O E And from (4.38) it follows that (4.89) E? ,R X? !iR HM C? H"u? \\ \\ \\ \\ coincides with a light-like component of the inverse Lorentz-vector harmonics. Using the de"ning harmonic relations (4.53) one can convince oneself that E? and E@M are \\ \O related by the Cartan}Penrose formula (2.87) d u? "v CC? v . \P \O OP \\ In Section 2.5.4 we have demonstrated that in D"10 the independent components of the matrix E@M parametrize the S sphere, while from Eq. (4.87) we see that E@M coincides with components of \O \O Lorentz harmonics v@M (which are new auxiliary variables introduced in the superembedding \O formulation). As we have mentioned, the harmonics (v\)@M "(v@M , v@M ) parametrize larger ? \O >OY (16-dimensional) coset space SO(1, 9) . SO(1, 1);SO(8) Therefore, there should be an additional local symmetry which allows one to reduce the number of independent components in (v\)@M from 16 to 8. ? This symmetry indeed occurs since half of the harmonics, namely v@M , do not enter the relations >OY de"ning the pullback of the target}space supervielbein (4.87), (4.89). Hence, we can vary the form of v@M in a way which keeps the harmonic condition (4.53) intact. The allowed variations have the >OY form of boost transformations K G dv@M "0, dv@M "KG (z)cG v@M , (4.90) \O >OY >> OOY \O u? , du? "0, du? "KG u? , du? "K \\ >> >> G G >>G \\ where KG (z) are eight independent parameters. Therefore, we see, from a somewhat di!erent >> point of view, that in the case of the N"1, D"10 superparticles and superstrings the commuting (harmonic or twistor-like) spinor variables parametrize the compact manifold S-sphere, which is
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realized as a coset space [92,40] SO(1, 9) S" , [SO(1, 1);SO(8)] K 1 G where the stability subgroup is the semidirect product of SO(1, 1);SO(8) with the boost transformations K (4.90). G We conclude that the doubly supersymmetric formulation of N"1 superparticles and superstrings, which we have developed in Sections 2 and 3 on dynamical grounds, completely "ts into the general geometrical picture of superembeddings. Let us now apply the general properties of superembedding to the description of M-branes.
5. M-theory branes M-theory [16] is a D"11 theory whose low energy limit is D"11 supergravity [155] and which also contains supermembranes [9,156] and super-5-branes [10] as part of its non-perturbative spectrum. The D"11 supergravity multiplet consists of the graviton e? (x), the gravitino t? (x) and the K K of the supervielbein antisymmetric 3-rank gauge "eld A (x), which are leading components JKL E? (X,H), E? (X,H) and the three-form A(X,H), respectively (a"0, 1,2, 10; a"1,2, 32). as superstrings minimally couple The supermembrane minimally couples to the gauge "eld A, to the Neveu}Schwarz "eld B (3.10). The Green}Schwarz-type supermembrane action is [9]
S "! dm(!det g # KL +
M
dZ*M dZ+ dZ,M A
,+*
(Z)
(5.1)
(g (m)"E? E , and the tension is put equal to one). KL K L? The super-5-brane minimally couples to a six-form gauge "eld A whose seven-form "eld strength F"dA!A dA is dual to the four-form "eld strength F"dA of A. The 5-brane also (nonminimally) couples to A via the extended self-dual "eld strength H(m)"db!A(Z(m))
(5.2)
of an antisymmetric "eld b (m) living in the worldvolume of the 5-brane [157]. In this sense the KL 5-brane is a dyonic object. The form of the 5-brane supergravity coupling is uniquely de"ned by local symmetries of the 5-brane worldvolume action which are responsible for the self-duality properties of b (m) [158]. The &Born}Infeld-like' super-5-brane action is [63,64] KL (!g 1 HHKLH )# A! dbA , (5.3) S " dm(!(!det (g #HH )# KL KL KL + 4 2 M
and v (m)"R a(m)/ where g (m)"E? E , H ,H vJ(m), HHKL,(1/(!g)eKLJJ J J v H KL KLJ J J J J J J K L? KL (!(Ra) is the normalized derivative of the auxiliary scalar "eld a(m) which ensures the covariance of the 5-brane action, its construction being based on a generic method for the covariant Lagrangian description of duality-symmetric "elds proposed in [159,160].
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A duality-symmetric component action for D"11 supergravity, which includes both gauge "elds, A and its dual A, and which thus couples to both the membrane and the 5-brane, was constructed in [161]. This action reduces to the Cremmer}Julia}Scherk action [155] when the "eld A is eliminated by solving for an algebraic part of the duality relations between F and F. The i-symmetry of the supermembrane and the super-5-brane worldvolume actions requires the superbranes to propagate in target}superspace backgrounds whose supergeometry is constrained to be that of D"11 supergravity. In the case of the superspace description of D"11 supergravity the constraints imposed on components of the torsion ¹ (Z)" E "dE #E M X , curvature R(Z) M "dX M #X!M X M , and on !M M the dual "eld strengths F(Z) and F(Z) have the following form [127,129]: ¹? "!iEM C? E ,
(5.4)
1 F E? (C@M 2@M !8d @M C@M 2@M )? E@M , ¹? " ? ? @M 288 @M 2@M
(5.5)
i 1 EM (C?@2 #24C d? d@M )EFA 2A # EBM EA R?@ , R?@"! A A A A
A A AB 144 2
(5.6)
i 1 F"dA" E? E@M EM C E# E? 2E? F 2 , @? ? ? 2 4!
(5.7)
1 F"dA! A dA 2 i 1 " E? 2E? EM C 2 E# E? 2E? F 2 , ? ? ? ? 5! 7! where the "eld strength components F 2 and F 2 are related by the Hodge duality ? ? ? ? 1 F 2 " e 2 2 F@M 2@M . ? ? 4! ? ? @M @M
(5.8)
(5.9)
Constraints (5.4)}(5.9) are on-shell D"11 supergravity constraints since they imply the D"11 supergravity equations of motion (which can be alternatively obtained from the D"11 supergravity component actions [155,161]). Note that in D"11 we deal with 32-component Majorana spinors WM "(W2)@M C where C is @M ? ? the charge conjugation matrix, and we can use C and its inverse (C\)? @M "!C? @M "!C to ? @M ? @M lower and raise the spinor indices. Then the following antisymmetric products of the C-matrices (5.10) C? ,C (C? )A , (CC? ? ) , (CC? 2? ) ? A ? @M ? @M ? @M @M are symmetric in the spinor indices. We are now in a position to describe the M-branes by applying the superembedding methods of Section 4.
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5.1. The supermembrane The superembedding of the D"11 supermembrane was "rst analyzed in [45]. is that of D"11 supergravity and the embedded supersurface The target superspace M M M , associated with the supermembrane worldvolume, is characterized by a local super vielbein basis e (z)"(e?, e?O) (a"0, 1, 2; a"1, 2; q"1,2, 8) transformed under the action of SO(1, 2);SO(8). The d"3 spinors are two}component Majorana spinors. An appropriate form of the D"11 C-matrices (4.42), (4.43), which re#ects the embedding, and of the charge conjugate matrix C? @M to raise indices, is
c? d C? " ?@ OP ? @M 0 0 CG " ? @M d@ cG ? OYO e?@d OP C? @M " 0
0
, a"0, 1, 2 , c??@d OYPY d? cG @ OOY , i"1,2, 8 , 0
0
(5.11)
(5.12)
. (5.13) !e d ?@ OYPY where cG ,cG are the SO(8) C-matrices, c? ,e c?A are symmetric, and the d"3 C-matrices OYO OOY ?@ ?A @ c?A were de"ned in Eq. (2.49). The d"3 spinor indices are raised and lowered by the unit @ antisymmetric matrices e and e?@, and the SO(8) indices are raised and lowered by the unit ?@ matrices d , d , so that there is no distinction between the upper and lower SO(8) indices. (Recall OP OYPY that the unprimed index q and the primed index q correspond to di!erent spinor representations of SO(8).) Induced supergeometry on M is described by the general superembedding conditions (4.36)}(4.39), (4.46)}(4.51), (4.56)}(4.59) and (4.65). We shall now analyze whether in the case under consideration their integrability puts additional restrictions on the form of the supervielbein pullback (4.49) and on components of the one-form XK G "e XK G in (4.58). Remember that XK G is related to the second fundamental form (4.28), (4.60) ? ? ? of the (super)surface, and if its components satisfy additional conditions, such as Eqs. (4.29) and (4.61), these reproduce (super)brane equations of motion, as Eq. (4.30) in the bosonic case. Consider in more detail Eq. (4.58). Making use of the harmonic relations (4.41), the form of the D"11 C-matrices (5.11), (5.12), and the equations (4.46), (4.49), we can rewrite Eq. (4.58) in the following form: e?XK G "e? E@M uG "e?(dE@M #EA X@M )uG ? @M ? ? A @M ? "!2ie?Oe@PcG hOY #2ie?e?OcG E@M v@OYe . (5.14) OOY ?P @ OOY ? @M ?@ (Recall that E@M "u@M , see Eq. (4.38).) Comparing the left- and right-hand sides of (5.14) we "nd that ? ? e?e@XK G "0PXK G "XK G " E@M uG , (5.15) @ ? @ ? ? @ @ ? @M XK G " E@M uG "2icG E@M v@OYe , (5.16) ?O ? ?O ? @M OOY ? @M ?@
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and cG hOY "!cG hOY . (5.17) OOY ?P @ POY @O ? Let us analyze Eq. (5.17). It implies that the matrix cG hOY is antisymmetric with respect to the pair OOY ?P @ of the indices (ar) and (bq). A general matrix h is decomposed in the basis of the elements of the SO(8) and SO(1,2) ?P @OY Cli!ord algebras as follows cH H H )#c? (h cH #h cH H H ) , "e (h cG #h ?@ ?H OYP ?H H H OYP ?P @OY ?@ H OYP H H H OYP where cH H H denotes the antisymmetric product of the SO(8) C-matrices c H cH cH . OYP OYO OPY PYP Substituting (5.18) into (5.17) we get h
(5.18)
cH H H )!(q r)"0 , cG (h cH #h OOY H OYP H H H OYP
(5.19)
and cG (h cH #h cH H H )#(q r)"0 . OOY ?H OYP ?H H H OYP Using the de"ning properties of the matrices cG and cG ,cG OOY OYO OOY cG cH #cH cG "2d , OOY OYP OOY OYP OP we "nd that Eqs. (5.19) and (5.20) imply that all components of the matrix (5.18) vanish
(5.20)
h
"0 . (5.21) ?P @OY Hence, in the case of the supermembrane Eq. (4.51) for the spinor components E@M of the ?O M supervielbein pullback reduces to M (5.22) E@M ,e+ R Z+ E@M "v@M (z) . ?O ?O + + ?O This, in particular, means that the induced superworldvolume torsion (4.57) obeys the standard d"3, n"8 supergravity constraint ¹? "!2id c? , (5.23) ?O @P OP ?@ which follows from (4.57), (4.41), (5.11) and (5.22). The vanishing of the h-matrix (5.21) also implies that, in view of the D"11 supergravity torsion constraint (5.5), the purely spinor components of the superworldvolume torsion ¹?O de"ned in (4.66) are zero ¹?O "0 . (5.24) @P AQ In turn, torsion constraints (5.23) and (5.24) imply that the covariant spinor derivatives D ,e+R #u #A of the superworldvolume satisfy the algebra + +D , D ,"2id c? D #R (z) , (5.25) ?O @P OP ?@ ? ?O@P where R (z) are components of the superworldvolume curvature (4.67), (4.68). ?O@P
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5.1.1. The fermionic equation Applying the covariant derivative D to Eq. (5.22), symmetrizing the result with respect to br @P and aq, and making use of the basic superembedding condition (4.39), the torsion constraint ¹@M "0 (5.5) and the superalgebra (5.25) we get ? A (5.26) D E@M #D E@M "2id c? E@M !vA X@M !vA X@M "D v@M #D v@M . ?O @P OP ?@ ? ?O @PA @P ?OA ?O @P @P ?O @P ?O Then multiplying Eq. (5.26) by vAOY and using the orthogonality properties of the spinor harmonics @M we obtain (5.27) 2id c? E@M vAOY"( v@M )vAOY#(aq br) . @P ?O @M OP ?@ ? @M (Remember that is the spinor component of the pullback of the external target-space di!eren@P tial "d#X and X@M "X?@(C )@M .) ?@ ? ? Making use of the harmonic relations (4.41), (4.53) and the representation (5.11), (5.12) of the C-matrices, one can show that ( v@M )vAOY"( u? )uG (c?)A (c )OY,XK G (c?)A (c )OY . (5.28) @P ?O @M @P ? ? ? GO @P ? ? G O Substituting (5.28) into (5.27) and taking the trace of its left- and right-hand side with respect to the indices (q, r) and (A ) we get @ (5.29) 16ic? E@M v@OY"XK G (c?)@(c )POY . @P ? ? G ?@ ? @M On the other hand, let us take the earlier found relation (5.16) and multiply it by (c?)@ and cOYP, the ? G result is (5.30) 16ic? E@M v@OY"!XK G (c?)@cPOY . @P ? ? G ?@ ? @M Comparing (5.30) with (5.29) we see that their right-hand sides have di!erent coe$cients and, hence, are zero. We thus get the additional restriction on components XK G of the second @P ? fundamental form (4.60) XK G (c?)?@(c )POY"(c?)? E@M v@OY"0 , @P ? G @ ? @M or, in view of (5.16),
(5.31)
XK G (c?)?@" E@M uG (c?)?@"0 . (5.32) @P ? @P ? @M Eq. (5.31) is nothing but the 16-component Dirac-type fermionic equation of motion of the D"11 supermembrane [9] written in the Lorentz-harmonic form [133] and promoted to the worldvolume superspace [45]. To see this let us rewrite (5.31) in a form similar to the fermionic equation which one derives from the supermembrane action (5.1). First of all notice that, with the use of the harmonic relations (4.53), the form of the C-matrices (5.11), and the embedding conditions E@M v?O"0 (4.47) and E? "u? (4.38), Eq. (5.31) can be presented in the following form: ? @M ? ? (c?)@E@M v "E@M (g?@E? C ) vA "0 . (5.33) ? ? @M @OY ? @ ? @M A ?OY
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Multiplying Eq. (5.33) by v?OY we get ? E@M (g?@E? C ) vA v?OY,g?@E@M [E? C (1!CM )] "0 @M ? ? @ ? @M A ?OY ? ? @ ? where, because of the projector
77
(5.34)
(1!CM )A ,v?OYvA , ? ?OY ? the l.h.s. of (5.34) identically vanishes under the action of the projector (5.35) (1#CM )? ,v?Ov? @M ?O @M in which one can recognize the i-symmetry projector of Eq. (4.80). Thus i-symmetry insures that among the 32 equations in (5.34), only 16 are independent. And these 16 equations reduce the number of the fermionic physical modes of the supermembrane down to 8. Using the Eqs. (4.38), (4.41), (4.53), (5.11) and (5.12) it is not hard to verify that the matrix CM de"ned in (5.35) has exactly the same form as in the i-symmetry projector and in the equations of motion of the Green}Schwarz formulation of the supermembrane [9] 1 CM " e?@AC ?@A 3!
where C "E? C . ? ? ?
(5.36)
(Note that the harmonics have disappeared.) Thus, at (gI"0) the worldvolume super"eld equation (5.34) (which is the consequence of (5.31)) reduces to the standard fermionic component equations of the supermembrane. To show this one should choose the Wess}Zumino gauge (see, for instance, [123]) for the components of the inverse supervielbein e+(z)" on the supersurface such that E eK(m) !tIP(m) ? , (5.37) e+(z)" " ? E 0 dI dP ? O and take into account the superembedding conditions E? "0 (4.39) and E? "v? (5.22). Then, ?O ?O ?O ,eK(m)E? (Z(m)) , E? " "eK(m)R Z+ E? " (5.38) ? ? E ? K + E K " "eK(m)R Z+ E? v " ,eK(m)E? (Z(m))v (m) . (5.39) E? v ? ? @OY E ? K + ? @OY E ? K ? @OY And the worldvolume gravitino "eld tIO(m) (5.37) is expressed in terms of E? (Z(m)) K ? (5.40) tIO(m)"E? vIO(m) , K ? ? due to the superembedding condition (4.47). Then the component equation of motion for hI (m) takes the form
gKLE@M [E? C (1!CM )] "0, @M ? K L ?
g (m)"e? e (m)"E? E@M g . KL K L? K L ?@
(5.41)
5.1.2. The bosonic equation Once the supermembrane fermionic equations are implied by superembedding, we can expect that (because of supersymmetry) also the bosonic equations should appear. As we have shown, the bosonic "eld equations (4.30) of a bosonic brane are encoded in the minimal embedding condition (4.29). We shall now "nd an inhomogeneous generalization of the minimal embedding condition,
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the inhomogeneity re#ecting the interaction of the supermembrane with the supergravity gauge "eld A(Z). It is a simple exercise to show that the bosonic "eld equations must hold provided that the fermionic equations (5.31) hold. This can be veri"ed in di!erent ways, one of which is to analyze the pullback components ¹? v@PY of the torsion constraint (5.5). We, however, "nd it here simpler to ? ?O ? look at the e?Oe@P components of the Codazzi equation (4.69) !2id c@ XK G !D XK G !D XK G "RG . (5.42) OP ?@ @ ? ?O @P ? @P ?O ? ?O@P ? Multiplying (5.42) by c??@ and dOP, and taking into account the fermionic equation (5.31) we get i . g?@XK G ,g?@( E? )uG "! c??@dOPRG ?O@P ? @ ? ? @ ? 32
(5.43)
We observe that, in comparison with the bosonic minimal embedding condition (4.29), the bosonic equation (5.43) acquires the r.h.s. If we now look at the supergravity curvature constraint (5.6) we realize that the r.h.s. of (5.43) contains components of the "eld strength F of the D"11 supergravity gauge "eld A. Hence, the r.h.s. of (5.43) describes the coupling of the supermembrane to A. Its form is the same as that obtained from the Wess}Zumino term of the supermembrane worldvolume action (5.1). To see this we should take the corresponding e?Oe@P components of the curvature pullback (5.6) and make use of the basic superembedding condition (4.39), the relation (5.22), the form of the C-matrices (4.42), (4.43) and the harmonic relations (4.41), (4.53). If we do this, and take the leading (g"0) component of Eq. (5.43) in the Wess}Zumino gauge (5.37)}(5.40), the r.h.s. of (5.43) takes the form 1 1 i " "! e?@AEBM EA E@M F? uG "! eJKLEBM EA E@M F? uG , ! c??@dOPRG ?O@P ? E ? @ A @AB ? J L L @AB ? 3! 32 3!(!g
(5.44)
where , g (m)"e? e (m)"E? E@M g . E M ,R Z+ E " KL K L? K K + E K L ?@ The g"0 component of the left-hand side of Eq. (5.43) is rewritten as follows: "g?@eK(m)( eL E? )uG (m)!g?@t?O(m)( E? )uG (m) g?@( E? )uG " ? K @ L ? ? ?O @ ? ? @ ? E 1
((!ggKLE? )uG !2igKLE? v?OE@M v (cG)PY " L ? K ? L @M ?PY O (!g K 1 "
((!ggKLE? )uG !2iE? v?O(c KcL ) E@M v@PY(cG) ?@ L @M OPY L ? K ? (!g K 1 2i "
((!ggKLE? )uG ! eJKLE? v?O(cJ) E@M v@PY(cG) ?@ L @M OPY L ? K ? (!g K (!g
"
i R ((!ggKLE? )#gKLE M E@M X? uG ! eJKLE@M EM C? E uG . K ? L K L @ J K @M L ? M (!g (!g 1
(5.45)
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When developing Eq. (5.45) we used the relations (5.37)}(5.40) and (5.16), the harmonic relations (4.41), the C-matrices (5.11) and (5.12), and the fermionic equation (5.31). Putting together (5.44) and (5.45) we "nally get
1 R ((!ggKLE? )#gKLE M E@M X? # eJKLE"M E!M E M F? uG "0 , K L K L @ J L L !" ? M 3!(!g (!g 1
(5.46)
where eJKLE"M E!M E M F? "eJKL[EBM EA E@M F? !iE@M EM C? E ] J K L !" J K L @AB J K @M L is the pullback onto the bosonic worldvolume of the constrained "eld strength (5.7). In the square brackets of (5.46) one can recognize the conventional scalar "eld equations of the supermembrane [9] obtained from the action (5.1). If we multiply them by E we may check that K? they are identically zero (modulo the fermionic equations (5.41)), which re#ects the d"3 worldvolume reparametrization invariance of the theory. Hence, indeed, only eight worldvolume scalar "eld equations are independent. To conclude this subsection we repeat the main expressions which describe the D"11 supermembrane in the superembedding approach. 5.1.3. Main superembedding equations for the M2-brane The superembedding conditions are E? (XK (z),HI (z))"e?u? (z) N E? "0 , ? ?O E? (XK (z),HI (z))"(1!CM )? e?E@M #e?Ov? (z) , @M ? ?O 1 1 (1!CM )? ,v?OYv? , CM " e?@AC , C "E? C . @M ?OY ?@A ? @M ? ? 3! 2
(5.47) (5.48)
This form of the projector was introduced in [94,53]. The supermembrane equations of motion (encoded in (5.47) and (5.48)) are ( E? )uG (c?)?@"0"(c?)? E@M v@OY , @ ? @M @P ? ? 1 g?@ E? # e?@AEBM EA E@M F? uG "0 . ? @ 3! ? @ A @AB ?
(5.49) (5.50)
In the static gauge X?"m?, H? v@P"g@P, in #at target superspace and in the #at (linearized) limit of ? the worldvolume supergeometry, where E? "D X? !iD HC? H, the superembedding condition ?O ?O ?O reduces to the super"eld constraint on the transversal oscillations of the supermembrane . (5.51) D XG"i(cG)OYh , h ,H? v ? ?OY ?O O ?OY ?OY This constraint describes an n"8, d"3 on-shell scalar supermultiplet. we transform the worldvolume scalar Note that acting on H? by the harmonics v@P and v ? ? ?OY h . When this is done, Eq. (5.49) becomes the fermions H? into the worldvolume spinors h@P and ?OY standard d"3 Dirac equation for h . Indeed, using the superembedding relations, it is a simple ?OY
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exercise to varify that in #at target superspace, where E? "dH? , Eq. (5.49) can be rewritten (modulo (5.43) or (5.50)) in the Dirac form (c?)?@D h "0 , (5.52) ? @OY where D "e+R #uA #APY is the worldvolume covariant derivative. ? ? + ?@ ?OY Thus, the supersymmetric "eld theory in the worldvolume of the supermembrane governed by Eqs. (5.49) and (5.50) is that of the n"8, d"3 scalar supermultiplet having eight bosonic and eight fermionic physical modes. 5.2. The super-5-brane We now turn to the most impressive example of the power of superembedding which has allowed one to get the on-shell description of the M-theory 5-brane without using the action principle [51,52]. The supersurface M associated with the super-5-brane worldvolume is characterized by a local supervielbein frame e (z)"(e?, e?O) (a"0, 1,2, 5, a"1,2, 4; q"1,2, 4) whose components form the vector and spinor representations of the group SO(1, 5);SO(5). Namely, the index a stands for a spinor representation of S;H(4)&Spin(1, 5) and the index q is that of the spinor representation of ;Sp(4)&Spin(5). The d"6 spinors are ;Sp(4) simplectic Majorana}Weyl spinors [117,120]. They are de"ned in the same way as the S;(2) simplectic Majorana}Weyl spinors, which we introduced in Section 2.5.4, Eqs. (2.73)}(2.74). : e ? "B? e@PC , (5.53) e?O " O @ PO where C is the antisymmetric ;Sp(4) invariant tensor, its inverse being (C\)OP"!COP,!C . OP OP The matrices COP and C can be used to raise and lower the ;Sp(4) spinor indices using the rules OP e "eOC , eO"COPe . P OP P The matrix B is de"ned by the conditions Bc?B\"(c?)H, BHB"!1 and * denotes complex conjugation. The 4;4 matrices (c?) are antisymmetric. We note that the ?@ matrix B? can be used to convert the dotted indices (of the complex conjugate representation) into @ undotted ones, so that one can always deal with only undotted indices, but there is no S;H(4) invariant tensor for lowering the S;H(4) spinor indices. Thus, the spinors t?O and tO have di!erent ? S;H(4) chiralities, one of them is chiral (Weyl) and another one is antichiral. The supersurface M under consideration is therefore a d"6, (2, 0) chiral superspace (where 2 stands for ;Sp(4)).
The N"1, D"6 superspace with one S;(2) simplectic Majorana}Weyl coordinate (discussed in Section 2.5.4) is also called D"6, (1, 0) chiral superspace, since the number of its Grassmann coordinates is half the number of those in the (2, 0) superspace.
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An appropriate form of the D"11 C-matrices (4.42), (4.43) and of the charge conjugation into M , is matrices C "C? @M , which re#ects the embedding of M ? @M M !c? C 0 ?@ OP C? " , a"0, 1,2, 5 , (5.54) ? @M 0 c??@COP
0 CG " ? @M !d@(cG)O ? P
d? (cG)P @ O , i"1,2, 5 , 0
(5.55)
0 d? dP @ O . (5.56) C "C? @M " ? @M !d@ dO 0 ? P In (5.55) (cG)O"COQ(cG)R C are ;Sp(4)&SO(5) gamma-matrices. The matrices (cG) "(cG)R C are P Q RP OP O RP antisymmetric. In (5.54) the S;H(4) matrices c? and c??@ are antisymmetric and de"ned by the following ?@ relations: c? c@A@#c@ c?A@"2d@g?@, tr(c?c@)"4g?@, c c? "!2e . (5.57) ?A ?A ? ??@ AB ?@AB (c ?c@cA ) ,c?@A"c?@A"!e?@ABCD(c ) . (5.58) ?@ ?@ @? BCD ?@ (5.59) (c ?c@cA )?@"e?@ABCD(c c c )?@ . B C D Eqs. (5.58) and (5.59) imply that the antisymmetric product of three SO(1, 5) c-matrices is symmetric in spinor indices and (anti)-self-dual in vector indices. The S;H(4);;Sp(4) splitting of the D"11 Lorentz-spinor harmonics v@M and their inverse is as ? follows: v@M "(v@P, v ), (v\)? "(v? , v@P ? ) . (5.60) ? ? @P ? @M @P The induced supergeometry on M is described by the general superembedding conditions (4.36)}(4.39), (4.46)}(4.51), (4.56)}(4.59), (4.65). As in the case of the supermembrane, our goal is to "nd further restrictions on components of XK G "e XK G required by the integrability of the ? ? superembedding conditions and to identify these restrictions with the 5-brane equations of motion. So we repeat the steps made in the previous subsection. We again analyze condition (4.58). Making use of the harmonic relations (4.41) and (5.60), the form of the D"11 C-matrices (5.54) and (5.55), and equations (4.46) and (4.49), we rewrite Eq. (4.58) in the following form: e?XK G "e? E@M uG "e?(du@M #uA X@M )uG ? @M ? ? A @M ? "!2ie?Oe@P(cG)Q h #2ie?e?O(cG)P E@M v . O ?P @Q O ? @M ?P Comparing the left- and the right-hand side of (5.61) we "nd that e?e@XK G "0PXK G "XK G " E@M uG , @ ? @ ? ? @ @ ? @M XK G " E@M uG "2i(cG)P E@M v , O ? @M ?P ?O ? ?O ? @M
(5.61)
(5.62) (5.63)
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and (cG)Q h "!(cG)Q h . (5.64) O ?P @Q P @O ?Q In the basis of the SO(1, 5);SO(5) c-matrices an arbitrary matrix h has the following ?O @P decomposition: c?@A] . (5.65) "C [h c? #h c?@A]#cGH [h c? #h OP GH ? ?@ GH ?@A ?@ ?O @P OP ? ?@ ?@A ?@ Substituting Eq. (5.65) into (5.64) we "nd that the only non-zero component of h is ?O @P hBCD , (5.66) h "C h "C h c?@A, h "e ?@A ?@ABCD ?O @P OP ?@ OP ?@A ?@ where h (z) is a self-dual tensor due to the c-matrix relations (5.58). The matrix h satis"es the ?@A ?@ identities h
h c?ABh "!2h?@Ah cB , (5.67) ?A B@ @AB ?@ c?Ah c?B@"0 , (5.68) ? AB tr(hc c c )"!8h , (5.69) ? @ A
?@A which we shall use below. We have thus observed that the integrability of the superembedding conditions reveals the presence of a self-dual tensor "eld on the embedded supersurface. Later on this self-dual tensor will be related to the "eld strength H"db!A of the 5-brane tensor gauge "eld which appeared in the M5-brane action (5.3). In view of Eq. (5.66) the general embedding condition (4.51) reduces to (5.70) E@M ,e+ R Z+ E@M "v@M (z)#h C v@P @M . ?O ?O + + ?O ?@ OP We may now derive the e?Oe@P components of the supersurface torsion (4.57) and (4.66). Using Eqs. (4.57) and (5.70), harmonic relations (4.41), the C-matrix decomposition (5.54) and identity (5.67) we get ¹? "!2iC c@ m? , m? ,d? !2h h?AB . (5.71) ?O @P OP ?@ @ @ @ @AB We observe that Eq. (5.71) di!ers from the standard supergravity torsion constraint (5.23) by the non-unit matrix m? . The reason for this is that (as in the case of the bosonic surface) we have chosen @ the bosonic embedding condition (4.36) such that the induced metric of the embedded surface is of the standard &Green}Schwarz' type (4.38) which is used in the construction of the M5-brane action (5.3). If we wanted to have the standard constraint on the torsion of the supersurface M (i.e. with d? instead of m? in (5.71)), we should rede"ne the M supervielbein e? in (4.36) as follows: @ @ E?,E@M u? (z)"e?(z)"e( @m? , @M @ The price for this would be that in the new frame e( ? the metric g( "e( ? e( on M does not KL K L? coincide with the induced metric of the Green}Schwarz type. So we prefer to work in the standard (induced) frame on M .
The e( ?-frame has been used for the description of the M5-brane superembedding in Ref. [51], and the transition to the Green}Schwarz frame has been discussed in [52].
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We now look at the e?Oe@P components of the torsion ¹AQ (4.66). Because of our choice of the supersurface connection (4.62), (4.65) and the presence of the tensor h(z), they are non-zero and have the following form: (5.72) ¹AQ "!C h ( vBR ? )vAQ#(aq) (br) . ? ?O @P OR ?B @P In view of (5.71) and (5.72) the anticommutator of the supersurface covariant derivatives D gets ?O modi"ed and acquires an additional term in comparison with that of the supermembrane (5.25) +D , D ,"2iC c? m@ D !¹AQ D #R (z) . ?O @P OP ?@ ? @ ?O @P AQ ?O @P
(5.73)
5.2.1. The tensor xeld equation We are now in a position to relate h (z) to the "eld strength H (m) of the worldvolume gauge ?@A JKL "eld b (m) of the M5-brane action. By de"nition, in the bosonic worldvolume (5.2) KL H"db!A . (5.74) The Bianchi identity for (5.74) is dH"!dA"F ,
(5.75)
where A and F are the pullbacks onto the bosonic worldvolume of the D"11 gauge "eld potential and of its "eld strength, respectively. To "nd the relation between H and h (m, g) we should promote the equations (5.74) and (5.75) ?@A to the supersurface M , i.e. to consider b(z) and H(z) as M super"elds. Since h (z) carries only vector indices it is natural to assume that the super"eld ?@A H "e* e+e,H (Z) , ! ! *+, also has only non-zero components with three vector indices H (z), i.e. ?@A H (z)"0 . (5.76) ?O ! Assumption (5.76) about the structure of the superform H is consistent with the Bianchi identity (5.75) extended to the supersurface, which allows one to express H (z) in terms of h (z) and to ?@A ?@A "nd the equation of motion of h (z) (or H (z)), as we shall do in a moment. ?@A ?@A We should stress in advance that Eq. (5.76) is not an additional constraint on the superembedding since, as the analysis has shown [51,52], the equations for h (z) which follow from the ?@A Bianchi identity (5.75) are equivalent to those which are contained in the basic superembedding condition E? "0 (4.39). ?O Let us analyze the Bianchi identity (5.75). To this end we rewrite it in the following form: dH"(De!)e e H #e"e!e e D H ! !" 1 1 1 " ¹!e e H # e"e!e e D H "! e"e!e e F . ! 6 " ! !" 2 4!
(5.77)
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In view of (5.76), of the torsion constraint (5.71) and of the D"11 "eld strength constraint (5.7), the "rst non-zero component of Eq. (5.77) is that of e?Oe@PeAeB (5.78) iC c? m@ H "!iE? (C ) E@M E? E@M . ?O ?@ ? @M @P A B OP ?@ ? @AB Making use of the harmonic relations (4.41) and (5.60), form (5.54)}(5.61) of the C-matrices, relation (5.70) for E? , and that E? "u? , we reduce (5.78) to ?O A A c? mB H "!(c c )A h !h (c c )A . (5.79) ?@ ? B@A @ A ? A@ ?A @ A @ Then, multiplying (5.79) by c@? and making use of identities (5.57) and (5.69), we "nally arrive at the ? relation between H and h ?@A ?@A mB H "4h 0H "4(m\)B h . (5.80) ? B@A ?@A ?@A ? B@A Using the self-duality of h and de"nition (5.71) of m@ "d@ !2h h@AB one can show that ?@A ? ? ?AB 1 1 (2d@ !m@ ), k@ ,h h@AB, k, tr(kk) . (5.81) (m\)@ " ? ? ?AB ? 1!k ? 6 It is then not hard to check that the r.h.s. of (5.80) is indeed totally antisymmetric because of the self-duality of h . For this one should notice that kB h is anti-self-dual ?@A ? B@A kBB hCD . (5.82) kB h "!e B ? B@A ?@ABCD The form of expressions (5.80) implies that though h obeys ordinary Hodge self-duality (5.66), ?@A the "eld strength H satis"es a much more complicated non-linear self-duality condition. Its ?@A explicit form can be derived from the M5-brane action (5.3) as an equation of motion of b KL [162,159,63,64], or directly from (5.80) [163]. We refer the reader to these papers for details on the di!erent forms of the generalized self-duality condition. Note that at the linearized level, when m? is replaced with d? , H becomes proportional to @ @ ?@A h and satis"es the ordinary Hodge self-duality condition. ?@A When the "eld strength of a gauge "eld is self-dual, its Bianchi identities are equivalent to the gauge "eld equations of motion. We shall now show how the Bianchi identities (5.75) produce the tensor "eld equations in the case under consideration. Consider the e?e@eAeB component of the Bianchi identity (5.77). It has the form 1 1 e?@ABCD(D H #3¹D H )"! e?@ABCDE"M E!M E M E M F , A BCD AB CDD A B C D !" 4! 6
(5.83)
1 1 e?@ABCDDK H "! e?@ABCDE"M E!M E M E M F , A BCD A B C D !" 6 4!
(5.84)
or
where, for convenience, we have introduced the covariant derivative DK
A DK "e+R #u( "e+R #(u !(¹ !¹ !¹ )) . (5.85) A A + A @? A + A @? A@ ? A? @ @? A Note that in the pure bosonic limit the connection u( is torsion free, i.e. for such a connection the @? components ¹K ? of the torsion ¹K ?"DK e? are zero. @A
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We now substitute H in (5.84) with (m\)D h (5.80), and use the relations (5.81), (5.71) and BCD B D CD (5.82) to get
1 1 mA hB?@ "! e?@ABCDE"M E!M E M E M F . (5.86) 4DK A B C D !" A 1!k B 4! Eq. (5.86) is the equation of motion for the self-dual tensor "eld h . We conclude that the Bianchi ?@A identity (5.75) with H de"ned by the relation (5.80) is equivalent to the "eld equations of the self-dual worldvolume tensor "eld. Let us now proceed with deriving the fermionic equation. 5.2.2. The fermionic equation To get the fermionic "eld HI (z) equation of the M5-brane we hit the left- and the right-hand side of Eq. (5.70) with the covariant derivative D , symmetrize with respect to the pairs of indices (aq) @P and (br), and multiply the resulting expression by v . We thus obtain @M AQ iC c? m@ E@M v #C h ( vBR ? )vNR E@M v !E? X@M v #(aq) (br) OR ?B @P ? NR @M AQ ?O @P ? @M AQ OP ?@ ? @ @M AQ (5.87) "D v@M v #(D h C vBR @M )v #(aq) (br) . @M AQ @P ?B OR @P ?O @M AQ To simplify Eq. (5.87) we should make use of expression (5.70), identity (5.67), relations (4.62) and (4.65) between the supersurface connection u and the target superspace connection X M , and to recall that due to the harmonic relations (4.59) and (4.76), and the form of the C-matrices (5.54) and (5.55) ( v@M )v "( u? )uG c? (c ) ,XK G c? (c ) , @P ?O @M AQ @P ? ? ?A G OQ @P ? ?A G OQ ( v?O @M )vAQ"( u? )uG c??A(c )OQ,XK G c??A(c )OQ . @M @P ? ? G @P ? G @P We thus reduce (5.87) to
(5.88)
(5.89) 2iC c? m@ E@M v "c@ m? XK G (c ) #D h C #(aq) (br) . @P ?A OQ OP ?@ ? @ @M AQ ?A @ @P ? G OQ Let us multiply Eq. (5.89) by COP and c@?cABA. Because of identity (5.68) the term with the covariant A derivative of h vanishes and we get ?A 16icBAm@?E@M v "!cBAm@?XK G (c )O . (5.90) @ ? @M AQ @ AO ? G Q Now take relation (5.63) and multiply it by (c )O and cB?m@?. We have GQ @ 10icBAm@?E@M v "!cBAm@?XK G (c )O . (5.91) @ ? @M AQ @ AO ? G Q Comparing (5.90) with (5.91) we see that the coe$cients on their left-hand sides do not match, and hence (also in view of (5.63)) (5.92) c?@m@?E@M v "0"c?@m@?XK G . ? @M @O @ @O ? @ We have thus arrived at the 16-component fermionic "eld equation of motion of the M5-brane similar to that of the supermembrane (5.31), the only di!erence being in the presence of the worldvolume self-dual tensor h (z) in the M5-brane equation via the matrix m? "d? !h h?AB. ?@A @ @ @AB
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Let us now rewrite Eq. (5.92) in the Green}Schwarz form similar to the supermembrane equation (5.34). First of all we use the harmonic relations (4.41) and (5.60), the form of the C-matrices (5.54), and the embedding conditions E@M v?O"0 (4.47) and E? "u? (4.38) to get ? @M ? ? (5.93) m@?E@M (E? C ) v?O A "0 . ? @ ? @M A Then, to &hide' the supersurface spinor indices, we multiply (5.93) by E ,v !v@Ph C . ? ?O ? @? PO ? ?O We get
(5.94)
(5.95) m@?E@M (E? C ) v?O A E ,m@?E@M [E? C (1!CM )] "0 . ? ?O @M ? ? @ ? @M A ? @ ? is prompted by the requirement that the resulting fermionic equation The choice of the matrix E ? ?O is invariant under the i-symmetry transformations (as the superbrane dynamical equations must always be). This means that this equation should be annihilated by the i-symmetry projector (4.80), (4.81) whose form is (5.96) (1#CM )? ,v@PE? "v@P(v? #C h vAQ ? ) . @M @P @M @M @P PQ @A in such a way that E? E ,0. So we have chosen E @P ? ?O ? ?O Finally we use the form (5.66) of h and the harmonic relations (4.41) to "nd the explicit form of @? the matrix CM of Eq. (5.95) in terms of antisymmetric products of the D"11 C-matrices (5.54), (5.55) 1 1 CM " e? 2? C 2 # h?@AC , ? ? 3 ?@A 6!
C "E? C . ? ? ?
(5.97)
Thus we have derived the Green}Schwarz-type fermionic equation of motion (5.95) of the super-5brane. In the Wess}Zumino gauge (5.37) its g"0 component has the form m@?eKeL E@M [E? C (1!CM )] "0 , @ ? K L ? @M ? where E M "R Z+ E (Z(m)) (A"b, a). K K + M M
(5.98)
5.2.3. The scalar equation To identify the equations of motion of the superworldvolume scalar "elds XK (z) we analyze the e?e?O components of the pullback of the target superspace torsion ¹? (5.5) multiplied by de"ned in (5.94). E ? @P ¹? E ,( E? )E "¹ E? E #e ( E? !u E? )E ? @P ? @P ? @P ? @P "¹@E? E #e ( E? !u E? )E (5.99) @ ? @P ? @P where ¹ "(¹@, ¹?O) is the supersurface induced torsion de"ned in (4.57) and (4.66). Note that the term with ¹?O has disappeared from (5.99) since by de"nition E? E ,0 (see Eqs. (5.94), (5.96)). ?O ? @P
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The e?e?O components of (5.99) are ¹? E "¹@ E? E !( E? #u@ E? )E !( E? )E . ? ?O ? @P ? ?O @ ? @P ?O ? ?O ? @ ? @P ? ?O ? @P
(5.100)
Using the de"nition of E? (5.70) and E (5.94), i.e. ?O ? @P E? "v? (z)#h C vAQ ? , E "v !vAQh C , ? @P ?O ?O ?A OQ ? @P ? A@ QP the orthogonality properties of harmonics (5.60) and relations (4.62), (4.64) and (5.88) for the supersurface and target superspace connections, the last term of (5.100) can be rewritten as ( E? )E "XK G mA c (c ) #D h C . ? ?@ OP ? ?O ? @P ? @ @ A?@ G OP
(5.101)
Thus Eq. (5.100) takes the following form: , XK G mA c (c ) #D h C "(¹@ E? !¹? )E !K ? ?O ? @P ?O ? @P ? @ @ A?@ G OP ? ?@ OP ? ?O @
(5.102)
where K ,( E? #u@ E? )E ?O ? @P ?O ? ?O ? @ ? @P are components of the second fundamental form introduced in (4.60). The equation of motion of the worldvolume scalar "elds XK (z) is obtained from (5.102) by multiplying the latter by m?Bc@?(c )PO B G m@Am?XK G ,m@Am?( E? )uG A ? @ A ? @ ? "m?Bc@?(cG)OP[(¹@ E? !¹? )E !K ]. B ? ?O @ ? ?O ? @P ?O ? @P
(5.103)
We observe that the l.h.s. of (5.103) has the form similar to the supermembrane bosonic equation (5.43) but with the matrix m@Am? replacing the Minkowski metric g?@. The complicated r.h.s. of A Eq. (5.103) describes the interaction of the super-5-brane with the D"11 gauge "eld strength F"dA. Note that since (5.102) contains the vector derivative of h , from (5.102) we can also derive ?@ another form of the equation of motion of the tensor "eld h . To this end we multiply (5.42) by ?@A COP and m?B(c c c )?@. Using the identity (5.69) we get B @ A
]. m?BD h "! COPm?B(c c c )?@[(¹@ E? !¹? )E !K B @ A
? ?O @ ? ?O ? @P ?O ? @P ? B@A
(5.104)
The form of Eq. (5.104) di!ers from that of Eq. (5.76), but a somewhat tedious analysis shows that the two equations are in fact equivalent [51,52] modulo the fermionic equation (5.95). The "ve scalar "eld equations (5.103) have a rather complicated structure of the right-hand side. Cumbersome manipulations using the form of the torsion constraints (5.4) and (5.5), the superembedding conditions E? "0 and (5.70), the harmonic relations (4.41), the form of the D"11 ?O C-matrices (5.54)}(5.56) and the identities (5.80)}(5.82) allow one to disentangle Eqs. (5.103) and to present their leading (g"0) components (which are the proper equations for xK (m)) in the
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following form: [GKL(D E? #E@M X? ) L K K L@M 1 1 eK 2K F? 2 ! (F? !E? FL )H uG "0 , ! ? L K K K K K K (!g(1!k) 6! K K (3!) K K K where all quantities depend only on the bosonic worldvolume coordinate mK,
(5.105)
GKL(m)"m@Am?eKeL (m) , A @ ? E M "R Z+ E (Z(m)), F 2 "E M N 2E M F 2 (Z(m)) (p"4,7) KN K K K + KN K N and D "R #CJ (m) K K KL is the covariant derivative with the Christo!el symbol CJ (m) determined by the worldvolume KL induced metric g (m)"E? E@M g . K L ?@ KL The expression in the square brackets of (5.105) is the same as the one obtained from the M5-brane action (5.3) as the xK (m) equation of motion. The equivalence of the M5-brane equations of motion (5.98) and (5.105), and of the &self-duality' relation (5.80) to the equations of motion yielded by the M5-brane action (5.3) has been demonstrated in [164]. The proof is not straightforward, since in the action one deals directly with the "eld strength H and not with h , and a projector (1#C), which appears in a natural way in KLJ ?@A the i-symmetry transformations of the action, di!ers from the i-symmetry projector (5.96), (5.97) of the superembedding approach. The two projectors are related to each other by the identities 1#CM 1#CM 1#C 1#C 1#C 1#CM " , " . 2 2 2 2 2 2 For details we refer the reader to the papers [63,164], and to [165] where the equivalence of the covariant energy-momentum tensors of the M5-brane in both approaches has been discussed. The two formulations of the M5-brane have been applied to studying various aspects of M-theory and its duals in [65,74,103,166}170,161,171}196]. To summarize, in the superembedding approach the M5-brane is described by the following main relations. 5.2.4. Main superembedding equations for the M5-brane The superembedding conditions are E? (XK (z), HI (z))"e?u? (z) N E? "0 , ? ?O E? (XK (z), HI (z))"(1!C )? e?E@M #e?O(v? (z)#h C v@P @M ) , @M ? ?O ?@ OP h "c?@Ah , ?@ ?@ ?@A 1 C " e? 2? C 2 , C "E? C . (1!C )? "v v@P ? , @M @M @P 6! ? ? ? ? ?
(5.106) (5.107)
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The M5-brane equations of motion encoded in (5.106) and (5.107) are the fermionic "eld equation c?@m@?E@M v "0"c?@m@? E@M uG , ? @M @O @ @O ? @M @ m? "d? !2k? , k? "h h@AB , @ @ @ @ ?AB the worldvolume scalar "eld equation
(5.108)
e? 2? 1 1 F? 2 ! F? H uG , m@Am?( E? )uG " A ? @ ? 1!k 6! ? ? (3!) ? ? ? ? ? ? ? where F? 2 "F? 2 E? N 2E? , (p"3, 6); and the self-dual tensor "eld equation ?N ? ? N ? ?N ? 1 1 dH"! E"M E!M E M E F , H "4(m\)B h , h " e hBCD . !" ?@A ? B@A ?@A 6 ?@ABCD 4!
(5.109)
(5.110)
As in the case of the supermembrane (see Eq. (5.51)) in the linearized limit the superembedding condition (5.106) reduces to the constraint on the super"elds corresponding to the transverse oscillations of the super-5-brane D XG"i(cG)P h , h ,H? v , i"0, 1,2, 5, q, r"1,2, 4 . ?O O ?P ?P ? ?P This constraint describes an n"(2, 0), d"6 on-shell tensor supermultiplet. Thus, the supersymmetric "eld theory in the worldvolume of the M5-brane governed by Eqs. (5.108)}(5.110) is that of the n"(2, 0), d"6 tensor supermultiplet which on the mass shell has "ve scalar, three (self-dual) tensor and eight fermionic physical modes.
6. Other developments and applications 6.1. The generalized action principle In Sections 2 and 3 we have constructed doubly supersymmetric super"eld actions for superparticles and superstrings which produce the basic superembedding condition E? "0 dynamically. ?O This has been possible because in these cases the superembedding condition does not contain dynamical "eld equations, and the latter should be obtained by imposing the minimal embedding conditions, or from an action. This method of constructing worldvolume super"eld actions can be used in other cases where the superembedding condition does not put the superbrane on the mass shell, as, for example, N"2, D"3, 4 and 6 superparticles [42,69], the N"2, D"3 superstring [42], the N"1, D"4 supermembrane [70] and an N"2, D"4 space-"lling D3-brane. As we have seen, in the case of the M-theory branes the superembedding condition implies the dynamical equations of motion. Therefore, if one tried to construct a worldvolume super"eld action for these branes using the prescription of Sections 2 and 3, i.e. by introducing the superembedding condition into the action with a Lagrange multiplier, this Lagrange multiplier super"eld would acquire dynamical equations and thus would contain redundant degrees of
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freedom, which are absent from the spectrum of the brane theories of interest. This is analogous, for instance, to the super"eld formulations of ten-dimensional Super-Yang}Mills theory and D"10 and D"11 supergravities whose super"eld constraints put the theories on the mass shell, and the way of constructing super"eld actions for them has not been found. In such cases one should either deal with component actions, as Green}Schwarz-type actions for the superbranes, or consider so called generalized actions, whose construction is based on a group manifold (rheonomic) approach which has been developed in application to supersymmetric "eld theories in [73] and in application to superbranes in [72] and, from somewhat di!erent perspective, in [70]. The main principles of the construction of the generalized actions (which we enumerate for superbranes [72,78]) are (i) In the superworldvolume of the superbrane with the number of bosonic dimensions d"p#1 embedded into a D-dimensional target superspace one constructs a p#1-superform ¸N> which is closed, d¸N>"0, modulo the superembedding condition E? "0 (or the one ?O which replaces it in the case of the space-"lling branes [75]). The superform is constructed from the pullbacks of the target-space supervielbeins, the harmonic variables (which are auxiliary worldvolume super"elds) and from the "eld strengths of the worldvolume gauge "elds, when present. It contains both the kinetic and the Wess}Zumino term of the superbrane Lagrangian. A systematic way to get this form (which can always be found) has been proposed in [70]. An example of such a form has been considered in the case of the heterotic string in Sections 3.2.2 and 3.2.3 (Eqs. (3.50) and (3.58)). (ii) The integral of this superform over an arbitrary p#1-dimensional bosonic submanifold M "(mK, gI(m)) of the superworldvolume is the generalized action N>
S"
¸N> .
(6.1)
M N>
So the generalized action is not a fully #edged worldvolume super"eld action. (iii) But, in spite of the fact that the integration is performed over a bosonic submanifold, action (6.1) is superdi!eomorphism invariant in the whole superworldvolume modulo the superembedding condition
dS"
M N>
d¸N>"
M N>
d(i ¸N>)# B
M
N>
i d¸N> , B
(6.2)
where i ¸N> " : (1/p!) dz*N 2dz* dz+ ¸ 2 . We see that the "rst term in the variation +* *N B (6.2) is a total derivative and the second term vanishes because of the condition i). (iv) The variation of the surface M in the action functional (6.1) is equivalent to the superN> di!eomorphisms dg(m) of the superworldvolume in the odd directions orthogonal to M , N> which leads to the condition d¸N>"0. As a result, because of (iii) the variation of action (6.2) with respect to the worldvolume super"elds yields the superembedding conditions and the super"eld equations of motion in the whole worldvolume superspace. When contained in the superembedding conditions, the equations of motion do not produce any new information in addition to the superembedding conditions.
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(v) When the integration surface is chosen to be (mK; gI"0) and we take the integral of the leading component ¸N>" of the superform, the generalized action (6.1) is reduced to a comEE ponent action, which can be then rewritten in the Green}Schwarz form. An example related to the generalized action taken at g"0 is the property of the Weil triviality of the term (3.65) of the doubly supersymmetric string action considered in Section 3.2.3, with the two-form (3.58) being closed up to the superembedding condition. Because of a solid geometrical ground the generalized action formalism can be useful, for example, for deriving component actions of supersymmetric models for which actions have been unknown. The generalized actions have been constructed for the ordinary super-p-branes [72,150], the super-D-branes [197,75] and (implicitly) for so called ¸-branes [198] whose physical modes form linear supermultiplets. For the M5-brane the generalized action which extends the Green} Schwarz-type action (5.3) to the worldvolume superspace and which produces the superembedding conditions is still unknown because of problems caused by the presence of the self-dual "eld. For further details on the generalized action approach to super"eld theories we refer the reader to original literature [73,72,197,78,150,75,70]. 6.2. D-branes, ¸-branes and branes ending on branes We have already mentioned that the superembedding approach is applicable to the description of all known superbranes, including the Dirichlet branes [50,197,75], and it has also been used to derive equations of motion and actions for a class of branes called ¸-branes [50,198] which contain on their worldvolumes antisymmetric gauge "elds dual to worldvolume scalars or vectors. As in the case of the M5-brane the worldvolume gauge "elds of the D-branes and ¸-branes (or more precisely their "eld strengths) show up in the superembedding condition (4.49) as the spin-tensor "eld h?Y (z). The analysis of the superembedding integrability conditions involving h?Y is ?O ?O made along the lines explained with the example of the M5-brane and reveals the Born}Infeld structure of equations of motion of the gauge "elds. ¸-branes can be obtained, by dualizing vector "elds of Dp-brane actions [199,158,177,207], and from standard super-p-branes by a direct dimensional reduction of the target superspace and the dualization of worldvolume scalars, corresponding to compacti"ed dimensions, into rank p!1 antisymmetric "elds, which enter the linear supermultiplet of the supersymmetric worldvolume "eld theory. In some cases, when, for example, there are eight supersymmetries in the worldvolume, the scalar supermultiplets are on the mass shell, while the corresponding dual linear supermultiplets are o!-shell, and the ¸-branes admit the o!-shell worldvolume super"eld description [198]. In the superembedding approach ¸-branes naturally appear when for a given supersurface and target superspace the basic superembedding condition yields the super"eld constraint of the linear supermultiplet [50]. The ¸-brane duals of D-branes also occur when one considers Dp-branes ending on D(p#2)-branes [74]. The consideration of open branes ending on another (host) branes from the point of view of superembedding is an interesting problem of its own, as, for instance, studying an M2-brane ending
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on an M5-brane. The investigation of such brane con"gurations has been carried out in [74], where it has been demonstrated that the basic principles of superembedding work perfectly well also in these cases. It has been shown that if the constraints on the superworldvolume of the open brane are imposed, the superembedding conditions determine the superworldvolume constraints for the host branes, and provide one with information about the dynamics of the boundary of the open brane in the host brane. For further details we refer the reader to the original literature cited above. 6.3. Non-linear realizations and superembeddings We have already discussed that the presence of the superbranes in the target superspaces (partially) breaks supersymmetry of the background vacuum. This supersymmetry breaking is spontaneous, since the worldvolume equations of motion of the superbrane are manifestly invariant under target-space supersymmetry, while their (classical vacuum) solutions preserve only a fraction of the supersymmetry transformations. The superbrane con"gurations whose worldvolume actions possess i-symmetry, with the number of independent parameters being half the number of target-space supersymmetries, include BPS states which preserve half the supersymmetry. An e!ective group-theoretical and geometrical method to describe theories with spontaneously broken symmetries is the method of non-linear realizations of symmetries (or the coset space approach) which is based on the Cartan theory of group manifolds and coset spaces. For the "rst time this method was applied in physics to the construction of phenomenological Lagrangians of particle interactions by Callan, Coleman, Wess and Zumino [200], and independently by Volkov [201]. It was then used for the construction of the "rst globally supersymmetric "eld theory [202] and supergravity [203]. Since branes naturally provide us with a geometrical mechanism of partial (super)symmetry breaking (which is nothing but the Goldstone}Higgs mechanism), it is natural to apply to the description of the "eld theory on the worldvolume of the brane the method of non-linear realizations. This has been done in a number of papers. A super-3-brane in N"1, D"6 superspace as a model of partial spontaneous n"2 supersymmetry breaking in d"4 was considered in [204,205]. An N"1, D"4 supermembrane and N"2, D"2 superparticles were discussed in [206]. In the static gauge a worldvolume n"1 super"eld Born}Infeld-type action for the D3-brane in N"2, D"4 superspace was "rst constructed in [71] as the non-linear action of the Goldstone}Maxwell supermultiplet for partially broken N"2 supersymmetry. And the gauge "xed super"eld action for its dual L3-brane was derived in [207]. The formalism of partial supersymmetry breaking has been further developed in application to branes in superstring and M-theory in [208,211]. Because of its nature, this method is applicable to the description of (super)branes propagating in the backgrounds with isometries, which have the geometry of coset (super)spaces, such as #at superspace or anti-de-Sitter superspaces. For instance, in the framework of the AdS/CFT (superconformal "eld theory) correspondence [212], the coset space approach has been used to get the explicit form of the supervielbeins of AdS ;S"\N\ superspaces and to construct gauge "xed N> superconformal actions for superstrings [176,213,216], a D3-brane [176,214,216] and M-branes [176,215,216] propagating in these superbackgrounds.
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As a simple example of the use of the method of non-linear realizations let us consider a supermembrane in N"1, D"4 #at superspace parametrized by coordinates X? and H? (a"0, 1, 2, 3; a"1, 2, 3, 4). From the point of view of the worldvolume "eld theory this is the model of spontaneous breaking n"2, d"3 supersymmetry down to n"1 [206,210]. The #at superspace is associated with the coset (actually supergroup) manifold of the N"1, D"4 translations whose element can be exponentially parametrized as K(X, H)"e 6? .? >H? /M ? , where P and Q are the supertranslation generators ? M ? +Q , Q ,"2iP (CC? ) , [Q , P ]"0 . ? ? @M M ? ? M ? M @M The coset element (6.3) can be multiplied by the Lorentz-group coset matrix [208,210]
(6.3)
(6.4)
SO(1, 3) ;" , SO(1, 2) which corresponds to the harmonic matrices (4.11) and (4.44) of the superembedding approach. The Cartan one-form K\ dK takes its values in the superalgebra (6.4) and yields the supervielbeins of the superspace under consideration 1 K\ dK"E? P #E? Q , ? i M ?
(6.5)
where E? "dX? !i dHM C? H, E? "dH? .
(6.6)
In the superembedding approach we have considered the pullbacks E (Z(z)) of the supervielbeins (6.6) onto the supermembrane worldvolume parametrized by coordinates z"(m?, g?) (a"0, 1, 2; a"1, 2), and imposed the superembedding condition E? "0 to specify the embedding ? corresponding to the dynamics of the brane. In the method of non-linear realizations one, from the beginning (i.e. already in (6.3)) identi"es part of the superspace coordinates X? and H? with the superworldvolume coordinates m? and g? m?"X? (a"0, 1, 2), g"(1#C)H .
(6.7)
The corresponding supertranslations P and Q"(1#C)Q generate n"1, d"3 supersym? M brane. metry which remains unbroken in the superworldvolume of the The coordinates X(m, g) and h(m, g)"(1!C)H transverse to the brane are associated with the Goldstone super"elds of spontaneously broken supertranslations generated by P and S"(1!C)Q. M perspective of the n"2, d"3 worldvolume "eld theory the N"1, D"4 Thus, from the superalgebra (6.4) looks as +Q , Q ,"2P c? , +S , S ,"2P c? , +Q , S ,"2e P , ? @ ? ?@ ? @ ? ?@ ? @ ?@ where P plays the role of the central charge.
(6.8)
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The Goldstone super"elds X(m, g) and h(m, g) are then subject to constraints which should reduce the number of their components to a suitable irreducible supermultiplet. (Such constraints can be found, for instance, with the help of a so called &inverse Higgs' e!ect [217].) In the case under consideration the constraint is D X"h , (6.9) ? ? which singles out a scalar n"1, d"3 supermultiplet describing physical degrees of freedom of the supermembrane. In some cases [205}211] the constraints can be solved in terms of super"elds which can be used to construct non-linear actions. Comparing this formalism with the superembedding approach we see that the choice (6.7) of the superbrane coordinates is nothing but the static gauge, which can be chosen in the superembedding approach to "x worldvolume superdi!eomorphisms, and the constraint (6.9) imposed on the Goldstone super"elds is similar to the superembedding condition (5.47), (5.51). Thus, the method of the non-linear realizations can be regarded as a gauge "xed version of the covariant superembedding description of superbranes, which can provide us with a way of explicit solving for the superembedding constraints and with an alternative method to construct (gauge "xed) superbrane actions. A detailed analysis of the relation between the two approaches in di!erent cases is an interesting subject for future study.
7. Concluding remarks and outlook We have given an introduction to generic features of the geometrical approach to the description of the theory of superbranes. This powerful approach uni"es on the grounds of supersurface theory various formulations of supersymmetric extended objects, such as the Green}Schwarz, twistor and Lorentz-harmonic formulation, and the method of non-linear realizations. Being manifestly supersymmetric in the worldvolume and in the target space, this approach also establishes (at the classical level) the link between spinning particles and superparticles, and between Neveu}Schwarz}Ramond and the Green}Schwarz superstrings. Superembedding explains the nature of the i-symmetry of the Green}Schwarz-type actions as odd superdi!eomorphisms (local supersymmetry) of the superbrane worldvolume. These properties of superembeddings allow one, in certain cases, to overcome the covariant quantization problem of superparticles and superstrings. As we have seen, in many cases the basic superembedding condition contains the full information about the dynamics of the superbrane, i.e. it produces the superbrane equations of motion. This is of particular importance for the description of new objects for which the use of other methods may encounter problems. Depending on whether or not the superembedding condition puts the superbrane on the mass shell, the approach gives a recipe for the construction of component, generalized or super"eld actions of the superbranes by the use of a closed (p#1)-superform which exists in the brane superworldvolume.
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As further applications of the superembedding approach, one may use it to search for new types of (dual) families of branes, study superembeddings which correspond to brane con"gurations preserving less than half target-space supersymmetry, as intersecting branes, carry out more detailed analysis of the relation between the superembedding approach and the method of non-linear realizations, and to use it for studying the dynamics of branes and gauge "xed brane actions in AdS superbackgrounds in connection with the AdS/CFT correspondence conjecture.
Acknowledgements The author is grateful to J. Bagger for the suggestion to write down this article and to N. Berkovits, S. Krivonos, P. Pasti, C. Preitschopf, H. Skarke and M. Tonin for interest to this work, useful discussions and comments, and especially to I. Bandos and D. Smith for reading the manuscript and numerous discussions which helped the author to make some points more transparent. The author acknowledges the "nancial support from the Alexander von Humboldt Foundation. This work was also partially supported by the European Commission TMR Programme ERBFMPX-CT96-0045 to which the author is associated, by the INTAS Grant 96-308 and under the project N 2.5.1/52 of the Ukrainian State Committee on Science and Technology.
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THE CONTROL OF CHAOS: THEORY AND APPLICATIONS
S. BOCCALETTI , C. GREBOGI, Y.-C. LAI, H. MANCINI , D. MAZA Department of Physics and Applied Mathematics, Institute of Physics, Universidad de Navarra, Irunlarrea s/n, 31080 Pamplona, Spain Institute for Plasma Research, Department of Mathematics, and Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742, USA Dept. of Math. and Electrical Engineering, Center for Systems Science and Engineering Research, Arizona State University, Tempe, AZ 85287
AMSTERDAM } LAUSANNE } NEW YORK } OXFORD } SHANNON } TOKYO
Physics Reports 329 (2000) 103}197
The control of chaos: theory and applications S. Boccaletti , C. Grebogi, Y.-C. Lai, H. Mancini , D. Maza Department of Physics and Applied Mathematics, Institute of Physics, Universidad de Navarra, Irunlarrea s/n, 31080 Pamplona, Spain Institute for Plasma Research, Department of Mathematics, and Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742, USA Dept. of Math and Electrical Engineering, Center for Systems Science and Engineering Research, Arizona State University, Tempe, AZ 85287, USA Received June 1999; editor: I. Procaccia Contents 1. Introduction 1.1. The control of chaos: exploiting the critical sensitivity to initial conditions to play with chaotic systems 1.2. From the Ott}Grebogi}Yorke ideas and technique to the other control methods 1.3. Targeting desirable states within chaotic attractors 1.4. The control of chaotic behaviors, and the communication with chaos 1.5. The experimental veri"cations of chaos control 1.6. Outline of the Report 2. The OGY method of controlling chaos 2.1. The basic idea 2.2. A one-dimensional example 2.3. Controlling chaos in two dimensions 2.4. Pole placement method of controlling chaos in high dimensions 2.5. Discussion 3. The adaptive method for control of chaos 3.1. The basic idea 3.2. The algorithm for adaptive chaos control 3.3. Application to high-dimensional systems
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4. The problem of targeting 4.1. Targeting and controlling fractal basin boundaries 4.2. The adaptive targeting of chaos 5. Stabilizing desirable chaotic trajectories and application 5.1. Stabilizing desirable chaotic trajectories 5.2. The adaptive synchronization of chaos for secure communication 6. Experimental evidences and perspectives of chaos control 6.1. Introduction 6.2. Nonfeedback methods 6.3. Control of chaos with OGY method 6.4. Control of electronic circuits 6.5. Control of chemical chaos 6.6. Control of chaos in lasers and nonlinear optics 6.7. Control of chaos in #uids 6.8. Control of chaos in biological and biomechanical systems 6.9. Experimental control of chaos by time delay feedback 6.10. Other experiments Acknowledgements References
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Abstract Control of chaos refers to a process wherein a tiny perturbation is applied to a chaotic system, in order to realize a desirable (chaotic, periodic, or stationary) behavior. We review the major ideas involved in the control of chaos, and present in detail two methods: the Ott}Grebogi}Yorke (OGY) method and the adaptive method. We also discuss a series of relevant issues connected with chaos control, such as the targeting problem, i.e., how to bring a trajectory to a small neighborhood of a desired location in the chaotic attractor in both low and high dimensions, and point out applications for controlling fractal basin boundaries. In short, we describe procedures for stabilizing desired chaotic orbits embedded in a chaotic attractor and discuss the issues of communicating with chaos by controlling symbolic sequences and of synchronizing chaotic systems. Finally, we give a review of relevant experimental applications of these ideas and techniques. 2000 Elsevier Science B.V. All rights reserved. PACS: 05.45.#b
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1. Introduction 1.1. The control of chaos: exploiting the critical sensitivity to initial conditions to play with chaotic systems A deterministic system is said to be chaotic whenever its evolution sensitively depends on the initial conditions. This property implies that two trajectories emerging from two di!erent closeby initial conditions separate exponentially in the course of time. The necessary requirements for a deterministic system to be chaotic are that the system must be nonlinear, and be at least three dimensional. The fact that some dynamical model systems showing the above necessary conditions possess such a critical dependence on the initial conditions was known since the end of the last century. However, only in the last thirty years, experimental observations have pointed out that, in fact, chaotic systems are common in nature. They can be found, for example, in Chemistry (Belouzov}Zhabotinski reaction), in Nonlinear Optics (lasers), in Electronics (Chua}Matsumoto circuit), in Fluid Dynamics (Rayleigh}BeH nard convection), etc. Many natural phenomena can also be characterized as being chaotic. They can be found in meteorology, solar system, heart and brain of living organisms and so on. Due to their critical dependence on the initial conditions, and due to the fact that, in general, experimental initial conditions are never known perfectly, these systems are instrinsically unpredictable. Indeed, the prediction trajectory emerging from a bonaxde initial condition and the real trajectory emerging from the real initial condition diverge exponentially in course of time, so that the error in the prediction (the distance between prediction and real trajectories) grows exponentially in time, until making the system's real trajectory completely di!erent from the predicted one at long times. For many years, this feature made chaos undesirable, and most experimentalists considered such characteristic as something to be strongly avoided. Besides their critical sensitivity to initial conditions, chaotic systems exhibit two other important properties. Firstly, there is an in"nite number of unstable periodic orbits embedded in the underlying chaotic set. In other words, the skeleton of a chaotic attractor is a collection of an in"nite number of periodic orbits, each one being unstable. Secondly, the dynamics in the chaotic attractor is ergodic, which implies that during its temporal evolution the system ergodically visits small neighborhood of every point in each one of the unstable periodic orbits embedded within the chaotic attractor. A relevant consequence of these properties is that a chaotic dynamics can be seen as shadowing some periodic behavior at a given time, and erratically jumping from one to another periodic orbit. The idea of controlling chaos is then when a trajectory approaches ergodically a desired periodic orbit embedded in the attractor, one applies small perturbations to stabilize such an orbit. If one switches on the stabilizing perturbations, the trajectory moves to the neighborhood of the desired periodic orbit that can now be stabilized. This fact has suggested the idea that the critical sensitivity of a chaotic system to changes (perturbations) in its initial conditions may be, in fact, very desirable in practical experimental situations. Indeed, if it is true that a small perturbation can give rise to a very large response in the course of time, it is also true that a judicious choice of such a perturbation can direct the trajectory to wherever one wants in the attractor, and to produce a series of desired dynamical states. This is exactly the idea of targeting.
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The important point here is that, because of chaos, one is able to produce an in"nite number of desired dynamical behaviors (either periodic and not periodic) using the same chaotic system, with the only help of tiny perturbations chosen properly. We stress that this is not the case for a nonchaotic dynamics, wherein the perturbations to be done for producing a desired behavior must, in general, be of the same order of magnitude as the unperturbed evolution of the dynamical variables. The idea of chaos control was enunciated at the beginning of this decade at the University of Maryland [1]. In Ref. [1], the ideas for controlling chaos were outlined and a method for stabilizing an unstable periodic orbit was suggested, as a proof of principle. The main idea consisted in waiting for a natural passage of the chaotic orbit close to the desired periodic behavior, and then applying a small judiciously chosen perturbation, in order to stabilize such periodic dynamics (which would be, in fact, unstable for the unperturbed system). Through this mechanism, one can use a given laboratory system for producing an in"nite number of di!erent periodic behavior (the in"nite number of its unstable periodic orbits), with a great #exibility in switching from one to another behavior. Much more, by constructing appropriate goal dynamics, compatible with the chaotic attractor, an operator may apply small perturbations to produce any kind of desired dynamics, even not periodic, with practical application in the coding process of signals. 1.2. From the Ott}Grebogi}Yorke ideas and technique to the other control methods It is reasonable to assume that one does not have complete knowledge about the system dynamics since our system is typically complicated and has experimental imperfections. It is better, then, to work in the space of solutions since the equations, even if available, are not too useful due to the sensitivity of the dynamics to perturbations. One gets solutions by obtaining a time series of one dynamically relevant variable. The right perturbation, therefore, to be applied to the system is selected after a learning time, wherein the dependence of the dynamics on some external control is tested experimentally. Such perturbation can a!ect either a control parameter of the system, or a state variable. In the former case, a perturbation on some available control parameter is applied, in the latter case a feedback loop is designed on some state variable of the system. The "rst example of the former case is reported in Ref. [1]. Let us draw the attention on a chaotic dynamics developing onto an attractor in a D-dimensional phase space. One can construct a section of the dynamics such that it is perpendicular to the chaotic #ow (it is called PoincareH section). This (D!1)-dimensional section retains all the relevant information of the dynamics, which now is seen as a mapping from the present to the next intersection of the #ow with the Poincare' section. Any periodic behavior is seen here as a periodic cycling among a discrete number of points (the number of points determines the periodicity of the periodic orbit). Since all periodic orbits in the unperturbed dynamics are unstable, also the periodic cycling in the map will be unstable. Furthermore, since, by ergodicity, the chaotic #ow visits closely all the unstable periodic orbits, this implies that also the mapping in the section will visit closely all possible cycles of points corresponding to a periodic behavior of the system. Let us then consider a given periodic cycle of the map, such as period one. A period one cycle corresponds to a single point in the PoincareH section, which repeats itself inde"nitely. Now, because of the instability of the corresponding orbit, this point in fact possesses a stable manifold and an unstable manifold. For stable (unstable) manifold we mean the collection of directions in phase space through which the trajectory
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approaches (diverges away from) the point geometrically. The control of chaos idea consists in perturbing a control parameter when the natural trajectory is in a small neighborhood of the desired point, such that the next intersection with the PoincareH section puts the trajectory on the stable manifold. In this case, all divergences are cured, and the successive natural evolution of the dynamics, except for nonlinearities and noise, converges to the desired point (that is, it stabilizes the desired periodic behavior). Selection of the perturbation is done by means of a reconstruction from experimental data of the local linear properties of the dynamics around the desired point. In some practical situations, however, it may be desirable to perform perturbations on a state variable accessible to the operator. This suggests the development of some alternative approaches. The "rst was introduced in Ref. [2]. It consists in designing a proper feedback line through which a state variable is directly perturbed such as to control a periodic orbit. This second method requires the availability of a state variable for experimental observation and for the perturbations. In such a case, a negative feedback line can be designed which is proportional to the di!erence between the actual value of the state variable, and the value delayed of a time lag ¹. The idea is that, when ¹ coincides with the period of one unstable periodic orbit of the unperturbed system, the negative feedback pushes to zero the di!erence between the present and the delayed dynamics, and the periodic orbit is stabilized. Furthermore, as soon as the control becomes e!ective, this di!erence goes e!ectively to zero, so that the feedback perturbation vanishes. Moreover, as before, a preliminary learning time is needed, for learning the periods of the unstable periodic orbits. In the above mechanism, the proportionality constant entering in the feedback loop is given in Ref. [3] where an adaptive technique has been introduced which automatically selects this constant by adaptively exploiting the local dynamics of the system. Many other techniques have been introduced with the aim of establishing control over chaos that will be referred to and described along this Report. Among the many available reviews, books, and monographies on this matter, here we address the reader the most recent ones, contained in Refs. [4}8]. In face of this huge number of theoretical studies, experimental realizations of chaos control have been achieved with a magnetoelastic ribbon [9], a heart [10,11], a thermal convection loop [12,13], a yttrium iron garnet oscillator [14], a diode oscillator [15], an optical multimode chaotic solid-state laser [16], a Belousov}Zhabotinski reaction di!usion chemical system [17], and many other experiments. While control of chaos has been successfully demonstrated experimentally in many situations, the control of patterns in space-extended systems is still an open question. This is the reason why most of the interest has moved actually from the control of periodic behaviors in concentrated systems, to the control of periodic patterns in space-extended systems, with the aim of controlling in"nite dimensional chaos, or even space}time chaos. The applications would be enormous, ranging from the control of turbulent #ows, to the parallel signal transmission and computation to the parallel coding-decoding procedure, to the control of cardiac "brillation, and so forth. 1.3. Targeting desirable states within chaotic attractors One of the major problems in the above process is that one can switch on the control only when the system is su$ciently close to the desired behavior. This is warranted by the ergodicity of chaos regardless of the initial condition chosen for the chaotic evolution, but it may happen that the small neighborhood of a given attractor point (target) may be visited only infrequently, because
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of the locally small probability function. Thus the unperturbed dynamics may take a long time to approach a given target, resulting in an unacceptably large waiting time for the operator to apply the control of chaos process. E$cient targeting methods can, instead, reduce the waiting time by orders of magnitude, and so they can be seen as a preliminary task for chaos control, independent of the particular control algorithm that one applies. In this Report, we devote Section 4 to the problem of targeting of chaos, since it is crucial for the realization of the control procedure, and summarize the di!erent proposed methods for directing chaotic trajectories to target points in the attractor.
1.4. The control of chaotic behaviors, and the communication with chaos Another section of this Report is devoted to the problem of the control of desired chaotic behaviors and its major applications. The critical sensitivity to initial conditions of a chaotic system can, indeed, be exploited not only to produce a large number of possible periodic behaviors (the di!erent unstable periodic orbit), but, much more generally, any desired behavior compatible with the natural evolution of the system. Therefore, one can imagine to select suitable perturbations to slave the chaotic system toward a particular `desireda chaotic behavior. Among the practically unlimited possible applications of the control of chaotic behavior, herewith we concentrate on two applications, which have attracted considerable attention in the scienti"c community over the past few years; namely the control of chaotic behavior for communicating with chaos and for the synchronization of chaotic systems for various communication schemes. In the "rst case, a chaotic system is conveniently perturbed, in order to give rise to a particular chaotic trajectory carrying a given message. In the second case, the process of chaos synchronization is applied to a communication line between a message sender and a message receiver, allowing the synchronization between them. There is a simple connection between chaos and communication theory. Chaotic systems can be viewed, indeed, as information sources that naturally produce digital communication signals. The formal connection between chaotic dynamics and information theory began with the introduction of the concept of measure-theoretic entropy in ergodic theory [18}20]. Chaotic systems are, indeed, characterized by having positive entropies and thus they are information sources. By assigning a discrete alphabet to the system state space using the formalism of symbolic dynamics, the chaotic system becomes a symbol source, and because it is a continuous-time waveform source, it is also a digital signal source. A chaotic system is, therefore, a natural source of digital communication signals. This concept has been recently shown to be more than formal [24]. Controlling the output of an oscillator via small guiding current pulses allows for the transmission of a desired message without e!ectively altering the time-evolution equations for the system. As an example, a very simple chaotic electrical oscillator can produce a seemingly random sequence of positive and negative (bipolar) voltage peaks [21]. If these bipolar peaks are assigned binary symbols 0 and 1, respectively, then the signal can be viewed as a binary communication waveform. We can furthermore encode any desired message into the waveform by using small perturbing pulses to control the sequence of peaks representing the symbols 0 and 1. More sophisticated waveforms and encodings are possible, but this example su$ces to convey the basic concept. In this Report, we also
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summarize the most relevant achievements in communicating with chaos, and we suggest some problems still unsolved. The control of chaotic behavior has another important application, namely, the synchronization of chaotic systems. If one consider two identical chaotic systems starting from di!erent initial conditions, then the critical sensitivity to initial conditions implies that their di!erence grows exponentially in time, and that they will evolve in an unsynchronized manner. The feeding of the right signal from one system to another can, however, reduce to zero such di!erence, and push the two systems into a synchronized manifold, wherein the chaotic motion is now developed so as the system are in step during the course of time. This proposal was intensively pushed forward at the beginning of this decade [22]. In the present Report, we simply summarize a possible application of synchronization of chaos, consisting in making secure the transmission of a signal between a message sender and a receiver along a communication line. 1.5. The experimental verixcations of chaos control Finally, we devote a section to summarize the most relevant experimental applications of the above ideas and techniques. Since it would be unrealistic to cover the whole body of experimental implementations of chaos control, herewith we limit ourselves to focus on few prototypical experiments, and we suggest to the interested reader to the most relevant literature. The OGY ideas found experimental applications in several di!erent "elds, such as mechanical oscillations (magnetoelastic ribbon), electronic circuits (diode resonator), chemical systems (Belouzov}Zhabotinski reaction), nonlinear optics (multimode laser). Di!erent control techniques were also experimentally tested on #uid dynamical systems leading to the control of convective instabilities, and on biomechanical systems for the control of the cardiac activity in a rabbit heart, and of the neuronal activity of an hippocampal slice. In every experimental example, we point out the relevance of the achievements, the di$culties for the practical realization of the theoretical proposals, and the perspective opened by such implementations. 1.6. Outline of the Report The present Report is organized as follows. In Section 2, the OGY method is illustrated with applications to one-dimensional and two-dimensional mappings. The pole placement technique is then discussed for the control of higher-dimensional situations. In Section 3, we discuss alternative schemes for chaos control, and we describe in detail the adaptive strategy with application to delayed dynamical systems, since it constitutes a bridge between concentrated and spatially extended systems. Section 4 is devoted to the discussion of the targeting problem. We show how the OGY criterion and adaptivity can provide suitable tools for directing the chaotic trajectories to desired targets. Furthermore, we show a possible application for the control of fractal basin boundaries. In Section 5, we discuss the issue of stabilizing desirable chaotic trajectories, and we point out two main applications: the communication with chaos, and communication through chaos synchronization. Section 6 summarizes the main experimental work in chaos control, and points out the perspective open in di!erent "elds by this process.
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2. The OGY method of controlling chaos 2.1. The basic idea Besides the occurrence of chaos in a large variety of natural processes, chaos may also occur because one may wish to design a physical, biological or chemical experiment, or to project an industrial plant to behave in a chaotic manner. The idea of Ott, Grebogi, and Yorke (OGY) is that chaos may indeed be desirable since it can be controlled by using small perturbation to some accessible parameter [1,23] or to some dynamical variable of the system [24]. The major key ingredient for the control of chaos [1,23] is the observation that a chaotic set, on which the trajectory of the chaotic process lives, has embedded within it a large number of unstable low-period periodic orbits. In addition, because of ergodicity, the trajectory visits or accesses the neighborhood of each one of these periodic orbits. Some of these periodic orbits may correspond to a desired system's performance according to some criterion. The second ingredient is the realization that chaos, while signifying sensitive dependence on small changes to the current state and henceforth rendering unpredictable the system state in the long time, also implies that the system's behavior can be altered by using small perturbations [1,23]. Then, the accessibility of the chaotic systems to many di!erent periodic orbits combined with its sensitivity to small perturbations allows for the control and the manipulation of the chaotic process. Speci"cally, the OGY approach is then as follows. One "rst determines some of the unstable low-period periodic orbits that are embedded in the chaotic set. One then examines the location and the stability of these orbits and chooses one which yields the desired system performance. Finally, one applies small control to stabilize this desired periodic orbit. However, all this can be done from data [1,23] by using nonlinear time series analysis for the observation, understanding and control of the system. This is particularly important since chaotic systems are rather complicated and the detailed knowledge of the equations of the process is often unknown. In what follows, we "rst give a pedagogical example of controlling chaos by using a simple one-dimensional map. We describe a general method for two-dimensional maps. Issues of noise and the average time of achieve control will also be discussed. We then describe the pole-placement method for controlling high-dimensional chaos, and we discuss how controlling chaos can be done when the equations of the systems are not available. 2.2. A one-dimensional example The basic idea of controlling chaos can be understood [25] by considering the following one-dimensional logistic map, one of the best studied chaotic systems: x "f (x , r)"rx (1!x ) , L> L L L
(1)
where x is restricted to the unit interval [0,1], and r is a control parameter. It is known that this map develops chaos via the period-doubling bifurcation route. For 0(r(1, the asymptotic state of the map (or the attractor of the map) is x"0; for 1(r(3, the attractor is a nonzero "xed point x "1!1/r; for 3(r(1#(6, this "xed point is unstable and the attractor is a stable period-2 $ orbit. As r is increased further, a sequence of period-doubling bifurcations occurs in which
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successive period-doubled orbits become stable. The period-doubling cascade accumulates at r"r +3.57, after which chaos can arise. Consider the case r"3.8 for which the system is apparently chaotic. An important characteristic of a chaotic attractor is that there exists an inxnite number of unstable periodic orbits embedded within it. For example, there are a "xed point x +0.7368 and a period-2 orbit with components $ x(1)+0.3737 and x(2)"0.8894, where x(1)"f (x(2)) and x(2)"f (x(1)). Now suppose we want to avoid chaos at r"3.8. In particular, we want trajectories resulting from a randomly chosen initial condition x to be as close as possible to the period-2 orbit, assuming that this period-2 orbit gives the best system performance. Of course, we can choose the desired asymptotic state of the map to be any of the in"nite number of unstable periodic orbits. Suppose that the parameter r can be "nely tuned in a small range around the value r "3.8, i.e., r is allowed to vary in the range [r !d, r #d], where d;1. Due to the nature of the chaotic attractor, a trajectory that begins from an arbitrary value of x will fall, with probability one, into the neighborhood of the desired period-2 orbit at some later time. The trajectory would diverge quickly from the period-2 orbit if we do not intervene. Our task is to program the variation of the control parameter so that the trajectory stays in the neighborhood of the period-2 orbit as long as the control is present. In general, the small parameter perturbations will be time dependent. We emphasize that it is important to apply only small parameter perturbations. If large parameter perturbations are allowed, then obviously we can eliminate chaos by varying r from 3.8 to 2.0 for example. Such a large change is not interesting. The logistic map in the neighborhood of a periodic orbit can be approximated by a linear equation expanded around the periodic orbit. Denote the target period-m orbit to be controlled as x(i), i"1,2, m, where x(i#1)"f (x(i)) and x(m#1)"x(1). Assume that at time n, the trajectory falls into the neighborhood of component i of the period-m orbit. The linearized dynamics in the neighborhood of component i#1 is then: Rf Rf x !x(i#1)" [x !x(i)]# *r L> Rr L Rx L "r [1!2x(i)][x !x(i)]#x(i)[1!x(i)]*r , (2) L L where the partial derivatives are evaluated at x"x(i) and r"r . We require x to stay in the L> neighborhood of x(i#1). Hence, we set x !x(i#1)"0, which gives L> [2x(i)!1][x !x(i)] L . (3) *r "r L x(i)[1!x(i)] Eq. (3) holds only when the trajectory x enters a small neighborhood of the period-m orbit, i.e., L when "x !x(i)";1, and hence the required parameter perturbation *r is small. Let the length of L L a small interval de"ning the neighborhood around each component of the period-m orbit be 2e. In general, the required maximum parameter perturbation d is proportional to e. Since e can be chosen to be arbitrarily small, d also can be made arbitrarily small. As we will see, the average transient time before a trajectory enters the neighborhood of the target periodic orbit depends on e (or d). When the trajectory is outside the neighborhood of the target periodic orbit, we do not apply any parameter perturbation, so the system evolves at its nominal parameter value r . Hence
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we set *r "0 when *r 'd. Note the parameter perturbation *r depends on x and is time L L L L dependent. The above strategy for controlling the orbit is very #exible for stabilizing di!erent periodic orbits at di!erent times. Suppose we "rst stabilize a chaotic trajectory around a period-2 orbit. Then we might wish to stabilize the "xed point of the logistic map, assuming that the "xed point would correspond to a better system performance at a later time. To achieve this change of control, we simply turn o! the parameter control with respect to the period-2 orbit. Without control, the trajectory will diverge from the period-2 orbit exponentially. We let the system evolve at the parameter value r . Due to the nature of chaos, there comes a time when the chaotic trajectory enters a small neighborhood of the "xed point. At this time we turn on a new set of parameter perturbations calculated with respect to the "xed point. The trajectory can then be stabilized around the "xed point [25]. In the presence of external noise, a controlled trajectory will occasionally be &kicked' out of the neighborhood of the periodic orbit. If this behavior occurs, we turn o! the parameter perturbation and let the system evolve by itself. With probability one the chaotic trajectory will enter the neighborhood of the target periodic orbit and be controlled again. The e!ect of the noise is to turn a controlled periodic trajectory into an intermittent one in which chaotic phases (uncontrolled trajectories) are interspersed with laminar phases (controlled periodic trajectories) [1,23]. It is easy to verify that the averaged length of the laminar phase increases as the noise amplitude decreases [25]. It is interesting to ask how many iterations are required on average for a chaotic trajectory originating from an arbitrarily chosen initial condition to enter the neighborhood e of the target periodic orbit. Clearly, the smaller the value of e, the more iterations that are required. In general, the average transient time 1q2 before turning on control scales with d as: 1q2&d\A ,
(4)
where c'0 is a scaling exponent. For one-dimensional maps, the probability that a trajectory enters the neighborhood of a particular component (component i) of the periodic orbit is given by
VG>C
o[x(i)] dx+2eo[x(i)] , (5) VG\C where o is the frequency that a chaotic trajectory visits a small neighborhood of the point x on the attractor. We have 1q2"1/P(e)&e\&d\, and therefore c"1. For higher-dimensional chaotic systems, the exponent c can be related to the eigenvalues of the periodic orbit to be controlled. A major advantage of the controlling chaos idea [1,23] is that it can be applied to experimental systems in which a priori knowledge of the system is usually not known. A time series found by measuring one of the system's dynamical variables in conjunction with the time delay embedding method [26,27], which transforms a scalar time series into a trajectory in phase space, is su$cient to determine the desired unstable periodic orbits to be controlled and the relevant quantities required to compute parameter perturbations [1,23]. The theoretical issue of using delay-coordinate embedding technique to control chaos will be detailed in Section 2.4. Another advantage of the OGY paradigm of controlling chaos is its #exibility in choosing the desired periodic orbit to be controlled, as we have detailed in the example of the logistic map. P(e)"
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2.3. Controlling chaos in two dimensions The general algorithm for controlling chaos for two-dimensional invertible maps (or threedimensional autonomous #ows that can be reduced to two-dimensional maps on the PoincareH surface of section) can be formulated in a similar way. Consider the following map: "F(x , p) , (6) L> L where x 3R, F is a smooth function of its variables, and p3R is an externally accessible control L parameter. We restrict parameter perturbations to be small: x
"p!p "(d ,
(7)
where p is some nominal parameter value, and d;1 de"nes the range of parameter variation. We wish to program the parameter p so that a chaotic trajectory is stabilized when it enters an e-neighborhood of the target periodic orbit. In the sequel, we "rst discuss the simple case of stabilizing a "xed point of the map F. We then give a method for stabilizing higher-period periodic orbits. 2.3.1. Stabilizing a xxed point 2.3.1.1. A linear control law. Let x (p ) be one of the "xed points of the map (6) at the nominal $ parameter value p that we wish to stabilize. In general, the location of the "xed point in the phase space depends on the control parameter p. Upon application of a small perturbation *p, we have p"p #*p. Since *p is small, we expect x (p) to be close to x (p ). We write $ $ x (p)+x (p )#u *p , (8) $ $ where the vector u is given by
x (p)!x (p ) $ + $ . (9) *p NN The vector u needs to be determined before a control law can be derived to stabilize the "xed point x (p ). $ The simplest way to formulate an applicable control law is to make use of the fact that the dynamics of any smooth nonlinear system is approximately linear in a small e-neighborhood of a "xed point. Thus, near x (p ), we can use the linear approximation for the map: $ [x !x (p)]+M[x (p)] ) [x !x (p)] , (10) L> $ $ L $ where M[x (p)] is the 2;2 Jacobian matrix of the map F(x, p) evaluated at the "xed point x (p), $ $ which is de"ned as follows: Rx u, $ Rp
RF M[x (p)]" $ Rx x
RM +M[x (p )]# $ Rp
*p . (11) N NN Note that *p&e and "[x !x (p)]"&e, where e is the size of the small neighborhood in which the L $ linear approximation (10) is valid. Writing x (p)+x (p )#u *p [from Eq. (9)], substituting this $ $ $
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relation and Eq. (11) into Eq. (10), and keeping only terms which are "rst-order in e, we obtain !x (p )+u *p#M[x (p )] ) [x !x (p )!u *p] . (12) L> $ $ L $ In Eq. (12), the Jacobian matrix M is evaluated at the "xed point x (p ) of the unperturbed system, $ which is the one to be stabilized. Since x (p ) is embedded in the chaotic attractor, it is unstable $ and it has one stable and one unstable direction [28]. Let e and e be the stable and unstable unit eigenvectors at x (p ), respectively, and let f and f be two unit vectors that satisfy $ f ) e "f ) e "1 and f ) e "f ) e "0, which are the relations by which the vectors f and f can be determined from the eigenvectors e and e . The vectors f and f are the contravariant basis vectors associated with the eigenspace e and e [1,23]. The Jacobian matrix M[x (p )] can then be $ written as: x
M[x (p )]"j e f #j e f , (13) $ where j and j are the stable and unstable eigenvalues in the eigendirections e and e , respectively. When the trajectory point x falls into the small e neighborhood of the desired "xed point x (p ) L $ so that Eq. (10) applies, a small parameter perturbation *p can be applied at time n to make the L "xed point shift slightly so that at the next iteration (n#1), x falls on the stable direction of L> x (p ). That is, we choose the parameter control *p such that $ L f ) [x !x (p )]"0 . (14) L> $ If x falls on the stable direction of x (p ), we can then set the control perturbation to zero, and L> $ the trajectory for subsequent time will approach the "xed point at the geometrical rate j . Thus for su$ciently small [x !x (p )], we can substitute Eq. (12) into Eq. (14) to obtain *p "c : L $ L L j f ) [x !x (p )] L $ ,C ) [x !x (p )] . (15) c " L $ L (j !1) f ) u We assume in the above that the generic condition u ) f O0 is satis"ed so that c &"x !x (p )", L L $ which is small. The considerations above apply only to a local small neighborhood of x (p ). $ Globally, one can specify the parameter perturbation *p by setting *p "0 if "c " is too large, since L L L the range of the parameter perturbation is limited by Eq. (7). Thus, practically, we can take *p to L be given by
if "c "(d , L (16) if "c "5d , L where in the de"nition of c in Eq. (15), it is not necessary to restrict the quantity "x !x (p )" to be L L $ small. Eqs. (15) and (16) are one formulation of the OGY-idea of controlling chaos [1,23]. There are several practical factors which must be considered when applying this formulation in practical situations. Among these are the following: c *p " L L 0
(a) The control situation is possible only if the quantity c de"ned in Eq. (15) satis"es "c "(d. This L L condition may be violated when the "xed point of the system is such that the vectors u and f are nearly orthogonal to each other. Such "xed points are therefore uncontrollable by using
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small perturbations. In principle, the magnitude of the control "c " can be made arbitrarily L small since "x !x (p )" can be made arbitrarily small by waiting for the trajectory x to be L $ L su$ciently close to the desired "xed point x (p ). However, the average waiting time for this to $ occur can be signi"cantly longer. There is thus a trade-o! between the magnitude of the perturbations and the average waiting time. There are also nonlinear corrections to Eq. (10) which are typically quadratic in [x !x (p )]. The quadratic corrections are important in L $ determining the average waiting time, which is, the average length of the chaotic transient experienced typically by trajectories before control is achieved. (b) In real systems there is often noise present. (c) There may be errors present in the measurements of the system states used in identifying the system. The location of the coordinates of the "xed point we wish to control may thus di!er from its true coordinates. Similarly, the quantities f , f , j , and j which are required to compute the parameter perturbations may contain some error. 2.3.1.2. Time to achieve control. Let m ,(x !x (p )). We note that the control in Eq. (16) is L L $ activated (i.e., *p O0) only when "c "(d, which in turn, determines a narrow strip region, in the L L phase space, along the stable direction of the "xed point to be stabilized: "m"(m , L H
(17)
where m,f ) m , m ,f ) m , and m can be determined by setting c "d. We obtain L L L L H L m "d"(1!j\)u ) f " . H
(18)
Thus, for small d, a typical trajectory resulting from a random initial condition will execute a chaotic motion, unchanged from the uncontrolled case, until m falls in the strip (Eq. (17)). Even L then, due to the nonlinearity not included in the linear expansion (10), the control may not be able to bring the trajectory to the desired "xed point. In this case, the trajectory will leave the strip and continue to wander chaotically as if there was no control. Since the trajectory on the uncontrolled chaotic attractor is ergodic, which means that the trajectory will visit the neighborhood of every point on the attractor in the course of time evolution, at some time the trajectory will eventually come su$ciently close to the desired "xed point to satisfy "c "(d so that control is achieved. Thus, L in ideal situations (no noise and no imperfect identi"cation in the system parameter), applying control creates a stable orbit, but, for a typical initial condition, it is preceded in time by a chaotic transient in which the orbit is similar to orbits on the uncontrolled chaotic attractor. The lengths of the chaotic transients are di!erent for di!erent initial conditions, and they can be regarded as realizations of a random variable q with an exponential probability distribution: P(q)&exp[!q/1q2] ,
(19)
where 1q2 is the average length of the chaotic transient. When d decreases, the size of the control strip (Eq. (17)) decreases, so the average time to achieve control, or 1q2, increases. To understand the exponential distribution (19), imagine we choose a large number of points on the chaotic attractor according to the natural measure [29,30]. Regarding the strip area surrounding the desired "xed point x (p ) as an `escapinga region, we see that under the dynamics, each $
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initial condition enters the escaping region in di!erent times. These times can be considered as the lifetimes for trajectories to stay outside the escaping region. To "nd the probability distribution for these lifetimes, we iterate the escaping region backward in time under the inverse dynamics. The inverse images intersect the chaotic attractor along the stable foliations, creating a set of in"nite number of gaps on the chaotic attractor (there are an in"nite number of inverse images) and leaving behind a fractal set of points which never enters the escaping region. This fractal set is nonattracting because a point in the vicinity of the set will eventually leave it and enter one of the gaps. Since almost all points on the chaotic attractor fall into gaps, we see that the lifetimes for those points are in fact the escaping times that the points leave the nonattracting fractal set. It is known that the distribution of escaping times from a nonattracting fractal set is exponential [31,32]. Apparently, di!erent choice of the control strip leads to di!erent fractal sets. There are, in fact, an in"nite number of nonattracting fractal sets embedded in the chaotic attractor. Such nonattracting fractal sets are also called chaotic saddles [33,34]. Following Refs. [1,23], we now derive the scaling law (4) and give a formula for the scaling exponent c. Dotting the linearized map for m , (Eq. (12)), with f and then with f , and using L> Eqs. (13) and (15) which is appropriate for "m"(m , we obtain the following two relations for the L H size of the controlling strip: m +0 , L> m +j m#j m , L> L L where
(20) (21)
j u)f . j ,(1!j ) (j !1)u ) f Eqs. (20) and (21) are linearizations, and typically the lowest-order corrections to them are quadratic. In particular, m is not restricted by "m"(m and thus may not be small when the L L H control condition is satis"ed. Thus we expect the correction that is quadratic in m to be most L signi"cant. Including such a correction in Eq. (20), we have m +i(m ) , (22) L> L where i is a constant. Thus, if "i"(m )'m , then "m "'m , and control is not achieved even L H L> H though "m"(m . Control is achieved when the trajectory falls in the small control parallelogram L H P given by "m"(m , (23) L H "m "((m /"i" . (24) L H For very small m , a trajectory will wander on the uncontrolled chaotic attractor for a long time H before it falls into the control parallelogram P . At any given time step, the probability of falling in P is approximately the natural measure [29,30] of the uncontrolled chaotic attractor contained in P . If one distributes a large number of initial conditions on the chaotic attractor according to the natural measure and then follows the trajectories resulting from these initial conditions, this probability k(P ) gives the rate at which these orbits fall into the control parallelogram. Let N(n) be the uncontrolled orbits remaining at time n. If the chaotic transient is long, we can treat the discrete
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time approximately as continuous. We obtain dN(n) "!k(P )N(n) . dn Thus k(P ) is the inverse of the exponential decay time for N(n) so that we have 1q2\"k(P ) . (25) To obtain the natural measure k(P ), we make use of the de"nition of the pointwise dimensions [29]. For the control parallelogram P , since its sizes along the stable and unstable directions of desired "xed point x (p ) are (m /"i" and m , respectively, we obtain the following scaling of k(P ) H $ H with m : H k(P )&(m )B ((m /"i")B &mB > B &(m )B > B , H H H H where d and d are the pointwise dimensions for the uncontrolled chaotic attractor at x (p ) in the $ stable and unstable directions, respectively. Since m &*p, we obtain the scaling relation (4), where H the scaling exponent is given by c"d #d . (26) Usually, the chaotic attractor is e!ectively smooth in the unstable direction [29], which implies d "1. The pointwise dimension in the stable direction is given in terms of the stable and unstable eigenvalues at x (p ) as, d "(ln"j "/(ln"j "\)). Thus $ 1 ln"j " . (27) c"1# 2 ln(1/"j ") The above scaling for the average time to achieve control is obtained under the assumption that the control is o! for "m"'m . It is possible, in some situations, to greatly reduce 1q2 by applying L H small controls to the orbit in the region outside the control parallelogram. This is an example of another general type of chaos control problem, the so-called target acquisition problem [35}38]. In particular, the goal is to apply small perturbations to a chaotic trajectory so as to bring it from its initial condition to a target region in which the desired "xed point (or periodic orbits) located in the shortest possible time. Note that in the above, the small neighborhood of the desired "xed point for turning on the control is assumed to have the shape of a parallelogram. One can also consider a small circular area around the desired "xed point. That is, the control is turned on when "m "(m . This again yields L H the algebraic scaling (4) but with a di!erent exponent: in this case the exponent is the pointwise dimension (d #d ). Note that this value of the exponent is larger than that given by Eq. (26) due to the fact that the area of the circular region "m "(m is much smaller than the control parallelogram L H P and is contained in P . Thus, the times to achieve control become longer and, hence, in practice it is more advantageous to consider the control parallelogram for turning on the control. 2.3.1.3. Ewect of noise. To study the e!ect of noise on control, we add a term ed to the right-hand L side of the linearized equations for m , (Eq. (12)), where d is a random variable and e is a small L> L parameter specifying the intensity of the noise. To mimic environmental noise in a typical experimental setting, we take the random variable d to have zero mean (1d 2"0), be independent L L
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(1d d 2"0 for mOn), have mean squared value unity (1d2"1), and have a probability density L K L independent of n (stationary random variable). Due to noise, the quantity m , which is the dot L> product between f and m (Eq. (12)), no longer vanishes when high-order nonlinear terms are L> neglected. Instead, we have m "ed , (28) L> L where d ,f ) d . If the noise is bounded, i.e., "d"(d , then the control will hardly be a!ected by L L L
noise if d is smaller than the width of the control parallelogram m , i.e., if
H ed (m . (29)
H If Eq. (29) is not satis"ed or if the noise is unbounded (d "R), then the noise can kick
a controlled trajectory which is in the control parallelogram P into the region outside P . A situation often encountered in practice is that the probability for large noise intensity is extremely low, such as the case where the noise distribution is Gaussiana. In this case, the loss of control due to noise-induced kick-outs (out of P ) are rare because they are caused by low probability tails on the probability density and, hence, the average time 1¹2 for a controlled trajectory to be kicked out of P will be long. Thus a trajectory will typically alternate between epochs of chaotic motion of average time 1q2 in which it is far from the desired "xed point, and epochs of average length 1¹2 in which the orbit is e!ectively controlled and lies in the control parallelogram. If 1¹2 is much greater than 1q2, then the trajectory spends most of its time in the controlled epochs, and the control can be regarded as being e!ective. The average time 1¹2 for which control is maintained can be computed as follows. Let the probability density of the random variable d be P(d). The quantity 1¹2 is then the time for d to L L L become larger than d ,m /e, which is the inverse of the probability for d to lie outside the strip H H L "d ". Thus, we have H \BH \ . (30) P(d) dd# P(d) dd 1¹2" L L L L \ BH If P(d) is Gaussian and if d is several times the standard deviation, then the kick-outs will L H correspond to d in the far tails of the probability density. In this case, 1¹2 will be large. L
2.3.2. Stabilizing a periodic orbit of higher period Let the desired period-m orbit be x(1, p )Px(2, p )P2Px(m, p )Px(m#1, p )"x(1, p ). The linearized dynamics in the neighborhood of component i#1 of the period-m orbit is !x(i#1, p )"A ) [x !x(i, p )]#B *p , (31) L> L L where *p "p !p , *p 4d, A is a 2;2 Jacobian matrix, and B is a two-dimensional column L L L vector: x
A"D F(x, p) x x , G NN (32) . B"D F(x, p) x x N G NN In two dimensions, there exist a stable and an unstable directions at each component (point) of an unstable periodic orbit. The stable (unstable) direction is a direction along which points
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approach (leave) the periodic orbit exponentially. (For higher-dimensional maps, there may be several stable and unstable directions.) Intuitively, the existence of both a stable and an unstable directions at each point of a periodic orbit can be seen as follows. Choose a small circle of radius e around an orbit point x(i). This circle can be written as dx#dy"e in the Cartesian coordinate system whose origin is at x(i). The image of the circle under F\ can be expressed as A dx #B dx dy #C dy "1, an equation for an ellipse in the Cartesian coordinate system whose origin is at x(i!1). The coe$cients A, B and C are functions of elements of the inverse Jacobian matrix at x(i). This deformation from a circle to an ellipse means that the distance along the major axis of the ellipse at x(i!1) contracts as a result of the map. Similarly, the image of a circle at x(i!1) under F is typically an ellipse at x(i), which means that the distance along the inverse image of the major axis of the ellipse at x(i) expands under F. Thus the major axis of the ellipse at x(i!1) and the inverse image of the major axis of the ellipse at x(i) approximate the stable and unstable directions at x(i!1). We note that typically the stable and unstable directions are not orthogonal to each other, and in rare situations they can be identical (nonhyperbolic dynamical systems) [34,39]. The stable and unstable directions at each point of an unstable periodic orbit can be computed using the algorithm developed in Ref. [34]. This algorithm can be applied to cases where the period of the orbit is arbitrarily large. To "nd the stable direction at a point x, one "rst iterates this point forward N times under the map F and obtains the trajectory F(x), F(x),2, F,(x). Now imagine that a circle of small radius e is placed at the point F,(x). If this circle is iterated backward once, it becomes an ellipse at the point F,\(x), with the major axis along the stable direction of the point F,\(x). One can iterate this ellipse backwards continuously, while at the same time rescaling the ellipse's major axis to be order e. When the ellipse is iterated back to the point x, it becomes very thin with its major axis along the stable direction at the point x, if N is su$ciently large. For a short period-m orbit, one can choose, say N"km, where k an integer. In practice, instead of using a small circle, one takes a unit vector at the point F,(x), since the Jacobian matrix of the inverse map F\ rotates a vector in the tangent space of F towards the stable direction. Hence one iterates a unit vector backward to the point x by multiplying by the Jacobian matrix of the inverse map at each point on the already existing orbit. The vector is rescaled after each multiplication to unit length. For su$ciently large N, the unit vector so obtained at x is a good approximation to the stable direction at x. Similarly, to "nd the unstable direction at point x, one "rst iterates x backward under the inverse map N times to obtain a backward orbit F\H(x) with j"N,2, 1. One then chooses a unit vector at point F\,(x) and iterates this unit vector forward to the point x along the already existing orbit by multiplying by the Jacobian matrix of the map N times. (Recall that the Jacobian matrix of the forward map rotates a vector towards the unstable direction.) The vector is rescaled to unit length at each step. The "nal vector at point x is a good approximation to the unstable direction at that point if N is su$ciently large. The above method is e$cient. For instance, the error between the calculated and real stable or unstable directions [34] is on the order of 10\ for chaotic trajectories in the HeH non map [40] if N"20. Let e and e be the stable and unstable directions at x(i), and let f and f be the G G G G corresponding contravariant vectors that satisfy the conditions f ) e "f ) e "1 and G G G G f ) e "f ) e "0. To stabilize the orbit, we require that the next iteration of a trajectory point, G G G G
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after falling into a small neighborhood about x(i), along the stable direction at x(i#1, p ): [x !x(i#1, p )] ) f "0 . (33) L> G> Taking the dot product of both sides of Eq. (33) with f and use Eq. (31), we obtain the following G> expression for the parameter perturbations: +A ) [x !x(i, p )], ) f L G> . (34) *p " L !B ) f G> The general algorithm for controlling chaos for two-dimensional maps can thus be summarized as follows: (a) Find the desired unstable periodic orbit to be stabilized. (b) Find a set of stable and unstable directions, e and e , at each component of the periodic orbit. The set of corresponding contravariant vectors f and f can be found by solving e ) f "e ) f "1 and e ) f "e ) f "0. (c) Randomly choose an initial condition and evolve the system at the parameter value p . When the trajectory enters the e neighborhood of the target periodic orbit, calculate parameter perturbations at each time step according to Eq. (34). 2.4. Pole placement method of controlling chaos in high dimensions We consider the following discrete-time high-dimensional dynamical system: "F(x , p ) , (35) L> L L where x 3R,, F is a smooth vector function, p is an accessible parameter that can be externally L L perturbed. Continuous dynamical systems can be regarded as discrete maps on the PoincareH surface of section. Periodically driven dynamical systems have a natural PoincareH surface of section at the period of the driver. However, for autonomous dynamical systems such a section may not exist, or it may be singular if some of the trajectories take arbitrarily long time to return to it. One might need then, in order to discretize the dynamical process, to select some other kind of section whose choice typically depends on the particular system. We conceive using only small controls, so we restrict p to lie in some small interval, as in the control of low-dimensional chaos: x
"p !p "(d , (36) L where p is a nominal parameter value. As in the low-dimensional case, if p is outside this interval, L we set p "p . Assuming that the dynamical system F(x , p ) possesses a chaotic attractor, the goal is L L to vary the parameter p within the range (p !d, p #d) in such a way that for almost all initial L conditions in the basin of the chaotic attractor, the dynamics of the system converges onto a desired time periodic orbit contained in the attractor. To do this we consider a small neighborhood of size comparable to d of the desired periodic orbit. In this neighborhood, the dynamics is approximately linear. Since linear systems are stabilizable if the controllability assumption is obeyed, it is reasonable to assume that the chosen periodic orbit can be stabilized by feedback control. The ergodic nature of the chaotic dynamics guarantees that the state trajectory enters the
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neighborhood. Once inside, we apply the stabilizing feedback control law to keep the trajectory in the neighborhood of the desired orbit. For simplicity we describe the method as applied to the case where the desired orbit is a "xed point of the map F. Consideration of periodic orbits of period larger than one is straightforward [41]. Let x (p ) be an unstable "xed point on the attractor. For values of p close to p and in the H L neighborhood of the "xed point x (p ), the map can be approximated by the linear map: H x !x (p )"A[x !x (p )]#B(p !p ) , (37) L> H L H L where A is the N;N Jacobian matrix and B is an N-dimensional column vector: A"Dx F(x, p) ,
(38) B"D F(x, p) . N The partial derivatives in A and B are evaluated at x"x and p"p . To calculate the timeH dependent parameter perturbation (p !p ), we assume that it is a linear function of x: L p !p "!K 2[x !x (p )] , (39) L L H where the 1;n matrix K 2 is to be determined so that the "xed point x becomes stable. H Substituting Eq. (39) into Eq. (37), we obtain !x (p )"(A!BK 2)[x !x (p )] , (40) L> H L H which shows that the "xed point will be stable if the matrix (A!BK 2) is asymptotically stable; that is, all its eigenvalues have modulus smaller than unity. The solution to the problem of determining K 2, such that the eigenvalues of the matrix (A!BK 2) have speci"ed values, is known from control systems theory as the `pole placement techniquea [42]. In this context, the eigenvalues of the matrix (A!BK 2) are called the `regular polesa. The following results give a necessary and su$cient condition for a unique solution of the pole placement problem to exist, and also a method for obtaining it (Ackermann's method) [42]: (1) The pole placement problem has a unique solution if and only if the N;N matrix x
C"(B$AB$AB$2$AL\B) , is of rank N, where C is the controllability matrix; and (2) the solution of the pole placement problem is given by K 2"(a !a ,2, a !a )T\ , , , where T"CW and
a ,\ a ,\ W" $ a 1
a ,\ a ,\ $
2 a 2 1
1
0
$
$ .
1
2
0
0
0
2
0
0
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Here +a ,2, a , are the coe$cients of the characteristic polynomial of A: , "sI!A""s,#a sL\#2#a , , and +a ,2, a , are the coe$cients of the desired characteristic polynomial (A!BK 2). , The condition for the matrix C to be of rank N is too strong as far as stabilizability of a closed loop system is concerned. In fact, the pole placement technique only requires a set of N points, placed symmetrically with respect to the real axis in the complex plane. Then there exists a feedback matrix K 2 such that the poles of the closed-loop system are the above set of points. It should be pointed out that there is a large class of control systems, in particular those arising in physical situations, which do not have a controllable linearization as indicated in Eq. (40). One has then to choose another control that obeys the controllability assumption if one wishes to use linear control. In particular, special care should be exercised when dealing with pole placement technique for nonautonomous systems. It should be noted that the control Eq. (39) is based on the linear Eq. (37) and therefore it is only valid in the neighborhood of the desired "xed point x (p ). The size H of this valid neighborhood is determined by the limitation in the size of the parameter perturbation d. Combining Eqs. (36) and (39), we obtain "K 2[x !x (p )]"(d . (41) L H This de"nes an invariant slab of width 2d/"K 2" in R,. We choose to activate the control according to Eq. (41) only when the trajectory falls into the slab, and we leave the control parameter at its nominal value p when the trajectory is outside this slab. It should also be noted that the matrix K 2 can be chosen in many di!erent ways. In principle, a choice of regulator poles inside the unit circle, which does not violate the controllability condition, works [41]. The OGY method consists of setting the unstable poles equal to zero while leaving the stable ones as they are. With the OGY choice of regulator poles, the trajectory approaches the "xed point geometrically along the stable manifold after the control is turned on. Since the control is turned on only when the trajectory enters the thin slab about the desired "xed point, one has to wait for some time for this to occur if the trajectory starts from a randomly chosen initial condition. Even then, because of nonlinearity not included in the linearized Eq. (37), the control may not be able to keep the trajectory in the vicinity of the "xed point. In this case the trajectory will leave the slab and continue to wander chaotically as if there was no control. Since a chaotic trajectory on the uncontrolled chaotic attractor is ergodic, at some time it will eventually reenter the slab and also be su$ciently close to the "xed point so that control is achieved. As a result, we create a stable orbit, which, for a typical initial condition, is preceded by a chaotic transient [32,43] in which the orbit is similar to orbits on the uncontrolled chaotic attractor. Of course, there is a probability zero Cantor-like set of initial conditions which never enters the slab. In Ref. [41], it is shown that the OGY choice for the regulator poles yields the shortest chaotic transient or, equivalently, the shortest average time to achieve control. As a typical higher-dimensional physical system, we consider the double rotor map which is a four-dimensional map describing the time evolution of a mechanical system known as the kicked double rotor [41,44]. The double rotor consists of two thin, massless rods as shown in Fig. 1. The "rst rod of length l pivots about P ("xed), and the second rod of length 2l , pivots about P which moves. A mass m is attached at P , and two masses m /2 are attached to each end of the second rod. The end of the second rod (P ) receives vertical periodic impulse kicks of period ¹ and
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Fig. 1. From Ref. [41]. The double rotor.
strength f . The motion is in the horizontal plane so that gravity can be neglected. The double rotor is also subject to friction at P and P which is proportional to the angular velocity dh (t)/dt and dh (t)/dt!dh (t)/dt with proportional constants l and l , respectively. Due to the periodic forcing, the set of di!erential equations describing the double rotor can be reduced to the following four-dimensional map by using the stroboscopic sectioning technique [41,44]:
X MY #X L> " L L , (42) Y LY #G(X ) L> L L> where X"(x, x)2, Y"(y, y)2, x and x are the angular positions of the rods at the instant of the kth kick, and y and y are the angular velocities of the rods immediately after the kth kick. L and M are constant 2;2 matrices de"ned by eHH ¹!1 , L" W eHH ¹, M" W H j H H H H a b d !b W " , W " , b d !b a
(43)
a"(1#(l /D)), d"(1!(l /D)), b"!l /D , "!(l #l $D), D"(l #4l . The function G(X) is given by j
c sin x G(X)" , c sin x
(44)
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Fig. 2. From Ref. [41]. Double rotor map: successive control of unstable "xed points embedded in the chaotic attractor. The arrows indicate the times of switching.
where c "f l /I, c "f l /I, and I"(m #m )l "m l . For illustrative purposes we "x l"¹"I"m "m "l "1 and l "1/(2. For f "9.0, the double rotor map has a chaotic attractor with two positive Lyapunov exponents. There are 32 unstable "xed points embedded in this chaotic attractor [41]. To apply the pole-placement technique, we choose f as the control parameter so it can be varied about its nominal value f "9.0. Let (X , Y ) be a "xed point to be stabilized. The quantities H H required in the application of the pole-placement technique are as follows:
I M , H(X ) L#H(X )M H H 0 f l cos xH , H(X )" H I 0 l cos x H B2"(0, 0, l sin x /I, l sin x /I) , H H C"(B$AB$AB$2$AL\B), T"CW , A"
a a W" a 1
a a 1
a 1
1
0
0
0
0
0
0
(45)
,
K 2"(a !a , a !a , a !a , a !a )T\ . Fig. 2(a) and (b) show how the method works to stabilize di!erent unstable "xed points embedded in the chaotic attractor. Control of the "rst "xed point was turned on when the trajectory enters its
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slab de"ned by (41) with switches to control other "xed points occurring at later times. In the "gures, the state variables x and x of an orbit are plotted. The times at which the control is switched from stabilizing one "xed point to stabilizing another are labeled by the arrows in the "gures. The magnitude of the parameter perturbation is chosen as d"1.0, which is roughly 10% of the nominal f value. A smaller d would increase the average time to achieve control. The "gures clearly illustrate the #exibility o!ered by the method in controlling di!erent periodic motions embedded in the attractor. 2.4.1. Use of delay coordinates In most experimental situations, a detailed knowledge of the system's equations is not known. One usually measures a time series of a single scalar state variable, say u(t), and then uses the delay coordinates [26] to represent the system state. A delay-coordinate vector in the m-dimensional embedding space can be formed as follows: x(t)"(u(t), u(t!t ), u(t!2t ),2, u(t!(m!1)t )) , " " " where t is the continuous time variable, and t is some conveniently chosen delay time. The " embedding theorem [26] guarantees that for m52N, where N is the phase-space dimension of the system, the vector x is generically a global one-to-one representation of the system state. Since we only require x to be one-to-one in the small region near the "xed point, the requirement for the embedding dimension is actually m"N!1 [1,23]. To obtain a map, one can take a PoincareH surface of section. For the often encountered case of periodically driven systems, one can de"ne a `stroboscopic surface of sectiona by sampling the state at discrete time t "n¹#t , where ¹ is L the driving period. In this case the discrete state variable is x "x(t ). L L As pointed out in Ref. [45], in the presence of parameter variation, delay coordinates lead to a map of a di!erent form than Eq. (1). For example, in the periodically forced case, since the components of x are u(t!it ) for i"0, 1,2, (m!1), the vector x must depend not only on p , L " L> L but also on all previous values of the parameter that are in e!ect during the time interval (t !(m!1)t )4t4t . In particular, let r be the smallest integer such that mt (r¹. Then the L " L " relevant map is in general of the form: "G(x , p , p , ,p ) . (46) L> L L L\ 2 L\P We now discuss how the OGY method can be applied to the case of delay coordinates. For simplicity we consider r"1. In this case, we have x
"G(x , p , p ). (47) L> L L L\ Linearizing as in Eq. (37) and again restricting attention to the case of a "xed point, we have x
!x (p )"A[x !x (p )]#B (p !p )#B (p !p ) , (48) L> H L H ? L @ L\ where A"Dx G(x, p, p ), B "D G(x, p, p ), B "D G(x, p, p ), and all partial derivatives in A, B , ? N @ NY ? and B are evaluated at x"x (p ) and p"p "p . One can now de"ne a new state variable with one @ H extra component by x
x
L>
"
x L> , p L
(49)
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and introduce the linear control law p !p "!K 2[x !x (p )]!k(p !p ) . L L H L\ Combining Eqs. (48) and (50), we obtain x
L>
(50)
!x (p )"(AM !BK 2)[x !x (p )] , H H
(51)
where
x (p )" H
x (p ) A B B K H @ , BM " ? , KM " , AM " . p 0 0 1 k
Since Eq. (51) is now of the same form as Eq. (40), the method of pole placement can be applied. A similar result holds for any r'1. Although the explicit form for the function G(x , p , p ) is not L L L\ known, the quantities required for computing the parameter perturbations in Eq. (51) can usually be extracted directly from the measurement [9]. The location of the periodic orbits are obtained by looking at recurrences in the embedded space [46,47]. The matrix A in Eq. (48) and the corresponding eigenvalues and eigenvectors are obtained by looking at the same recurrences about the desired periodic orbit and "tting an a$ne transformation x "Ax #b, since the dynamics is approxL> L imately linear close to the periodic orbit. The vectors B and B in Eq. (48) are obtained by ? @ perturbing the control parameter of the system [1,23,45,48]. 2.5. Discussion The OGY method described above applies to invertible maps. In general, dynamical systems that can be described by a set of "rst-order autonomous di!erential equations are invertible, and the inverse system is obtained by letting tP!t in the original set of di!erential equations. Hence, the discrete map obtained on the PoincareH surface of section also is invertible. Most dynamical systems encountered in practice fall into this category. Noninvertible dynamical systems possess very distinct properties from invertible dynamical systems [49,50]. For instance, for two-dimensional noninvertible maps, a point on a chaotic attractor may not have a unique stable (unstable) direction. A method for determining all these stable and unstable directions is not known. If one or several such directions at the target unstable periodic orbit can be calculated, the OGY method can in principle be applied to noninvertible systems by forcing a chaotic trajectory to fall on one of the stable directions of the periodic orbit. The transient phase where the orbit wanders chaotically before locking into a controlled orbit can be greatly shortened by applying a `targetinga technique [35}38] so that a trajectory can be rapidly brought to a target region on the attractor by using small control perturbations. The idea is that, since chaotic systems are exponentially sensitive to perturbations, careful choice of even small control perturbations can, after some time, have a large e!ect on the trajectory location and can be used to guide it. Thus the time to achieve control can, in principle, be greatly reduced by properly applying small controls when the orbit is far from the neighborhood of the desired periodic orbit. We have considered the case where there is only a single control parameter available for adjustment. While generically a single parameter is su$cient for stabilization of a desired periodic orbit, there may be some advantage to utilizing several control variables. Therefore, the single
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control parameter p becomes a vector. In particular, the added freedom in having several control parameters might allow better means of choosing the control so as to minimize the time to achieve control, as well as the e!ects of noise. We emphasize that full knowledge of the system dynamics is not necessary in order to apply the OGY idea [1,23]. In particular, we only require the location of the desired periodic orbit, the linearized dynamics about the periodic orbit, and the dependence of the location of the periodic orbit on small variation of the control parameter. Delay-coordinate embedding has been successfully utilized in experimental studies to extract such information purely from observations of experimental chaotic orbits on the attractor without any a priori knowledge of the equations of the system, and such information has been utilized to control periodic orbits [9]. The OGY idea of controlling chaos gives #exibility. By switching the small control, one can switch the time asymptotic behavior from one periodic orbit to another. In some situations, where the #exibility o!ered by the ability to do such switching is desirable, it may be advantageous to design the system so that it is chaotic. In other situations, where one is presented with a chaotic system, the method may allow one to eliminate chaos and achieve greatly improved behavior at relatively low cost. Finally, we point out that the OGY method is not restricted to the control of unstable periodic orbits. The success of the method relies on the existence of distinct stable and unstable directions at trajectory points. It can be applied to stabilizing any desirable chaotic trajectory embedded in a chaotic attractor [51}53] and, consequently, it is also applicable to pseudo-periodic orbits which are chaotic trajectories coming arbitrarily close to some unstable periodic orbits. This observation has a number of potential applications. One can use this to synchronize two chaotic systems [51], to convert transient chaos into sustained chaos [52], to communicate with chaos by controlling symbolic dynamics [21,24,54}57], and to select a desirable chaotic state from intermittent chaotic signal [53], etc.
3. The adaptive method for control of chaos 3.1. The basic idea Many alternative approaches to the OGY method have been proposed for the stabilization of the unstable periodic orbits (UPO) [47] of a chaotic dynamics. In general the strategies for the control of chaos can be classi"ed into two main classes, namely: closed loop or feedback methods and open loop or non feedback methods. The "rst class includes those methods which select the perturbation based upon a knowledge of the state of the system, and oriented to control a prescribed dynamics. Among them, we here recall (besides OGY) the so called occasional proportional feedback (OPF) simultaneously introduced by Hunt [15] and Showalter [17], the method of Huebler [58], and the method introduced by Pyragas [2], which apply a delayed feedback on one of the system variables. All these methods are model independent, in the sense that the knowledge on the system necessary to select the perturbation can be done by simply observing the system for a suitable learning time. The second class includes those strategies which consider the e!ect of external perturbations (independent on the knowledge of the actual dynamical state) on the evolution of the system.
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Periodic [59] or stochastic [60] perturbations have been seen to produce drastic changes in the dynamics of chaotic systems, leading eventually to the stabilization of some periodic behavior. These approaches, however, are in general limited by the fact that their action is not goal oriented, i.e. the "nal periodic state cannot be decided by the operator. In what follows, we will try to summarize one of the possible closed-loop approach for the stabilization of UPO. Although it belongs to the same class of OGY, it can be considered as alternative, insofar as it consists in perturbing a state variable of the system, instead of a control parameter. This may result preferable in all cases in which the control parameters are strongly in#uenced by the environmental conditions, and a variation of them is not easy to be performed. 3.2. The algorithm for adaptive chaos control In the following we will then discuss the general problem of forcing a dynamical system x "F(x, k)
(52)
toward a desired goal dynamics u(t), by the use of a feedback perturbation ;(t) acting on one of its state variables. Here dot denotes temporal derivative, k is a vector of control parameters, x(t),(x (t), x (t),2, x (t)) is a m-dimensional vector of the state variables, F is a suitable K nonlinear function, and u(t),(g (t), g (t),2, g (t)) is the m-dimensional desired dynamics. The K further hypothesis here is that it exists a scalar state variable (e.g., x ) out from x that is accessible for measurements and perturbations. The problem is to select a suitable additive perturbation to the x equation so that the whole system evolves asymptotically to yield "x(t)!u(t)""0. In general u(t) can be any desired dynamics compatible with the natural evolution of the system. However, along this section, we will consider u(t) to equal a particular UPO of period ¹(u(t),x(t!¹)) so that the above condition corresponds to the stabilization of that UPO. The "rst attempt to solve this problem was provided by Pyragas [2], who introduced an additive feedback perturbation in the equation for the observable variable x given by ;(t)"K(g (t)!x (t))"K(x (t!¹)!x (t)) , (53) K being a suitable parameter setting up the weight of the perturbation. This perturbation is nothing but a time-delayed continuous feedback on the state variable x , with the e!ect of forcing x (t) to follow g (t). In practice, the method transform a system of ordinary di!erential equations into a delayed dynamical system. This implies to increase its dimensionality so as the desired UPO (which was unstable in the original ordinary di!erential equation system) becomes now stable in the new delayed dynamical system. The two crucial parameters in Ref. [2] are ¹ and K. The "rst can be experimentally detected by observing the unperturbed evolution of the system for a while, and applying e.g. the standard topological techniques for the detection of UPOs [61]. Once ¹ has been properly detected, the operator begins with di!erent K values in order to extract the range for which the desired UPO comes out to be stabilized (the so called controllability range). The above technique has been later elaborated in Refs. [62] and by the same Pyragas in Ref. [63], by rede"ning the correction perturbation ;(t) as a sum of the contributions at all previous multiple p¹ (p integer) of the UPOs period.
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An improvement of the Pyragas' technique has been recently o!ered, based on an adaptive recognition [64] and control [3] method, allowing a natural selection of the weighting factor K in Eq. (53), which becomes now function of time and it is selected exploiting the local information of the dynamics. The method, originally introduced for the stabilization of UPOs in chaotic systems, has been later extended for solving the problem of chaos synchronization [65], targeting of chaos [66], "ltering noise from experimental chaotic data sets [67], and for the control of defects and space-like structures in delayed dynamical systems [68]. The adaptive algorithm can be summarized as follows. In Eq. (52), at each time t at which the L perturbation must be applied, one measures the distance d(t ) between the dynamics of the L observable x (t ) and the goal dynamics g (t ) L L d(t )"x (t )!g (t ) . (54) L L L Then, one evaluates the local variation rate
d(t ) L , (55) j(t )"log L d(t ) L\ measuring how the distance between actual and desired dynamics evolves in time, and selects the perturbation as ;(t )"K(t ) (g (t )!x (t )) , L L L L where
(56)
1 1 " (1!tanh(pj(t ))), p'0, K '0 . (57) L K(t ) K L The adaptive nature of the algorithm is clear when one considers that the strength of the perturbation in Eq. (56) now adaptively depends on the local dynamics of the system. Namely, when x (t) naturally tends to shadow the goal pattern g (t), this is re#ected by a temporal decreasing behavior of d(t), implying a negative value of j(t) and therefore a shrinking of the weighting factor K(t) in Eq. (57). On the contrary, whenever the natural evolution of the dynamics tends to push the system away from the goal dynamics, this is re#ected by an increasing process for K(t). In other words the perturbation is adapted to the local dynamics, since the far (close) the system is to the goal dynamics, the big (small) is the weight given to the perturbation. It should be remarked that the limit pP0 of the above algorithm recovers the Pyragas' control method of Ref. [2], implying a constant weighting factor K in Eq. (57), which then equals Eq. (53). While this process solves the problem of encountering the controllability range (the weighting factor K is now not constant as far as pO0 and it is automatically selected by the adaptive algorithm), there are here two crucial parameters, namely p and K . We have already pointed out that pP0 leads the adaptive algorithm to be equivalent to the Pyragas' method. On the other hand, a too big p may cause stability problems in Eq. (57). In order to have a reasonable K dynamics, avoiding undesirable saturation e!ects of the function tanh, one immediately see that p should be selected as to be su$ciently smaller than the maximum j, in order that the product pj(t ) lies within the linear region of the function tanh for all times t . L L
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Therefore, a preliminary observation of the unperturbed dynamics may be in order so as to have statistics of the typical j values occurring in the unperturbed dynamics, leading to the natural choice of the sensitivity parameter. As for K , while there are no upper limit in the choice, a lower limit should be taken into account. Indeed, Eq. (57) allows variations of K(t) in the range from 0 (when j"#R) to 2K (when j"!R). Therefore one should assure that this range overlaps conveniently the controllability range, through a judicious choice of K . Even though one could then be tempted to take a very large K , this can result in an undesirably large initial perturbation, leading to practical problems in the implementation of the method. In practice, the optimal K for the adaptive technique can be always easily individuated. A further remark is in order. Eq. (57) can be rewritten as K(t ),1/q(t ) L L (58) q(t )"q (1!tanh(pj(t ))) . L L Since the perturbation is additive on the right-hand side of the model equation, q(t ) and q must L be time intervals. The original adaptive method (Ref. [64]) was indeed introduced for the recognition of chaos, and it consisted in selecting the appropriate time interval in order to minimize the second variations of the dynamics. In that case, the evolution equation for q(t ) was L q(t )"q(t ) (1!tanh(pj(t ))) , (59) L L\ L that is, substituting in Eq. (58) a "xed reference time interval q with a recursive search in the time intervals. The obtained irregular sequence of time intervals was used to extract suitable chaotic indicators, such as the periods of the UPOs [3] and a discriminator between chaos and stochasticity [67]. It is important to remark that the recursive formula implies a direct choice of the optimal q in Eq. (58), thus of the optimal K in Eq. (57), while the selection of a "xed reference time interval q implies an overconstraints to the search for the optimal perturbation. However, the former mechanism can be only applied when the switch on of the process is done close to the UPO to be controlled. In practice, this would imply a preliminary learning task for an initial targeting procedure. We will come back to this point in the following of the present Report. When, instead, one wants to switch on the control on an arbitrarily selected region of the attractor, then the latter strategy would be preferable, insofar as the former one can have limitations due to initial large #uctuation of the q which can eventually drive the q sequence away L from the controllability range. This, of course, can be cured by a reliable choice of the sensitivity, but the whole process may result quite complicate to implement. This is the main reason for which we will focus on the latter strategy, which, though being less complete, has the merit of being easily implementable. 3.3. Application to high-dimensional systems While applications to low-dimensional dynamics are straightforward, and we here address the reader to the available literature contained in Ref. [3], in the following we provide an example of the reliability of the above algorithm when used for the stabilization of periodic motions in
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high-dimensional systems. The natural framework for this is the application to delayed dynamical systems (DS), which was provided in Ref. [68], and represents an intermediate stage between the control of concentrated systems (CS) and that of space-extended systems (ES). DS are systems ruled by y "F(y, y ) , (60) where y"y(t) is a m dimensional real vector, dot denotes temporal derivative, F is a nonlinear function, and y ,y(t!¹), ¹ being a time delay. Experimental evidence of the analogy between DS and ES was provided for a CO laser with delayed feedback [69] and supported by a theoretical model [70]. Most of the statistical indicators for DS, such as the fractal dimensions, are extensive parameters proportional to ¹, which thus plays a role analogous to the size for the extended case [71]. The conversion from the DS to ES is based on a two variable time representation, de"ned by t"p#h¹ ,
(61)
where 04p4¹ is a continuous space-like variable and the integer h plays the role of a discrete temporal variable [69]. By such a representation the long-range interactions introduced by the delay are reinterpreted as short-range interactions along the h direction, since now y ,y(p, h!1). In this framework, the formation and propagation of space}time structures, as defects and/or spatiotemporal intermittency can be identi"ed [69,70]. When ¹ is su$ciently larger than the oscillating period of the system, the behavior of a delayed system is analogous to that of a one-dimensional extended system. In particular, it may display phase defects, i.e. points where the phase suddenly changes its value and the amplitude goes to zero. For sake of exempli"cation, let us make reference to the following particular delayed dynamics: AQ "eA#b A(t!¹)A#b A(t!¹)A , (62) k (63) e "k S! e!kA . k Here, all quantities are real. A is an order parameter, e is a time-dependent linear gain, b , b , k , k are suitable "xed parameters, k is a measure of the ratio between the characteristic time scales for A and e, and S is a measure of an external pumping to the system. Eq. (63) are rather general. For instance, when ¹"0, S(0, b '0, b (0, k '0, k '0, k'0, they model an excitable system, producing the so called Leontovitch bifurcation, evidence of which has been shown experimentally on a CO laser with intracavity saturable absorber [72]. For ¹O0, they are similar to the models already used to describe self-sustained oscillations of con"ned jets [73], or memory induced low frequency oscillations in closed convection boxes [74], or even the pulsed dynamics of a fountain [75]. It should be here remarked that control of Eq. (63) with the OGY method can be complicated because of the extreme high dimensionality of this particular situation, requiring a very high dimensional embedding space wherein reconstructing the local stability properties of the UPOs. This is therefore the typical case in which alternative approaches may provide a less costly strategy for the stabilization of UPOs. Eq. (63) has been found also a good model for the temperature evolution in a well controlled time-dependent convection experiment [76], done with a cylindrical layer of silicon oil heated from
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Fig. 3. (a) Cross-section of the experimental setup. A hot drop (thermal) is dragged by the #ow and then reinjected into the heating region after having completed a round trip of the cell in a mean time ¹. P indicates the point where temperature is measured. (b) Experimental time behavior of the temperature at the point P. Vertical axis reports the temperature in arbitrary units, horizontal axis reports the time in seconds (¹"330 s). (c) Expanded view of the signal within the arrows which exhibits a phase jump (solid line) and reference signal translated by ¹ (dashed line).
below by a square heater limited to the central part of the container (see Fig. 3a for a scheme of the system). The heater is surrounded by the same insulating material of the vessel. A convective instability driven simultaneously by buoyancy and temperature-dependent surface tension, called BeH nard}Marangoni convection, grows as the heating is increased. A steady state is reached and a stationary pattern composed of four convective cells appears in the hot region. Additional details on this experiment can be found in Ref. [76]. By further increasing the heating, a time-dependent regime arises consisting in spatio-temporal modulations, or thermals, generated at the bottom boundary layer and then dragged by the #ow along the cell as can be seen in Fig. 3a. This con"guration provides a natural delayed interaction insofar as it reiterates at each position the local value of the order parameter after a delay ¹, corresponding to the time lag necessary for the trip of the cell. In this situation, an experimental measurement of the temperature at the point P of Fig. 3a yields the data of Fig. 3b. The vertical axis (temperature) can be taken as representative of the order parameter A. The main feature of this experiment consists of trains of modulated oscillations, interrupted by localized events (phase defects), wherein the phase of the signal changes suddenly and the amplitude decreases to zero. Therefore, the relaxation oscillations are represented by the normal form of a Hopf bifurcation (Eq. (63)), wherein the saturating terms are delayed to account for the transport of the convective cell. Eq. (63) represents the slow evolution (k (1) of the control parameter e, which is enhanced by
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the external pump S and depressed by the convective motion (!kA) which would naturally uniformize top and bottom temperatures. Let us now see how phase defects emerge in Eq. (63). If one adjusts pump and delay parameters (S and ¹) so that the system enters the chaotic region, one realizes that, in fact, chaos appears in the system in two di!erent forms. For low ¹ values, chaos is due to a local chaotic evolution of the phase, whereas no appreciable amplitude #uctuations are observed. This regime was called phase turbulence (PT). By increasing ¹, one observes a transition toward amplitude turbulence (AT), wherein the dynamics is dominated by the amplitude #uctuations, and a large number of defects is present. Both PT and AT have counterparts in a one-dimensional complex Ginzburg}Landau equation. Here the parameter space shows a transition from a regime of stable plane waves toward PT (Benjamin}Fair instability), followed by another transition to AT with evidence of space}time defects [77]. We now apply the adaptive method to this particular case. Here, the algorithm can be conveniently reduced to an easily workable form. The application of the method can be summarized by using Eqs. (54)}(57) and substituting x (t) with A(t) and g (t) with A(t!¹ ) (¹ being & & the Hopf period). The perturbation ;(t) is then applied as additive term to the right-hand side of the "rst of Eq. (63). A nice approximation holds in this case. Let us refer to Eq. (58), and let 1q2 denote the average of the +q , set, then the second of Eq. (58) can be written as L q K1q2(1!pj ) (64) L L where (i) q has been replaced with its ensemble average, and (ii) the tgh function has been L linearized. Point (i) corresponds to "xing once forever a reference time scale for the process under study, while point (ii) corresponds to selecting a conveniently small g to keep gj always within L> the linear region of the tgh function. In the same way, the equation for j can also be linearized as AQ (t)!AQ (t!¹ ) & . j(t)K A(t)!A(t!¹ ) & Combining Eqs. (64) and (65) into Eq. (57), this reduces to
(65)
;(t)"K (A(t!¹ )!A(t))#K (AQ (t!¹ )!AQ (t)) (66) & & with K "1 and K "g/1q2. The consequences of this approximation are interesting. First of all, it is now evident that the case K "0 (p"0) recovers the Pyragas control method [2]. However, in the present case, K and K can be independently selected, and this introduces an extra degree of freedom with respect to Ref. [2]. Now, the control is more active when the error is increasing and vice versa, so reducing oscillations. Indeed, Eq. (66) performs as a proportional derivative controller, the more usual action for stabilizing feedback linear systems, due to its e!ect which consists in increasing the phase of the compensated system in a suitable frequency band. This approximation constitutes the order-one approximation of the adaptive method, and it is very easy to implement, consisting in a double feedback line, one in the observable variable, and one in its derivative. We should point out, however, that not always this approximation leads to positive results, and in some cases (as for example the control of patterns in extended media), one should instead apply the whole adaptive strategy, without any approximation. It is straightforward
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Fig. 4. Space (p)}time (h) representation of the controlling process for Eqs. (63). b "1, b "!1/16, k "0.8, k "1, k"11, S"5.5, ¹ "1.95. (a) ¹"15, PT regime. The chaotic dynamics results in a local turbulent phase of the Hopf & oscillation which is corrected by the controlling algorithm. K "K "0.2. Arrow indicates the instant at which control is switched on. (b) ¹"50, AT regime. The dynamics is dominated by amplitude #uctuations, with the presence of defects. The algorithm (K "K "0.2) suppresses the defects and restores the regular oscillation. Arrow indicates the instant at which control is switched on.
Fig. 5. ¹"50, AT with 10% noise (a) and 20% noise (b). Control with K "K "0.2. Same stipulations and parameters as in the caption of Fig. 4. Arrows indicate the instant at which control is switched on.
to understand that the range of validity of such approximation is having small j values, which implies to be su$ciently close to the UPO to be controlled so as linear properties are valid. In Fig. 4 we report the application of our method to Eq. (63). The desired oscillation, which in the space}time representation gives rise to a roll set, is controlled in PT (Fig. 4a) and in AT (Fig. 4b). Finally, let us discuss the robustness of our procedure against external noise. For this purpose, we add white noise to the measured A values before the onset of the adaptive feedback control. Notice that the noise does not act additively, insofar as it is reinjected into the nonlinear equations through the control feedback, hence a!ecting dynamically the evolution of the system. A relevant result is that our method is robust against large amounts of noise. In Fig. 5 the control is achieved within AT for 10% noise (Fig. 5a) and for 20% noise (Fig. 5b). The controlled UPO is slightly distorted by the action of the noise fed back into the system.
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4. The problem of targeting 4.1. Targeting and controlling fractal basin boundaries 4.1.1. Introduction It is common for nonlinear dynamical systems to exhibit multiple coexisting attractors, each with its own basin of attraction [78}83]. The basin of attraction of an attractor is the set of initial conditions in the phase space that asymptote to the attractor. In practical applications, when one of the attractors according to some criteria would yield superior systems performance over the others, it is important to be able to drive most trajectories to the desirable attractor in an e$cient and economic way. That is, one wishes to drive trajectories to the desirable attractor rapidly by using only small feedback control to an accessible parameter or state of the system. Previous work has demonstrated that in periodically driven dynamical systems, multiple basins of attraction can be eliminated by replacing the periodic driving by some appropriately chosen, but somewhat largeamplitude chaotic driving [84]. We review a method to drive most trajectories to a desirable attractor by using only small feedback control [85]. We emphasize the need to use small feedback control [1,23] since: (1) we do not wish to alter the system substantially; and (2) large perturbations to the system may be costly. As such, it is only possible to alter the fate of the trajectories resulting from initial conditions in the vicinity of basin boundaries because, for a trajectory deep in the basin of an undesirable attractor, small perturbations cannot change the attractor to which the trajectory is asymptoting. In this regard, it is necessary to distinguish between smooth and fractal basin boundaries [78}83]. Imagine there is an N-dimensional chaotic system. Consider a phase space region that contains part of the basin boundary. Assume that only small perturbations of magnitude e (e;1) to an accessible system parameter or state are allowed. If the boundary is smooth, the dimension of the boundary is D"N!1. Thus, the fraction of trajectories whose asymptotic attractors can be altered by small e perturbation is on the order of magnitude of e,\""e, which is also very small. If, on the other hand, the basin boundary is fractal with box-counting dimension (capacity) D, where D is a fractional number that satis"es (N!1)(D(N, the fraction of trajectories whose fate can be manipulated using small perturbation is f (e)&e?, where a"N!D(1 is the uncertainty exponent. Thus, if a;1, f (e) can be large. Fractal basin boundaries with a(1 are common in dynamical systems [78}83], and a;1 are particularly common in high-dimensional systems [87}89] or in systems with riddled basins [90}96]. Therefore, although the presence of fractal basin boundaries with a(1 poses a fundamental di$culty to predict the asymptotic attractor of the system because of the inevitable error in the speci"cation of initial conditions or system parameters, these boundaries o!er a possibility for us to greatly increase the probability that
The relation between the uncertainty exponent and the box-counting dimension, a"N!D, was rigorously proven for Axiom-A systems [86]. It was conjectured that the same relation holds for more general dynamical systems [78,79]. It should be noted that a riddled basin has dimension equal to N and its basin boundary is the basin itself. Thus, the boundary also has dimension equal to N. However, its uncertainty exponent is not equal to zero, although in most cases it is close to zero. Therefore, strictly speaking, the relation between the basin boundary dimension and the uncertainty exponent D"N!a does not hold for riddled basins.
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typical trajectories can be driven to the desirable attractor by using arbitrarily small perturbations, provided that we are able to harness the system in an intelligent way. It was demonstrated in Ref. [85] that this is indeed possible. In the sequel, we discuss method of control and present numerical examples for a system with fractal basin boundaries and a system with riddled basins. 4.1.2. Method of control The setting of the problem is as follows. Let the dynamical system be described by an N-dimensional #ow dx/dt"F(x, p) or an N-dimensional map x "M(x , p), where p is an L> L accessible system parameter. For concreteness, assume there are two distinct attractors for the range of system parameter values of interest. Furthermore, assume that the coexistence of the two attractors is structurally stable, i.e., small change in the parameter changes the behavior of attractors and their basin structures only slightly. Denote the two distinct attractors by A and B. For a given region R in the phase space that contains part of the basin boundary, a fraction of initial conditions f will yield trajectories that asymptote to attractor A, and the remaining initial conditions, a fraction of f ,(1!f ), asymptote to attractor B. Without loss of generality, assume that f and f are on the same order of magnitude. Suppose that one of the two attractors yields much superior system performance than the other. We thus wish to increase f as much as possible so that most initial conditions asymptote to the attractor with better system performance. This will not occur if no external perturbations to the system are applied. Our goal is to devise an algorithm to increase substantially the fraction of initial conditions that asymptote to the desirable attractor, given that p can be adjusted "nely around a nominal value p : p3[p #*p, p !*p], where *p/p ;1. The idea is to build a hierarchy or `tree-likea structure of paths to the desirable attractor [36,85]. Speci"cally, let A be the desirable attractor. We "rst randomly choose an initial condition in R such that it generates a trajectory to A. Call this trajectory the `roota path 1 to A and denote it by X , X ,2, X , where X is a point on A (or a point in the vicinity of A). We then choose a second trajectory to A from an arbitrary initial condition Y in R. But for this second path, we examine if it approaches to A directly without coming close to root path 1, in which case we call it root path 2. It is also likely that a point on this trajectory Y call fall into a suitably small neighborhood of some L point along root path 1 before it comes close to A. In this case, we store Y together with the path of L (n!1) points leading to Y . We call Y , Y ,2, Y the secondary path of the root path 1. This L L procedure can be repeated for initial conditions chosen on a uniform grid of size d in R. Of course, if a trajectory goes to an undesirable attractor, we simply disregard this trajectory in the tree-building process. Finally, with suitably chosen d, a hierarchy of paths to A in R can be built with, say, N root paths. On each root path i, there can be some secondary paths, and on each secondary 0 path there can be third-order paths, etc. We therefore obtain a tree of paths to A in a region that contains the basin boundary, as schematically shown in Fig. 6. In fact, since there can be many root paths, this is more like a `busha of paths leading to the desirable attractor. A remaining question is This bush-like structure of paths is somewhat di!erent from the tree structure of paths used in the targeting procedure in Ref. [36]. In that case, there is only one root path to the target point on the attractor, and all paths to the target live on the same chaotic attractor.
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Fig. 6. From Ref. [85]. A schematic illustration of the hierarchy of paths (bush) to the desirable attractor in the phase space.
how "ne the grid from which initial conditions are chosen should be. Clearly, the size of the grid d should be comparable to the magnitude of the allowed parameter perturbation *p, which is approximately the size of controlling neighborhood around each point on the bush of paths. If d<*p, most trajectories that originally go to the undesirable attractors will not come close to bush of paths and therefore will not be controlled. If d;*p, the bush of paths may have contained too many details and therefore may have used too much computer memory that is unnecessary for realizing the control. To control a trajectory to direct it to the desirable attractor after it comes close to a path on the bush, we employ a simple feedback scheme. For simplicity we consider the N-dimensional map x "M(x , p). Suppose a trajectory originated from a random initial condition x falls into an L> L e-neighborhood of a point y on the bush at some later time n, i.e., "x !y "4e. Let y , y , ,y L L L L L> 2 be the path on the bush that starts at y and ends at y which is in the e-neighborhood of the L desirable attractor. In the vicinity of y , we have the following linearized dynamics: L *x "DM(x , p)* x #(RM/Rp)*p , where * x "x !y , *p "p !p , and the Jacobian L> L L L L L L L L matrix DM(x , p) and the vector RM/Rp are evaluated at x "y and p "p . Choosing a unit L L L L vector u in the phase space and letting u ) * x "0, we obtain for the required parameter L> perturbation: !u ) DM(x , p) ) * x L L . *p " L u ) (RM/Rp)
(67)
In principle, the unit vector u can be chosen arbitrarily provided that: (i) it is not orthogonal to x ; and (ii) the denominator in Eq. (67) is not close to zero. In practice, we de"ne a maximum L> allowed magnitude for the parameter perturbation *p &e. If the computed "*p " exceeds *p ,
L
we set dp "0. Doing this would cause lost of control occasionally. But it was found in numerical L
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Fig. 7. From Ref. [85]. Basins of attraction for Eq. (68). Black and blank regions are the basins of the A and > A attractors, respectively. Parameter setting is: a"1.32, b"0.9, and J "0.3. \
experiments that robust control can still be achieved since setting *p "0 is done only rarely [85]. L Because * x is small, *p is also small. In the sequel, we present two numerical examples to L L illustrate the control method. 4.1.3. Example 1: controlling fractal basin boundaries We consider the following two-dimensional map [78,79]: h
"h #a sin(2h )!b sin(4h )!x sin(h ) , L> L L L L L (68) x "!J cos(h ) , L> L where x can be regarded as the radial distance from the center of an annulus, h is an angle variable so that h and h#2p are equivalent, and a, b and J are parameters. The system is invariant under the symmetry hP2p!h. The determinant of the Jacobian matrix is J sinh(1 (for J (1). At the following parameter setting, a"1.32, b"0.9, J "0.3, there are two attractors, located at x"!0.3, h"0 (denoted by A ) and x"0.3, h"p (denoted by A ), respectively. The bound\ > aries between basins of the two attractors are fractal, as shown in Fig. 7, where black dots represent the basin of the A attractor. The dimension of the basin boundary is approximately 1.8, > corresponding to an uncertainty exponent of a+0.2 [78,79]. Now assume that the attractor A corresponds to a better system performance so that it is the > desirable attractor. Without control, the fraction of initial conditions that asymptote to A is > about 50% for the phase-space region in Fig. 7. Assume a is an accessible parameter which can be perturbed slightly around its nominal value a "1.32. We "rst build a bush of paths to A by > using a grid of 100;100 initial conditions in the region (04h4p, !0.54x40.5) (corresponding to grid size d+3.3;10\). We arbitrarily choose u"(1/(2)(1, 1) to compute the parameter perturbation *a from Eq. (67). Fig. 8a shows a controlled trajectory (solid line) to the desirable L attractor A , where both the size of the controlling neighborhood e and the maximal allowed > parameter perturbation *p are set to be 10\. The trajectory would asymptote to the un desirable attractor without control, as shown by the dotted line in Fig. 8a. When control is applied
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Fig. 8. From Ref. [85]. (a) A controlled trajectory to the desirable attractor A , where the size of the controlling > neighborhood e and the maximum allowed parameter perturbation *p are both 10\. The trajectory would
asymptote to the undesirable attractor A without control. (b) At d&10\, the fraction f > of initial conditions that \ asymptote to A versus log e. (c) The three-dimensional plot of f > versus log d and log e. We see that the optimal > e value for which f > reaches maximum is about 10\. (d) The maximum value f > versus log d.
using these values of e and p , about 70% of the initial conditions in the region
(04h4p, !0.54x40.5) asymptote to A , increased by 20% as compared with the case > without control. Let d be the size of the covering when the bush is built. Clearly, the fraction of initial conditions that asymptote to A depends on both d and the size of the control> ling neighborhood e. Fig. 8b shows, with the same bush of paths to A as in Fig. 8a, f > > versus e (*p "e) for d "xed at about 3.3;10\ and 10\4e410\. In the "gure, for each
value of e, N "90 000 (300;300) initial conditions uniformly distributed in the region (04h4p, !0.54x40.5) are chosen and the number of controllable initial conditions N , i.e., > those asymptote to A via control, are recorded. The fraction f > is approximated by N /N . > > Since the grid size for building the bush is d&10\, we see that when e;d, essentially no improvement in f > is achieved because it is unlikely for initial conditions originally asymptoting to A to fall in the vicinity of points along the bush. When e&10\, maximum increase in f > is \ achieved because in this case, it is easy for trajectories to come close to the bush and to be
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controlled. However, if e'd, control may be lost for some initial conditions because the linearized dynamics used to derive the parameter perturbation Eq. (67) no longer holds at large values of e, although in this case it is easy for trajectories to fall in the e-neighborhood of the bush. Thus, we see that f > starts to decrease as e increases about 10\. To more clearly see the dependence of f > on both d and e, we compute f > for systematically chosen d and e values. Fig. 8c shows the threedimensional plot of f > versus d and e for 10\(d(10\ and 10\(e(10\. Because of the two-dimensional phase space region used to construct the bush, decreasing d to values below 10\ leads to huge number of points on the bush and thus to numerical di$cults. Nonetheless, it is clear from Fig. 8c that for the range of d values chosen, maximum improvement in f > occurs at e&10\. To understand why the optimal improvement in f > occurs at e&10\, we note that the maximum value f > of f >, as e changes, depends on d. Generally, f > is small if d is too large
because in this case, the basin boundaries are not adequately covered. As d decreases, f > increases.
But if d becomes so small that the entire basin boundaries are covered by the bush, decreasing d further does not help to increase f > . Thus, f > saturates as d decreases through a critical value.
This behavior is shown in Fig. 8d, where we see that f > saturates at d+10\ . The saturated
value of f > is about 0.706. At this d value, there are approximately N "17 300 points on the
A bush. These are the required points to cover the basin boundaries adequately. We ask, how many of these points can be in#uenced by perturbations of magnitude e? The answer is &e?N because the A fraction of basin boundary points that are uncertain with respect to perturbation e scales like e?. Since these N points on the bush provides a good covering of the basin boundaries, we have A e? N e &1, which gives e &N\ >?+1.2;10\. This agrees with the numerical A A observation in Figs. 8b and c. 4.1.4. Example 2: controlling riddled basins We "rst brie#y review the concept of riddled basins. Riddled basins usually occur in dynamical systems with a simple type of symmetry. The existence of symmetry often leads to invariant subspace in the phase space. The description of riddled basins was introduced in Ref. [90] where it was shown that for certain class of dynamical systems with an invariant subspace: (i) if there is a chaotic attractor in the invariant subspace; (ii) if there is another attractor in the phase space; and (iii) if the Lyapunov exponent transverse to the subspace is negative, then the basin of the chaotic attractor in the invariant subspace can be riddled with holes belonging to the basin of the other attractor. That is, for every initial condition that asymptotes to the chaotic attractor in the invariant subspace, there are initial conditions arbitrarily nearby that asymptote to the other attractor. Rigorous results on the dynamics of riddled basins for discrete maps were presented in Refs. [90,91]. The dynamics of riddled basins was subsequently investigated in [92] using a more realistic physical model. A more extreme type of basin structure referred to as `intermingled basinsa in which the basins of more than one chaotic attractors are riddled, was also studied using both discrete maps [90] and a more realistic physical system [93]. Riddled basins have been veri"ed in experiments conducted using coupled electrical oscillators [94,95]. The mechanism for riddling to occur, and the basin structure associated with the riddling, were investigated by Ashwin et al. [95,96]. We consider the following two-dimensional map [85]: x "g(x )#by , L> L L y "ax y #y , L> L L L
(69)
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Fig. 9. From Ref. [85]. (a) Riddled basin of the y"0 attractor for Eq. (69). The parameter setting is r"3.8, a"1.7 and b"0.1. (b) For d"10\, f > versus log e. (c) The three-dimensional plot of f > versus log d and log e. (d) The maximum value f > versus log d.
where g(x) is a chaotic map, b and a are parameters. The invariant subspace is the one-dimensional line de"ned by y"0 since if y "0, then y "0 for n51. For simplicity we choose g(x) to be L the logistic map g(x)"rx(1!x) with a chaotic attractor. The transverse Lyapunov exponent is given by
Ry 1 * ln L> " axo(x) dx , K " lim , Ry ¸ L WL * L
(70)
where o(x) is the invariant density of the chaotic attractor produced by the logistic map. We choose r"3.8 and obtain a +1.725 where K 50 for a5a and K (0 for a(a . For a(a , there A , A , A A are two attractors, one is y"0 and another is "y""R. The basin of the y"0 attractor is riddled.
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Fig. 9a shows part of the basin of the y"0 attractor (black dots) for a"1.7(a and b"0.1, A where a grid of 600;600 initial conditions is chosen in the region 04x41 and 0(y40.2. Examination of the "gure on "ner and "ner scales reveals that there are white regions (basin of the y"R attractor) near every black dots, a typical feature of riddling. Now assume that the y"0 chaotic attractor is the desirable attractor and the y"R attractor is the undesirable one. To facilitate numerical computation, we choose to control trajectories starting from initial conditions on a line, say, y"0.1. Without control, about 56% of the initial conditions on this line go to the desirable attractor. We build a bush of trajectories starting from y"0.1 with size d, i.e., we use d\ points on y"0.1 to determine the points that asymptote to the desirable attractor. A di$culty here is that it typically takes many iterations for a trajectory starting at y"0.1 to reach the desirable attractor (numerically a trajectory is regarded as having y"0 if it says within 10\ of y"0 for certain prescribed number of iterations). Thus, for small grid size the number of points on the bush can be very large. It then becomes computationally di$cult to determine whether a trajectory point is close to the bush. To make the computation feasible, we adopt the following strategy. For a random initial condition chosen from the line at y"0.1, we examine whether it falls in an e-neighborhood of a starting point of a path on the bush. If yes, we control it. Otherwise we let it evolve without control. Assume r in the logistic map is the accessible parameter to be perturbed. The parameter perturbations can be computed from Eq. (67). Fig. 9b shows f > versus log e for d"10\, where f > is the fraction of initial conditions that asymptote to the desirable attractor. The plot exhibits similar feature to that of Fig. 8b. We see that no improvement in f > is achieved if e is too small because there are almost no points that come close to the bush. If e is too large, although many trajectories would fall in the e-neighborhood of the bush, control can get lost because Eq. (67) is only a linear control law. The optimal e value for which f > reaches maximum is about 10\ , and the maximum possible value of f > is about 0.86, a substantial improvement in f > compared with the case of no control. Fig. 9c shows the three-dimensional plot of f > versus d and e. We see that for d(10\, f > reaches maximum at e +10\ . Fig. 9d shows f > versus log d. For 10\(d(10\, f > is about the same as
if there were no control. It then increases rapidly as d is decreased from 10\ and starts to increase slowly as d decreases through 10\. Compared with the example of controlling fractal basin boundaries, we see that the maximum value of f > can be higher. This is due to the feature of
riddled basins where the uncertainty exponent a is close to zero. The reason that we obtain e +10\ can be understood by noting that for d+10\, there are about 5560 points on y"0.1 that belong to the bush. Thus we have e>? N&1. Since a+0, this gives e &10\ . 4.1.5. Discussions We have reviewed an algorithm [85] to drive trajectories to a desirable attractor by using small feedback control for dynamical systems with multiple coexisting attractors. The basic idea is to build a bush-like structure of paths to the target attractor and to stabilize a trajectory around one of the many paths on the bush so that the trajectory asymptotes to the desirable attractor. Such a structure of paths, in principle, can be built up even in more realistic applications. For instance, an experimentalist could run the system "rst, measure time series resulting from many initial conditions, and build the bush of paths to the desirable attractor in the reconstructed phase space by using the delay-coordinate embedding technique. One can then use techniques such as the direct proportional feedback control [97,98] to compute the required parameter perturbations as it may
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be a formidable task to apply Eq. (67) in practice. But of course, at present there is no guarantee that the method can be applied to practical applications. We stress, however, that from a more general theoretical point of view, the success of the method relies on the region in the phase space to which the bush extends. As such, the method is particularly e!ective when there are fractal basin boundaries with large values of fractal dimension (or small values of the uncertainty exponent a) in the phase space region of interest. In contrast, there is no uncertainty exponent a) in the phase space region of interest. In contrast, there is no appreciable increase in the probability for a trajectory to be driven to the desirable attractor if the basin boundaries are smooth. One could, therefore, deliberately build into the system fractal basin boundaries or riddled basins in order to drive most initial conditions to the desirable attractor. While there is a great uncertainty in determining the asymptotic attractor for individual initial conditions when there are fractal basin boundaries or riddled basins, the uncertainty is greatly reduced for a path that consists of a large number of points in the phase space. Therefore, insofar as a trajectory can be stabilized around a path on the bush, the fate of the trajectory is almost certain, i.e., the desirable attractor. Theoretically, there is no reason for restricting the control to a bush. For instance, one may obtain more optimal results in the following way. Assume that for a map M we have basin of attraction D for a desirable attractor A. For perturbation of magnitude e, let B (D) be the union of C all e-balls around all points in D (the e-parallel body of D). Consider the union of all the preimages of the B (D). This union would give the largest possible domain (always an open set) for which there C exist e-pseudo orbits of the map that eventually asymptote to the desirable attractor A. If a practical method can be devised to cover this union and to drive trajectories in this union to A, we would expect to achieve an absolute maximum size for the basin of attraction of A under arbitrarily small perturbations. At present, how to cover such a union and how to devise a control algorithm to achieve this theoretical maximum remain unknown. The central problem in controlling dynamical systems with multiple coexisting attractors is how to maximize the probability of being able to control an arbitrary initial condition. Thus, it is important to assess how this probability varies with the maximum allowed perturbation and the dimension of the basin boundaries. To obtain this information, we now imagine an `ideala controller. We restrict to situations where only the initial conditions near basin boundaries are accessible to control. For a given initial condition, the ideal controller would evolve the system to see if the asymptotic attractor is the desirable one. If not, small parameter perturbation e is applied and the system is evolved from the same initial condition. The controller would then check if the initial condition yields the desirable attractor. It could repeat this procedure for a given number of time, insofar as the asymptotic attractor is not the desirable one. In this case, the probability for driving an arbitrary initial condition to the desirable attractor is proportional to the fraction of uncertain initial conditions, which scales with the perturbation as e?. Thus, we see that for "xed a, increasing e would increase the desired probability. For "xed e(1, increasing the dimension of the basin boundary, which is equivalent to decreasing the uncertainty exponent a, would increase the desired probability. This, of course, holds only for the ideal controller. In more practical situation, we see that there exists an optimal e value for achieving the desired probability (see Figs. 8 and 9). This optimal e value depends on many factors including the dimension of the basin boundaries. Nonetheless, high desired probability can be achieved if the dimension of the basin boundaries is large (or a is small). In cases where the basin-boundary dimension is close to the phase-space
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dimension (or a is close to zero, such as in riddled basins), one expects to achieve higher desired probability as in such a case, most initial conditions are near the basin boundaries. Finally, we emphasize that the control method reviewed here represents only one possible approach to solve the general problem of controlling dynamical systems with multiple basins of attraction. There will undoubtedly be better methods that await for future investigation. 4.2. The adaptive targeting of chaos Targeting of chaos means judiciously perturb a chaotic system with the aim of directing the orbit emerging from a given point to a neighborhood of some other prespeci"ed point (called target) on the attractor within a "nite and speci"ed time (called target time). As already mentioned in the Introduction, even though ergodicity assures that all point on the attractor are shadowed regardless on the initial conditions chosen for the chaotic evolution, in many cases a small neighborhood of a given attractor point may be visited infrequently; thus, the unperturbed dynamics may take a long time to approach a given target. Thence, the necessity of implementing e$cient targeting methods, which can reduce strongly the waiting time [35,36,99}101]. The targeting procedure may be seen as a preliminary task for chaos control, because, as we have already pointed out, the control algorithms (see, e.g. [1}3]) use linearizations of the dynamics that are valid only in a rather small neighborhood of the desired saddle point, and therefore need the system to target such a small neighborhood before the switch on. The "rst targeting method was introduced by Shinbrot et al. [35], who have suggested to use the exponential sensitivity of a chaotic process to tiny perturbations in some accessible control parameter. This technique was successfully applied to one-dimensional mappings both theoretically [99] and experimentally [100] and then extended to three-dimensional chaotic #ows [101]. Later, Kostelich et al. [36] faced the problem of targeting hyperchaos, that is extending the above procedure to cases where there is more than one positive Lyapunov exponent associated with typical orbits on the attractor. (See Ref. [102] for a review of these procedures.) The basic algorithm in [36] applies tiny perturbations for performing two successive changes of a control parameter (or one change of two parameters). The perturbations are selected to move the image of the initial condition onto the stable manifold of the target. The robustness of this method against the presence of a small amount of noise or a small modeling error has been proved, and further developments have pointed out how it can help in switching between controlled unstable periodic orbits even in higher-dimensional chaotic situations [103]. However, there are two main limitations for the application of such a technique, namely: (1) the above method is only applicable to invertible mappings, and (2) it needs full a priori information on the stable and unstable manifolds of the target. The latter requirement can give rise to serious drawbacks in all cases in which the target corresponds to an attractor point whose neighborhood is rarely visited by the natural evolution of the system, insofar as one needs a long data acquisition time to obtain points whose orbits closely visit the target. Alternative methods have been proposed [104] to increase the number of visits to a target by making small perturbations of the state variables of the system. The application of the adaptive technique for the targeting of chaos was realized in Ref. [66], either for the case in which all state variables are accessible for detections and perturbations, and
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for the case in which the operator may rely on a single state variable, whereas the others are hidden, and not accessible for measurements and perturbations. We summarize herebelow the main results of Ref. [66]. We start from the general problem of considering a chaotic process ruled by x "f (x, k) ,
(71)
where x is a D-dimensional vector (D53), f is a nonlinear function of x, dot denotes temporal derivative, and k is a vector of parameters. The targeting strategy consists of two distinct parts: (1) an algorithm that slaves the chaotic dynamics x(t) to a given goal dynamics u(t) with the only use of small perturbations, and (2) an algorithm detecting and constructing a goal dynamics u (t) that 2 brings the trajectory to a small neighborhood of the target within the desired target time starting from a given initial condition u (0)"u in a way compatible with the unperturbed evolution of the 2 system. The point u is chosen so as to lie on the attractor, and typically its neighborhood is visited frequently by the unperturbed dynamics. While for point (1) one can naturally rely on the existent slaving techniques (and here we will use the adaptive methods of Eqs. (54)}(57)), point (2) has found a solution in Ref. [66]. The main improvement o!ered by Ref. [66] with respect to other targeting techniques, is that there the extraction of the goal dynamics is done with a single visit of the target. This way, one minimizes the learning time, in all cases in which the target is rarely visited by the unperturbed dynamics (which are, indeed, the most interesting cases, since when the target is frequently visited by the natural dynamics, the whole targeting procedure would not be needed). Let us then consider the following two cases: (i) the system allows detection and perturbation of all its state variables; and (ii) only a single state variable is available for observation. Let us start with case (i) and discuss the problem of constructing the goal dynamics u (t) from 2 a preliminary observation of the unperturbed behavior of the system. Following Ref. [66], we will describe the targeting procedure with reference to the threedimensional RoK ssler system [105] x "!z!y,
y "x#ay,
z "b#z(x!c) ,
(72)
with a"b"0.2 and c"5.7. One "rst constructs a partition of the three-dimensional phase space in parallelograms of sides e,(e , e , e ). For that purpose, one de"nes I(x , y , z , e)"+(x, y, z): x (x(x #e , y (y(y #e and z (z(z #e , . (73) By letting system (72) evolve from the initial condition x(0)"y(0)"1, z(0)"4, a chaotic set is realized, a portion of which is contained in the parallelogram I (4.655146,!6.691886, 0.013528, e), 2 where e,(0.205382, 0.186303, 0.228361). It is important to remark that this choice of e corresponds to a box whose sides are 10\ as long as the corresponding sides of the smallest parallelogram containing the attractor for t'9743.658203 (from now on t"t "9743.658203 will be the time at which the unperturbed 2 trajectory "rst enters I ). 2 By registering the natural evolution of the system from the above initial conditions up to the "rst visit to the target, and by using this portion of the natural trajectory, one constructs a web of paths compatible with the unperturbed dynamics, each of which connecting di!erent parallelograms to
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Fig. 10. Procedure for the construction of the goal dynamics u(t). Each bubble represents the neighborhood of a point in the phase space. I : target; P(I ): unique preimage of the target; P (P(I )), j"1, 2, 3: multiple preimages of P(I ); I : 2 2 H 2 2 $ most frequently visited neighborhood. The selected path is shown as a thick line.
Fig. 11. (a) An (x, y) projection of the unperturbed RoK ssler dynamics (dots) and path followed by the perturbed dynamics to reach the target (thick dashed line). The path is inside the chaotic attractor. (b) Zoom of (a): the path (solid line) moves from high probability regions of the attractor toward lower probability regions, until reaching I (indicated as Target in 2 the "gure). Initial conditions and control parameters as in the text.
I . Precisely, one follows few trajectories for a given observation interval. Suppose to call P(I ) the 2 2 preimage of I . Since the recorded trajectory visits the target only once, P(I ) is univocally 2 2 determined. One then records the portions of the observed trajectories that lie in I and P(I ) and 2 2 determines successive preimages of P(I ), which, in most cases, have been visited previously by the 2 portions of the observed trajectories. Going backward in time, one selects from the observations a path starting from the most frequently visited parallelogram I and leading to the box I . Fig. 10 schematically illustrates the $ 2 obtained web of paths. At the end of this process, one can make use of the observed path from I to $ I , as goal function for the application of the adaptive control procedure. Since the natural 2 measure of I is large (that is, it is frequently visited by the unperturbed dynamics), the target can be $ reached quickly regardless of the initial conditions. Fig. 11a reports the results of applying the adaptive method of Eqs. (54)}(57) to the system of Eqs. (72), using as goal dynamics the reconstructed path. I is reached by the unperturbed $ dynamics for the "rst time when t"30.9 s, and then I is reached by the perturbed dynamics only 2
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1.6 s later. Thus, the total waiting time required to reach I is reduced from 9743 to 32.5 s, implying 2 a total speedup of two orders of magnitude. Fig. 11b illustrates the mechanism that leads the system to the target: the trajectory followed by the perturbed evolution moves from high probability sections of the attractor toward lower and lower probability sections, up to the target. What described above requires that all state variables be accessible for measurements and perturbations. Therefore, its applicability can be seriously limited in experimental situations, where often only a single state variable of the system is accessible, whereas the others are in general hidden, or not accessible. It is convenient then to reformulate the adaptive targeting strategy in the more realistic case, in which measurements can be done only to one of the state variables of the RoK ssler system (say the x variable) and perturbations can be applied only to the "rst of Eqs. (72). In this case, one immediately realizes that the problem is to retrieve a suitable scalar goal dynamics g(t) from the observations compatible with the unperturbed evolution of the system and coming at least once within a suitable neighborhood of the target. To do this, one can make use of the time delay embedding technique [26], allowing to reconstruct the attractor from a time series of measurements of a single variable, say x(t), from Eqs. (72). The above problem was solved again in Ref. [66], by selecting a suitable delay time q, and considering the D-dimensional embedding space of the vectors x(t)"(x(t), x(t!q),2, x(t!(D!1)q)) (D"3 in the present case). One of the main feature of the embedding technique is to retain the basic metric properties of the original phase space description. In other words, this means that points that are neighbors in the original phase space with respect to a given metric M remain neighbors in the embedding space with respect to some new metric M . q"5.71157 0 # was chosen in Ref. [66], corresponding to the inverse frequency of the largest peak in the power spectrum of the signal x(t). The target point in the original phase space is now mapped into the point x "(x (t ), x (t !q), x (t !2q))"(4.727415, 4.295067, 4.929038). The idea is to retrieve 2 2 2 2 2 2 2 a scalar goal dynamics g(t) with the following properties: g(t )"x (t !2q), g(t #q)" 2 2 x (t !q), and g(t #2q)"x (t ). Here t is the instant at which the unperturbed x(t) "rst 2 2 2 2 satis"es x (t !2q)!e /2(x(t)(x (t !2q)#e /2 (e "0.205382). 2 2 2 2 By these requirements, one immediately realizes that the perturbations move the trajectory to the target within the target time t #2q, regardless on the particular initial conditions for the evolution of the dynamics. The simplest choice of the goal function would be the recorded unperturbed evolution of x from t !2q to t . But this choice is not the optimal one. Indeed, since in the present case the 2 2 observations are limited to a one-dimensional subspace, there is no certainty that at t"t the other hidden variables are within a su$ciently small distance from their values at t !2q. 2 The process could therefore result in an unacceptably large initial perturbation, and another choice of g would be necessary. For instance, the evolution of x(t) could be exploited more thoroughly by constructing two successive webs of 1-dimensional paths, the "rst connecting x (t !2q) to 2 2 x (t !q), and the second connecting x (t !q) to x (t ). 2 2 2 2 2 2 But here the selection of a goal dynamics is further complicated by the requirement that paths reach from a given point to another given point within a speci"ed time. This constraint motivates In most cases, and also in ours, M and M both coincide with the metric de"ning the Euclidean distance between 0 # points.
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Fig. 12. From Ref. [66]. (a) An (x, y) projection of the unperturbed RoK ssler dynamics (dots) and path followed by the perturbed dynamics to reach the target (thick dashed line). In this case the perturbation acts only on the x variable of Eqs. (72). Again, the path is inside the chaotic attractor; thus it is compatible with the natural evolution of the system and it goes from higher to lower probability regions. (b) Temporal evolution of the perturbation during the targeting process. The range spanned by ; is less than 1% of the range spanned by the x dynamics. Initial conditions and control parameters as in the text.
the choice of q in Ref. [66] as the reciprocal of the frequency of the main peak in the power spectrum. This time is, more or less equivalent to the return time of the system onto its PoincareH section. For su$ciently long observations, ergodicity of chaos assures a covering of all the states and guarantees the existence of at least one path from any initial state to any "nal state within observation interval. However, these concerns are largely obviated by a suitable choice of pO0 in the adaptive control algorithm. In the present case, it is su$cient to choose g(t) as the unperturbed dynamics from t !Nq to t (N'2). While the integer N should be selected as small as possible to minimize 2 2 the waiting time, larger values of N improve the robustness of the method. Fig. 12a reports the new phase space results for N"6. The system is left unperturbed from t"0 (same initial conditions as before) until t"t "12.9. Here t is the instant at which the unperturbed dynamics "rst enters the e interval containing x(t !6q). The adaptive scalar perturbation assures a convergence to the target within a target 2 time of t #6q, which again is more than two orders of magnitude smaller than t . 2 It is important to visualize accuracy of the adaptive method in targeting the desired I in the real 2 phase space, even with this simple choice of g. Fig. 12b shows the range of #uctuations of the perturbations, and Fig. 12a shows the range spanned by the unperturbed x dynamics.
5. Stabilizing desirable chaotic trajectories and application 5.1. Stabilizing desirable chaotic trajectories 5.1.1. Overview We consider the following situation: suppose there is a nonlinear dynamical system whose trajectories lie on a chaotic attractor. Suppose further that one of the uncountably in"nite number of chaotic orbits embedded in the chaotic attractor corresponds to a desirable operational state of
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the system. Our goal is to apply only small feedback control to keep trajectories originating from random initial conditions in the vicinity of the desirable chaotic orbit. In what follows, we present a general method to achieve this goal and demonstrate that the idea can be applied to synchronization of chaotic systems. Our method to stabilize a desirable chaotic orbit is based on OGY's method of controlling chaos. We "rst select the desirable orbit (the target) according to our needs. Di!erent methods can be used to construct such a target orbit in di!erent contexts. We then stabilize a trajectory originated from a random initial condition around the target orbit. This can in fact be achieved if the target chaotic orbit is a trajectory generated by the evolution equations of the dynamical system. Such target orbit possesses a local hyperbolic structure of stable and unstable directions at almost all points, which is rather typical for chaotic orbits. Finally, we apply small feedback control to stabilize the target chaotic orbit. The construction of such a target orbit is, therefore, a crucial step in the controlling method. This will be detailed in numerical examples. 5.1.2. Method for stabilizing a desirable chaotic orbit Our method for stabilizing a desirable chaotic orbit is based on the OGY idea of stabilizing a desirable unstable periodic orbit. Intuitively, the orbit to be stabilized has a period equal to the length of the orbit. Consider chaotic systems described by two-dimensional maps on the PoincareH surface of section: "F(x , p) , (74) L> L where x 3R, p is an externally controllable parameter. In the spirit of the OGY ideas, we require L that the parameter perturbations be small: x
"*p","p!p "(d , (75) where p is some nominal parameter value, d is a small number de"ning the range of parameter perturbations. Let +y , (n"0, 1, 2,2, N) be the target chaotic orbit. Now generate a trajectory L +x , to be stabilized around the target orbit. Randomly pick an initial condition x , assume that L the trajectory point x (n50) falls in a small neighborhood of the point y of the target orbit at L I time step n. Without loss of generality, we set k"n on the target orbit. In this small neighborhood, linearization of Eq. (74) is applicable. We have (p )!y (p )"J ) [x (p )!y (p )]#K *p , (76) L> L L> L L L where *p "p !p , "*p "4d, J is the 2;2 Jacobian matrix and K is a two-dimensional column L L L vector: x
, K"D F(x, p)"x yL . (77) J"Dx F(x, p)"x yL NN NN N Without control, i.e., *p "0, the trajectory x (i"n#1,2) diverges from the target orbit L G y (i'n) geometrically. The task is to program the parameter perturbations *p so that G L "x !y "P0 for subsequent iterates i'n. G G For almost all points on the target orbit, there exist both a stable and an unstable direction [34]. These directions can be calculated by using the numerical method in Ref. [34]. The calculated stable and unstable directions are stored together with the target orbit, and this information is used
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to compute the parameter perturbations applied at each orbit point for i'n. Let e and e L L be the stable and unstable directions at y , and f and f be two dual vectors orthogonal L L L to e and e , respectively. The vectors f and f satisfy f ) e "f ) e "1 and L L L L L L L L f ) e "f ) e "0. To stabilize +x , around +y ,, we require the next iteration of x , after L L L L L L L falling into a small neighborhood around y , to lie on the stable direction at y (p ): L L> [x (p )!y (p )] ) f "0 . (78) L> L L> L> Substituting Eq. (76) into Eq. (78), we obtain the following expression for the parameter perturbation: +J ) [x (p )!y (p )], ) f L L L> , (79) *p " L !K ) f L> where if *p 'd, we set *p "0. L L In stabilizing unstable periodic orbits, the average transient (`waitinga) time to achieve the control scales with the maximum allowed parameter perturbation d as q&d\A, where the scaling exponent c can be computed in terms of the stable and unstable eigenvalues of the unstable periodic orbits [1,23]. For cases where c'1, the transient time can be signi"cantly reduced if somewhat larger parameter perturbations are allowed. The problem of transient time is much less severe here, since the target orbit is long. In principle, when the trajectory enters the neighborhood of any one of the points on the target orbit, parameter control based on Eq. (79) can be applied. Thus, even if the size of every neighborhood around the target orbit is small, the transient time required can be signi"cantly reduced by increasing the length of the target orbit. 5.1.3. Synchronization of low-dimensional chaotic systems by control Chaos is characterized by a sensitive dependence of system's dynamical variables on initial conditions. Trajectories starting with slightly di!erent initial conditions diverge from each other geometrically. Consequently, synchronization seems unlikely even for two perfectly identical chaotic systems, if trajectories start from initial conditions that di!er slightly. Moreover, in practical applications the existence of noise (both external and internal) and system imperfect identi"cation makes the hope of synchronizing two chaotic systems even more remote. Nonetheless, it was demonstrated by Pecora and Carroll [22] that synchronization of chaotic dynamical systems is not only possible but it is believed to have potential applications in communication [22,106}108]. In the feedback control approach to synchronize chaotic systems [51], as opposed to the open loop synchronization method in Refs. [22,106}108], it is not required that the system under study be divided into subsystems, and both noise and a small amount of system parameter mismatch are allowed. Speci"cally, the OGY strategy is extended to stabilize a chaotic trajectory of one system about a chaotic orbit of the other system to achieve synchronization of the two systems. It should be noted that the idea of stabilizing chaotic orbits by using OGY method was also proposed by Metha and Henderson [109]. Their approach is to construct an arti"cial dynamical system evolving errors between the system's output and the target chaotic orbit. If the arti"cial system has a zero "xed point, parameter perturbations based on the OGY algorithm are then applied to stabilize the arti"cial system around its zero "xed point, which means that the original system's output is brought to the desired chaotic orbit. They illustrated their method by using one-dimensional
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Fig. 13. From Ref. [51]. A schematic illustration of the strategy to synchronize two chaotic systems. Some dynamical variables of two systems are measured, based on which temporal parameter perturbations are calculated and applied to the system B. We assume that before the synchronization, some information about the geometrical structure of the chaotic attractor (e.g., the Jacobian matrices along a long chaotic trajectory that practically covers the whole attractor) has been obtained.
maps. Construction of the arti"cial map for more general dynamical systems may be nontrivial. In Ref. [51], on the other hand, parameter perturbations are applied directly to the original dynamical system and the method makes use of the geometrical structure of the chaotic trajectory. To synchronize two chaotic systems which we call A and B, we imagine that some parameter of one system (assume B) is externally adjustable. The strategy is illustrated schematically in Fig. 13, where we assume that some state variables of both systems A and B can be measured. Based on this measurement and our knowledge about the system (we can, for example, observe and learn the system "rst), when it is determined that the state variables of A and B are close, we calculate a small parameter perturbation based on the OGY algorithm and apply it to system B. Two systems can then be synchronized, although their trajectories are still chaotic. Under the in#uence of external noise, there is a "nite probability that the two already synchronized trajectories may lose synchronization. However, with probability one (due to the ergodicity of chaotic trajectories), after a "nite amount of transient time, the trajectories of A and B will get close and are synchronized again. In this sense, the synchronization method is robust against small external noise. We consider two almost identical chaotic systems that are described by two-dimensional maps on the PoincareH surface of section: "F(x , p ) [A], y "F( y , p) [B] (80) L> L L> L where x , y 3R, F is a smooth function in its variables, p for system A is a "xed parameter value L L and, p for system B is an externally controllable parameter. For the purpose of synchronization, we require that the dynamics should not be substantially di!erent for systems A and B. Equivalently, we require that the parameter perturbations be small, i.e., "p!p "(d, where d is a small number de"ning the range of parameter variation. Suppose that the two systems start with di!erent initial conditions. In general, the resulting chaotic trajectories are completely uncorrelated. However, due to ergodicity, the two trajectories can get arbitrarily close to each other at some later time n . A Without control, the two trajectories will separate from each other exponentially again. We then program the parameter p using the method in Section 2 so that " y !x "P0 for n5n , which L L A means that A and B are synchronized for n5n . A We now illustrate the synchronization algorithm by using the HeH non map: (x, y)P(a!x# 0.3y, x), where a is the control parameter. Consider two such HeH non systems. One has "xed x
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Fig. 14. From Ref. [51]. Synchronizing two HeH non systems [(x, y)P(a!x#0.3y, x)]. In system A, the parameter a is "xed at a "1.4. In system B, a is allowed to vary in [1.39,1.41]. (a) The uncorrelated and synchronized chaotic trajectories of the two systems before and after the parameter control is turned on and, (b) part of the time series of the di!erence *x"x !x corresponding to (a). The synchronization neighborhood is chosen to be a circle of radius 0.01 (see text).
parameter value (a"a "1.4) which serves as the `targeta and, in the other system we adjust a in a small range (1.39, 1.41) according to Eq. (6). At time t"0, we start two systems with di!erent initial conditions: (x , y )"(0.5,!0.8) and (x , y )"(0.0, 0.0). The two systems then move in completely uncorrelated chaotic trajectories. At time step 2534, the trajectory points of the two systems come close to each other within a circle of radius of 0.01. When this occurs, we turned on the parameter perturbations calculated from Eq. (6). Note that the radius 0.01 above can be changed slightly (without a!ecting the synchronization) depending on how we de"ne the `synchronization neighborhooda in which the two trajectories are considered to be close together. In general, the size of such a neighborhood should be chosen to be proportional to d, the maximum allowed parameter perturbation. Fig. 14a shows part of a time series of the uncorrelated and synchronized chaotic trajectories before and after the control is turned on, respectively, where the crosses and diamonds denote values of x for the two chaotic trajectories. Clearly, after the control is turned on, crosses and diamonds overlap each other, indicating the two chaotic HeH non trajectories evolve completely in phase (synchronization), although they are still chaotic. Fig. 14b shows a time series of *x(t)"x (t)!x (t), where we see that *x(t)"0 after the control is applied. In the presence of noise, the two synchronized trajectories can go uncorrelated again (x is `kickeda out of the neighborhood of x by the noise). When *x(t) exceeds a critical value, say 0.01, we turn o! the control and let the two systems evolve by themselves. Due to ergodicity, the two trajectories will come close again and be synchronized. To model the e!ect of noise, we add a term ep(t) to the x-component of the two HeH non systems, where p is a random variable with Gaussian probability distribution of zero mean and unit standard deviation and, e characterizes the noise amplitude. Figs. 15a and b show part of the time series of *x(t) for e"3.8;10\ and e"4.18;10\, respectively. Clearly, the smaller the noise amplitude is, the longer the two systems are expected to remain synchronized. In stabilizing unstable periodic orbits, the average transient time to achieve the control is shown to scale with the maximum allowed parameter perturbation d as q&d\A, where c is given in terms
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Fig. 15. From Ref. [51]. The in#uence of noise [of the form ep(t), where p(t) is a Gaussian random variable having zero mean and unit standard deviation and, e is the noise amplitude] on synchronized orbits. (a) e"3.8;10\ and (b) e"4.18;10\. It is clear that noise can make the synchronized orbits uncorrelated by kicking one orbit out of the neighborhood of the other orbit.
Fig. 16. From Ref. [51]. Average time to achieve synchronization q versus the size of the synchronization neighborhood d on a log}log plot. Note that q&d\A, where c is the absolute value of the straight line in the "gure.
of the stable and unstable eigenvalues (j and j ) of the unstable periodic orbit by [1,23]: c"1!log "j "/log "j " ,
(81)
if the controlling neighborhood is chosen to be a circle and the process is two dimensional. In the case of synchronization, such a scaling relation still holds, as shown in Fig. 16 for the standard HeH non map, where we plot the average time (with respect to 200 random pairs of initial conditions) to achieve synchronization versus d on a logarithmic scale. The absolute value of the slope of the
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line is the scaling exponent c, which is approximately 1.23 for Fig. 16. Following the same argument as in Ref. [1,23], it is easy to see that c is still given by Eq. (81), except that now j and j are the stable and unstable Lyapunov numbers of a typical chaotic trajectory. For the standard HeH non map, we found that c+1.27 in terms of Eq. (81), which agrees reasonably well with the value obtained from the linear "tting in Fig. 16. Note that the average time to achieve synchronization increases algebraically as d is decreased. For d&10\ in the HeH non map, q&10 [see Figs. 14a, b and 16]. For stabilizing unstable periodic orbits, it has been demonstrated that the average time to achieve control can be greatly reduced by applying small controls to the orbit outside the control neighborhood. This technique is known as `targetinga [35]. Note that in such a case, the target (the unstable periodic orbit) is always "xed. While in the synchronization problem, the target moves chaotically because both trajectories wander on the chaotic attractor and, the actual location where the two trajectories get close to each other depends sensitively on the pair of initial conditions and the size of the synchronization neighborhood. We remark that one advantage of the OGY method is that it does not require complete knowledge of the system equations [1,23], although it is necessary to `learna from data to obtain enough knowledge about the unstable periodic orbits to be stabilized in order to control. Here by `knowledgea we mean the Jacobian matrices J [note that f can be calculated in terms of J ] and L vector K in Eq. (79). A nonlinear time series of the process is enough to extract the necessary parameter perturbations to stabilize a chaotic trajectory around the unstable periodic orbit. In the synchronization problem, the orbit to be stabilized is chaotic. Nonetheless, one can still run the system for enough long time to estimate both J and K at many trajectory points, enough points to practically cover the whole chaotic attractor. Indeed, Newell et al. [97,98] successfully demonstrated that the synchronization method [51] can be realized in experiments where a detailed knowledge of the system's equations is not available. The experiments involved two almost identical chaotic laser diodes. Synchronization was readily achieved when small feedback control was applied to one of them [97,98]. In principle, not only two such lasers can be synchronized, but also an array of almost identical chaotic lasers. This may be potentially useful in engineering applications. 5.1.4. Synchronization of spatiotemporal chaotic systems by control Spatiotemporal chaotic systems are high-dimensional dynamical systems. Consider such a system that consists of a spatial network of chaotic elements. For the Pecora}Carroll type of synchronism [22] to occur, it may be necessary to use a large number of driving variables spatially distributed among chaotic elements. Nonetheless, it is often the case that the subsystem obtained by excluding only a few driving variables is still chaotic to a similar degree as the original system. That is, the subsystem still has a comparable number of positive Lyapunov exponents as the original system. To illustrate this, consider the coupled logistic map lattice [110] (to be described later) with 20 spatial sites (a 20-dimensional system). In certain parameter regimes, there are 8 positive Lyapunov exponents. Linking one arbitrarily chosen dynamical variable yields a 19dimensional subsystem that still has 7 positive Lyapunov exponents. While synchronizable nonchaotic subsystem can be obtained by linking su$cient number of dynamical variables, they are di$cult to identify due to the high dimensionality of the system. It is di$cult to extend the control strategy proposed in Ref. [51] to high-dimensional systems because its success depends on the existence of one stable and an unstable directions at each trajectory point. Spatiotemporal chaotic
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systems usually have many unstable and stable directions at each trajectory point in the phase space. One strategy [111] is to combine the Pecora}Carroll idea [22] and the control idea in Ref. [51] to synchronize two nearly identical spatiotemporal systems. Speci"cally, by using a certain number of driving variables and by applying appropriately designed feedback controls, synchronization can be achieved for the two systems. The choice of driving variables can be done arbitrarily and their number can be as few as only one. The feedback control is applied to one of the two systems to be synchronized. The magnitude of the feedback control required can, in general, be very small. The design of the feedback control in Ref. [111] is based on the principle of KalmaH n "lter [112] which tracks the system state by measuring a single scalar function of the system state. The KalmaH n "lter is optimal for linear systems. For nonlinear or chaotic systems, a modi"ed technique was developed in Ref. [113] to deduce and track the state of the system from limited observation. Our design of the synchronization scheme is a direct application of this modi"ed technique. Consider two identical spatiotemporal systems described by the following maps: "F(x ), x( "F(x( ) , (82) L> L L> L where x and x( are N-dimensional state vectors. Following Pecora and Carroll [22], we decompose the system state into two parts: one is the N -dimensional driving system which we denote z and z( , B and the other is the N -dimensional subsystems to be synchronized denoted by y and y( , where N ;N . In general, we allow the subsystems y and y( to be chaotic. By de"nition of `drivinga, B z and z( are identi"ed, i.e., z"z( . The equations for y, y( and z are as follows: x
"F ( y , z ), y( "F ( y( , z ), z "F ( y , z ) , (83) L> W L L L> W L L L> X L L where F"[F , F ]. In cases where the full system F is chaotic, Pecora and Carroll argued that W X when the subsystem F has all negative Lyapunov exponents, y and y( can be synchronized. W L L Subsystems having only negative Lyapunov exponents are, however, hard to identify when Eq. (82) is spatiotemporally chaotic with many positive Lyapunov exponents and, in principle, we do not know how many. To achieve synchronization of y and y( , we apply the following feedback control L L to one of the subsystems y( : L y( "F ( y( , z )!C ) [F ( y( , z )!F ( y , z )] , (84) L> W L L L X L L X L L where C is an N ;N control matrix to be evaluated at each time step. The synchronization L B scheme is schematically shown in Fig. 17. The feedback control !C ) [F ( y( , z )!F ( y , z )] is L X L L X L L applied only when y and y( are close. The linearized dynamics in the neighborhood of y can L L L therefore be written as y
dy "y( !y "F ( y( , z )!F ( y , z )!C ) [F ( y( , z )!F ( y , z )] L> L> L> W L L W L L L X L L X L L "[DF ( y , z )!C ) DF ( y , z )] ) dy ,A ) dy , (85) W L L L X L L L L L where DF and DF are the N ;N and N ;N Jacobian matrices of F and F , respectively, W X B W X evaluated at y and z . Since F is chaotic, y( will diverge from y exponentially without L L W L> L> control. Our goal is to design the control matrix C so that dy P0 as nPR. To achieve this L L we assume that the subsystem F has N positive and N negative Lyapunov exponents, where W N #N "N . Furthermore, we do not assume hyperbolicity for the subsystems y( and y. We
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Fig. 17. From Ref. [111]. The scheme of synchronizing two spatiotemporal chaotic systems by driving and feedback control.
require that almost every point but not all on the asymptotic attractor of F has N unstable and W N stable directions, the stable and unstable subspaces are bounded away from zero [34,39]. The feedback control so designed thus applies to nonhyperbolic dynamical systems [111,113]. Let eG (i"1,2, N ) be the set of base column vectors in the unstable space at y . If we restrict the L L control matrix C to the unstable space of F at y : L W L> , , C " bG " [CG eG ) * #2#CG B eG ) * B ] , (86) L L L> , L> , G G where +* , ( j"1,2, N ) are a complete set of row vector that span the driving system F , and H B X CG , CG ,2, CG B (i"1,2, N ) are the set of N ;N control coe$cients, then it can be shown B , [113] that the matrix A ["DF ( y , z )!C ) DF ( y , z )] reduces to the following upper trianguL W L L L X L L lar form:
U A " L L 0
W L , (87) S L where U (S ) is an N ;N (N ;N ) matrix that evolves a vector in the unstable (stable) space at L L y into a vector in the unstable (stable) space at y , and W is an N ;N matrix that takes L L> L a vector in the stable space at y into a vector in the unstable space at y . In order to have L L> "dy "P0 as nPR, it is required that all eigenvalues of the product matrix A A 2A vanish as L L L\ nPR. Since, L L U W G\ S G HG> H G I I , (88) S S 2S L L\ i.e., the product matrix A A 2A is upper triangular and, since the matrices S are already in L L\ L the stable space along the trajectory (eigenvalues of the matrix product S S 2S P0 as L L\ nPR), the stability of the product A A 2A depends solely on the stability of U U 2U . L L\ L L\ One way to make the product U U 2U stable is to let U be lower triangular and be stable, i.e., L L\ G all diagonal elements of U are eigenvalues of U and are less than 1. In this way, the product G G UU 2U is still lower triangular and has vanishing diagonal elements (eigenvalues) [113]. L L\ Now de"ne a set of contravariant row vectors f G (i"1,2, N ) in the unstable space at L> y such that f G ) eH "d , where d "0 if iOj and d "1. The matrix elements of U are L> L> L> GH GH GG L
A A 2A " L L\
U U 2U L L\ 0
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given by ; "f G ) A ) eH [113]. In order to make the matrices U lower triangular, we look at GH L> L L G elements ; of the matrix U , which can be expressed as follows: GH L ; "f G ) A ) eH "f G ) [DF ( y , z )!C ) DF ( y , z )] ) eH GH L> L L L> W L L L X L L L "f G ) DF ( y , z ) ) eH !f G ) bG ) DF ( y , z ) ) eH , (89) L> W L L L L> L X L L L where f G ) bH "0 for iOj has been used. In order to have ; "0 for j'i, So et al. [113] L> L GH suggested the following procedure for choosing the unstable base vectors: je "[DF ( y , z )] ) e , L L> W L L L je "[DF ( y , z )!b ) DF ( y( , z )] ) e , L L> W L L L X L L L (90) 2"2 , \ j, e, " DF ( y , z )! bG ) DF ( y( , z ) ) e, , L L> L L X L L W L L G where jG (i"1,2, N ) are a set of numbers which can be related to the stretching rate of L in"nitesimal vectors along the unstable direction eG . It can then be shown that elements of the L matrix U are given by L ; "0, j'i , GH ; "jG !f G ) bG ) DhG , (91) GG L L> L L ; "!f G ) bG ) DhH , j(i , GH L> L L where DhH ,DF ( y , z ) ) eH . To make the eigenvalues of the matrix U less than one, we can adjust L X L L L L the N ;N free control parameters CG (i"1,2, N , j"1,2, N ) such that all diagonal ele H B ments of U are less than one. But this only provides N conditions, and there are still N (N !1) L B free control parameters we must set. The simplest choice is to set CG "0 for j'1. Then setting the H diagonal elements in Eq. (91) zero gives, CG "jG /[* ) DhG ] (i"1,2, N ) and consequently, the L L control matrix is given by
jG , L eG ) * . (92) C " L * ) DhG L> L G In practice, the set of numbers jG and the set of unstable base vectors eG can be computed by L L randomly initializing a set of base vectors eG and evolving them in terms of Eq. (90). After a period of transient, the set of vectors so obtained converge to the real unstable directions. To assure that only small perturbations are applied, it is necessary to monitor the magnitude of the term in the denominator of Eq. (92). When "* ) DhG " is below some small threshold, we set C "0. This will not L L result in the loss of control provided it is done only occasionally. We stress that the feedback control is derived under the applicability of linearized dynamics and, hence, the control is applied only when trajectories y( and y are su$ciently close. No control is applied when they are not close. Also note that the control law Eq. (92) has been derived under the condition of hyperbolicity at almost every point along the trajectory, while there is no guarantee that spatiotemporal chaotic systems are hyperbolic. Nonetheless, as we illustrate below, the control works for spatiotemporal systems modeled by coupled map lattices.
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Fig. 18. From Ref. [111]. The Lyapunov spectrum for N"10 and e"0.7, (a) of the full system, and (b) of the reduced system by using one site as the driving signal.
To illustrate the applicability of the control method, we consider the following system of di!usively coupled logistic maps [110] as a phenomenological model for spatiotemporal chaotic systems: x (i)"(1!e) f [x (i)]#e+ f [x (i#1)]#f [x (i!1)],, i"1,2, N , L L L> L
(93)
where i and n denote discrete spatial sites and time, respectively, N is the total number of maps coupled in the lattice, e denotes the coupling strength, and f (x) is the one-dimensional logistic map f (x)"ax(1!x). We assume periodic boundary condition: x (N#1)"x (1). Eq. (93) exhibits L L extremely rich dynamical phenomena seen in real spatiotemporal systems and it is perhaps the most extensively studied model spatiotemporal system so far. In examples, a"4 was chosen [111], the parameter value for which the logistic map has a chaotic attractor. Our "rst example is for N"10 and e"0.7. At this e value, there are three positive Lyapunov exponents for Eq. (93). Fig. 18a shows the corresponding Lyapunov spectrum for the full system Eq. (93), in which j versus the index k (k"1,2, N, j 5j 525j ) is plotted. To synchroI , nize two such systems, we choose one of the x(i) (i"1,2, 10) as the driving variable. Choosing a di!erent x(i) does not change the result due to symmetry of Eq. (93) with respect to site index i. The subsystems to be synchronized are therefore nine-dimensional, and still possesses 3 positive Lyapunov exponents, as shown by the corresponding Lyapunov spectrum in Fig. 18b. Thus, the Pecora}Carroll type synchronism will not occur for the nine-dimensional subsystem. The control neighborhood is set to be " y( !y"4r "0.015. The control Eq. (92) is applied only when "* ) DhH "510\ ( j"1, 2, 3). With these control parameter settings, most randomly chosen initial L conditions can be controlled. In general, the smaller the control neighborhood, the larger the probability that trajectories resulting from two randomly chosen initial conditions can be synchronized. In cases where one set of initial conditions fails to be synchronized, we disregard them and choose another set of initial conditions. Figs. 19a and b show, when trajectories of the two subsystems resulting from a pair of randomly chosen initial conditions are within r , the error D , L
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Fig. 19. From Ref. [111]. Synchronization of two logistic map lattices (N"10, e"0.7). (a) The synchronization error log D versus n, and (b) the required feedback control magnitude log ("C ") versus n. The control neighborhood is set L L to be 0.015.
de"ned as D "" y( !y " , L L L and the control magnitude, de"ned as
(94)
"C """C ) [F ( y( , z )!F ( y , z )]" , (95) L L X L L X L L versus the time step n after the control is turned on. Clearly, the two trajectories rapidly approach each other to within computer roundo! error (&10\) after the control is applied, and the required feedback control decreases correspondingly to extremely small values. Under the in#uence of small random noise, the degree to which two subsystems can be synchronized, or the value of "dy " is proportional to the amplitude of the noise. Figs. 20a and b L show D and "C " versus time step n for the parameter setting of Fig. 19 when a noise term modeled L L by hpG is added to each site of the lattice, where h"10\ is the noise amplitude and pG is L L a Gaussian random variable with zero mean and unit variance. In general, minimum values of D and "C " have the same order of magnitude as h. Occasionally both D and "C " can have values L L L L larger than 10\, indicating that the degree of synchronization decreases signi"cantly at these time steps. Eventually, the systems lose synchronization due to large amplitude noise in the tail of the Gaussian distribution. When this occurs, we turn o! the control and let the systems evolve by themselves. Due to the ergodicity of the chaotic attractor, at some later time the two trajectories will come close to each other and can be controlled again. To demonstrate the applicability of the control algorithm in higher dimensions, we have performed control using N"20. In this case, we found that for e"0.5, there is a unique chaotic attractor with 8 positive Lyapunov exponents. The subsystem obtained by using a driving signal x (i), where i can be any number between 1 and 20, has 7 positive Lyapunov exponents, as shown in L Fig. 21. In this case, the control neighborhood needs to be smaller for synchronization to occur. Besides, the quantity N used in the control algorithm needs to be slightly larger than the actual number of unstable directions. We found that using N "10 su$ces. Figs. 22a and b show D and L "C " versus n, where the control is applied only when "dy "45;10\. L L
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Fig. 20. From Ref. [111]. (a) The synchronization error log D , and (b) the required feedback control magnitude L log ("C ") versus n when a noise term 10\pG is added to each site of the lattice, where pG is a Gaussian random variable L L L with zero mean and unit variance.
Fig. 21. From Ref. [111]. The reduced Lyapunov spectrum for N"20 and e"0.5, where one site of the lattice is used as the driving signal.
As we have demonstrated with Eq. (93), the control neighborhood needs to be reduced as the number of unstable directions increases. Going from 3 unstable directions (Fig. 19, the N"10 case) to 7 unstable directions (Fig. 22, the N"20 case) requires almost three orders of magnitude decrease in the size of the control neighborhood. As the size of control neighborhood is decreased, the average transient time for two trajectories to get close increases algebraically with a scaling exponent determined by the Lyapunov spectrum of the chaotic attractor. Thus, even for spatiotemporal systems with moderate sizes, the transient time required may be very long. The reason that extremely small control neighborhood is needed is not clear, but may be related to the noninvertibility and nonhyperbolicity of the coupled logistic map lattice. For instance, for noninvertible dynamical systems, there may not be unique stable and unstable spaces at every trajectory points, whereas the control algorithm is designed under the assumption that the dynamical systems possess unique and distinct stable and unstable spaces (invertibility and hyperbolicity).
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Fig. 22. From Ref. [111]. (a) The synchronization error log D , and (b) the required feedback control magnitude L log ("C ") versus n for N"20 and e"0.5. The control neighborhood needs to be reduced to 5;10\ in order to achieve L the control.
5.1.5. Remarks Stabilizing a chaotic orbit has other applications as well. Here we brie#y discuss an application to select a desirable chaotic phase from a chaotic attractor that contains two interconnected components. In the study of chaotic systems, there are situations where orbits switch intermittently between distinct chaotic phases. For example, a dynamical system in parameter regime after a bifurcation called the `interior crisisa [32,114] exhibits such intermittent chaotic behavior. The phenomenology of interior crisis is as follows. Before the crisis, there is a chaotic attractor and a coexisting nonattracting chaotic saddle in the phase space. The chaotic attractor and the chaotic saddle are separated from each other and, hence, trajectories originating from almost all initial conditions eventually asymptote to the chaotic attractor. At the crisis, the chaotic saddle collides with the chaotic attractor so that the original nonattracting chaotic saddle becomes part of the combined attractor, whose phase-space extent is larger than the original chaotic attractor. After the crisis, trajectories wander on the whole combined larger attractor, in such a way that the trajectories visit both parts, which correspond to the original chaotic attractor and the chaotic saddle, in an intermittent fashion. As a consequence, time series recorded from such a trajectory exhibits distinct intermittent chaotic phases. It has been demonstrated [53] that it is possible to keep trajectories in one of the chaotic phases by applying only small parameter perturbations to the system. We remark that similar intermittent chaotic signals also arise in biomedical systems [53].
5.1.6. Encoding digital messages using chaos control Recent development in nonlinear dynamics and chaos has led to the idea of encoding digital information by using chaos [21,24,54}57]. In particular, it has been demonstrated both theoretically and experimentally by Hayes et al. [21,24] that a chaotic system can be manipulated, via arbitrarily small time-dependent perturbations, to generate controlled chaotic orbits whose symbolic representation corresponds to the digital representation of a desirable message. Imagine
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a chaotic oscillator that generates a large amplitude signal consisting of an apparently random sequence of positive and negative peaks. A possible way to assign a symbolic representation to the signal is to associate a positive peak with a one, and a negative peak with a zero, thereby generating a binary sequence. The use of small perturbations to an accessible system parameter or variable can then cause the signal to follow an orbit whose binary sequence encodes a desirable message which one wishes to transmit [21,24]. One advantage of this type of message-encoding strategy is that the nonlinear chaotic oscillator that generates the waveform for transmission can remain simple and e$cient, while all the necessary electronics controlling encoding of the signal can remain at low-powered microelectronic level most of the time. A central issue in any digital communication device concerns with the channel capacity [115,116], a quantity that measures the amount of information that the device can encode. For a chaotic system, channel capacity is equivalent to the topological entropy because it de"nes the rate at which information is generated by the system [117]. To give a concrete example, consider symbol sequences consisting of a string of n symbols generated by the dynamics. For a completely random process, one expects to be able to observe all 2L possible symbol sequences. In this case, the topological entropy is ln 2L "ln 2 , h " lim 2 n L which is the maximum possible value for processes de"ned by two symbols. A deterministic chaotic system is, however, not purely random. Thus, if its symbolic dynamics requires only two symbols, the topological entropy of the chaotic attractor is in general less than ln 2. In a digital communication scheme, it is highly desirable to have the channel capacity as large as possible to maximize the amount of information that can be encoded. It has been pointed out recently [118] that in nonlinear digital communication, it is generally more advantageous to use transient chaos naturally arising in wide parameter regimes of nonlinear systems as information sources from the standpoint of channel capacity. Dynamically, transient chaos is generated by nonattracting chaotic saddles in the phase space [119,120]. A general observation is that, typically, a nonlinear system can generate chaotic attractors (corresponding to sustained chaos) and chaotic saddles in di!erent parameter regimes. As a system parameter changes, a chaotic attractor can abe converted into a chaotic saddle via a dynamical event such as crisis [32]. The orbital complexity associated with trajectories on a chaotic saddle can be greater than that of trajectories on a chaotic
The basic principle that makes digital encoding with chaos possible lies in the fundamental link between chaos and information. The evolution of a chaotic system is unpredictable in long terms. In communication, it was realized by Shannon in 1948 that a sequence of events conveys information if the events are not fully predictable [115,116]. Thus, the fundamental unpredictability of chaos implies that chaotic systems can be regarded as sources that naturally generate digital communication signals. By manipulating a chaotic system in an intelligent way, digital information can be encoded. There are chaotic attractors with topological entropies ln 2, such as the logistic map f (x)"rx(1!x) at r"4. But such situations are rare.
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attractor because, crisis is generally a complexity-increasing event. As such, if one measures the topological entropy of the system as a single parameter changes through the crisis point, one usually "nds that the entropy is a nondecreasing function of the system parameter. For a symbolic dynamics of two symbols observed in typical low-dimensional chaotic systems, the maximum allowed value of the topological entropy, ln 2, is often realized in a parameter regime where there is transient chaos. Thus, it is highly desirable to design a chaotic system operating in a transient chaotic regime for digital encoding. In what follows, we "rst develop a theoretical and numerical framework for encoding digital information using transient chaos. We provide numerical evidence that the topological entropies of chaotic saddles are usually greater than those of chaotic attractors. We then detail a procedure for encoding digital messages into trajectories that live on chaotic saddles. We argue that digital encoding with chaotic saddles can be robust against environmental noise, thereby signi"cantly reducing the probability of bit error in encoding. Finally, we describe message encoding using two-dimensional symbolic dynamics. 5.1.7. The channel capacity In general, the topological entropy associated with transient chaos is greater than that with permanent chaos. Although at present there is no rigorous proof for this statement, is was observed in several chaotic systems: (1) Hamiltoniian maps [121,122]; and (2) the He`non map [123]. For illustrative purpose, we demonstrate how transient chaos can be utilized to encode digital information by using the one-dimensional logistic map: x "f (x )"rx (1!x ). The map L> L L L exhibits chaotic attractors and stable periodic attractors for r (r4r "4, where r +3.58 is the $ A $ Feigenbaum point of the transition to chaos via the route of period-doubling bifurcations [124]. At r"r , crisis occurs [32] so that the chaotic attractor is converted into a chaotic repeller. Thus, for A r'r , what is typically observed is transient chaos. That is, a trajectory starting from a random A initial condition in x3[0, 1] behaves chaotically for a period of time and then asymptotes to x"!R. The average transient time depends on the parameter di!erence (r!r ) and scales with A it algebraically: q&(r!r )\ [32]. A The existence of a chaotic repeller for r'4, which is in fact a fractal Cantor set in the unit interval, can be easily seen in the map function in Fig. 23a. Letting s,(r/4!1) we see that there is a primary gap of size (s/(1#s). Initial conditions from this gap maps out of the unit interval in
A crisis that destroys a chaotic attractor and converts it into a nonattracting chaotic saddle is called a boundary crisis [32]. It is triggered by the collision of the attractor with the basin boundary. Dynamically, a crisis is induced by a homoclinic or a herteroclinic tangency of stable and unstable manifolds of some unstable periodic orbits in the phase space. In the case of boundary crisis, the two colliding sets are, (1) the chaotic attractor which lives in the closure of the unstable manifold of an unstable periodic orbit on the basin boundary, and (2) the basin boundary which is the stable manifold of the periodic orbit on the boundary. Accompanying a homoclinic or a heteroclinic tangency and the subsequent homoclinic or heteroclinic crossing of stable and unstable manifolds is a horseshoe type of dynamics, which creates an in"nite number of new unstable periodic orbits, and consequently increases the complexity of the resulting chaotic set. Chaotic repellers are one-dimensional analogies of chaotic saddles in two-dimensional maps or in three-dimensional #ows, because one-dimensional noninvertible maps can be regarded as the limiting case of two-dimensional invertible maps when the determinant of the Jacobian matrix tends to zero (e.g., logistic map versus the He`non map).
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Fig. 23. (a) The logistic map for r'4. (b) For r"4.1, a numerical trajectory of 10 000 points on the chaotic repeller. Apparently, the chaotic repeller exhibits a fractal structure.
one iteration and goes to x"!R. There are two (2L) preimages of the primary gap in which initial conditions map out of the unit interval in two [(n#1)] iterations. Taking the limit nPR, we see that almost all initial conditions in the unit interval eventually escape from it except for a set of Lebesgue measure zero. This set, by construction, is a fractal Cantor set. Fig. 23b shows, for r"4.1, a numerical trajectory of 10 000 points on the Cantor set [33]. The fractal structure of the set is apparent from the "gure. A symbolic dynamics for the logistic map can be de"ned by setting the symbolic partition at the critical point x "0.5. A trajectory point x bears a symbol 0 if x(x and a symbol 1 if x'x . A A A A trajectory in the phase space thus corresponds to a symbol sequence in the symbolic space. The topological entropy h quanti"es how random such a symbol sequence can be. To compute h , 2 2 we generate a large number of symbol sequences of length n from many trajectories in the phase space and count N(n), the number of possible symbol sequences. In general, N(n) scales with n as N(n)&eF2 L and, hence, h is given by 2 ln N(n) . (96) h " lim 2 n L In practice, we plot ln N(n) versus n for say, 14n416. The slope of such a plot is approximately h . 2 We now discuss the topological entropy of the logistic map in di!erent parameter regimes. For r(r , the asymptotic invariant sets are stable periodic orbits so that h "0 because there is no $ 2 randomness in the trajectory. As r is increased from r , chaos can arise so that the topological $ entropy starts increasing from zero. For r"4, the logistic map is topologically equivalent to the tent map whose symbolic dynamics is a Bernoulli shift, the topological entropy of which is ln 2. Thus, as r is increased from r to r "4, the topological entropy h continuously increases from $ A 2 zero to ln 2 except when r falls in one of the in"nite number of parameter intervals of periodic windows. In each window, there is an attracting set (periodic or chaotic) and a chaotic repeller. The topological entropy of the chaotic repeller remains constant in the window, where the constant is the value of h at the beginning of the window. Since ln 2 is the maximally realizable value of the 2
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Fig. 24. For the logistic map, the topological entropy h (r) versus r for 3.5(r(4.1. We see that for r'4, h remains at 2 2 ln 2, the maximum possible value for a symbolic dynamics of two symbols.
topological entropy for a symbolic dynamics of two symbols, and since a crisis occurs at r , we see A that for r'4, the entropy remains at ln 2. These behaviors are shown in Fig. 24, a plot of h (r) 2 versus r for 3.5(r(4.1. Thus, for the logistic map, whose dynamical behaviors are seen in a large class of deterministic chaotic systems, the largest possible value of the topological entropy, or the channel capacity, is achieved in a parameter regime of transient chaos where the invariant sets are chaotic repellers. This can be quite advantageous because message encoding, an essential task in any communication scheme, becomes quite straightforward for transient chaos as there are no forbidden words associated with the symbolic dynamics. In communication terminology, such a communication channel is unconstrained. 5.1.8. Message encoding, control scheme, and noise immunity To encode an arbitrary binary message into a trajectory that lives on the chaotic repeller, it is necessary to use small perturbations to an accessible system parameter or a dynamical variable. For the logistic map we choose to perturb the state variable x. Say we wish to apply only small perturbations on the order of 2\K. Our procedure is as follows. First, we convert the message into a binary sequence by using the ASCII code and store the sequence into a symbol registor. Next, we choose an initial condition whose trajectory stays near the chaotic repeller for certain number, say, n (n 'm) iterations. This is practically feasible as one can run the system and pre-determine the A A phase space regions, from which initial conditions chosen yield trajectories whose lifetimes (the times trajectories spend near the chaotic repeller) are at least n . We then determine all m symbols A corresponding to m points on the trajectory starting from x and check to see if the mth symbol agrees with the "rst message bit in the symbol registor. If yes, we iterate x once to get x and determine the mth symbol from x [equivalently the (m#1)th symbol from x ] to see if it matches the second message bit in the symbol registor. If no, we apply a small perturbation to x so that the mth symbol from it matches the "rst message bit. This process continues until all the message bits in the symbol registor are encoded into the chaotic trajectory.
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Fig. 25. The coding function R(x) for the logistic map at r"4.1. We see that indeed, R can assume any value between 0 and 1, but there are many gaps on the x-axis, due to the fact that the chaotic repeller is a fractal Cantor set on the x-axis.
To compute the parameter perturbation, we make use of the coding function [24,21], which can be determined as follows. We divide the unit interval in x into N bins of size dx"1/N, where dx;1/2K and 1/2K is the maximally allowed perturbation. We then choose a point from each bin, iterate it for m times, and determine the corresponding symbol sequence of length m: a a 2a , K where a can be either zero or one. If a point leaves the unit interval in less than m iterations, we G simply disregard it. For those points x for which a symbol sequence of length m can be de"ned, we compute the following symbolic value: K R" a /2G , (97) G G where 04R41. This thus gives the coding function R(x) for points on the chaotic repeller. Since the chaotic repeller has a topological entropy ln 2, R can, in principle, have any value between 0 and 1. Fig. 25 shows the coding function for the logistic map at r"4.1, where dx"2;10\. We see that indeed, R can assume any value between 0 and 1, but there are many gaps on the x-axis, due to the fact that the chaotic repeller is a fractal Cantor set on the x-axis. Given the coding function, the determination of the state perturbations is quite straightforward. Let the natural m-bit symbol sequence from x be a a 2a a (produced by iterating the K\ K map directly) and let the "rst message bit to be encoded be b . We compare the natural symbol sequence a a 2a a with the desirable symbol sequence a a 2a b and compute K\ K K\ dR"(a !b )/2K. From the coding function R(x), we can then compute the perturbation dx. This K is done by locating pairs of points with same values of dR in the computer representation of the coding function R(x) and choosing the one that yields the smallest value of dx. Thus, by applying dx to the initial condition x , the trajectory point after m iterations will correspond to a symbol which is the "rst message bit. Note that if a is identical to the message bit b, no perturbation is necessary. K To encode the next message bit, we iterate the perturbed initial condition once to obtain x . Let x "x . The natural m-bit symbol sequence of x is a a 2b a , where a "a , a "a ,2, and K a is the binary symbol corresponding to the trajectory point f K(x ). We now compare a and K K
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Fig. 26. Encoding the word `TIGERa into a trajectory on the chaotic repeller at r"4.1 for the logistic map. The binary representation of the word is shown on the top of the "gure. Shown is a time series where small control is initiated at n"1 and the "rst binary bit of the message is encoded into the trajectory at n"8. Time dependent perturbations are applied at subsequent iterations so that the entire message `TIGERa can be encoded into the trajectory. The magnitudes of the control perturbations required are shown in (b).
b to determine the next perturbation to be applied to x . Continuing this procedure, we can encode an arbitrary message into the chaotic trajectory +x ,. L Fig. 26a shows an example of encoding the word `TIGERa into a trajectory on the chaotic repeller at r"4.1 for the logistic map. The binary (ASCII) representation of the word is shown on the top of the "gure. Assuming that perturbations of magnitude of 2\ are to be applied, we generate a set of initial conditions whose lifetimes in the unit interval under the map are at least 8. Shown in Fig. 26a is a time series where small control is initiated at n"1 and the "rst binary bit of the message is encoded into the trajectory at n"8. Time dependent perturbations are applied at subsequent iterations so that the entire message `TIGERa can be encoded into the trajectory. Fig. 26b shows the magnitude of the control perturbations applied at di!erent time steps. We see that the perturbations required are small. No control perturbation is required for the "rst six time steps because for this initial condition, the natural symbols corresponding to the trajectory points from n"8 to n"13 happen to coincide with the "rst six bits of the message.
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Some features of the above control scheme are as follows. Note that since the channel capacity of the chaotic repeller is ln 2, there are no forbidden symbol sequences. Thus, in the above encoding scheme, any binary sequences can be produced by a typical trajectory near the chaotic repeller. Since we use the coding function R(x) to compute the perturbation dx, we see that once the perturbation is turned on, the trajectory is then automatically con"ned in the vicinity of the chaotic repeller because the coding function is de"ned with respect to trajectories on the chaotic repeller. Suppose that small perturbations on the order of 2\K are to be applied. To encode a message, we only need to identify a set of initial conditions which can stay near the chaotic repeller for m iterations. Since the typical value of m is, say, 10, it is actually fairly easy to identify a large number of such initial conditions. In practice, before encoding, we can run the system to produce a set of initial conditions whose lifetimes are greater than m. Together with the coding function which also needs to be determined beforehand, one can in principle encode any binary sequences into a dynamical trajectory on the chaotic repeller. We mention that utilizing the symbolic dynamics as described above may in turn provide an alternative way to control transient chaos, which has been a challenging problem in the study of controlling chaos. Besides possessing the maximum topological entropy ln 2, the chaotic repellers of the logistic map for r'4 also have the property of strong noise immunity. To see this, we contrast a chaotic repeller, such as the one shown in Fig. 23b, with the chaotic attractor at r"4. For the chaotic repeller, we see that there is a gap of size &(s, where s"r/4!1, about the partition point x "1/2. For the chaotic attractor there is no such gap. A trajectory on the chaotic attractor A can then come arbitrarily close to the partition point. In a noisy environment, this may cause a bit error. Say the trajectory point is to the immediate right side of x . This point thus has a symbol 1. A Due to noise, the trajectory can be kicked through x and thus assumes a wrong symbol 0. For A a trajectory on the chaotic repeller, this situation is much more improved. In so far as the noise amplitude is smaller than the size of the gap across the partition point x , the symbolic dynamics is A immune to noise. This is of tremendous value to practical implementation of communication with chaos [56,57]. Since all chaotic repellers for r'4 in the logistic map have the same topological entropy ln 2, it appears that it is more advantageous to use chaotic repellers at large r because they possess larger gaps across x and thus their corresponding symbolic dynamics are more robust against noise. We A note, however, as r increases, the sizes of the gaps increase so that it becomes more di$cult to generate trajectories that can stay near the chaotic repeller for su$ciently long time. In general, when choosing an optimal chaotic repeller for digital encoding, there is a trade-o! between the degree of di$culty of generating a trajectory near the chaotic repeller and the noise immunity [56,57]. 5.1.9. Message encoding using two-dimensional symbolic dynamics We consider message encoding by controlling symbolic dynamics in two-dimensional maps (equivalently three-dimensional #ows) [125]. Our motivation comes from the fact that, although the principle of utilizing chaotic symbolic dynamics for communication is quite general, most examples illustrating this idea exclusively utilize chaotic systems whose dynamics can be approximated by one-dimensional maps [24,25]. Many chaotic systems encountered in practice, however,
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cannot be described by one-dimensional dynamics. It is thus of interest to study whether communicating using controlled symbolic dynamics can be realized in higher dimensions. The major di$culty when two-dimensional maps are utilized for communication is to locate a generalized partition so that a good symbolic dynamics can be de"ned. This di$culty arises due to nonhyperbolicity. In smooth, noninvertible two-dimensional maps such as those arising on the PoincareH surface of section of three-dimensional autonomous #ows, nonhyperbolicity is typically characterized by the existence of an in"nite number of points embedded in a chaotic attractor at which the stable and unstable directions coincide } the set of tangency points. Analogous to the critical point in one-dimensional chaotic maps [e.g., x "1/2 in the logistic map f (x)"rx(1!x)], A which is naturally the generating partition point for de"ning symbolic dynamics, in two dimensions the generating partition is a zig-zag curve, the curve connecting all primary tangency points in the chaotic attractor [126,127]. It is generally quite di$cult to locate precisely the partition curve even for well studied two-dimensional systems such as the HeH non map [40]. To overcome this di$culty, in Ref. [125], a general solution is proposed: one exploits various hyperbolic chaotic invariant sets embedded in the nonhyperbolic chaotic attractors. Due to hyperbolicity, it is straightforward to locate a generating partition for trajectories restricted to these saddles. One can choose the chaotic saddles so that the symbolic dynamics are robust against small random noise. Thus, utilizing hyperbolic chaotic saddles for communication also provides a solution to overcome the in#uence of noise [56,57]. We brie#y describe symbolic partitions in two dimensions. The fundamental requirement that quali"es a chaotic system for communication is whether a good symbolic dynamics can be de"ned which faithfully represents the dynamics in the phase space. That is, there should be a one-to-one correspondence between points in the phase space and those in the symbolic space. To generate a symbolic dynamics, one "rst partitions the phase space into cells C (i"1, 2,2, m) covering G the entire attractor and then assigns symbols s to cells C , respectively. Consider a point x in the G G cell C , together with a segment of its unstable manifold. Let a and b be the two intersecting points G of the unstable manifold segment with the cell boundaries, as shown in Fig. 27. A primary condition for a good symbolic dynamics is that the images of a and b under the map F(x), denoted by F(a) and F(b), respectively, should still be at the cell boundaries within which the curve F(a)F(b) lies, as shown in Fig. 27a. However, for an arbitrary partition, situation may arise where one of the end points, say b, is no longer on a cell boundary, thus creating a `dangling enda, as shown in Fig. 27b. Dangling ends may also occur for the stable manifold of x under the inverse map F\. In both cases, there is no one-to-one correspondence between points in the phase space and those in the symbolic space. Such an ill-de"ned symbolic representation of phase-space points is not desirable for communication application, as ambiguities will arise when one attempts to assign symbols to di!erent cells. Nonetheless, if the chaotic attractor is hyperbolic, the partition into cells can be chosen in such a way that the situation of dangling ends depicted in Figs. 27b does not occur [128]. Such partitions are called Markov partitions [129], the dynamics of which is schematically illustrated in Fig. 27a. The partition is generating if every in"nitely long symbol sequence created by the partition corresponds to a single point in the phase space [126,127].
Strictly speaking, the hyperbolic subsets do not cover the entire attractor and, hence, the corresponding `generating partitiona is not the generating partition for the original map but for a `truncateda map.
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Fig. 27. From Ref. [125]. (a) The forward dynamics of a Markov partition. (b) For an arbitrary partition, a `dangling enda of the unstable manifold. This dangling end destroys the one-to-one correspondence between the phase space and the symbolic space.
Since chaotic attractors arising in most two-dimensional maps are nonhyperbolic, the key issue becomes how to "nd hyperbolic subsets embedded in the attractor. To illustrate the principle, we use the HeH non map [40]: (x, y)P(1.4!x#0.3y, x), which is widely believed to admit a chaotic attractor. The partition is a zig-zag curve connecting all primary tangency points in the phase space, which lies near y"0 [126,127]. Trajectory points above the curve bear symbol 1 and those below bear symbol 0. This curve is a generating partition but it is di$cult to compute. To overcome this di$culty while preserving the generating partition, we look for chaotic saddles embedded in the attractor with a gap region, or a forbidden region, de"ned by y"$s/2, which covers the partition curve. Due to the gap, a trajectory restricted to the chaotic saddle will never visit the vicinity of the zig-zag partition curve. The partition for the chaotic saddles thus becomes easy to locate: it is the gap itself. In particular, trajectory points above the gap bear symbol 1 and those below bear symbol 0, and this partition is generating. Fig. 28a shows such a chaotic saddle with gap size s"0.2. The chaotic saddles are numerically computed by the Proper-Interior-Maximum triple (PIM-triple) procedure [33]. Since the forbidden region contains all the primary tangency points, the chaotic saddle in Fig. 28a is apparently hyperbolic: it does not contain any tangency points between the stable and unstable manifolds. As such, a Markov partition can be de"ned for such a hyperbolic saddle, which naturally admits a good symbolic dynamics. In fact, there are in"nitely many gap sizes s which correspond to di!erent hyperbolic chaotic saddles embedded in the attractor. The hyperbolic chaotic saddle shown in Fig. 28a is a subset embedded in the chaotic attractor and, hence, its topological entropy cannot be larger than that of the attractor. A question is then, how severe is the reduction in the topological entropy. This question is important for communication because the topological entropy of a chaotic set characterizes, quantitatively, how much information can be encoded into the trajectories on the set (the channel capacity) [24,55}57,115]. To address this, we compute the topological entropy h (s) of the chaotic saddle as the gap size s 2
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Fig. 28. From Ref. [125]. (a) A hyperbolic chaotic saddle embedded in the HeH non attractor with gap size s"0.2. (b) The topological entropy h (s) of the chaotic saddle as a function of the gap size s. 2
is increased from zero [56,57], as shown in Fig. 28b for 04s(s +0.42, where the dashed
horizontal line at h "0.466 de"nes the minimum gap size s above which the generating 2
partition for the chaotic saddle is simply y"0. We see that as s increases from 0, h decreases 2 slowly at "rst, and then faster. The slowly decreasing behavior warrants a relatively large regime s(s +0.3 in which h decreases only slightly (less than 10% reduction in h ). The key A 2 2 implication is that utilizing chaotic saddles with gap size smaller than s but larger than s seems A
to be practically bene"cial in communication applications: the speci"cation of the symbolic dynamics is straightforward, yet the channel capacity is close to that obtained when one utilizes the original chaotic attractor. It was conjectured [125] that the function of h versus s is a devil's 2 staircase, a statement that can be made rigorous for certain one-dimensional maps [56,57].
When the noise-gap size is small so that the zigzag generating partition curve cannot be covered entirely, utilizing y"0 as the partition line for the symbolic dynamics leads to an error *N in N(n), the number of possible symbol sequences of length n. The values of h for s"0 (the chaotic attractor) were compared [125]: (i) by counting N(n), with 2 x-axis as the partition line; and (ii) by using a procedure developed by Newhouse and Pignataro [130]. The counting method (i) yields h (s"0)+0.500, while the Newhouse}Pignataro algorithm (ii) gives h (s"0)+0.466. This suggests 2 2 that when a chaotic saddle has entropy less than about 0.466, its noise-resisting gap has already covered the zigzag generating partition of the attractor. This, in turn, gives an estimation for the value s in Fig. 28b, the minimum gap size
for which a good symbolic dynamics can be de"ned by simply using y"0 as the partition.
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In general, it is advantageous to use chaotic saddles, such as the ones depicted in Fig. 28a, for communication, because the symbolic dynamics on the chaotic saddle are immune to small noise. If the system is in a noisy environment, and the original chaotic attractor is used to encode messages, then a bit error (i.e., 0 becomes 1 or vice versa) may occur whenever the trajectory comes close to the partition curve, because noise can kick the trajectory over the curve in both directions. However, trajectories on the chaotic saddles do not come close to the partition point because of the forbidden region. Thus, the possibility for bit error due to noise can be substantially reduced when a chaotic saddle is utilized to encode messages if the noise amplitude is smaller than s . Generally,
there is a trade-o! between the channel capacity and noise resistance. We now give an example of coding a speci"c message. Suppose we wish to encode the message `BEAT ARMY!a into a trajectory in the chaotic saddle in Fig. 28a. The message `BEAT ARMY!a has the following ASCII representation: ! # 2 0 + 7 CDE CDE CDE CDE CDE CDE CDE CDE CDE CDE 1000010 1100101 1100001 1110100 0100000 1000001 1110010 1101101 1111001 0100001. If the chaotic saddle were equivalent to the fullshift grammar symbolic dynamics, i.e., no grammatical restrictions, then we could simply "nd a trajectory in the x, y plane such that its y itinerary exactly follows the above digital message. However, the symbolic dynamics of the chaotic saddle are subshift-type because its topological entropy is less than ln 2. Dynamics on the saddle is representable by the Bernoulli shift map, on a bi-in"nite symbol space R of two symbols [126,127,131]. A bi-in"nite symbol sequence is a point in the symbolic space: p"2p p p ) p p p 23R, where p "0 or 1, and p is the position of (x , y )3R, relative \ \ G G G G to a partition curve, on the ith (pre)iterate for (i(0) i50. Shifting the decimal to the right represents a forward iteration, and shifting the decimal to the left represents an inverse iteration. To quantify the correspondence between a point x in the phase space and a point (points) in the symbolic space, it is necessary to use a vector function (the so-called coding function [24], corresponding to the `symbolic planea discussed in Ref. [131]): G"(d, c), where d and c are determined by d"1!0.d d 2d ,1! d 2\I , I I c"0.c c 2c , c 2\I , (98) I I where d " I (1!a ) mod(2) and c " I a mod(2). The phase-space dynamics can then be \G I I G G G represented by the following map in the coding space: (d ,c )"D(d , c ), where L> L> L L D(d, c)"+(1!d)/2, 2c, if c(1/2 and D(d, c)"+(1#d)/2, 2!2c, if c51/2. A trajectory of 50 000 points in the symbolic plane corresponding to the chaotic saddle in Fig. 28a is shown in Fig. 29. The forbidden points (blank regions) in the symbolic plane is generated by the pruning front [131]. Fig. 29 thus determines, completely, the grammar on the chaotic saddle, from which a controlling scheme can be derived to encode messages into the trajectories in the chaotic saddle. We note that the pruning front of an embedded chaotic saddle must be ordered less than or equal to the pruning front of the full chaotic attractor, following the fact that the subshift grammar of the chaotic saddle
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Fig. 29. From Ref. [125]. The symbolic plane for the hyperbolic chaotic saddle in Fig. 28a.
must be a subset of the subshift grammar of the attractor. Furthermore, the pruning front must be a monotone nonincreasing curve (i.e., receding), as a function of the increasing gap. This corresponds to the fact that we observe a monotone nonincreasing topological entropy. In what follows we present a practical method to learn the grammar and then to encode digital messages. In physical or numerical experiments, only "nite precision can be achieved and, hence, it is reasonable to choose an n-bit precision approximation (subshift of "nite type). A way to represent the transitions between the allowed n-bit words is to use the directed-graph method in Ref. [55] which was originally discussed for one-dimensional noninvertible chaotic maps (with an in"nite shift space). The directed-graph representation is, however, more general: two-dimensional invertible maps (with a bi-in"nite shift space) requires little modi"cation, for n-bit words, or truncations of the bi-in"nite symbol sequences, which represent n/2 pre-iterates and n/2 future iterates. The main point of a symbol dynamics representation is that each (x, y) state in phase space occupies a neighborhood which corresponds to an n-bit code, labeled as a node on the graph. There are two possible situations: (1) either a 0 or a 1 may be shifted into the n-bit register, and this choice means that one of the message bits may be controlled; or alternatively, (2) only a 0 or a 1 exclusively may be shifted in to the bit register, and this must be a nonmessage bearing `bu!er-bita even if the bit happens to coincide with the next message bit because according to Shannon's information theory [115], an event only carries information if that event is not pre-determined. It is exactly this time spent transmitting the bu!er-bits which causes decreased channel capacity, as measured by the topological entropy. The more of the n-bit words which have the two possible outcomes, 0 or 1, the higher the channel capacity. In numerical experiments, we approximate the symbolic dynamics of the chaotic saddle by using 12-bit words. By using the method outlined in Ref. [55], we encode the message into a trajectory on the chaotic saddle, where the actual phase-space trajectory is shown in Fig. 30a and the corresponding time series y is shown L
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Fig. 30. From Ref. [125]. Example of encoding a message into a chaotic saddle: (a) controlled trajectory in the phase space; and (b) the corresponding time series.
in Fig. 30b. The receiver can completely recover the original message, given the time-series, the location of the symbol partition y"0, and the grammar in the form of the 2L list of transitions. We remark that an alternative method to record the grammar of a two-dimensional map is to use the pruning fronts [131] which was orginally developed for the HeH non map as the analogy in the symbol plane to the one-dimensional kneading theory of Milnor and Thurston [132]. Both theories give a partial order for the symbol representation of a given point, relative to a `maximuma grammatically allowed word. Given a particular n-bit sequence, it is only necessary to check whether both possibilities are grammatically permitted. In the case of the kneading theory, one checks, in the Gray-code ordering, whether both shifting in a 0 and a 1 give new n-bit words which are also below the kneading sequence, which is the maximum sequence corresponding to the symbolic code of the critical point. In the case of the two-dimensional pruning-front theory, one must check that both 0 and 1 lead to symbolic codes ordered `belowa the pruning front; if either shift, say a 0 (or 1), is greater than the pruning front, then that word is grammatically forbidden on the chaotic saddle, and therefore the alternative shift, say the 1 (or 0), is determined. In either case, just as with the directed-graph method of book-keeping the grammar, information theory demands that when the two possible outcomes are permitted, the message bit can be transmitted, but when one of the alternatives (0 or 1) leads to a symbolic code ordered larger than the pruning front, the transmitted bit must be a non-message bearing bu!er bit. 5.1.10. Discussions Nonlinear digital communication using chaos has become a "eld of recent interest. There are two di!erent approaches to the problem. One is to use the principle of synchronous chaos to embed and transmit digital information. Another is to extend the principle of controlling chaos [1,23] to dynamical systems with well-de"ned symbolic dynamics to encode information [21,24,54}57]. The latter approach makes explicit use of the fundamental principle that chaotic For an overview of the "eld of chaos control and synchronization, see the focus issue of Chaos 7(4), 1997.
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systems are natural information sources. By manipulating the symbolic dynamics of a chaotic system in an intelligent way, the system produces trajectories in which digital information is embedded. The methodology reviewed here is within the scope of the second approach. Speci"cally, we address the problem of controlling the symbolic dynamics of chaotic systems. We argue, by explicit computation of the topological entropy, that a dynamical system operating in parameter regimes of transient chaos on nonattracting chaotic saddles usually possess a greater information-bearing capability as compared with the same system in regimes of sustained chaos on attractors. We detail a procedure, which is applicable to chaotic systems described by onedimensional maps, to encode digital information into typical trajectories wandering in the vicinity of the chaotic saddle. Using transient chaos for message encoding has the advantage of strong noise immunity. This notion has been recently pointed out from the perspective of "nding an optimal coding scheme for communicating using chaotic attractors [56,57]. In such a case, a code is necessary for message encoding in order to accommodate the grammar of the chaotic attractor whose topological entropy is typically less than ln 2. It is argued [56,57] that a code always yields a chaotic saddle embedded in the chaotic attractor. The topological entropy of such a chaotic saddle is smaller than that of the original chaotic attractor, although the symbolic dynamics restricted to the chaotic saddle also has noise immunity. Thus, in digital encoding using sustained chaos on attractors, the trade-o! between channel capacity and noise immunity becomes a critical issue. The chaotic saddles considered here are those naturally arising in wide parameter regimes of a nonlinear system. The characteristic di!erence between a natural chaotic saddle and one embedded in a chaotic attractor is that the former usually has the maximally allowed value of the topological entropy. As such, coding is not necessary for communicating with these natural chaotic saddles, while strong noise immunity can be achieved. As transient chaos arises commonly in many situations of physical interest [119,120], we expect the main point and its implications reviewed here to be a major considering factor in designing practical communication schemes. The observation that the topological entropy is greater for transient chaos also seems to hold for some well studied high-dimensional chaotic systems. For instance, it is known that for the HeH non map [40]: (x, y)P(a!x#by, x), the topological entropy increases as a system parameter changes from regimes with chaotic attractors to regimes with transient chaos (see Fig. 2 in Ref. [123]). Communicating with high-dimensional chaotic symbolic dynamics is itself an interesting problem. The main di$culty is to de"ne a generating partition in the phase space so that a good symbolic dynamics can be de"ned [126,131]. Here we wish to point out that utilizing transient chaos may greatly simplify the task of symbolically partitioning the phase space. Take the HeH non map as an example. When the parameter b is "xed at 0.3, it is believed that the map generates a chaotic attractor at a"1.4. In this case, the generating partition is a zigzag curve connecting all primary tangencies between the stable and unstable manifolds, a good computation of which is highly nontrivial [126]. In contrast, for a"3.0, the dynamical invariant set is a hyperbolic chaotic saddle. The symbolic dynamics can be described by two symbols on a full shift, and the generating partition is simply the straight line y"0. With the advantage of having the maximum topological entropy ln 2, the hyperbolic chaotic saddles can serve as excellent information sources for digital encoding in noisy environment. We also remark that the procedure of controlling symbolic dynamics can be e!ectively utilized for controlling transient chaos [133}136], which is still a challenging problem in the "eld of chaos
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control. By encoding an arbitrary but random symbol sequence, the trajectory of the system remains in the vicinity of the chaotic saddle. Insofar as the encoding continues, the chaotic trajectory is stabilized. This may be of value to the important problem of maintaining chaos [52,136}140]. Finally, we have reviewed the feasibility of utilizing two-dimensional symbolic dynamics for communicating with chaos. The main di$culty for chaotic attractors in two-dimensional invertible maps, arising from three-dimensional #ows, is that due to nonhyperbolicity, the generating partition for de"ning a good symbolic dynamics is extremely di$cult to compute. Our idea is that there typically exists an in"nite number of hyperbolic chaotic saddles embedded in the chaotic attractor for which the generating partition can be easily speci"ed. The hyperbolic chaotic saddles have the additional property that their symbolic dynamics are immune to small environmental noise. When chosen properly, the topological entropies of the chaotic saddles can be close to that of the original attractor. These advantages make dynamical systems described by two-dimensional invertible maps potential candidates for nonlinear digital communication. 5.2. The adaptive synchronization of chaos for secure communication In the previous section, we have described how the OGY idea can be implemented for the control of chaotic behaviors, with applications to the process of communicating with chaos. Here, we show that the adaptive technique introduced for the stabilization of UPOs can, in fact, be successfully applied to the problem of synchronization of chaos. The process of synchronizing two identical chaotic systems starting from di!erent initial conditions [22] consists in linking the trajectory of one system to the same values as the other so that they remain in step with each other, through the transmission of a signal. We have already described in details the possibility of encoding a message within a chaotic dynamics [21], and we have pointed out that synchronization of chaos provides a good tool for communicating between a sender and a receiver. In this section, we highlight that the use of a chaotic carrier for the transmission of a message can give rise to security in the communication. Several problems arise in order to assure security. The main one is due to the fact that the sender must transmit to the receiver a part of the information on the dynamical system (in the Pecora}Carroll case [22] this is realized by means of the transmission of one of the system variables). As a result, a clever spy intercepting the communications can try to reconstruct the whole dynamics, hence decoding the message. To prevent reconstructions of the message, Cuomo and Hoppenheim [106] have proposed to use chaos to hide messages, by transmitting a signal which consists in the sum of a chaotic signal and of a given message. Later, Perez and Cerdeira [107] have shown that messages masked by low-dimensional chaotic processes can be intercepted and extracted. Therefore, the attention was directed to the implementation of the Pecora and Carrols (PC) idea to higher dimensional systems [141] with the hope that increased unpredictability could improve security in the communication. Another problems using the PC procedure is that the subsystem to be synchronized must show negative subLiapunov exponents [22]. In other words, synchronization occur in a linear sense, and, because of this requirement, an additive signal used to hide the real message should be an in"nitesimal perturbation of the signal itself, while masking messages with large signals could lead to problems in the synchronization.
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Enrichments of the PC method has been provided an alternative approaches to synchronization based on nonreplica subsystems have been proposed [142], but still the problem of security in the communication is an hot issue. Ref. [65] describes the application of the adaptive scheme for chaos synchronization, and shows how some of the above di$culties can be overcome, leading to a reliable level of security against external interceptions even in the case of low dimensional chaotic systems. The communication scheme in Ref. [65] consists of a message sender (Alice), a receiver (Bob) and a spy (James) ready to intercept and decode all communications between Alice and Bob. Alice consists of two identical chaotic systems x 2 "f (x2 , k) , (99) x 1 "f (x1 , k), where k is a set of control parameters chosen in such a way as to produce chaos, x1 , x2 are two D-dimensional vectors (D53) and f is a nonlinear function. Bob consists of a third identical system (100) x 3 "f (x3 , k) . The three systems start from di!erent initial conditions, and therefore produce unsynchronized dynamics. Ref. [65] considers the Lorenz system, where the vectors xj "(x , y , z ), ( j"1, 2, 3) obey H H H the equations: x "p(y !x ), y "rx !y !x z , z "!bz #x y . (101) H H H H H H H H H H H H The scheme for the communication is reported in Fig. 31. With this scheme, one supposes to have encripted a message within the vector x1 (with the use, e.g. of the encoding techniques described in Section 5.1). The point is how to transmit the message to Bob in a secure way, so as to avoid a possible reconstruction by James. The "rst step is to produce synchronization between x2 and x3 . Bob sends to Alice the variables y (t) which is replaced into the equations for x and z . Synchronization (in the PC sense) is here assured by the fact that the sub-Liapunov exponents for the subsystem (x , z ) are both negative (for p"10, b" and r"60 they are !2.67 and !9.99, respectively [22]). This way, Alice knows the actual dynamical state of Bob and can transmit the perturbation ;(t) to be applied to the x equation in order for Bob to synchronize the system x3 to x1 . Alice makes use of the adaptive method to slave the system x2 to the goal dynamics x1 . Precisely, the same algorithm of Eqs. (54)}(57) is used with actual dynamics x (t) and goal dynamics x (t). The perturbation ;(t) is then given by ;(t)"K(t) (x (t)!x (t)) . (102) The e!ectiveness of this scheme is illustrated in Fig. 32, which reports the temporal behavior of *x""x !x ", measuring the synchronization between Alice and Bob for p"10, b" and r"60. Similar results hold also for "y !y " and "z !z ". As a consequence, the message encoded within x1 is received by Bob. Let us move to discuss the problem of security. James intercepts the two communication signals ;(t) and y (t). No information on x1 can be retrieved from ;(t) since ;(t) vanishes as soon as Alice and Bob reach synchronization, and the weighting factor K(t) is not decided a priori, but it is continuously changed by the adaptive algorithm. This is equivalent to have a time variable decoding key.
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Fig. 31. The scheme for adaptive synchronization. Bob sends to Alice the variable y to synchronize x2 and x3 . Alice sends to Bob the adaptive correction ;(t) to be added to the evolution equation for x . James intercepts both ;(t) and y . Fig. 32. From Ref. [65]. Temporal evolution of the quantity log ("x !x ") measuring the synchronization between x1 and x3 . p"10, b", r"60, K "10, p"0.011.
However, from the knowledge of y , James can reconstruct the whole chaotic attractor corre sponding to x3 through standard embedding techniques, and can easily decode the message once x3 and x1 become synchronized. To prevent for this, Ref. [65] introduces the following trick. Alice and Bob agree on a given accuracy h in the message reception. Once such an accuracy is reached. (Alice can test on it), Bob stops sending y for a given while ¹ . During this time lag the two systems x2 and x3 evolve separately. After ¹ Bob starts again sending y to Alice. Now, if ¹ exceeds the decorrelation time q of the system (which is de"ned as the reciprocal of the maximum Liapunov exponent K), then the e!ective signal sent by Bob results in the sum of uncorrelated temporal subsequences, and the embedding technique fails. Therefore, no reconstruction of x3 is possible by James in this case. Fig. 33 reports the results for ¹ "1 and h"10\ (notice that in the present case KK1.41, hence ¹ 'qK0.71). The synchronization scheme maintains the stipulated accuracy (Fig. 33a) even when the signal sent by Bob is a!ected by large holes (Fig. 33b) preventing external reconstructions of x3 (t). Finally, Fig. 33c shows the controlling signal which remains con"ned within a range negligible with respect to the x dynamics (x variations from !28 to 28). 6. Experimental evidences and perspectives of chaos control 6.1. Introduction The large body of the theoretical proposal on chaos control has stimulated di!erent applications in experimental systems whose natural behavior showed chaos. The stabilization of a dynamical
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Fig. 33. From Ref. [65]. (a) Temporal evolution of log ("x !x ") for h"10\ and ¹ "1'1/KK0.71. The stipulated accuracy in the transmission is perserved in time even though (b) the syncronization signal Bob sends to Alice is a!ected by large holes which prevent any reconstruction of the message, and (c) the controlling signal ;(t) is kept within a range negligible with respect to the dynamics. Other parameters as in Fig. 5.
system toward a state of controlled periodicity allows a lot of possible technological applications, thus motivating the interest for the experimental demonstration of the reliability of the di!erent theoretical techniques, in practical situations where often a mathematical model for the system is unknown, or very detailed dynamical features are practically impossible to be extracted from the outputs. Stabilization and control are old engineering problems [42,143]. Open loop methods and closed loop negative feedback approaches has been developed long time ago and are still extensively used. All experimental systems need some kind of stabilization to give safe operation and outputs. To avoid irregular #uctuations in a system, normal procedure include adding dissipativity to lower the gain, or redesigning the operating parameters so as to "nd a safe operating point, or "ltering with narrow band the desired output, or other similar techniques. In particular, in nonlinear devices where irregular #uctuations are normally obtained, there is a strong practical interest in obtaining an output intensity or a frequency free of irregular
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behaviors. This is the main reason why a lot of di!erent stabilization and control techniques have been used. Stabilization and control in our context means achieving safe operations of a chaotic system onto some unstable orbits giving the possibility of choosing (and changing) a controlled, complex, multiperiodic desired behavior without hard changes in the parameter space of the system. In general, we cannot claim that a control method is better than others for practical experimental implementations. Rather, the performances of the di!erent proposed techniques depend on the particular situation under study. What can be surely a$rmed is that, in order to exploit all the advantages of a control chaos method, one needs an algorithm which takes pro"t of the possibilities o!ered by the chaotic attractor. Therefore, if the control will be performed on a variable, a parameter, modulating the pump or adding negative feedback, mainly depends on the particular problem to be faced. In the following, we will consider some relevant experimental achievements that, in our opinion, may guide the reader toward a family of other works, which use similar or related techniques. We are aware that it would be unrealistic to pretend covering all the body of experimental works which are o!ered in the literature nowadays, and it has been necessary to concentrate our treatment only on few prototypic experiments, duly referring to the other literature on similar matter. Along this line here we want to illustrate both experiments designed to verify some theoretical aspects, and others that intend, within the state of art, to solve some real technological problems. Under this methodology, we will attempt to keep the chronological order. 6.2. Nonfeedback methods Before the OGY method, Lima and Pettini [59] proposed a perturbative technique of stabilizing the chaotic system toward a periodic state. In this case, the periodicity is "xed by the frequency of a control signal perturbing the parameter space. Such a technique was called `suppression of chaosa by the same authors. Its implementation, however, can be complicated by the fact that it needs a preliminary learning task of the system response to possible perturbations of variable amplitude. The e!ects of parametric perturbations was "rst studied by Azevedo and Rezende in Ref. [14], in a control experiment with a system of spin waves excited by microwaves. The experimental system consisted in a sample of yttrium iron garnet (YIG) localized within a waveguide, which was syntonized with the mode ¹E . In these conditions, the system behavior becomes chaotic as far as an external "eld H perturbs the YIG sample in the direction perpendicular to the "eld h of the cavity. The authors did not develop a systematic study of the system dynamics as a function of control parameters. Rather, they limited to the introduction of a periodic modulation in H (H"H #dH cos(2pft)), with amplitude dH about four orders of magnitude smaller than the continuous component H . In this case, the operator suitably scans a large frequency band in f, in order to individuate those forcing frequencies leading to a global periodic behavior of the system. Once the frequencies have been selected, the operator acts on the respective amplitudes in order to get the best control conditions. This experiment should be considered as pioneering, insofar as it demonstrated for the "rst time that a suitable periodic perturbation may produce mode-locking, thus restoring a periodic state for control parameter values much above the threshold of the
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Fig. 34. Experimental setup of the magnetoelastic ribbon control experiment.
appearance of chaos. In other words, it is here experimentally demonstrated that a synchronization mechanism between a chaotic attractor and a periodic perturbation can be considered as a reliable control method in various circumstances. Among the various experimental application of this technique, we here mention the results by Fronzoni et al. [144], who have shown the elimination of chaotic oscillations of a bistable magnetoelastic system, actually represented by the Du$ng}Holmes equation. Recently this technique has been also applied for stabilizing periodic orbits in a single mode CO laser with modulated losses [146]. The authors applied a slow nonresonant (or near-resonant) parametric modulation to the cavity detuning. The addition of the `control signala is able to reduce the periodicity of the periodic orbit (e.g. from period four to period two and to period one) and even to control unstable periodic orbits [147]. 6.3. Control of chaos with OGY method The "rst experimental application of the OGY control method was realized by Ditto et al. at the end of 1990 [9]. In this paper, the authors obtained the stabilization of period one oscillations in the dynamics of a magnetoelastic ribbon. The experimental setup consisted of an amorphous magnetoelastic ribbon of thickness 25 lm, width 3 mm and length 100 mm. The "rst 35 mm were "xed at the basis, allowing only the remaining part of the ribbon to perform horizontal displacements. The choice of the material was carefully done, since amorphous materials allow large variations of their Young modulus for relatively small changes in the applied magnetic "eld H. The system was put inside three Helmoltz reels (perpendicular to each other) in order to compensate for the earthly magnetic "eld. The setup is reported in Fig. 34. In order to force the appearance of the oscillations, a forcing "eld H"H #H cos(2pft) was applied, where H (H ) is the amplitude of a continuous (alternate) magnetic "eld. The deter
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Fig. 35. From Ref. [9]. (a) Time series of X "<(t ) before and after the control was switched ON. (b) The "rst return L L map (X versus X ) for the uncontrolled system (in gray) and the controlled output when the control is turned ON L> L (in black).
mination of the dynamics was done by measuring the horizontal displacements of the ribbon by means of a photonic sensor pointing at a height of 6 mm with respect to the basis of the system. The dynamics of such displacements shows interesting chaotic features. For a detailed discussion on the structure of the chaotic attractor, we here address the reader to Ref. [145]. The application of the OGY to such a system required the localization of a saddle "xed point within the attractor set, with a neighborhood wherein the local dynamics could be considered linear in the perturbations introduced in some control parameter. Ditto et al. then constructed experimentally the time series X "<(t ) (t being the natural sampling time corresponding to the L L L forcing period) of the voltages out from the photodetector. In Fig. 35, this time series was used to construct the return map, for 2350 points, H "0.112 Oe, H "2.050 Oe and f"0.85 Hz. In this framework the period one orbit is individuated by the condition X "X , which was L> L said to be ful"lled within the experimental resolution (in the present case, the error in the measure of the voltage was $0.005 V). For each pair of points, it is possible to construct a local linear map M, and to extract the values of stable (j ) and unstable (j ) eigenvalues, and the corresponding eigenvectors (e , e ). By slightly varying a control parameter (in this case it was the continuous magnetic "eld H ), the authors were able to estimate the variation in the "xed point position, and by the use of OGY technique, to control the desired "xed point. In the present case, the whole execution time of the control line was about three orders of magnitude smaller than the characteristic time of the system oscillations, thus allowing a real time on line control. Within the parameter range of Fig. 35, the authors were able to stabilize the period one orbit for more than 200 000 iterations (about 64 h), using a maximum perturbation of 9% of the unperturbed dynamics. By using the second return map of the experimental time series, the authors were also able to stabilize the period two orbit, with the same limitation in the perturbation strength. This experiment constituted the "rst robust evidence of the reliability of the OGY technique, insofar as it was shown that the control over chaos persisted inspite the presence of experimental
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Fig. 36. From Ref. [15]. (a) First return map (I versus I ) for the uncontrolled system. Bright points correspond to L> L the controlled orbit. (b) The current through the resonator versus time, with the control window `Wa. The control signal (lower trace) is turned ON only when a peak of the signal enters within the window.
noise, which a!ects the determination of the map M, thus of the estimate of the eigenvalues and eigenvectors. 6.4. Control of electronic circuits The two experiments by Azevedo and Rezende [14] and Ditto et al. [9], even though realized with di!erent methods, have a common underlying idea, which is that of forbidding large parameter variations in the control procedure. A qualitative change in this framework was performed by Hunt in Ref. [15], where the perturbations were allowed to be quite large, thus permitting the control of more complicated periodic behaviors, associated with higher periodic orbits of period as large as 23 times the characteristic oscillation period. In this experiment, a simple electronic circuit is used, based on a diode resonator. A p}n junction is excited by an harmonic voltage, and the system shows a series of period doubling bifurcation, eventually entering a chaotic regime. Because of its extreme simplicity, the system comes out to be well described by a bidimensional mappings, which can be directly constructed by means of the experimental observations of the maxima of the current traveling through the junction (see Fig. 36a). By exploiting such a preliminary learning process, the system is fed back with a control signal proportional to the distance between the measured maximum and the expected one for the periodic
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orbit. Due to this procedure, the method itself was called Occasional Proportional Feedback (OPF). Even though the basic concept does not di!er from the OGY one, here the author does not consider explicit limitations for the perturbation strength (which, instead, are peculiar of the formal OGY technique). As a result, the chaotic attractor comes out to be slightly distorted as a consequence of the large variations in the control parameters. Precisely, Hunt de"nes a window in the neighborhood of the current value corresponding to the desired orbit. The width of this window also de"nes the maximum strength of the perturbation. This process is simply realized by the use of a current}voltage converter, which sends the output signal to an electronic comparator, which eventually delivers the control pulse. The duration of the pulse can be adjusted to achieve robustness in the control, but it is always limited to be a rather small fraction of the characteristic oscillation time. The results of Ref. [15] show that small perturbations (about 0.5%) are able to stabilize lower periodic orbits, whereas the process leading to the control of higher periodic orbits is associated with a larger perturbation strength (about 10%). Such large perturbations may induce important modi"cations in the controlled orbits, as it appears evident in Fig. 36b, where a period 5 controlled orbit is shown. If one compares the coordinates of the controlled orbits with those of the unperturbed signal in the "rst return map, the former appear to be quite displaced, thus meaning that a slight deformation of the orbit has been produced. The robustness of the method is highlighted by the control of very large periodic orbits, as large as period 21. In this case, the perturbation is even stronger, and the consequent deformation is more evident. The relevance of such an experiment is due to the "rst evidence of robust control of very high periodic behaviors. 6.5. Control of chemical chaos Occasional Proportional Feedback method was introduced independently by Peng et al. [148] in order to control a model of chemical reactions. Chemical chaos generally correspond to a unpredictable variation in the concentration of some components that enter an oscillatory reaction. The most popular chaotic chemical system is the Belousov}Zhabotinsky (BZ) reaction, where a cerium ion-catalyzed oxidation malonic acid is produced by acidi"ed bromate [149] in a continuous-#ow stirred-tank reactor. The "rst control experiment in chemical chaos was implemented in a BZ reaction by the group of Showalter [17]. The authors here applied the so called single map based algorithm to an oscillatory BZ reaction continuously excited, feedbacking the control signal `*ka of the cerium and bromide solutions entering the reaction tank. The perturbation in the control signal *k"(A !A )/g can be directly calculated as a function L Q of A (the potential in a Bromide electrode), and A (the "xed point value obtained by the "rst L Q return map), while the weighting factor g can be evaluated by measuring the horizontal distance between two maps extracted from a suitable model of the reaction (Fig. 37a) [148,150]. This calculation allows a very precise estimate of the perturbation value necessary for the system to stabilize the desired periodic orbit. In Fig. 37b the potential of bromide electrode as a function of time is presented. The control algorithm was switched on to control a period one orbit (t"27 800 s until t"29 500 s) and a period two orbit (t"30 000 s until t"32 100 s). The control range was set at $30 mV.
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Fig. 37. Reprinted by permission from Nature 361, p. 242, copyright (1993) Macmillan Magazines Ltd. (a) First return map of the concentrations used in the experiment calculated from the GyoK ryi-Field model. The system is directed to the stable "xed point applying a change *k"(A !A )/g. (b) Bromide electrode potential versus time. The control L Q algorithm was switched on at t"27 800 s unit t"29 500 s for controlling a period 1 orbit (p1) and from t"30 000 s until t"32 100 s for the control trial of a period 2 orbit (p2). The control range was set at $30 mV.
The map-based control algorithm has been also implemented for tracking unstable periodic orbits in the same BZ reaction [151]. 6.6. Control of chaos in lasers and nonlinear optics The control of stability in lasers and nonlinear optical devices is frequently necessary. Indeed, from the beginning of laser industry a crucial point was to improve the performance in frequency and intensity, motivated by many problems inherent to the presence of several longitudinal and transverse modes (pulling and pushing of laser modes), couplings in nonlinear internal devices, nonlinearities in ampli"ers and other issues related to the generation of laser radiation or due to interaction of laser with matter. A crucial issue is connected with the important irregular intensity #uctuations that appear in the generation of second harmonic with an intracavity nonlinear device. The nonlinear couplings between modes that are produced in nonlinear crystals give rise to irregular #uctuations in the optical cavity. These #uctuations are ampli"ed from the beginning by the quality factor Q of the laser cavity and by the presence of the laser ampli"er media. Strong #uctuations appear then in the laser intensity. This is clearly a nondesirable situation for practical applications. To give an example of what discussed above, the second harmonic generation (green light) in a diode-pumped intracavity doubled Nd : YAG laser (1.06 lm) [152], is normally accompanied with strong intensity #uctuations. The irregular behavior in this kind of systems was largely investigated [153] and attributed to the destabilization of relaxation oscillations, always present in this kind of lasers, due to the nonlinear coupling of longitudinal modes. Several experimental works have been devoted to eliminate such a chaotic behavior [154]. In the context of chaos control, Roy et al. [16], in a very important experiment, used the Occasional Proportional Feedback method for selecting a series of perturbations of limited durations (`kicksa) to the driver of the Nd : YAG pump diodes. The feedback pulses drive the
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injection current of the pump diodes at periodic intervals and they are proportional to the di!erence between the chaotic output signal and a given reference value. The interval ¹ between successive kicks was adjusted to be roughly the period of the relaxation oscillations. The control parameters of the experiments are, together with the duration of the kick, its amplitude and the reference level. The results are remarkable in the case of a weak chaotic regime. The authors "rst prepare the laser as to operate in that regime, by an appropriated orientation of the nonlinear crystal (KTP). The results can be regarded in Fig. 38. The control of period 2, 4 and 9 periodic orbits was obtained in all cases with small amplitude perturbations (a few percents) operating near threshold. A weak chaotic regime means small output intensity generated in the green frequency (second harmonic). If a signi"cative amount of green light is generated, the regime becomes strongly chaotic and the system is unable to stabilize the output. The authors reported that, by adjusting the reference level to the mean of the chaotic #uctuations and by adjusting the period ¹ to the relaxations oscillations period, the dynamics comes out to realize a stable output. The control voltage #uctuations become very small once the steady state is achieved. The strong green output case represents a new situation, insofar as here if some parameter (like pump intensity) is changed after the steady state is reached, then the control is lost. The control variables need to be adjusted to stabilize the new operation conditions. Several attempts have been made to overcome this problem by means of a systematic tracking of periodic orbits [155]. By tracking and stabilization techniques [156], a stable output is presented in a pump range very far from threshold (as far as three time above threshold). This implies a strong improvement, even though the results are presented here on the 1.06 lm output (that is on the infrared range). Besides the interest of the obtained results, there are two qualitative di!erences between the experiments realized by Ditto et al. and those realized by the group of Roy. The "rst one is the number of degrees of freedom involved in the dynamics of the system under control. In the case of the magnetoelastic ribbon, the dynamic is conveniently described by few strongly localized modes, whereas in the case of the multimode laser, the system is itself highly dimensional. The second di!erence is the order of magnitude of temporal scales on which control must be performed. While in the "rst case the characteristic time scale of oscillations was few seconds, a time su$cient to allow a complicated o! line control scheme, in the second case the time scale is of the same order, or even smaller, than the algorithmic time necessary for the calculation of the control signal. The same group of Roy has later proved experimentally the possibility of synchronizing chaotic lasers [157], with possible application to digital communication [158]. More recently, experiments on communicating with chaos has been carried out in the same group, showing the possibility of encoding and decoding messages with chaotic lasers [159], and the possibility of transmitting a desired message in a very fast way using high-dimensional chaotic waveforms [160,161]. An exhaustive review of di!erent experimental setups for the communication of chaos with timedelayed optical systems is available in Ref. [162]. 6.7. Control of chaos in yuids The "rst control experiments on a convective #uid were reported by Singer and Bau [12,13]. In these experiments, a #uid is con"ned in a toroidal cell in a vertical position. The lower half of the cell is surrounded by a heater, whereas the upper part is maintained at a lower temperature
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Fig. 38. From Ref. [16]. Temporal traces of the output intensity, control signal and FFT corresponding to (a) no control, (b) period 1, (c) period 4 and (d) period 9.
through a thermal bath. This particular system give rise to a dynamics which can be reduced to the Lorenz model [163]. The chaotic regime is constituted by jumps between two possible basins of attraction, corresponding to the two possible orientations in the rotation direction of the #ux inside the convective cell.
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Fig. 39. Temperature di!erence between positions 3 and 9 o'clock around the loop. the change in the sign corresponds to an inversion in the #ow direction.
Chaotic oscillations of the temperature arise associated to the convective #ux. The control parameter for this system is the power supplied to the heater. The used control algorithm consisted in a negative feedback acting on the same heater by a perturbation proportional to the temperature di!erence between the regions where the #ux passes from the upper to the lower part of the cell. In Fig. 39 one can clearly see that the chaotic regime is realized by means of oscillations of increasing amplitude intermittently jumping between the two basins of attraction due to the change in the direction of the main convective #ux. Such chaotic oscillations are suppressed when the control is turned on. Another relevant experiment on convective systems was reported by Petrov et al. [164]. In this case, the authors controlled the oscillations inside of a liquid bridge of 3 mm width su!ering a temperature di!erence of about 153. The relevance of this work is due to the fact that the authors show the e!ectiveness in the control of an unstable isolated solution in the phase space. This is the case of toroidal trajectories corresponding to a quasiperiodic attractor. This particular solution is encountered in the unperturbed system for su$ciently high temperature di!erence across the bridge. The control was implemented in two stages. First of all, the authors study the system response against localized temperature perturbations inside the liquid gap. In the second stage, this learning process is used to select the right perturbation able to drive the system toward the desired state. Fig. 40 shows the toroidal regime and the controlled periodic state. In this case it has been shown how a low-dimensional space}time chaotic dynamics can be conveniently controlled by means of localized perturbations. 6.8. Control of chaos in biological and biomechanical systems Another application of the techniques for chaos control was developed for biomechanical systems. The "rst experiment on chaos control in this framework was reported by Gar"nkel et al. [10,11]. Here, the authors use the OGY method on a preparation of rabbit cardiac muscle. The peculiarity of this case is that the perturbation, instead of acting on the amplitude of the signal, is
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Fig. 40. From Ref. [164]. Second return map constructed from the experimental time series of the temperature measured approximately at the medium of the liquid bridge and near to the free surface. In the "gure can be seen the quasiperiodic regime, and the controlled unstable periodic orbit.
applied to vary the interbeat intervals. Fig. 41 shows the control of period 3 periodic orbit. It should be remarked that the intrinsic di$culty here is constituted by the fact that the perturbation must act unidirectionally, insofar as the operator can reduce but cannot increase the natural duration of the pulse, due to the presence of refractoriness in the experimental system. Another example of control (or anticontrol) of chaos in biological system is the control of the neuronal activity of hippocampal slices realized by Schi! et al. [165]. An interesting suggestion here is that one can, in fact, try to maintain the chaotic state, which sometimes can be preferable. For example, since the epileptic processes are associated with a periodic synchronization of the neuronal tissue, in order to avoid the primer of epilepsis, one can actually implement a technique similar to OGY in order to anticontrol the chaos, that is in order to maintain a chaotic state wherever the system would naturally be pushed onto a periodic one. 6.9. Experimental control of chaos by time delay feedback Another important branch of experimental control of chaos is constituted by di!erent implementations of the method originally introduced by Pyragas [2], that is by the application of a continuous time delayed feedback. The "rst experimental evidences of this type of control was reported by Pyragas and Tamasevicius [166]. The experimental setup was an externally driven nonlinear oscillator with a tunnel diode as negative resistance device. Bielawski et al. [167] use essentially the same diode resonator which was used in the experiment by Hunt [15], but with a higher frequency (about 10.3 MHz). The control signal is selected by comparing the output signal with the same signal delayed by a time q corresponding to the period of the desired orbit. In these conditions, the authors were able to stabilize orbits of di!erent periodicity in the system.
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Fig. 41. Reprinted with permission from Science 257, p. 1230, copyright (1992) American Association for the Advancement of Science. (a}c) Interbeat interval I versus the beat number during the chaotic phases. (d}f ) Corresponding L PoincareH sections.
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The same technique was later successfully used for the control of laser systems [168], namely in the case of a CO laser with intracavity electro optical modulator. Another interesting application of feedback control is the implementation of the so called washout xlter, which was used for the control of the chaotic dynamics generated by both autonomous [169,170] and nonautonomous [171] systems. In both cases the system under control was a CO laser with cavity losses modulated by an electro-optical crystal driven by an external sinusoidal voltage. The control signal is represented by a feedback on such a crystal, with amplitude of the order of few percent of the driving signal. In chemical systems, the implementation of Pyragas' technique has been provided for the Belouzov}Zhabotinski reaction [172,173] and in enzymatic reactions [174]. 6.10. Other experiments It should be here mentioned the large body of experimental realizations of chaos control in laser systems, posterior to the "rst evidence by the group of Roy mentioned in Section 6.5. Control of chaos was realized by means of weak parametric perturbations [175], negative feedback of subharmonic components [171,176,177], proportional feedback with delay [178], addition of a weak second periodic signal in a modulated multimode laser [179}181]. Other recent experiments in lasers are reported in Refs. [182,183]. In mechanical systems, the possibility of improving the OGY method has been shown by the use of a Neural Network for the optimization of the interpretation of the experimental results with the aim of "nding the di!erent unstable periodic orbits [184]. The OGY method has also been optimized by In et al. [185] through an adaptive technique. Furthermore, OGY has been also used by Kiss et al. [186] for the control of the electrodissolution of a rotating copper disk in a phosphoric acid electrolite. A recent electrochemical experiment is reported in Ref. [187]. Moreover, control of chaos was realized in ionization waves that appear in a periodically excited neon glow discharge [188,189]. More recently, chaos control in experimental high-dimensional systems was realized, namely in a double pendulum [190] and in a two coupled diode resonators [191]. In electronics, the most recent achievements are reported in Refs. [192}195]. Finally, two recent review papers contain discussions and References on some experimental applications of the control of chaos [196,197].
Acknowledgements The authors are grateful to F.T. Arecchi, E. Barreto, G. Basti, E. Bollt, A. Farini, R. Genesio, A. Giaquinta, S. Hayes, E. Kostelich, A.L. Perrone, F. Romeiras and T. TeH l for many fruitful discussions. SB acknowledges "nancial support from the EEC Contract no. ERBFMBICT983466. CG was supported by DOE and by a joint Brasil-USA grant (CNPq/NSF-INT). YCL was supported by AFOSR under Grant No. F49620-98-1-0400 and by NSF under Grant No. PHY9722156. HM and DM acknowledge "nancial support from Ministerio de Educacion y Ciencia (Grant N. PB95-0578) and Universidad de Navarra, Spain (PIUNA).
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STATISTICAL PHYSICS OF VEHICULAR TRAFFIC AND SOME RELATED SYSTEMS
Debashish CHOWDHURY, Ludger SANTEN, Andreas SCHADSCHNEIDER Institut fu( r Theoretische Physik, Universita( t zu Ko( ln, D-50923 Ko( ln, Germany
AMSTERDAM } LAUSANNE } NEW YORK } OXFORD } SHANNON } TOKYO
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Statistical physics of vehicular tra$c and some related systems夽 Debashish Chowdhury, Ludger Santen, Andreas Schadschneider* Institut fu( r Theoretische Physik, Universita( t zu Ko( ln, D-50923 Ko( ln, Germany Received November 1999; editor: I. Procaccia Contents 1. Introduction 2. Fundamental and practical questions 2.1. Some fundamental questions 2.2. Some practical questions 3. Some empirical facts and phenomenological explanations 3.1. Acceleration noise 3.2. Formation and characterization of tra$c jams 3.3. Flux}density relation 3.4. Microscopic states of tra$c #ow and phase transitions 3.5. Time- and distance-headways 4. Fluid-dynamical theories of vehicular tra$c 4.1. Lighthill}Whitham theory and kinematic waves 4.2. Di!usion term in Lighthill}Whitham theory and its e!ects 4.3. Greenshields model and Burgers equation 4.4. Navier}Stokes-like momentum equation and consequences 4.5. Fluid-dynamical theories for multi-lane highways and city tra$c 4.6. Some recent results of the #uid-dynamical theories and their physical implications
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5. Kinetic theories of vehicular tra$c 5.1. Prigogine model 5.2. Paveri}Fontana model 5.3. Derivation of the phenomenological equations of the macroscopic #uiddynamical theories from the microscopic gas-kinetic models 6. Car-following theories of vehicular tra$c 6.1. Follow-the-leader model 6.2. Optimal velocity models 7. Coupled-map lattice models of vehicular tra$c 7.1. The model of Yukawa and Kikuchi 7.2. The model of Nagel and Herrmann 7.3. The model of Krauss, Wagner and Gawron 8. Nagel}Schreckenberg cellular automata model of vehicular tra$c on highways 8.1. Relation with other models 8.2. Limiting cases of the NaSch model 8.3. Analytical theories of the NaSch model with periodic boundary conditions 8.4. Spatio-temporal organization of vehicles; is there a phase transition? 8.5. Exact solution of the NaSch model with v "1 and open boundary conditions
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夽 Partially supported by SFB 341 (KoK ln-Aachen-JuK lich). * Corresponding author. E-mail addresses:
[email protected] (D. Chowdhury),
[email protected] (L. Santen),
[email protected] (A. Schadschneider) On leave from Physics Department, IIT, Kanpur 208016, India. Present address: CNRS-Laboratoire de Physique Statistique, Ecole Normale SupeH rieure, 24, rue Lhomond, 75231 Paris Cedex 05, France.
0370-1573/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 1 1 7 - 9
D. Chowdhury et al. / Physics Reports 329 (2000) 199}329 9. Generalizations and extensions of the NaSch model 9.1. Single-lane highways 9.2. Multi-lane highways 9.3. Bidirectional tra$c 10. E!ects of quenched disorder on tra$c 10.1. Randomness in the braking probability of drivers and Bose}Einstein-like condensation 10.2. Random v
10.3. Randomly placed bottlenecks on the roads and the maximum #ux principle 11. Other CA models of highway tra$c 11.1. Fukui}Ishibashi model 11.2. Galilei-invariant vehicle}vehicle interaction and metastability 11.3. CA versions of the optimal-velocity model 11.4. CA from ultra-discretization 12. Cellular automata models of city tra$c and road networks 12.1. Biham}Middleton-Levine model and its generalizations 12.2. Marriage of the NaSch model and the BML model; a `uni"eda CA model of city tra$c 12.3. Practical applications of the models of vehicular tra$c; on-line simulation of tra$c networks 13. Some related systems, models and phenomena 13.1. Generalizations of the TASEP
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13.2. Surface growth, KPZ equation and Bethe Ansatz 13.3. Protein synthesis 13.4. Granular #ow 13.5. The bus route model 13.6. Mobile directional impurities 13.7. Computer networks 14. Summary and conclusion Acknowledgements Appendix A. De"nition of update orders Appendix B. TASEP Appendix C. Naive site-oriented mean-"eld treatment of the NaSch model Appendix D. Paradisical mean-"eld theory Appendix E. Equations of car-oriented theory of NaSch model and COMF approximation Appendix F. The matrix-product Ansatz for stochastic systems Appendix G. Two schemes for solving the mean"eld approximation of the DTASEP Appendix H. Self-consistent equations for v and V v in the mean-"eld approximation W of the BML model Appendix I. Derivation of the equations in the microscopic theory of the BML model References
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Abstract In the so-called `microscopica models of vehicular tra$c, attention is paid explicitly to each individual vehicle each of which is represented by a `particlea; the nature of the `interactionsa among these particles is determined by the way the vehicles in#uence each others' movement. Therefore, vehicular tra$c, modeled as a system of interacting `particlesa driven far from equilibrium, o!ers the possibility to study various fundamental aspects of truly nonequilibrium systems which are of current interest in statistical physics. Analytical as well as numerical techniques of statistical physics are being used to study these models to understand rich variety of physical phenomena exhibited by vehicular tra$c. Some of these phenomena, observed in vehicular tra$c under di!erent circumstances, include transitions from one dynamical phase to another, criticality and self-organized criticality, metastability and hysteresis, phase-segregation, etc. In this critical review, written from the perspective of statistical physics, we explain the guiding principles behind all the main theoretical approaches. But we present detailed discussions on the results obtained mainly from the so-called `particle-hoppinga models, particularly emphasizing those which have been formulated in recent years using the language of cellular automata. 2000 Elsevier Science B.V. All rights reserved. PACS: 45.70.Vn; 02.50.Ey; 05.60.!k Keywords: Cellular automata; Complex systems; Nonequilibrium physics
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1. Introduction The concepts and techniques of statistical physics are being used nowadays to study several aspects of complex systems [1] many of which, till a few decades ago, used to fall outside the traditional domain of physical systems [2]. Physical-, chemical-, earth-, biological- and social-sciences as well as technology meet at this frontier area of inter-disciplinary research. Flow of vehicular tra$c and granular matter [3,4], folding of proteins [5], formation and growth of bacterial colonies [6], biological evolution of species [7] and transactions in "nancial markets [8] are just a few examples of exotic phenomena in such systems. Most of these systems are interesting not only from the point of view of Natural Sciences for fundamental understanding of how Nature works but also from the points of view of applied sciences and engineering for the potential practical use of the results of the investigations. Our review of the current status and future trends of research on the theory of vehicular tra$c (and some related systems) will, we hope, convince you that, indeed, the results of recent studies of complex systems have been `a conceptual revolution, a paradigm shift that has far reaching consequences for the very de"nition of physicsa [9]. For almost half a century physicists have been trying to understand the fundamental principles governing the #ow of vehicular tra$c using theoretical approaches based on statistical physics [3,4,10}13]. The approach of a physicist is usually quite di!erent from that of a tra$c engineer. A physicist would like to develop a model of tra$c by incorporating only the most essential ingredients which are absolutely necessary to describe the general features of typical real tra$c. The theoretical analysis and computer simulation of these models not only provide deep insight into the properties of the model but also help in better understanding of the complex phenomena observed in real tra$c. Our aim in this review is to present a critical survey of the progress made so far towards understanding the fundamental aspects of tra$c phenomena from the perspective of statistical physics. There are two di!erent conceptual frameworks for modelling tra$c. In the `coarse-graineda #uid-dynamical description, the tra$c is viewed as a compressible #uid formed by the vehicles but these individual vehicles do not appear explicitly in the theory. In contrast, in the so-called `microscopica models of vehicular tra$c attention is explicitly focused on individual vehicles each of which is represented by a `particlea; the nature of the interactions among these particles is determined by the way the vehicles in#uence each others' movement. In other words, in the `microscopica theories vehicular tra$c is treated as a system of interacting `particlesa driven far from equilibrium. Thus, vehicular tra$c o!ers the possibility to study various fundamental aspects of the dynamics of truly nonequilibrium systems which are of current interest in statistical physics [14}17]. In order to provide a broad perspective, we describe both `macroscopica and `microscopica approaches although we put more emphasis on the latter. Sometimes the phenomenological equations of tra$c #ow in the `macroscopica models can be obtained from microscopic considerations in the same spirit in which macroscopic or phenomenological theories of matter are derived from their molecular-theoretic description. At present, even within the conceptual framework of `microscopica approach, there are several di!erent types of mathematical formulations of the dynamical evolution of the system. For example, the probabilistic description of vehicular tra$c in the kinetic theory is developed by
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appropriately modifying the kinetic theory of gases. On the other hand, a deterministic description of the motion of individual vehicles is provided by the so-called car-following theories which are based on the basic principles of classical Newtonian dynamics. In contrast, the so-called particlehopping models describe tra$c in terms of a stochastic dynamics of individual vehicles. We explain the guiding principles behind all these formulations. But we discuss in detail the results obtained mainly from the investigations of the recently developed `particle-hoppinga models which are usually formulated using the language of cellular automata (CA) [18]. At present, there is no tra$c model yet which can account for all aspects of vehicular tra$c. In this review we consider a wide variety of CA models which describe various di!erent types of tra$c phenomena. We map the particle-hopping models of vehicular tra$c onto some other model systems; these mappings indicate the possibility of exploiting powerful techniques, used earlier for other systems, to study tra$c models and, sometimes, enable us to obtain results for tra$c models directly from the known results for models of other systems. We present pedagogical summaries of the statistical mechanical treatments of the CA models of tra$c. We critically examine the regimes of validity of the approximation schemes of analytical calculations which we illustrate with explicit calculations in those limiting cases where these usually yield exact results. The results of the theoretical analysis of these models are compared with those obtained from computer simulations and, wherever possible, with the corresponding empirical results from real tra$c. We also compare vehicular tra$c with many other similar physical systems to show the ubiquity of some physical phenomena. Computer simulations are known to provide su$ciently accurate quantitative results when analytical treatments require approximations which are too crude to yield results of comparable accuracy. In this review we demonstrate how computer simulations often help in getting deep insight into various phenomena involved in tra$c and in qualitative understanding of the basic principles governing them thereby avoiding potentially hazardous experiments with real tra$c. Computer simulations of the `microscopica models of tra$c have not only attracted the attention of a growing number of statistical physicists in the recent years, but have also been received positively by many tra$c engineers. The ongoing research e!orts to utilize computer simulations of the microscopic models for practical applications in planning and design of transportation networks have been reviewed very recently by Nagel et al. [19]. Our review is complimentary to those published in the recent years by Helbing [13] and by Nagel et al. [19]. A large number of important papers on tra$c published in recent years are based on the particle-hopping models. But these works have received very little attention in [13]. We discuss the methods and results for the particle-hopping models in great detail in this review after explaining the basic principles of all the theoretical approaches. Moreover, we focus almost exclusively on the fundamental principles from the point of view of statistical physics while Nagel et al. [19] emphasize practical applications which are directly relevant for tra$c engineering. At this point a skeptic may raise a serious question: `can we ever predict tra$c phenomena with statistical mechanical theories without taking into account e!ects arising from widely di!erent human temperaments and driving habits of the individual drivers?a We admit that, unlike the particles in a gas, a driver is an intelligent agent who can `thinka, make individual decisions and `learna from experience. Besides, the action of the driver may also depend on his/her physical and mental states (e.g., sorrow, happiness, etc.). It is also true that the behavior of each individual driver does not enter explicitly into the `microscopica models of tra$c. Nevertheless, as we shall show in
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this review, many general features of tra$c can be explained in general terms with these models provided the di!erent possible behavioural e!ects are captured collectively through a probabilistic description which requires only a few phenomenological parameters. Similar strategies have been suggested also for capturing the behavioural e!ects of some individual traders in "nancial markets collectively through probabilistic descriptions [1]. These probabilistic descriptions make the dynamics of the models intrinsically stochastic. Throughout this article the terms vehicle (or, car) and driver are used interchangeably, although each of these terms usually refers to the composite unit consisting of the vehicle and the driver. The main questions addressed by physicists are posed as problems in Section 2. As a motivation we present some relevant empirically observed general features of real tra$c as well as their plausible phenomenological explanations in Section 3. The following sections review the di!erent theoretical approaches. The classi"cation of the models into di!erent classes is not unique. Mostly, we choose a classi"cation according to the use of discrete or continuous space, time and state variables. The conceptual basis of the older theoretical approaches, namely, the #uid-dynamical theories, the kinetic theories and the car-following theories, are explained, in Sections 4, 5 and 6, respectively, where the corresponding recent developments are also summarized. These model classes are continuous in space, time and state variables. Some coupled map lattice models of tra$c are considered in Section 7. They are discrete in time. CA models are discrete in space, time and state variables. The model suggested by Nagel and Schreckenberg (NaSch) [20] is the minimal model of tra$c on highways; the theoretical results on various aspects of this model are discussed in Section 8 where the nature of the spatio-temporal organization of vehicles are also investigated and the fundamental question of the (im-)possibility of any dynamical phase transition in the NaSch model is addressed. Various generalizations and extensions of the NaSch model (including those for multi-lane tra$c) are reviewed in Section 9. The occurrence of self-organized criticality in the so-called cruise-control limit of the NaSch model is pointed out. It is demonstrated how additional `slow-to-starta rules of CA can give rise to metastability, hysteresis and phase-separation in the generalized NaSch models, in qualitative agreement with empirical observations. In Section 10 the formation and `coarseninga of platoons of vehicles are investigated in an appropriate generalization of the NaSch model with one type of quenched randomness; your attention is drawn to the formal analogy between this phenomenon and the Bose}Einstein condensation. In this section the e!ects of other kinds of quenched disorder on the nature of the steady states of the NaSch model are also considered. In Section 11 we present some other CA models of highway tra$c which are not directly related to the NaSch model. The Biham}Middleton}Levine (BML) [21] model is the earliest CA model of tra$c in idealized networks of streets in cities; it exhibits a "rst order phase transition. A critical review of this model is presented in Section 12 together with a list of its generalizations which have been reported so far. Furthermore, a marriage of the NaSch description of tra$c and the BML model, which has led to the development of a novel model of city tra$c, is explained. A brief status report of the ongoing e!orts to make practical use of the theoretical models for tra$c engineering is also presented. The similarities between various particle-hopping models of tra$c and some other systems far from equilibrium are pointed out in Section 13 followed by the concluding Section 14 where your attention is also drawn towards challenging open questions. Several Appendices deal mostly with more technical aspects of some important calculations, but are not necessary for an understanding of the main text.
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2. Fundamental and practical questions The aim of basic research in trazc science is to discover the fundamental laws governing tra$c systems. The main aim of trazc engineering is on planning, design and implementation of transportation network and tra$c control systems. Statistical physicists have been contributing to tra$c science by developing models of tra$c and drawing general conclusions about the basic principles governing tra$c phenomena by studying these models using the tools of statistical physics. Moreover, using these models, statistical physicists have also been calculating several quantities which may "nd practical applications in tra$c engineering. Furthermore, several groups of statistical physicists are currently also engaged in developing strategies for fast on-line simulation and tra$c control so as to optimize tra$c #ow; signi"cant contributions to this traditional domain of tra$c engineering can reduce the "nancial burden on the governments. 2.1. Some fundamental questions Because of the apparent similarities between the `microscopica models of tra$c and macroscopic samples of ionic conductors in the presence of external electric "eld, 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 tra$c from the `microscopica models is a highly di$cult problem because (apart from the human element involved) (a) the vehicles interact with each other and (b) 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. 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., #uctuation 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 [22] although a condition of pairwise balance [23] holds for some special systems driven far from equilibrium. 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. (i) What are the various dynamical phases of tra$c? Does tra$c exhibit phase-coexistence, phase transition, criticality [24,25] or self-organized criticality [26,27] and, if so, under which circumstances? (ii) What is the nature of yuctuations around the steady states of tra$c? Analogous phenomenon of the #uctuations around stable states in equilibrium is by now quite well understood. (iii) If the initial state is far from a stationary state of the driven system, how does it evolve with time to reach a truly steady state? Analogous phenomena of equilibration of systems evolving from metastable or unstable initial states through nucleation (for example, in a supersaturated
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vapour) or spinodal decomposition (for example, in a binary alloy) have also been studied earlier extensively [28]. (iv) What are the e!ects of quenched (static or time-independent) disorder on the answers of the questions posed in (i)}(iii) above? 2.2. Some practical questions Let us "rst de"ne some characteristic quantitative features of vehicular tra$c. The yux J, which is sometimes also called yow or current, is de"ned as the number of vehicles crossing a detector site per unit time [29]. The distance from a selected point on the leading vehicle to the same point on the following vehicle is de"ned as the distance-headway [29]. The time-headway is de"ned as the time interval between the departures (or arrivals) of two successive vehicles recorded by a detector placed at a "xed position on the highway [29]. The distributions of distance-headways and time-headways are regarded as important characteristic of tra$c #ow. For example, larger headways provide greater margins of safety whereas higher capacities of the highway require smaller headways. Let us now pose some questions which are of practical interest in tra$c engineering. (a) What is the relation between density c and #ux J? In tra$c engineering, this relation is usually referred to as the fundamental diagram. (b) What are the distributions of the distance-headway and time-headway? (c) How should on- and o!-ramps be designed? (d) Does an additional lane really lead to an improvement? (e) What are the e!ects of a new road on the performance of the road network? (f ) What type of signalling strategy should be adopted to optimize the tra$c #ow on a given network of streets and highways? (g) The generalized travelling salesman problem: Is the shortest trip also the fastest?
3. Some empirical facts and phenomenological explanations For several reasons, it is di$cult to obtain very reliable (and reproducible) detailed empirical data on real tra$c. First of all, unlike controlled experiments performed in the conventional "elds of research in physical sciences, it is not possible to perform such laboratory experiments on vehicular tra$c. In other words, empirical data are to be collected through passive observations rather than active experiments. Secondly, unambiguous interpretation of the collected data is also often a subtle exercise because tra$c states depend on several external in#uences, e.g. the weather conditions. The systematic investigation of tra$c #ow has a quite long history [30}32]. Although we now have a clear understanding of many aspects of real tra$c several other controversial aspects still remain intellectual challenges for tra$c scientists. In this section we give an overview of some of the well-understood experimental "ndings, which are relevant for our theoretical analysis in the following sections. Moreover, wherever possible, we provide phenomenological explanations of these empirically observed tra$c phenomena. Furthermore, we shall also mention some of the more recent empirical observations for which, at present, there are no generally accepted explanations.
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3.1. Acceleration noise In general, because of the di!erent human temperaments and driving habits, di!erent drivers react slightly di!erently to the same conditions on a highway, even when no other vehicle in#uences its motion. Consequently, even on an empty stretch of a highway, a driver can neither maintain a constant desired speed nor accelerate in a smooth fashion. In addition to the type of the highway (i.e., the surface conditions, frequency of the curves, etc.) the driver-to-driver yuctuation of the acceleration also depends on the density of vehicles on the highway. 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 since mid-1950s and are well documented [33,34]. 3.2. Formation and characterization of trazc jams Tra$c jam is the most extensively studied tra$c phenomenon. Tra$c jams can emerge because of various di!erent reasons. Most often tra$c jams are observed at bottlenecks, e.g. lane-reductions or crossings of highways [35]. At bottlenecks the capacity of the road is locally reduced thereby leading to the formation of jams upstream tra$c. Downstream the bottleneck, typically, a free-#ow region is observed. In addition, tra$c accidents, which also lead to a local reduction of the capacity of the highway, can give rise to tra$c jams. Fig. 1 shows as an example empirical data of the velocity at a three-lane highway close to an onand o!-ramp [32]. The data show that downstream the bottleneck (at detector C) no slow vehicle has been recorded. In contrast, in the merging regime near detector A vehicles often have to move slowly. In between the on- and o!-ramps the vehicles move with larger velocities compared to those in location A although the number of vehicles passing detector B is maximal. Therefore, the on-ramp causes a local reduction of the capacity of the highway. Perhaps, what makes the study of tra$c jams so interesting is that jams often appear, as if, from nowhere (apparently without obvious reasons) suddenly on crowded highways; these so-called `phantom jamsa are formed by spontaneous #uctuations in an otherwise streamlined #ow. Direct empirical evidence for this spontaneous formation of jams was presented by Treiterer [36] by analyzing a series of aerial photographs of a multi-lane highway. In Fig. 2 the picture from [36] 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 tra$c jam. In the beginning of the analysed time vehicles are well separated from each other. Then, due to #uctuations, a dense region appears which, "nally, leads to the formation of a jam. The jam remains stable for a certain period of time but, then, disappears again without any obvious reason. This "gure clearly establishes not only the spontaneous formation of tra$c jam but also shows that such jams can propagate upstream (opposite to the direction of #ow of the vehicles). Moreover, it is possible that two or more jams coexist on a highway. A more detailed analysis of tra$c jams in absence of hindrances has been given by Kerner and Rehborn [37}39], who pointed out the following characteristic features of wide jams. They found `Stau aus dem Nichtsa in german. The discontinuous trajectories correspond to vehicles changing the lane.
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Fig. 1. Experimental #ow speed-diagrams near on- and o! ramps of a three lane highway. The upper part of the "gure shows the empirical results. Each dot represents a 5-min average of the local measurement. The lines serve merely as guide to the eyes. The lower part of the "gure shows the location of the detectors (from [32]). Fig. 2. Trajectories of individual vehicles on a single lane of a multi-lane highway. The trajectories were drawn from aerial photographs. During the analyzed time-interval the spontaneous formation, propagation and dissolution of a jam has been observed (from [36]).
that the upstream velocity and, therefore, the out#ow from a jam is approximately constant. The out#ow from a jam and the velocity of the jam fronts are now regarded as two important empirical parameters of highway tra$c which can be used for calibrating theoretical models. 3.3. Flux}density relation Obviously, tra$c #ow phenomena strongly depend on the occupancy of the road. What type of variation of #ux and average velocity 1v2 with density c can one expect on the basis of intuitive arguments? So long as c is su$ciently small, the average speed 1v2 is practically independent of c as the vehicles are too far apart to interact mutually. Therefore, at su$ciently low density of vehicles, practically `free #owa takes place. However, from the practical experience that vehicles have to move slower with increasing density, one expects that at intermediate densities, d1v2/dc40 ,
(1)
when the forward movement of the vehicles is strongly hindred by others because of the reduction in the average separation between them. A faster-than-linear monotonic decrease of 1v2 with increasing c can lead to a maximum [29] in the #ux 1J2"1cv2 at c"c ; for c(c , increasing
c would lead to increasing 1J2 whereas for c'c sharp decrease of 1v2 with increase of c would
lead to the overall decrease of 1J2. However, contrary to this naive expectation, in recent years
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some nontrivial variation of #ux with density have been observed. The nature of the variation of the #ux with the density is still not clearly understood [40] since the details of the complex experimental setup can strongly in#uence the empirical results. Fig. 3 shows typical time-averaged local measurements of the density and #ow which have been obtained from the Queen Elizabeth Way in Ontario (Canada) [40]. At low densities the data indicate a linear dependence of the #ow on the density. In contrast strong #uctuations of the #ow at large densities exist which prevents a direct evaluation of the functional form at high densities. In order to use the empirical results for a theoretical analysis it is often more convenient to use the mean values of the #ow at a given density. Fig. 4 shows a collection of possible forms of averaged fundamental diagrams consistent with empirical data [40]. While the discontinuity of the fundamental diagram now seems to be well established [41] no clear answer can be given to the question on the form of the diagram in the free-#ow or high-density regime. In the low-density regime linear as well as nonlinear functional forms of the fundamental diagrams have been suggested. For the high-density branch no consistent picture for the high-density branch exists. Here the results strongly depend on the speci"c road network. In several situations it has been observed that J does not depend uniquely on c in an intermediate regime of density; it indicates the existence of hysteresis ewects and meta-stable states. In the context of tra$c #ow, hysteresis e!ects have the following meaning: if a measurement starts in the free-#ow regime, an increase of the density leads to an increase of the #ow. However, beyond a certain density, a further increase of the density leads to a discontinuous reduction of the stationary #ow (`capacity dropa) and jams emerge. The corresponding fundamental diagram
Fig. 3. Empirical data for #ow and occupancy. The data have been collected by counting loops on a Canadian highway. Both the occupancy and the #ow have been directly measured by the detector. Each point in the diagram corresponds to an average over a time interval of 5 min (from [40]).
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Fig. 4. Schematic representation of fundamental diagrams consistent with empirical investigations (from [40]).
(Fig. 4A and B) then has the so-called &inverse-j form'. Fig. 5 shows an experimental veri"cation of a hysteresis loop [40] at a transition from a free #ow to a congested state. Recent empirical observations which have been obtained near a crossing of highways exhibit [41] a #at plateau (i.e., a density-indepedent #ux) over an intermediate regime of density of the vehicles. 3.4. Microscopic states of trazc yow and phase transitions The results for the #ux}density relation already suggest the existence of at least two di!erent dynamical phases of vehicular tra$c on highways, namely a free-#ow phase and a congested phase. In the free-#ow regime all vehicles can move with high speed close to the speed limit. The nature of the congested tra$c is still under debate. Careful empirical observations in the recent years indicate the existence of two di!erent congested phases, namely, the synchronized phase and the stop-andgo tra$c phase [38,39,42}44]. Vehicles move rather slowly in the synchronized states, as compared to the free-#ow states, but the #ux in the synchronized states can take a value close to the optimum
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Fig. 5. Time-traced measurements of the #ow (from [40]). Each data point corresponds to a time-averaged value of the #ow and density obtained from a local measurement. The average was performed over 5 min intervals.
value because of relatively small headways. Besides, nontrivial strong correlations between the density on di!erent lanes exist in the synchronized state [38,44,45] which actually motivates the notation synchronized tra$c. The stop-and-go tra$c di!ers from the synchronized states in the sense that every vehicle inside the jams come to a complete halt for a certain period of time. Following Kerner three di!erent types of synchronized tra$c can be distringuished by the time-dependent behaviour of the density and #ow [38,39,44]. In synchronized tra$c of type (i) constant values of density and #ow can be observed during a long period of time. In synchronized tra$c of type (ii) patterns of density and #ow quite similar to free #ow states have been observed. The di!erences between synchronized states of type (ii) and free #ow are given by the reduced average velocities and the alignment of the speeds on di!erent lanes in synchronized tra$c. Moreover, irregular patterns of time-traced measurements of the #ow have been found in synchronized tra$c (see Fig. 6) of type (iii). 3.5. Time- and distance-headways The #ux J can be written as J"N/¹ where ¹" , t is the sum of the time-headways G ¹G "(1/N) t is the average TH. recorded for all the N vehicles. Hence, J"1/¹ where G G Therefore, the TH distribution contains more detailed informations on tra$c #ow than that available from the #ux alone. With the variation of density c of the vehicles, ¹ exhibits a minimum at c"c where the #ux is maximum [29].
The results discussed in the preceding subsections are based on time-averaged local measurements. But it is also very useful to analyze the single-vehicle data directly [46}48,41]. The single-vehicle data allow calculation of the time-headway distributions [41]. All the time-headway
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Fig. 6. Fundamental diagrams corresponding to di!erent types of synchronized #ow. Measurements which correspond to synchronized tra$c of type (i) are localized in a narrow region of the #ow density plane, while synchronized states of type (ii) are similar to the results of free-#ow states. In contrast for synchronized tra$c of type (iii) the time-traced measurements show an unsystematic behaviour.
Fig. 7. Time-headway distribution for di!erent density regimes. Top: In free-#ow tra$c the *t-distribution is dominated by two peaks at 0.8 s and 1.8 s. Middle: In synchronized tra$c cars with narrow time gaps are to "nd as well as a dominating peak. Bottom: In stop-and-go tra$c short time headways are surpressed. The peak at 1.8 s remains since vehicles are leaving the jam with a typical temporal headway of approximately 2 s.
distributions in the free-#ow regime show a two peak structure. The "rst peak at *t"0.8 s represents the global maximum of the distribution (see Fig. 7). On a microscopic level these short time-headways correspond to platoons of some vehicles travelling very fast } their drivers are taking the risk of driving `bumper-to-bumpera with a rather high speed. These platoons are the reason for the occurrence of high-#ow states in free tra$c. The corresponding states exhibit
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Fig. 8. Speed-distance relation in free and congested tra$c. It turns out that the typical velocities at large distance headways strongly depend on the present tra$c state (from [41]).
metastability, i.e. a perturbation of "nite magnitude and duration is able to destroy such a high-#ow state [39]. Additionally, a second peak emerges at *t"1.8 s which can be associated with a typical drivers' urge to maintain a temporal headway of +2 s (which is the safe distance recommended in driving schools). Surprisingly, the small time headways have much less weight in congested tra$c. Only the peak at *t"1.8 s is recovered, where the time headway distribution now takes the maximum value. But nevertheless, the small time headways (*t(1.8 s) contribute signi"cantly in synchronized tra$c. In stop-and-go tra$c only the 1.8 s-peak remains and short time-headways are surpressed. The asymptotic behavior is rather unsystematic and re#ects the dynamics of vehicles inside the jams. Another important result characterizing the microscopic states is the dependence of the velocity of individual vehicles on the distance headway (see Fig. 8). This function is also of great importance for theoretical approaches, e.g. it is used as input for the so-called optimal velocity model. In the free-#ow regime it is evident that the asymptotic velocity is reached already for small distance-headways. The slope of the velocity function is much lower than that in the free-#ow regime. Surprisingly, the asymptotic velocity depends strongly on the microscopic state as well as on the density, e.g., in dense tra$c low velocities of the vehicles are also observed even when large distance headways are available.
4. Fluid-dynamical theories of vehicular tra7c When viewed from a long distance, say, an aircraft, #ow of fairly heavy tra$c appears like a stream of a #uid. Therefore, a `macroscopica theory of tra$c can be developed, in analogy with the hydrodynamic theory of #uids, by treating tra$c as an e!ectively one-dimensional compressible #uid (a continuum) [49]. We follow the convention that tra$c is #owing from left to right.
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Suppose, c(x; t) and J(x; t) are the `coarse-graineda density and #ux at an arbitrary location x at an arbitrary instant of time t. The equation of continuity for the #uid representing tra$c is ( ( Rc(x; t) RJ(x; t) # " a (x!x ; t)! b (x!x ; t) , (2) G G H H Rx Rt G H where the "rst and the second terms on the right-hand side take care of the sources and sinks, respectively, at the J on-ramps situated at x (i"1, 2,2, J ) and J o!-ramps situated at G x ( j"1, 2,2, J ). We can write a (x!x ; t) and b (x!x ; t) as H G G H H a (x!x ; t)"a(t) (x!x ) and b (x!x ; t)"b(t) (x!x ) , (3) G G G G G H H H H H where (x!x ) and (x!x ) describe the spatial distribution of the incoming and outgoing #ux, G G H H respectively, while a(t) and b(t) account for the corresponding temporal variations. G H In the following subsections, for simplicity, we shall consider a given stretch of highway with no entries or exits. In such special situations the equation of continuity reduces to the simpler form [50] Rc(x; t)/Rt#RJ(x; t)/Rx"0 .
(4)
One cannot get two unknowns, namely, c(x; t) and J(x; t) (or, equivalently, v(x; t)) by solving only one equation, namely (4), unless they are related to each other. In order to proceed further, one needs another independent equation, say, for v(x; t); we shall write down such an equation later in Section 4.4. An alternative possibility, which Lighthill and Whitham [50] adopted in their pioneering work, is to assume that J(x; t) is determined primarily by the local density c(x; t) so that J(x; t) can be treated as a function of only c(x; t). Consequently, the number of unknown variables is reduced to one as, according to this assumption, the two unknowns c(x; t) and J(x; t) are not independent of each other. 4.1. Lighthill}Whitham theory and kinematic waves As a "rst approximation, let us begin with Lighthill}Whitham assumption that J(x; t)"j(c(x; t)) ,
(5)
where j(c) is a function of c. The functional relation (5) between density and #ux cannot be calculated within the framework of the #uid-dynamical theory; this must be either taken as a phenomenological relation extracted from empirical data or derived from more microscopic considerations. In general, the #ux}density curve implied by Eq. (5) need not be identical with the fundamental diagram in the steady state. Under the assumption (5), the x-dependence of the local #ux J(x; t) arises only from the x-dependence of c(x; t). Alternatively, since J(x; t)"c(x; t)v(x; t), assuming v(x; t) to depend only on c(x; t) the x-dependence of v(x; t) arises only from the x-dependence of c(x; t). Using (5) the equation of continuity (4) can be expressed as
dv Rc(x; t) Rc(x; t) Rc(x; t) Rc(x; t) # v(x; t)#c(x; t) " #v "0 , Rx Rx dc Rt Rt
(6)
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where v "dJ/dc. Note that Eqs. (5) and (6) form the complete system of dynamical equations governing tra$c #ow in this "rst approximation. However, Eq. (6) is nonlinear because, in general, v "dJ/dc"v(c)#c dv(c)/dc depends on c. If v were a constant v , independent of c, Eq. (6) would become linear and the general solution would be of the form c(x; t)"f (x!v t) where f is an arbitrary function of its argument. In that case, the solution of any particular problem would be found by merely matching the function f to the corresponding given initial and boundary conditions. Such a solution describes a density wave motion as an initial density pro"le would get translated by a distance v t in a time interval t without any change in its shape. However, the nonlinearity of Eq. (6) gives rise to subtleties which are essential to capture at least some aspects of real tra$c. The solution of the nonlinear equation (6) is of the general form c(x; t)"F(x!v t) , (7) where F is an arbitrary function of its arguments. If we de"ne a wave to be `recognizable signal that is transferred from one part of a medium to another with a recognizable velocity of propagationa [51] then the solutions of the form (7) can be regarded as a `density wavea. There are several similarities between the density wave and the more commonly encountered waves like, for example, acoustic or elastic waves. But, the acoustic or elastic waves are solutions of linearized partial di!erential equations whereas Eq. (6) is nonlinear and, hence, v is c-dependent. Besides, the waves of the type (7) are called kinematic waves [50}52] to emphasize their purely kinematic origin, in contrast to the dynamic origin of the accoustic and elastic waves. From the initial given density proxle c(x; 0) the pro"le c(x; *t) at time *t can be obtained by moving each point on the initial pro"le a distance v (c)t to the right; obviously, the distance moved is di!erent for di!erent values of c. The time-evolution of the density pro"le can be shown graphically [50,51] on the space}time diagram (i.e., the x}t plane) where an arbitrary point x on the t"0 axis moves along a straight line of slope v (c) if the initial density at x is c. These straight lines are referred to as characteristics; di!erent characteristics corresponding to di!erent c have di!erent slopes v (c). The speed v (c) of the density wave should not be confused with v(c), the actual speed of the continuum #uid representing tra$c. In fact, at any instant of time v(x; t) can be obtained from the corresponding density pro"le c(x; t) by using the relation v(x; t)"j(c(x; t))/c(x; t). Moreover, since v "v(c)#c dv(c)/dc and since dv(c)/dc(0, the speed of the density wave is less than that of the #uid. Therefore, the density wave propagates backward relative to the trazc and the drivers are thereby warned of density #uctuations ahead downstream. Furthermore, the density wave moves forward or backward relative to the road, depending on whether c(c or c'c where
c corresponds to the maximum in the function j(c).
When J(c) is convex, i.e., dJ/dc(0, we have dv /dc(0; consequently, higher values of c propagate slower than lower values of c thereby distorting the initial density pro"le. On the other hand, when dv /dc'0 higher values of c propagate faster and the distortion has the opposite tendency as compared to the case of dv /dc(0. In both the situations the distortion of the initial density pro"le is caused by the c-dependence of v which arises from the nonlinearity of Eq. (6). The distortion of the density pro"le with time can also be followed on the space}time diagram. If dv /dc(0, in regions of decreasing density (i.e., c(x )'c(x ) for x (x ) the characteristics move away from each other whereas, in regions of increasing density, the characteristics move towards each other.
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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 de"nition, a shock represents a mathematical discontinuity in c and, hence, also in v. The speed of a shock wave is given by v "(J(c>)!J(c\))/(c>!c\) (8) where c> and c\ are, respectively, the densities immediately in front (downstream) and behind (upstream) the shock while J(c>) and J(c\) represent the corresponding downstream and upstream #uxes, respectively. Note that the shock wave moves downstream (upstream) if v is positive (negative). Often the shock is weak in the sense that the relative discontinuity (c>!c\)/c\ is small and in such cases the shock wave speed tends to v "dJ/dc. As a shock separates a section of high and low densities of the model, it corresponds to a section of a highway where a free-#ow and a congested regime is present. In particular for large di!erences between c> and c\ the velocity of the shock can be interpreted as the velocity of a backwards moving jam. One advantage of the kinematic approach outlined above over any dynamic approach is that the dynamical equation, which will be given in Section 4.4, is di$cult to derive from basic "rst principles and usually involve quite a few phenomenological parameters and even a phenomenological function. On the other hand, the only input needed for the kinematic approach is the phenomenological function J(c) which can be obtained from empirical data. 4.2. Diwusion term in Lighthill}Whitham theory and its ewects Let us now make improvement over the original Lighthill}Whitham theory, which is based on the "rst approximation (5). We now assume that the local #ux J(x; t) is determined not only by the local density c(x; t) but also by the gradient of the density. In other words, we replace the assumption (5) by J(c)"j(c)!D Rc/Rx
(9)
where D is a positive constant. Note that, for "xed c(x; t) (and, hence, "xed j(c)), a positive (negative) density gradient leads to a lower (higher) #ux as the drivers are expected to reduce (increase) the speed of their vehicles depending on whether approaching a more (less) congested region. Using relation (9) in the equation of continuity (4) we now get Rc(x; t)/Rt#v Rc(x; t)/Rx"D Rc(x; t)/Rx (10) where v (c)"dj(c)/dc. Eq. (10) reduces to the equation (6) when D"0. The nonlinearity and di!usion have opposite e!ects: the term v (c) Rc/Rx leads to `steepeninga and ultimate `breakinga of the wave whereas the term DRc/Rx smoothens out the pro"le. Nonvanishing D also leads to a nonzero width of the shock wave. 4.3. Greenshields model and Burgers equation So far in the preceding subsections we have not considered any speci"c form of the function j(c) relating #ux with density. One can start with the simplest (di!erentiable) approximation capturing
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the basic form of the fundamental diagram, J"v c(1!c) . (11)
Note that v in (11) is a phenomenological parameter and it is interpreted to be the maximum
average speed for cP0. In tra$c science and engineering, one usually uses 1!c/c instead of
1!c in Eq. (11) and the corresponding form of the relation between J and c is known as the Greenshields [30] model. Substituting (11) into Eq. (10) we get Rc(x; t) Rc Rc(x; t) Rc(x; t) #v !2v c "D .
Rx
Rx Rx Rt
(12)
Introducing the linear transformation of variables x"v t !x , t"t ,
one gets Rc(x; t) Rc(x; t) Rc #2v c "D
Rx Rt Rx
(13)
(14)
which is the (deterministic) Burgers equation [51,53]. Note that the transformation (13) takes one from the space-"xed coordinate system (x, t) to a coordinate system (x , t ) that moves with uniform speed v ; so, vehicles moving with speed v 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 the theory of tra$c #ow is that the Burgers equation (14) can be transformed further into a di!usion equation, thereby getting rid of the nonlinearity, through a nonlinear transformation called the Cole}Hopf transformation [51]. Since it is straightforward to write down the formal solution to the di!usion equation, one can see clearly the role of the coe$cient D and the nature of the solutions in the limit DP0. If Eq. (6) is assumed to be the only equation governing tra$c #ow then an inhomogeneous initial state can lead to a shock wave but the amplitude of the shock wave decreases with time and eventually the shock wave fades out leading to a homogeneous steady state in the limit tPR. Leibig [54] has studied how a random initial distribution of steps in the density pro"le evolves with time in this theory. No tra$c jam forms spontaneously from a state of uniform density at this level of sophistication of the #uid-dynamical approach. 4.4. Navier}Stokes-like momentum equation and consequences Corresponding to the assumption (9) we can write a velocity equation D Rc(x; t) , v(x; t)"v(c(x; t))! c Rx
(15)
where v(c)"j(c(x; t))/c(x; t). In the kinematic approach discussed so far in the preceding subsections it is implicitly assumed that, following any change in the local density (and density gradient) is followed by an immediate response (without delay) of the velocity "eld. For a more realistic description, the local speed should be allowed to relax after a nonzero delay time t . So, it seems
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natural to treat the right-hand side in (15) as a desired local velocity at x and write the totalderivative dv/dt of v with respect to time as [55}58] Rv Rv c Rc(x; t) c #v " [v (c)!v]!D (16) Rt Rx t Rx where the function v (c) is identical to v(c). Note that v (c) is a monotonically decreasing function of c, i.e. dv /dc(0. Eq. (16) is an additional dynamical equation describing the time dependence of the velocity v(x; t). Now let us interpret the two terms on the right-hand side of (16). The phenomenological function v (c) gives the safe speed, corresponding to the vehicle density c, that is achieved in time independent and homogeneous tra$c #ow and t is the corresponding average relaxation time. Next, note that the term D Rc(x; t)/Rx takes into account the natural tendency of the drivers to accelerate (decelerate) if the density gradient is negative (positive), i.e. if the density in front becomes smaller (larger); therefore, it can be interpreted as proportional to the pressure gradient in the #uid describing tra$c. In addition to these terms, another term proportional to Rv/Rx is also added to the right-hand side of the velocity equation; this tends to reduce spatial inhomogeneities of the velocity "eld and is usually interpreted as the analogue of the viscous dissipation term in the Navier}Stokes equation. Thus, "nally, in the #uid-dynamical approach, a complete mathematical description of the vehicular tra$c on highways is provided by two equations, namely, the equation of continuity (2) and the Navier}Stokes-like velocity equation [59}62]
Rv Rv Rc Rv c #v "!D #k # [v (c)!v] , Rt Rx Rx Rx t where D, k and t are phenomenological constants. c
(17)
4.5. Fluid-dynamical theories for multi-lane highways and city trazc One can describe the tra$c on two-lane highways [63] by two equations each of the same form (2) and where the source term in the equation for lane 1 (lane 2) takes into account the vehicles which 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 tra$c has been formulated recently [64]. This #uiddynamical model is motivated by the CA model, developed by Biham et al. [21], which will be discussed in detail later in this review. Instead of generalizing the Navier}Stokes equation (17) a simpler form of the velocity equation has been assumed. 4.6. Some recent results of the yuid-dynamical theories and their physical implications The #uid-dynamical model of vehicular tra$c has been studied numerically by discretizing the partial di!erential equations (2) and (17) together with appropriate initial and boundary conditions. Both periodic boundary conditions and open boundary conditions with time-independent external #ux a(t)"b(t)"c have been considered. G G
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In the #uid-dynamical theory, based on the equation of continuity (2) and the Navier}Stokeslike equation (17) tra$c jams can appear spontaneously, even if the initial density pro"le c(x; 0) deviates very little from the homogeneous state (Fig. 9). In order to understand the physical mechanism of the formation of local cluster of vehicles, let us consider a local increase of the density *c(x)'0 at some location x. Since dv /dc(0, the local increase of the density leads to a decrease of v . This decrease in the safe velocity forces drivers to reduce their average velocity v sharply if "dv /dc" is large enough. On the other hand, it follows from the equation of continuity that the local decrease of v gives rise to further increase of c around x and, consequently, further subsequent decrease of v(x) in this location. This avalanche-like process, which tends to increase the amplitude of the local #uctuation of the density around the homogeneous state, competes against other processes, like di!usion and viscous dissipation, which tend to decrease inhomogeneities. To our knowledge, the "rst attempt to understand the physical mechanism of synchronized tra$c within the framework of the #uid-dynamical formalism was made by Lee et al. [62]. On a "nite stretch of highway, of length ¸, they installed an on-ramp and an o!-ramp on the model highway with a separation of ¸/2 between them. They chose the spatial distribution of the external #ux (x) in Eq. (3) as
(x)"(2pp)\ exp(!x/2p)
(18)
with p"56.7 m. They also assumed the form v (c)"v (1!c/c( )/[1#E(c/c( )F] for the safe velocity with adjustable parameters v , E, h, c( . Lee et al. [62] "rst allowed the system to reach the steady state after applying a weak time-independent #ux a(t)"b(t)"c and simulating the G G time evolution of the tra$c by solving simultaneously the equation of continuity (2) and the Navier}Stokes-like equation (17) with a speci"c set of chosen values for the parameters t , D, k, v , E, h, c( , etc. Since they chose c(c , an initially homogeneous tra$c reaches a steady `free-#owa where homogeneous regions with di!erent densities are separated from each other by narrow `transition layersa near the ramps (no stable `free-#owa exists if c5c ). Then, they applied a pulse of additional #ux dq at the on-ramp for a short duration dt. After a transient period, which depends on the parameters of the model, the system was found to settle in a limit cycle in which the local density and local #ux oscillate periodically and the oscillations are localized near the on-ramp. The discontinuous change of the spatio-temporally averaged velocity induced by the
Fig. 9. Time evolution of a density pro"le in the KuK hne}Kerner}KonhaK user model [57}59] consisting of Eqs. (2) and (17) with a special choice of v (c). Initially an approximately constant density pro"le is assumed. The "gure shows that the homogeneous state is not stable, already for small deviations from a #at pro"le (from [13]).
A brief summary of that work can be found in [65].
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localized perturbations of "nite amplitude, associated hysteresis e!ects and the stability of the limit cycle were found to be qualitatively similar to some of the empirically observed characteristics of synchronized #ow in real tra$c; therefore, Lee et al. [62] identi"ed the limit cycle observed in their theoretical investigation as the synchronized state of vehicular tra$c. They drew analogy between this state and a `self-excited oscillatora [62]. However, this mechanism of the synchronized state is not yet accepted as the true and only possible explanation of the phenomena associated with the synchronized state observed empirically. Meanwhile similar results have been obtained for a gas-kinetic-based tra$c model [66], also using on- and o!-ramps in order to explain the transition from free-#ow to synchronized states. This work was completed in a recent paper, where a phase diagram was calculated, which depends on the on-ramp activity and the #ow on the highway [67]. Summarizing the recent results of the macroscopic tra$c models, there seems to be evidence that on- and o!-ramps play an important role for a theoretical explanation of synchronized tra$c. Nevertheless, some experimental features are still not captured by these approaches. E.g. the empirical results show that for synchonized tra$c of type (iii) no correlations between density and #ow exist [41], in contrast to the regular patterns of the oscillating states found in simulations of the macroscopic models. Despite its success in capturing many aspects of tra$c #ow the #uid-dynamical approach has its limitations; for example, viscosity of tra$c is not a directly measurable quantity. Nevertheless, the #uid dynamical approach is being pursued not only by some physicists but also by several members of the tra$c engineering community [68,69].
5. Kinetic theories of vehicular tra7c In the kinetic theory, tra$c is treated as a gas of interacting particles where each particle represents a vehicle. The various di!erent versions of the kinetic theory of vehicular tra$c [12,70}76] have been developed by modifying the kinetic theory of gases. Recall that in the kinetic theory of gases [77] f (r, p; t)dr dp denotes the number of molecules which, at time t, have positions lying within a volume element dr about r and momenta lying within the momentum-space element dp about p. The Boltzmann equation, which describes the time-evolution of the distribution f (x, v; t), is given by
p Rf Rf # ) #F ) f (r, p; t)" N Rt m P Rt
,
(19)
where the symbols and denote gradient operators with respect to r and p, respectively, while P N F is the external force. The term (Rf/Rt) represents the rate of change of f, with time, which is caused by the mutual collisions of the molecules. In the "rst of the following two subsections we present the earliest version of the kinetic theory of vehicular tra$c which was introduced by Prigogine and co-workers [70,12] by modifying some of the key concepts in the kinetic theory of gases and by writing down an equation analogous to the Boltzmann equation (19). In the subsequent subsection we discuss the kinetic theory developed later by Paveri-Fontana [71] to cure the defects from which the Prigogine theory was found to su!er.
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5.1. Prigogine model Suppose 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 co-workers [70,12] introduced a desired distribution f (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 in#uences, e.g., road conditions, weather conditions or interaction with other vehicles, etc. They also argued that some of these in#uences 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 f over a relaxation time q provided mutual interactions of the vehicles is negligibly small. On the basis of these arguments, Prigogine and co-workers [70,12] suggested that the analogue of the Boltzmann equation for the tra$c should have the form
Rf Rf Rf #v " Rt Rx Rt
#
Rf Rt
(20)
where (Rf/Rt) accounts for the relaxation of f towards f in the absence of mutual interactions of the vehicles while (Rf/Rt) accounts for the changes of f arising from mutual interactions among the vehicles. Note that the term (Rf/Rt) on the right-hand side of (20) may be interpreted as the analogue of the term (Rf/Rt) in Eq. (19) whereas the term (Rf/Rt) in Eq. (20) may be interpreted as the counterpart of the term F ) f (r, p; t) in Eq. (19). N Prigogine and co-workers wrote down an explicit form for the term (Rf/Rt) by generalizing that for the term (Rf/Rt) 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 Eq. (20) they assumed that (i) the collective relaxation, which would cause the actual distribution to tend towards the desired distribution, involves only a single relaxation time q so that (Rf/Rt) "!( f!f )/q (21) and (ii) the desired speed distribution F (v) remains independent of the local concentration c(x; t) so that f (x, v; t)"c(x; t)F (v) . (22) Therefore, a more explicit form of the Boltzmann-like equation (20) in the Prigogine theory is given by
Rf f (x, v; t)!c(x; t)F (v) Rf Rf # . (23) #v "! Rx q Rt Rt 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 f (x, v; t) and this scenario of collective relaxation of f towards f has subsequently come under severe criticism [71]. Analyzing a set of `ideal experimentsa in the light of the Prigogine theory, Paveri-Fontana [71] showed that the results obtained from the Boltzmann-like equation (23) are physically unsatisfactory. More recently, Lehmann [72] has attempted to revive the Prigogine approach by reformulating it as a semi- phenomenological theory where the distribution f (x, v; t) is assumed to follow the
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simpler form Rf f!f (v, c) Rf #v "! (24) Rx q Rt and the e!ects of the interactions are taken into account implicitly through a density-dependent desired distribution function f (v, c) which has to be determined empirically. 5.2. Paveri}Fontana model In order to remove the conceptual as well as mathematical drawbacks of the Prigogine model of the kinetic theory of vehicular tra$c, Paveri-Fontana [71] argued that each vehicle, in contrast to the molecules in a gas, has a desired velocity towards which its actual velocity tends to `relaxa in the absence of `interactiona with other vehicles. Thus, 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, g(x, v, v ; t) dx dv dv 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 v and v #dv . The one-vehicle actual velocity distribution function
f (x, v; t)" dv g(x, v, v ; t)
(25)
describes the probability of "nding 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
f (x, v ; t)" dv g(x, v, v ; t)
(26)
describes the probability of "nding a vehicle between x and x#dx having desired velocity between v and v #dv . The local density of the vehicles c(x; t) at the position x at time t can be obtained from
dv dv g(x, v, v ; t) . Similarly, the corresponding average actual speed 1v(x; t)2 and the average desired speed 1v (x; t)2 are de"ned as c(x; t)"
dv dv vg(x, v, v ; t) , 1v(x; t)2" c(x; t) dv dv v g(x, v, v ; t) . 1v (x; t)2" c(x; t) Finally, the local #ux J(x; t) is de"ned as J(x; t)"c(x; t)1v(x; t)2. Now let us assume that the desired velocity of each individual driver is independent of time, i.e., dv /dt"0. Of course, the drivers may also adapt to the changing tra$c environment and their
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desired velocities may change accordingly. In principle, these features can be incorporated into the kinetic theory 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"(v !v)/q where q is a relaxation time. The Boltzmann-like kinetic equation for g(x, v, v ; t) can be written as R Rg R R v !v #v g# g " . (27) Rt q Rt Rx Rv In order to write down an explicit form of the `interaction terma we have to model the interactions among the vehicles. First of all, 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!P where P 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. (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 threevehicle (or multi-vehicle) interactions. (vi) The postulate of `vehicular chaosa, which is the analogue of the postulate of `molecular chaosa in the kinetic theory of gases, holds, so that the two-vehicle distribution function g (x, v, v , x , v , v ; t) can be approximated as a product of two one-particle distributions g(x, v, v ; t) and g(x , v , v ; t), i.e., g (x, v, v , x , v , v ; t)Kg(x, v, v ; t)g(x , v , v ; t). Thus, Eq. (27) can be written explicitly as
R R R v !v #v g# g "f (x, v; t) dv (1!P )(v !v)g(x, v , v ; t) Rt q Rx Rv T T (28) !g(x, v, v ; t) dv (1!P )(v!v ) f (x, v ; t) , where the form of the `interaction terma on the right-hand side of Eq. (28) follows from the assumptions (i)}(vi) above. The "rst term on the right-hand side of (28) describes the `gaina of probability g(x, v, v ; 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, v ; 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, v ) is, by de"nition, 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-di!erential equation (28) by solving it analytically even for the simplest possible choice of the desired distribution function although numerical solutions [74,76] provide some insights into the regimes of validity of Eq. (28) and gives indications as to the directions of further improvements of the Paveri-Fontana model. For example, the "nite sizes of the vehicles must be taken into
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account at high densities. Besides, the assumption (iv) of instantaneous relaxation has also been relaxed in a more recent extension [74]. 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 appropriately, as quasi-one-dimensional. These neither deal explicitly with g (x, v, v ; t) for the individual lanes G (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 tra$c to deal with diwerent types of vehicles on multi-lane highways. Suppose g?(x, v, v ; t) G is the distribution for vehicles of type a on the ith lane of the highway. Obviously, the Boltzmannlike equations for the di!erent lanes are coupled to each other. However, one needs additional postulates to model the lane-changing rules [78,79]. Very little work has been done so far on developing kinetic theories of two-dimensional tra$c #ow which would represent, for example, tra$c in cities. Suppose, for simplicity, that the network of the streets consists of north}south and east}west streets and that east}west streets allow only east-bound tra$c while only north-bound tra$c #ow takes place along the north}south streets. Let P (P ) denote the probabilities of an east-bound vehicle passing another east-bound (northVV VW bound) vehicle. Similarly, suppose, P (P ) denote the probabilities of a north-bound vehicle WV WW passing an east-bound (north-bound) vehicle. Let g (x, y, u, u ; t) and g (x, y, v, v ; t) represent V W the distributions for the east-bound and north-bound vehicles, respectively, where u and v refer to the actual their actual velocities whereas u and v refer to the corresponding desired velocities. The Boltzmann-like equations governing the time evolutions of these distributions are given by [80]
R Rg R R R u !u V #u #v g # g " V Rt q Rt Rx Ry V Ru
V
#
Rg V Rt
,
(29)
W
where
Rg V Rt
V
"f (x, y, u; t) V
S
du (1!P )(u !u)g (x, y, u , u ; t) VV V
!g (x, y, u, u ; t) V
S
du (1!P )(u!u )f (x, y, u ; t) VV V
(30)
and
dv f (x, y, v ; t) du (1!P )u g (x, y, u , u ; t) W VW V W (31) !g (x, y, u, u ; t) dv (1!P )uf (x, y, v ; t) . VW W V The "rst term on the right-hand side of Eq. (30) describes gain of population of east-bound vehicles with velocity u because of interaction with other east-bound vehicles with velocity u 5u while the second term describes the loss of population of east-bound vehicles because of interaction with east-bound vehicles of velocity u (u. The right-hand side of Eq. (31) is based on the assumption Rg V Rt
"d S
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that when an east-bound vehicle interacts with a north-bound vehicle at a crossing, it either passes or stops. 5.3. Derivation of the phenomenological equations of the macroscopic yuid-dynamical theories from the microscopic gas-kinetic models In this section we discuss the results of the attempts to derive the phenomenological equations of tra$c #ow in the macroscopic #uid-dynamical theories from the microscopic gas-kinetic models. Several attempts have been made so far to derive the equation of continuity and the Navier}Stokes-like equation for tra$c from the corresponding Boltzmann-like equation in the same spirit in which the derivations of the equation of continuity and Navier}Stokes equation for viscous #uids from the Boltzmann equation have been carried out. However, because of the postulate of `vehicular chaosa, Eq. (28) is expected to be valid only at very low densities where the correlations between the vehicles is negligibly small whereas tra$c is better approximated as a continuum #uid at higher densities! Let us de"ne the moments
m l (x; t)" dv dv vIvl g(x, v, v ; t) . I
(32)
Note that c"m , 1v2"m . Integrating the Boltzmann-like equation (28) over the actual velocities we get R R f (x, v ; t)# [v (x, v ; t) f (x, v ; t)]"0 , Rx Rt where v (x, v
(33)
; t) is de"ned as
dv vg(x, v, v ; t) . (34) ; t)" f (x, v ; t) Eq. (33) is an equation of continuity for each desired speed v separately; it is consequence of the assumption that dv /dt"0, i.e., no driver changes the desired speed. Using the Boltzmann-like equation (28) and the de"nition (32) we can get separate partial di!erential equations for the moments of v, moments of v and the mixed moments of v and v . Unfortunately, these lead to a hierarchy of moment equations where each evolution equation for moment of a given order involves also moments of the next higher order. In order to close this system of equations, one needs to make appropriate justi"able assumptions. v (x, v
6. Car-following theories of vehicular tra7c In the car-following theories [10,11,81] one writes, for each individual vehicle, an equation of motion which is the analogue of the Newton's equation for each individual particle in a system of interacting classical particles. In Newtonian mechanics, the acceleration may be regarded as the
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response of the particle to the stimulus it receives in the form of force which includes both the external force as well as those arising from its interaction with all the other particles in the system. Therefore, the basic philosophy of the car-following theories [10,11,81] can be summarized by the equation [Response] J[Stimulus] , L L
(35)
for the nth vehicle (n"1, 2,2). Each driver can respond to the surrounding tra$c conditions only by accelerating or decelerating the vehicle. Di!erent forms of the equations of motion of the vehicles in the di!erent versions of the car-following models arise from the di!erences in their postulates regarding the nature of the stimulus (i.e., `behavioural forcea or a `generalized forcea [82]). The stimulus may be composed of the speed of the vehicle, the di!erence in the speeds of the vehicle under consideration and its lead vehicle, the distance-headway, etc., and, therefore, in general, xK "f (v , *x , *v ) , (36) L L L L where the function f represents the stimulus received by the nth vehicle. Di!erent versions of the car-following models model the function f di!erently. In the next two subsections we discuss two di!erent conceptual frameworks for modelling f . 6.1. Follow-the-leader model In the earliest car-following models [83,84] the di!erence in the velocities of the nth and (n#1)th vehicles was assumed to be the stimulus for the nth vehicle. 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 xK (t)"(1/q)[x (t)!x (t)] , (37) L L> L where q is a parameter that sets the time scale of the model. Note that 1/q in Eq. (37) can be interpreted as a measure of the sensitivity coe$cient S of the driver; it indicates how strongly the driver responds to unit stimulus. According to such models (and their generalizations proposed in the 1950s and 1960s) the driving strategy is to follow the leader and, therefore, such car-following models are collectively referred to as the follow-the-leader model. Pipes [84] derived Eq. (37) by di!erentiating, with respect to time, both sides of the equation *x (t)"x (t)!x (t)"(*x) #qx (t) (38) L L> L L which encapsulates his basic assumption that (a) the higher is the speed of the vehicle the larger should be the distance-headway, and, (b) in order to avoid collision with the leading vehicle, each driver must maintain a safe distance (*x) from the leading vehicle. It has been argued [85] that, for a more realistic description, the strength of the response of a driver at time t should depend on the stimulus received from the other vehicles at time t!¹ In the following we label the vehicles in driving direction such that the (n#1)th vehicle is in front of the nth vehicle.
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where ¹ is a response time lag. Therefore, generalizing Eq. (37) one would get [85] xK (t#¹)"S[x (t)!x (t)] , (39) L L> L where the sensitivity coe$cient S is a constant independent of n. According to Eqs. (37) and (39), 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 oversimpli"ed equation, fails to account for the clustering of the vehicles observed in real tra$c. Moreover, since there is no density dependence in this dynamical equation, the fundamental relation cannot be derived from this dynamics. In order to make the model more realistic, we now assume [86] that the closer is the nth vehicle to the (n#1)th the higher is the sensitivity of the driver of the nth car. In this case, the dynamical equation (39) is further generalized to i [x (t)!x (t)] , (40) L (t)!x (t)] L> L> L where i is a constant. An even further generalization of the model can be achieved [87,88] by expressing the sensitivity factor for the nth driver as xK (t#¹)" L [x
i[v (t#q)]K L , (41) S " L [x (t)!x (t)]l L> L where l and m are phenomenological parameters to be "xed by comparison with empirical data. These generalized follow-the-leader models lead to coupled nonlinear di!erential equations for x . L Thus, in this `microscopica theoretical approach, the problem of tra$c #ow reduces to problems of nonlinear dynamics. So far as the stability analysis is concerned, there are two types of analysis that are usually carried out. The local stability analysis gives information on the nature of the response o!ered by the following vehicle to a #uctuation in the motion of its leading vehicle. On the other hand, the manner in which a #uctuation 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 tra$c we know that drivers often observe not only the leading vehicle but also a few other vehicles ahead of the leading vehicle. For example, the e!ect of the leading vehicle of the leading vehicle can be incorporated in the same spirit as the e!ect of `next-nearest neighboura in various lattice models in statistical mechanics. A linear dynamical equation, which takes into account this `next-nearest neighboura within the framework of the follow-the-leader model, can be written as [81] xK (t#¹)"S[x (t)!x (t)]#S[x (t)!x (t)] , (42) L L> L L> L where S and S are two phenomenological response coe$cients. The weakest point of these theories is that these involve several phenomenological parameters which are determined through `calibrationa, i.e., by "tting some predictions of the theory with the corresponding empirical data. Besides, extension of these models to multi-lane tra$c is di$cult since every driver is satis"ed if he/she can attain the desired speed!
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6.2. Optimal velocity models We can express the driving strategy of the driver in the car-following models in terms of mathematical symbols by writing xK (t)"(1/q)[< (t)!v (t)] , (43) L L L where < (t) is the desired speed of the nth driver at time t. In all follow-the-leader models L mentioned above the driver maintains a safe distance from the leading vehicle by choosing the speed of the leading vehicle as his/her own desired speed, i.e., < (t)"x . L L> An alternative possibility has been explored in recent works based on the car-following approach [89}104]. This formulation is based on the assumption that < depends on the distanceL headway of the nth vehicle, i.e., < (t)"<(*x (t)) so that L L xK (t)"(1/q)[<(*x (t))!v (t)] (44) L L L where the so-called optimal velocity function <(*x ) depends on the corresponding instanL taneous distance-headway *x (t)"x (t)!x (t). In other words, according to this alternative L L> L driving strategy, the nth vehicle tends to maintain a safe speed that depends on the relative position, rather than relative velocity, of the nth vehicle. In general, <(*x)P0 as *xP0 and must be bounded for *xPR. For explicit calculations, one has to postulate a speci"c functional form of <(*x). Car-following models along this line of approach have been introduced by Bando et al. [89,90]. For obvious reasons, these models are usually referred to as optimal velocity (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 xrst-order di!erential 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 di!erential equations (for the positions of the vehicles) with respect to time [89,90]. The simplest choice for <(*x) is [91,92] H(*x!d) , (45)
where d is a constant and H is the Heavyside step function. According to this form of <(*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 v . A somewhat more realistic choice
[92,96] is <(*x)"v
0
<(*x)" f*x
for *x(*x , for *x 4*x4*x , for *x (*x .
(46)
v
The main advantage of the forms (45) and (46) of the OV function is that exact analytical calculations, e.g. in the jammed region, are possible [92]. Although (45) and (46) may not appear very realistic, they capture several key features of more realistic forms of OV functions [89], e.g., <(*x)"tanh[*x!*x ]#tanh[*x ]
(47)
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for which analytical calculations are very di$cult. For the convenience of numerical investigation, the dynamical equation (44) for the vehicles in the OV model has been discretized and, then, rewritten as a diwerence equation [104]. The main question addressed by the OV model is the following: what is the condition for the stability of the homogeneous solution x"bn#ct , (48) L where b"(*x) "¸/N is the constant average spacing between the vehicles and c is the constant velocity. It is not di$cult to argue that, in general, in the OV models the homogeneous #ow becomes unstable when R</R*x" '2/q [89]. V@ One can distinguish "ve di!erent density regimes with respect to the stability of microscopic states (see Fig. 10). At low and at high densities the homogeneous states are stable. For intermediate densities one can distinguish three regimes where 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 e!ects have been observed. Thus, the OV model is able to reproduce many aspects of experimental "ndings. Modi"ed Korteweg}de Vries (KdV) equation has been derived from Eq. (44) in a special regime of the parameters [93] and the relations between its kink solutions and tra$c congestion have been elucidated [99]. A generalization to two-lane tra$c can be found in [105]. In order to account for tra$c consisting of two di!erent types of vehicles, say, cars and trucks, Mason et al. [97] generalized the formulation of Bando et al. [89] by replacing the constant q by q so that L xK (t)"(1/q )[<(t)!v (t)] , (49) L L L L where q now depends on whether the nth vehicle is a car or a truck. Since a truck is expected to L take longer to respond than a car we should assign larger q to trucks and smaller q to cars. Some other mathematically motivated generalizations of the OV model have also been considered [95,98,100].
Fig. 10. Fundamental diagram of the OV model. The solid line shows the OV function and the dots simulation data. One can distinguish "ve di!erent density regimes with respect to the stable stationary state (from [92]).
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As mentioned earlier, drivers often receive stimulus not only from the leading vehicle but also a few other vehicles ahead of the leading vehicle. One possible way to generalize the OV models for taking into account such multi-vehicle interactions [102] is to write the dynamical equations as
K x !x L !v , xK " S < L>H (50) L H L j H where S are sensitivity coe$cients; one of the commonly used explicit forms of the optimal H velocity function, for example (47) can be chosen for that of the function < in (50). Before concluding our discussions on the OV models let us mention a very simple model [94] where one assumes that
a for *x 5*x , L A xK (t)" (51) L !a for *x (*x . L A with a'0. Obviously, *x may be interpreted as the safety distance. Moreover, a restriction A v 4v 4v is imposed on the allowed velocities of the vehicles by introducing the allowed
L
minimum and maximum velocities v and v , respectively. Note that in this oversimpli"ed
model xK depends on the corresponding distance-headway and, therefore, has some apparent L similarities with the OV models. But, unlike the more general OV models, in this model xK does not L depend on its instantaneous velocity. The main reason for considering such an oversimpli"ed scenario is that velocity of the propagation of jams can be calculated analytically. Some ideas of the OV model has been utilized by Mahnke and Pieret [106] in their master equation approach to the study of jam dynamics. They assumed that, at a time, only one vehicle can go into or come out of a jam; this, naturally, does not take into the merging or splitting of jams. Under this assumption, the master equation (see e.g. Eq. (F.1)) governing the probability distribution P(n; t) of the jam sizes n is given by dP(n; t) "= (n!1)P(n!1; t)#= (n#1)P(n#1; t)! [= (n)#= (n)]P(n; t) , > \ > \ dt
(52)
where = and = , are the `growtha and `decaya transition rates, respectively. It has been argued > \ [106,107] that = (n)"1/q"constant since a vehicle would require a constant average time q to \ come out of a jam. However, = (n) would depend on <(*x) since the time taken by a free> #owing vehicle immediately behind a jam to get into the jam would depend on *x as well as on <(*x) although the actual expression of = (n) may be complicated in a reasonable ansatz [107]. > Moreover, several assumptions of the model will have to be relaxed before the results of this approach can be compared with those from real tra$c. Before concluding this section we would like to emphasize that while formulating the dynamical equations for updating the velocities and positions of the vehicles in any `microscopica theory the following points should be considered: (i) in the absence of any disturbance from the road conditions and interactions with other vehicles, a driver tends to drive with a desired velocity v ; if the actual current velocity of the vehicle v(t) is smaller (larger) than v , the vehicle accelerates (decelerates) so as to approach v .
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(ii) In a freely-#owing tra$c, even when a driver succeeds in attaining the desired velocity v , the velocity of the vehicle #uctuates around v rather than remaining constant in time. (iii) The interactions between a pair of successive vehicles in a lane cannot be neglected if the gap between them is shorter than v ; in such situations the following vehicle must decelerate so as to avoid collision with the leading vehicle. Clearly, the reliability of the predictions of the OV model depends on the appropriate choice of the optimal velocity function.
7. Coupled-map lattice models of vehicular tra7c Recall that, 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 di!erential equations of the car-following models, one needs to discretize the continuous variables with appropriately chosen grids. In contrast, in the coupled-map lattice approach [108], one starts with a discrete time variable and, 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 tra$c conditions at the preceding time step t!1. Keeping in mind the general points raised at the end of the preceding section regarding the formulation of the dynamical equations for updating the velocities and positions of the vehicles, the general forms of the dynamical maps in the coupled-map lattice models can be expressed as v (t#1)"Map [v (t), v , *x (t)] , (53) L L L L x (t#1)"v (t)#x (t) , (54) L L L where v is a desired velocity. In general, the dynamical map Map[v (t), v , *x (t)] takes into L L account the velocity v (t) and the distance-headway *x (t) of the nth vehicle at time t for deciding L L the velocity v (t#1) at time t#1. The e!ects of the interactions among the vehicles enter into the L dynamical updating rules (53), (54) only through the distance-headway *x . L 7.1. The model of Yukawa and Kikuchi Yukawa and Kikuchi [109}111] have studied coupled-map models based on the map
v(t#1)"F(v(t)) " : cv(t)#b tanh
v$!v(t) #e c
(55)
for the unin#uenced motion of a single vehicle. v$ is the preferred velocity of the vehicle and b, c, d and e are parameters. For c close to 1 the map becomes chaotic, but acceleration and deceleration are approximately constant far from v$. Their magnitude is determined by the parameter b. e controls the di!erence of the acceleration and deceleration capabilities. Although the model is
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deterministic, #uctuations in the velocity are introduced through deterministic chaos. These #uctuations around v$ are determined by the parameter d. 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 [109]. In model A, the deceleration map describes a sudden braking process. If the front-bumper to front-bumper distance *x to the next vehicle ahead is less than the current velocity v (t) of the following vehicle, then L L the velocity is reduced to *x !l where l is the length of the vehicles. The corresponding map is L B(*x (t))"*x (t)!l. L L Model B has a more complex deceleration map: F(v (t))!v (t) L L [*x (t)!l!v (t)] v (t#1)"G(*x (t), v (t)) " : L L L L L (a!1)v (t) L (for v (t)4*x (t)!l4av (t)) . (56) L L L The parameter a determines the range within which the deceleration map G(*x, v) is used. For headways less than av (t) the map G is used instead of F(v). Note that G(*x, v"*x!l)"*x!l L and G(*x, v"(*x!l)/a)"F(v"(*x!l)/a), 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(v (t), v$) for av (t)4*x (t) , L L L L (57) Map (v (t), *x (t))" G(*x (t), v (t)) for v (t)4*x (t)4av (t)) , L L L L L L L L B(*x (t)) for *x (t)4v (t)) . L L L Local measurements of the #ow for a system of vehicles with di!erent preferred velocities v$ produce a fundamental diagram of inverse-j shape (see Fig. 4A) [111]. Here the non-uniqueness L of the #ow has a simple explanation. Due to the di!erent v$, platoons form behind the slowest L vehicles. Whenever such a platoon passes the measurement region, a #ow value on the lower branch is recorded. Otherwise, the #ow corresponds to the upper branch. Measurements of the power spectral density of temporal density #uctuations, i.e. the Fourier transform of the time-series of local densities, show a 1/f ?-behaviour with a+1.8 in the free-#ow regime. Due to the deterministic dynamics, the system evolves into a state with power-law #uctuations. In [110,111] it has been suggested that the origin of the 1/f ?-#uctuations is the power-law distribution (J1/(*x) ) of the headways *x, since these are related to density waves. The occurrence of jams destroys long-time correlations since vehicles lose their memory of current #uctuations when they are forced to stop in a jam [110]. Therefore, no 1/f ?-behaviour can be observed in the jammed regime. In [112] a coupled-map model based on optimal-velocity functions has been introduced by discretizing the time variable of the OV model (see Section 6.2). 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)#a(< (*x)!v(t))](1#f m) , where m3[!1/2,1/2] is a uniform random variable and f the noise level.
(58)
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7.2. The model of Nagel and Herrmann Nagel and Herrmann (NH) [113] have introduced a coupled-map model which is related to the continuum limit of the Nagel}Schreckenberg cellular automata model (see Section 8). A generalization of the NH model has later been presented in [114]. Vehicles are characterized by a maximal velocity v and a safety distance a. The velocity map for the NH model is given by
max(*x (t)!d, 0) for v (t)'*x (t)!a , L L L (59) v (t#1)" L min(v (t)#a, v ) for v (t)(*x (t)!b . L
L L In the velocity update step, vehicles which have a headway *x smaller than the safety distance a decelerate. The headway distance after deceleration is determined by the parameter d. Vehicles which have a large enough headway, on the other hand, accelerate. The acceleration coe$cient a is determined by a"a max(1, *x (t)/c).
L Since the dynamics of the model is deterministic, the behaviour depends strongly on #uctuations of the initial state [114]. For equidistant starting positions of the vehicles the fundamental diagram consists of two linear branches with maximum #ow f (c )"v c at density c "1/(v #b).
For homogeneous starting positions the system is free-#owing up to a critical density c . Beyond this density free-#owing and congested areas coexist.
7.3. The model of Krauss, Wagner and Gawron Krauss et al. [115,116] introduced a whole class of stochastic models by considering necessery conditions for the collision-free motion of vehicles. The models are continuous in space and discrete in time. The vehicles are characterized by a maximum velocity v , their acceleration and deceleration
capabilities a(v) and b(v), respectively, and their length l which will be taken to be l"1 in the following. Then the update rules for the velocity v and the space coordinate x of each vehicle are as follows: Step 1: Determine desired velocity; v "min[v , v#a(v), v ] .
Step 2: Randomization; v"max[0, rand(v !a, v )] . Step 3: Vehicle movement; xPx#v . Here rand(v , v ) denotes a random number uniformly distributed in the interval [v , v ) and v is a velocity which guarantees collision motion of the vehicles. It is given explicitly by v "v #b(v( )(g!v )/(v( #b(v( )) , (60) where v is the velocity of the preceding vehicle located at x and g"x !x!1 is the headway, i.e. the distance to the preceding vehicle.
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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 behaviour of the model can be classi"ed in three di!erent families (see Fig. 11). The three families of models sketched schematically in Fig. 11 can be characterized as follows: Class I: High acceleration. Here no spontaneous jamming exists. For aPv and b<1 the
behaviour is similar to that of a cellular automata model without velocity memory introduced in [117] which is closely related to the Kasteleyn model [118] of statistical physics. It can also be interpreted as 5-vertex model [119]. Another model belonging to this class has been introduced by Fukui and Ishibashi [120] (see Section 11.1). Class II: High acceleration}low deceleration. The out#ow from a jam is identical to the maximal possible #ow. The jamming transition is not a true phase transition, but rather a crossover. The limit bPR, a"1 corresponds to a continuum version of the Nagel}Schreckenberg cellular automata model [20] which will be introduced in the next section. Class III: Low acceleration}low deceleration. These models exhibit phase separation and metastability. The jamming transition is of "rst order. The out#ow from a jam is not maximal. For a;v
and b;v the model is closely related to the Gipps model [88] discussed in Section 6.1. Other
models belonging to this class are the Kerner}KonhaK user model (see Section 4.4), the optimalvelocity model (Section 6.2) and the models with slow-to-start rules which will be introduced in Section 9.1.2. On a macroscopic level, classes I, II and III can be distinguished by the ordering of the densities c and c , where c is the density of the out#ow from a jam and c is the density where D A D A homogeneous #ow becomes unstable [116]. For c 'c the out#ow from a jam is stable and A D the system phase-separates into free-#ow and jammed regions. Furthermore, metastable states can be found. This is the type of the behaviour found in class III. For c (c , on the other hand, the A D out#ow from a jam is unstable and no metastable states or phase separation can be found. This is the typical behaviour of classes I and II. These classes can further be distinguished since in class I one does not "nd any structure formation, like spontaneous jamming, in contrast to class II. A related model has been studied before by Migowsky et al. [121]. In this model vehicles are also characterized by a maximum velocity v and a bounded acceleration capability
(!b (xK (a ) which determines the safety distance d necessary to avoid accidents. The
investigations in [121] focused on the e!ect of so-called driving strategies. These strategies are
Fig. 11. Schematic sketch of the three di!erent classes of models in the KWG model.
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characterized by a vector ( f , f , f ), where f , f and f are the fraction of the vehicle's maximal velocity, acceleration and safety distance actually used, respectively. This can lead to the possibility of accidents and allows to study the number of crashes as a function of the driving strategies. In [121] di!erent strategies have been compared. Furthermore, dynamical changes of strategies can be introduced which allow the drivers to adapt to the local tra$c conditions. In general this leads to a decrease in the number of accidents, jams and fuel consumption, but at high densities the #ow is reduced compared to the case of "xed strategies. 8. Nagel}Schreckenberg cellular automata model of vehicular tra7c on highways In general, CA are idealization of physical systems in which both space and time are assumed to be discrete and each of the interacting units can have only a "nite number of discrete states. Note that for a discretization of di!erential equations, e.g. those of the hydrodynamic approach, space and time variables are discrete, but the state variable still is continuous. The concept of CA was introduced in the 1950s by von Neumann while formulating an abstract theory of self-replicating computing machines [122]. However, it received the attention of a wider audience in the 1970s through Conway's game of life [123]. The family of one-dimensional CA was studied systematically, in the 1980s, from the point of view of dynamical systems and popularized by Wolfram [18,124]. Since then the concept of CA has been applied to model a wide variety of systems [125}127]. To our knowledge, the "rst CA model for vehicular tra$c was introduced by Cremer and Ludwig [128]. In the CA models of tra$c the position, speed, acceleration as well as time are treated as discrete variables. In this approach, a lane is represented by a one-dimensional lattice. Each of the lattice sites represents a `cella which can be either empty or occupied by at most one `vehiclea at a given instant of time (see Fig. 12). At each discrete time step tPt#1, the state of the system is updated following a well de"ned prescription (a summary of various possible di!erent schemes of updating is given in Appendix A). The computational e$ciency of the discrete CA models is the main advantage of this approach over the car-following and coupled-map lattice approaches. In the NaSch model, the speed v of each vehicle can take one of the v #1 allowed integer
values v"0, 1,2, v . Suppose, x and v denote the position and speed, respectively, of the nth
L L vehicle. Then, d "x !x , is the gap in between the nth vehicle and the vehicle in front of it at L L> L time t. At each time step tPt#1, the arrangement of the N vehicles on a "nite lattice of length ¸ is updated in parallel according to the following `rulesa:
Fig. 12. A typical con"guration in the NaSch model. The number in the upper right corner is the speed of the vehicle.
In other words, the maximum velocity is given by vL "f Lv , etc.
T
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Step 1: Acceleration. If v (v , the speed of the nth vehicle is increased by one, but v remains L
L unaltered if v "v , i.e., L
v Pmin(v #1, v ) . L L
(U1)
Step 2: Deceleration (due to other vehicles). If d 4v , the speed of the nth vehicle is reduced to L L d !1, i.e., L v Pmin(v , d !1) . L L L
(U2)
Step 3: Randomization. If v '0, the speed of the nth vehicle is decreased randomly by unity with L probability p but v does not change if v "0, i.e., L L v Pmax(v !1, 0) with probability p L L
(U3)
Step 4: Vehicle movement. Each vehicle is moved forward according to its new velocity determined in Steps 1}3, i.e., x Px #v . L L L
(U4)
The NaSch model is a minimal model in the sense that all the four steps are necessary to reproduce the basic features of real tra$c; however, additional rules need to be formulated to capture more complex situations. Step 1 re#ects the general tendency of the drivers to drive as fast as possible, if allowed to do so, without crossing the maximum speed limit. Step 2 is intended to avoid collision between the vehicles. The randomization in step 3 takes into account the di!erent behavioural patterns of the individual drivers, especially, nondeterministic acceleration as well as overreaction while slowing down; this is crucially important for the spontaneous formation of tra$c jams. Even changing the precise order of the steps of the update rules stated above would change the properties of the model. E.g. after changing the order of steps 2 and 3 there will be no overreactions at braking and thus no spontaneous formation of jams. The NaSch model may be regarded as stochastic CA [18]. In the special case v "1 the deterministic limit of the NaSch
model is equivalent to the CA rule 184 in Wolfram's notation [18] and some abstract extensions of this CA-184 rules [129] have been studied in the more general context of complex dynamics and particle #ow. Why should the updating be done in parallel, rather than in random sequential manner, in tra$c models like the NaSch model? In contrast to a random sequential update, parallel update can lead to a chain of overreactions. Suppose, a vehicle slows down due the randomization step. If the density of vehicles is large enough this might force the following vehicle also to brake in the deceleration step. In addition, if p is larger than zero, it might brake even further in Step 3. Eventually this can lead to the stopping of a vehicle, thus creating a jam. This mechanism of spontaneous jam formation is rather realistic and cannot be modeled by the random sequential update. The update scheme of the NaSch model is illustrated with a simple example in Fig. 13. Space}time diagrams showing the time evolutions of the NaSch model demonstrate that no jam is present at su$ciently low densities, but spontaneous #uctuations give rise to tra$c jams at higher densities (Fig. 14(a)). From Fig. 14(b) it should be obvious that the intrinsic stochasticity of
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Fig. 13. Step-by-step example for the application of the update rules. We have assumed v "2 and p"1/3. Therefore,
on average one-third of the cars qualifying will slow down in the randomization step.
the dynamics [20], arising from nonzero p, is essential for triggering the jams [20,113]. For a realistic description of highway tra$c [20], 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 v "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 tra$c, namely the reaction time of the drivers. Almost all the models of tra$c considered in this review, including the NaSch model, have been formulated in such a way that no accident between successive vehicles is possible. However, accident of the vehicles is possible if the condition for safe driving is relaxed. For example, Boccara et al. [130] replaced the update rule of the NaSch model by the rule if v
(t)'0 then x (t#1)"x (t)#v (t#1)#*v , (61) L> L L L where *v is a Bernoulli random variable which takes the value 1 with probability p and zero with the probability 1!p . The probability P of accident per vehicle per time step is a non-monotonic function of the vehicle density c [130,131].
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Fig. 14. Typical space}time diagrams of the NaSch model with v "5 and (a) p"0.25, c"0.20, (b) p"0, c"0.5.
Each horizontal row of dots represents the instantaneous positions of the vehicles moving towards right while the successive rows of dots represent the positions of the same vehicles at the successive time steps.
Fig. 15. ASEP with periodic boundary conditions.
8.1. Relation with other models 8.1.1. Relation with totally asymmetric simple exclusion process Now we point out the similarities and di!erences between the v "1 limit of the NaSch model
and the totally asymmetric simple exclusion process (TASEP) which is the simplest prototype model of interacting systems driven far from equilibrium [14}16]. In the TASEP (Fig. 15) a randomly chosen particle can move forward, by one lattice spacing, with probability q if the lattice site immediately in front of it is empty. It corresponds to a Kawasaki dynamics [132] for exchange of a charged particle and hole on nearest-neighbour lattice sites at in"nite temperature and in the presence of an in"nite electric "eld [133] (Appendix B for some technical aspects of TASEP). Several di!erent generalized variants of the TASEP have been considered. For example, in the k-hop model [134] a particle can exchange its position with the nearest-hole on its right with probability q, provided the separation of the two sites under consideration is not more than k lattice spacings. The k-hop model reduces to the TASEP in the special case k"1. Note that in the NaSch model with v "1 every vehicle moves forward with probability
q"1!p in the time step t#1 if the site immediately in front of it were empty at the time step t; this is similar to TASEP. But, updating is done in parallel in the NaSch model whereas that in the
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TASEP is done in a random sequential manner. Nevertheless, the special case of v "1 for the
NaSch model achieves special importance from the fact that so far it has been possible to derive exact analytical results for the NaSch model only in the special limits (a) p"0 and arbitrary v (which we have already considered), and (b) v "1 and arbitrary p.
8.1.2. Relation with surface growth models and the phenomenological yuid-dynamical theories of trazc The NaSch model with v "1 can be mapped onto stochastic growth models of one dimensional surfaces in a two-dimensional medium, the single-step model [135]. Corresponding to each con"guration +p , of the NaSch model in the site-oriented description, one can obtain H a unique surface pro"le +H , through the relation H " (1!2p ) [135,136]. Pictorially one I H H HXI can interpret this mapping as shown in Fig. 16. Particle (hole) movement to the right (left) correspond to local forward growth of the surface via particle deposition. In this scenario a particle evaporation would correspond to a particle (hole) movement to the left (right) which is not allowed in the NaSch model. 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 [137]. The surface growth model described above is known to be a discrete counterpart of continuum models of growing surfaces whose dynamics are governed by the so-called Kardar}Parisi}Zhang equation [135,136]. Since the Kardar}Parisi}Zhang equation can be mapped onto the Burgers equation [53] using the Cole}Hopf transformation [135], it is not surprising that several features of vehicular tra$c are described by the NaSch model at the microscopic level and by the noisy Burgers equation for the coarse-grained continuum of the #uid-dynamical theory [138]. 8.2. Limiting cases of the NaSch model In spite of the fact that the deterministic limits p"0 and 1 of the NaSch model do not capture some of the most essential features of vehicular tra$c it may be instructive to examine these limits to gain insight into the features of this simpler scenario. Another limiting case which exhibits a surprisingly complex behaviour is the case v "R.
Fig. 16. The schematic pictorial representation of the mapping of the NaSch model with v "1 onto a stochastic
model of surface growth.
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8.2.1. NaSch model in the deterministic limit p"0 The NaSch model, a stochastic CA, 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 v (t#1)"min[v , v (t)#1, d (t)!1] , (62) L
L L x (t#1)"x (t)#v (t#1) L L L which can lead to two types of steady states depending on the density c [113]. At low densities, the system can self-organize so that d 'v for all n and, therefore, every vehicle can move with v , L
i.e., v (t)"v , giving rise to the corresponding #ux cv . This steady state is, however, possible L
only if enough empty cells are available in front of every vehicle, i.e., for c4c"(v #1)\
and the corresponding maximum #ux is J "v /(v #1). On the other hand, for c'c,
d (t)!14min[v (t)#1, v ] and, therefore, the relevant steady states are characterized by L L
v (t)"d (t)!1, i.e., the #ow is limited by the density of holes. Since the average distance-headway L L is 1/c!1, the fundamental diagram in the deterministic limit p"0 of the NaSch model (for any arbitrary v ) is given by the exact expression
J"min(cv ,1!c) . (63)
Note that the result v "1/c!1 is identical with Greenshields ansatz v "1/c!1/c if we L L
identify c "1.
8.2.2. NaSch model in the deterministic limit p"1 Are not the properties of the NaSch model with maximum allowed speed v , in the determinis tic limit p"1, exactly identical to those of the same model with maximum allowed speed v !1?
Although this expectation may seem to be consistent with the observation that J"0 for all c in the special case v "1"p, the answer to the question posed above is: NO. To understand the
subtle features of the deterministic limit p"1 one has to consider v '1. You can easily
convince yourself that if, for example, v "2, then, for c'1/3, all stationary states correspond to
J"0 because at least one vehicle will have only one empty cell in front (i.e. d "2) and it will never L succeed in moving forward. For v "2 and p"1, although there are stationary states corre sponding to JO0 for all c41/3, such states are metastable in the sense that any local external perturbation leads to complete breakdown of the #ow. 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 v and c!
8.2.3. NaSch model in the limit v "R
The limit v "R has been introduced in [139]. One has to be aware that there are several
possible ways of performing this limit since only "nite systems of length ¸ can be treated in computer simulations. In [139] the case v "¸ has been investigated, but other limiting
procedures are also possible, e.g. v J¸? with a'0 or even v "R independent of the system
size. In principle, these di!erent limiting procedures could lead to di!erent results, but up to now See also [113], where the case p"0 was studied.
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Fig. 17. Fundamental diagram of the NaSch model with v
241
"¸ for a system size of ¸"1000 (from [139]).
no systematic study has been performed. We therefore restrict ourselves to the case v "¸
studied in [139]. Surprisingly one "nds that the fundamental diagram has a form quite di!erent from that of the case of a "nite v . The #ow does not vanish in the limit cP0 since already one single car
produces a "nite value of the #ow, J(cP0)"1. Due to the hindrance e!ect of other cars, J(c) is a monotonically decreasing function of the density c (see Fig. 17). Another characteristic feature of the fundamental diagram is the existence of a plateau at #ow J where the value J depends on the . . randomization p, but not on the system size ¸. The length of the plateau, on the other hand, increases with ¸. What is the microscopic structure of the stationary state leading to such a fundamental diagram? At low densities, where #ow J is larger than the plateau value J , the cars tend to be uniformly . distributed just as in the deterministic case p"0 (see Section 8.2.1). 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 phase transition at c"0 between a jamless phase with J"1 and a phase with one jam and #ow J [139]. Increasing the density further, more jams . develop and the plateau ceases. Note that this behaviour is completely di!erent from the prediction of mean-"eld theory in that limit [140] (see Section 8.3.1) showing the importance of correlations. 8.3. Analytical theories of the NaSch model with periodic boundary conditions CA are, by design, ideal for large-scale computer simulations. However, proper interpretations of the numerical data obtained from computer simulations are not always quite straightforward because of the "nite-size e!ects and `numerical noisea. One cannot deny the importance of exact analytical results in providing a testing ground for the computer codes. On the other hand, the parallel updating makes exact analytical solution of CA models very di$cult. Nevertheless, even in those situations where exact solutions are not possible, a combination of approximate analytical treatments and computer simulation often turns out to be very powerful method of analysis of a problem. This approach has been quite successful in recent years in the studies of the NaSch model and its generalizations.
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Before we proceed with the analytical theories in the nondeterministic NaSch model, we would like to point out that the fundamental diagram J(c) is known exactly for arbitrary v and p in the
two limits cP0 and cP1. In the former case, JKcv where v "v !p is the free-#ow velocity. $ $
On the other hand, in the latter case, JK(1!p)(1!c) as #ow is determined by holes moving backwards at a speed 1!p. 8.3.1. Site-oriented naive mean-xeld theory for the NaSch model In the `site-orienteda theories one describes the state of the "nite system of length ¸ by completely specifying the state of each site, i.e., by the set (p , p ,2, p ) where p ( j"1, 2,2, ¸) * H can, in principle, take v #2 values one of which represents an empty site while the remaining
v #1 correspond to the v #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 [140] as this merely shifts the starting step and, therefore, does not in#uence the steady-state properties of the model. The advantage of this new sequence is that, in a site-oriented theory, the variable p H can now take v #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 de"nition: 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 of the speed). Obviously, the space-average of n(i; t) is the density of the vehicles, i.e., n(i; t)/¸"c. Suppose, G c (i; t) is the probability that there is a vehicle with speed v (v"0, 1, 2,2, v ) at the site i at the T
time step t. Obviously, c(i; t)" T c (i; t) is the probability that the site i is occupied by a vehicle at T T 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 site-oriented mean-"eld (SOMF) approximation for the NaSch model one writes down the equations relating c (i; t#1) (v"1,2, v ) with the corresponding probabilities at time T
t and, then, solves the equations in the steady-state (see Appendix C for a detailed derivation of these equations for arbitrary v ). In the simplest case of v "1 and periodic boundary
conditions one gets [140] c (i; t#1)"c(i; t)c(i#1; t)#pc(i; t)d(i#1; t) ,
(64)
c (i; t#1)"qc(i!1; t)d(i; t) .
(65)
Eq. (64) expresses the simple fact that at the time step t#1 the speed of the vehicle at the ith site can be zero either because the site i#1 was occupied at time t or because of random deceleration (if the site i#1 was empty at time t). Similarly, Eq. (65) implies that the speed of the vehicle at the site i can be 1 at time t#1 if at the time step t the site i was empty while the site i!1 was occupied by a vehicle which did not decelerate during the random deceleration stage of updating. In the steady state, c (i, t) are independent of t. Besides, if periodic boundary conditions are T imposed, the i-dependence of c (i) also drops out in the translational-invariant steady state. T Therefore, in the steady state J"c "qc(1!c) .
(66)
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It turns out [140] that the naive SOMF underestimates the #ux for all v . Curiously, if instead of
parallel updating one uses the random sequential updating, the NaSch model with v "1
reduces to the TASEP for which the equation (66) is known to be the exact expression for the corresponding #ux (see, e.g., [20])! 8.3.2. Paradisical mean-xeld theory of the NaSch model What are the reasons for these di!erences arising from parallel updating and random sequential updating? There are `Garden of Edena (GoE) [141] states (dynamically forbidden states) [142] of the NaSch model which cannot be reached by the parallel updating whereas no state is dynamically forbidden if the updating is done in a random sequential manner. For example, the con"guration shown in Fig. 18 is a GoE state 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, discussed in the preceding subsection, does not exclude the GoE states. On the other hand, results of the paradisical mean-"eld (pMF) theory are derived by repeating the calculations of the naive SOMF theory excluding all the GoE states from consideration. The exact expression, given in the next subsection, for the #ux in the steady state of the NaSch model with v "1 is obtained in the pMF theory (see Appendix D for detailed calculations), thereby
indicating that the only source of correlation in this case is the parallel updating [142]. But, for v '1, there are other sources of correlation because of which exclusion of the GoE states merely
improves the naive SOMF estimate of the #ux (Fig. 19) but does not yield exact results [142,143]. 8.3.3. Site-oriented cluster-theoretic approach to the NaSch model The site-oriented cluster theoretic approach leads to a systematic improvement of the naive SOMF theory of the NaSch model. We de"ne a n-cluster to be a collection of n successive sites and denote the probability of "nding an n-cluster in the state (p , p ,2, p ) in the steady state of the L system by the symbol P (p , p ,2, p ). In the general n-cluster approximation [140], one divides L L the lattice into `clustersa of length n such that two neighbouring clusters have n!1 sites in common (see Fig. 20); an n-cluster is treated exactly and the cluster is coupled to the rest of the system in a self-consistent way, as we shall show in this subsection. Even without any calculation, one would expect that, for a given v , the n-cluster approximation should yield more accurate
results with increasing n and should give exact results in the limit nPR. Fortunately, in the special case v "1 exact results are obtained already for n as small as 2, i.e., the results of 2-cluster
calculations are exact for v "1 [140].
Fig. 18. A GoE state for the NaSch model with v 52.
The con"guration shown in Fig. 12 is also a GoE state!
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Fig. 19. Fundamental diagram for v "2 and p"0.5 (left) and p"0.1 (right). Comparison of paradisical MFT
(full line) with results from computer simulations (䢇) and the naive MFT (dotted line).
Fig. 20. Decomposition of a lattice into (a) 1-clusters, (b) 2-clusters, and (c) 3-clusters in the SOMF theory.
Let us "rst explain the key concepts involved in the cluster theory. It is straightforward to verify, for example, in the special case of v "1, that the state of the 2-cluster p , p at time t#1
G G> depends on the state of the 4-cluster (q , q , q , q ) at time t. In general, in the n-cluster G\ G G> G> approximation for an arbitrary v one has to take into account the vehicles that can enter an
n-cluster from one of the v cells to its left and can leave it to occupy one of the v cells to its
right. Therefore, in general, the state p , p ,2p of an n-cluster at time t#1 depends on the H H> H>L\ , ,q at time t. Therefore, in the state of a n#2v cluster q ,2q , q ,2, q H H> H>L\ 2 H>L\>T
H\T special case of v "1, the master equations
P (p , p ; t#1)" =(p , p "q , q , q , q )P (q , q , q , q ; t) (67) G G> G G> G\ G G> G> G\ G G> G> OH
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governing the time evolution of the 2-cluster probabilities P (p , p ) involve the 4-cluster G G> probabilities for all those con"gurations (q , q , q , q ; t) which can lead to the 2-cluster G\ G G> G> con"guration (p , p ; t#1) under consideration as well as the corresponding transition probabilG G> ities =(p , p "q , q , q , q ). Similarly, the master equation governing the time evolution of G G> G\ G G> G> the 4-cluster probabilities P (q , q , q , q ) involve 6-cluster probabilities, and so on. In order G\ G G> G> to obtain a closed set of equations one has to truncate this hierarchy in an appropriate manner; in the n-cluster approximation one expresses the (n#2v )-cluster probabilities in terms of products
of n-cluster probabilities. The n-cluster approximation represented geometrically in Fig. 20 for n"1 can be expressed mathematically as OH> (68) P(q , q , q , q , q )" P (q ) . G H\ H\ H H> H> GOH\ Thus, 1-cluster approximation is equivalent to the naive SOMF approximation. The 2-cluster approximation represented geometrically in Fig. 20 can be expressed mathematically as [140] P(q , q , q , q , q , q )JP (q , q )P (q , q )P (q , q ) H\ H\ H H> H> H> H\ H\ H\ H H H> P (q , q )P (q , q ) H> H> H> H>
(69)
or, more precisely, P(q , q , q , q , q , q )"P (q "q )P (q "q )P (q , q ) H\ H\ H H> H> H> H\ H\ H\ H H H> P (q "q )P (q "q ) , H> H> H> H>
(70)
where P (q , q ) H\ H (71) P (q "q )" H\ H H\ P (q , q ) H\ H O are 2-cluster conditional probabilities. Similarly, the 3-cluster approximation consists of the approximate factorization P(q , q , q , q , q , q , q )"P (q "q , q )P (q "q , q ) H\ H\ H H> H> H> H> H\ H\ H H\ H H> ;P (q , q , q )P (q , q "q )P (q , q "q ) . (72) H H> H> H> H> H> H> H> H> Analoguous factorizations hold for an arbitrary number of sites on the left-hand-side of (68), (70) and (72). Let us now illustrate the scheme of the cluster calculations for the NaSch model by carrying out the calculation for the simplest case, namely, the 2-cluster calculations for v "1. For conveni ence, one follows the sequence 2}3}4}1, instead of 1}2}3}4 of the stages of updating so that, for v "1, a two-state variable p is adequate to describe the state of a lattice site; p"0, 1
correspond, respectively, to an empty site and a site occupied by a vehicle with speed 1. Thus, in the special case of v "1 one would need only four 2-cluster probabilities, namely,
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P (0, 0), P (1, 0), P (0, 1), P (1, 1). Interestingly, the constraints P (1, 0)#P (1, 1)"c and P (0, 0)#P (0, 1)"1!c together with the particle-hole symmetry
(73)
(74)
P (1, 0)"P (0, 1) (75) leave only one of the four 2-cluster probabilities, say, P (1, 0), as an independent variable which one needs to calculate by solving the corresponding master equation. For general v , the n-cluster
approximation on the right hand side of the master equation leads to (v #1)L nonlinear
equations; the number of independent equations gets reduced by the so-called Kolmogorov consistency conditions [144]. Using (70) one factorizes the 4-cluster probabilities on the right-hand side of (62) for P (1, 0) in terms of 2-cluster conditional probabilities. In the "rst column of the table in Fig. 21 we list all those con"gurations (q , q , q , q ; t) which can lead to the con"gurations, shown in the G\ G G> G> second column, which is the exhaustive list of the 4-cluster con"gurations each having p "1, p "0; the corresponding transition probabilities =(1, 0"q , q , q , q ) are given in G G> G\ G G> G> the third column. Using the con"gurations at t and t#1 as well as the corresponding transition probabilities given in table in Fig. 21 the master equation (67) for P (1, 0) reduces to the quadratic algebraic equation qy!y#c(1!c)"0 ,
(76)
Fig. 21. The four circles in the "rst two columns represent the states of the sites of a 4-cluster; the open and "lled circles correspond, respectively, to empty sites and sites occupied by vehicles with speed v"1.
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where we have used the shorthand notation y"P (1, 0). Solving this quadratic equation we get [140] (see also [145]) P (1, 0)"(1/2q)[1!(1!4qc(1!c)]
(77)
and, hence, P (1, 1), P (0, 0), P (0, 1) from Eqs. (73)}(75). Moreover, the expression (77) establishes that P (1, 0)5P (1)P (0)"c(1!c), which indicates an e!ective particle-hole attraction (par ticle}particle repulsion) in the NaSch model with v "1. Furthermore, from Eq. (77) one gets the
expression J(c, p)"qP (1, 0)"[1!(1!4qc(1!c)]
(78)
which can be proved [140] to be the exact expression for the corresponding #ux. It is not di$cult to carry out 2-cluster calculations for higher values of v , but one gets only approximate results for
v '1 [140].
An interesting feature of expression (78) is that the #ux is invariant under charge conjugation, i.e., under the operation cP(1!c) which interchanges particles and holes. Therefore, the fundamental diagram is symmetric about c"1/2 when v "1 (see Fig. 22(a)). Although this symmetry breaks
down for all v '1 (see Fig. 22(b)), the corresponding fundamental diagrams appear more
realistic. Moreover, for given p, the magnitude of c decreases with increasing v as the higher is
the v the longer is the e!ective range of interaction of the vehicles (see Fig. 22). Furthermore, for
v "1, #ux merely decreases with increasing p (see Eq. (78)), but remains symmetric about
c"1/2"c . On the other hand, for all v '1, increasing p not only leads to smaller #ux but
also lowers c (Fig. 23).
For v '1, one needs to carry out higher order cluster calculations [140,143] to get more
accurate results than those obtained in the 2-cluster approximation. For v "2, the fundamental
diagrams obtained from the n-cluster approximation (n"1, 2,2, 5) are compared in Fig. 24 with the Monte Carlo data. This comparison clearly establishes a rapid convergence with increasing n and already for n"4 the di!erence between the cluster calculation and MC data is extremely
Fig. 22. The fundamental diagram in the NaSch model for (a) v "1 and (b) v '1, both for p"0.25. The data for
all v '1 have been obtained through computer simulations.
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Fig. 23. The fundamental diagrams in the NaSch model with v
"5 are plotted for a few values of p.
Fig. 24. The fundamental diagrams in the NaSch model with v "2 in the n-cluster approximation (n"1, 2,2, 5).
small. In [143] the cluster probabilities for v "2 have been obtained from computer simula tions. The results suggest that the n-cluster approximation for n53 becomes asymptotically exact in the limit pP0. 8.3.4. Car-oriented mean-xeld theory of the NaSch model In the `car-orienteda theories the state of the tra$c system is described by specifying the positions and speeds of all the N vehicles in the system [146]. Suppose, P (t) is the probability to "nd at time L t exactly n empty sites immediately in front of a vehicle. Another auxiliary quantity, which turns out to be very convenient to use in several di!erent calculations, is g(t), the probability at time t that a vehicle moves in the next time step. These two sets of quantities, namely, P (t) and g(t) are related. L For example, in the NaSch model with v "1, a vehicle will move in the next time step if there is
P (t)) and if it does not decelerate in at least one empty cell in front of it (probability LY L P (t)]"q[1!P (t)]. Hence, the the randomization step (probability q); therefore, g(t)"q[ L #ux J(c, p) can be obtained from J(c, p)"cg"cq[1!P ]. LY
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The essence of the car-oriented mean-"eld (COMF) approximation [146] is to neglect the correlations between the gaps in front of the successive cars. The equations describing the time evolution of the probabilities P (t), under this approximation (see Appendix E for the derivation of L these equations), can be solved in the steady state using a generating function technique [146]. Following this approach, in the special case v "1, one recovers the exact expression (78) for
J(c, p)"cq[1!P ]. For v "2 one has to distinguish between P (v"1) and P (v"2). Moreover, one has to
L L generalize the quantity g to g , the probability that the vehicle moves a cells (a"1, 2) in the next ? time step. Applying the same generating function techniques as for v "1, one can also solve the
coupled sets of steady-state equations for P (v"1) and P (v"2) for v "2 but gets only L L
approximate results [146]. Interestingly, "nite size of the system a!ects the equations for v "2 in a much more dramatic
way [143] than those for v "1 thereby revealing the intrinsic qualitative di!erences in the
nature of correlations in the NaSch model for v "1 and v '1.
Comparisons with Monte Carlo simulations show that in contrast to the 3-cluster approximation for v "2 COMF does not become asymptotically exact in the limit pP0. This implies that
even in this limit correlations between the headways are not negligible. It is interesting, however, that for the fundamental diagram one "nds an excellent agreement between MC simulations and the predictions of COMF [146] for pP0. The reason is that in the deterministic limit many con"gurations exist which produce the same #ow. COMF is not able to identify the dominating structures correctly, but nevertheless can predict the correct current. 8.3.5. Microscopic structure of the stationary state As we have seen MFT underestimates the #ow in the stationary state considerably. Deviations become larger for higher velocities v . This shows the importance of correlations. As described
above a particle-hole attraction exist. Using the 2-cluster probabilities for v "1 this attraction
can be expressed in mathematical form as P (1, 0)'P (1)P (0)"c(1!c). For v "1 all improvements of MFT (2-Cluster, COMF and pMFT) are exact. Here only
correlations between neighbouring cells are important. The fact that pMFT is exact shows that no `truea correlations exist. All correlations have their origin in the existence of GoE states. This also helps to understand why for random-sequential dynamics already MFT is exact for v "1 and
the stationary state is uncorrelated. The reason is simply that for random-sequential dynamics no GoE states exist! The situation changes for higher velocities v '1. Here pMFT is no longer exact. Therefore
`truea correlations exist. This corresponds to the observation made in [140] that the NaSch model shows a qualitatively di!erent behaviour for v "1 and v '1. Furthermore, it explains why
so far the exact determination of the stationary state for v '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 8.2.1 that for densities c41/(v #1) the
A similar approach, the so-called interparticle distribution functions technique, is used for studying reactiondi!usion systems [147]. Note that for random-sequential dynamics also MFT is no longer exact!
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vehicles arrange themselves in such a way that all headways are at least v . 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 since overreactions are not possible. The behaviour in this limit can be interpreted as coming from a kind of `repulsive interactiona between the vehicles. The behaviour for p"1 is a little bit di!erent. Here we have seen in Section 8.2.2 that metastable states with "nite #ow exist for c41/3 and v '1.
For 0(p(1 the microscopic structure interpolates between these two limiting cases. This can be seen by analysing the 3-cluster probabilities obtained from Monte Carlo simulations [143]. For small p the microscopic structure of the stationary state is determined by the `repulsive interactionsa between vehicles. With increasing p one "nds a tendency towards phase separation into jammed and free-#ow regions. A standing vehicle is able to induce a jam even at low densities since the restart probability is small. The jams formed are typically not compact, but of the form `.0.0.0.a since 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 `"xed pointsa p"0 and p"1. 8.4. Spatio-temporal organization of vehicles; is there a phase transition? The density c corresponding to maximum #ux is an obvious "rst candidate for a critical density
separating the regimes of free-#ow and congested #ow in the NaSch model. We shall show in this subsection that this, indeed, is a critical point provided p"0. However, in spite of strong indications that, probably, a noise-induced smearing of the transition takes place when pO0, rigorous proofs are still lacking. 8.4.1. Order parameter For a proper description of a phase transition one should introduce an appropriate order parameter which can distinguish the two phases because of its di!erent qualitative behaviour within the two phases [24,25]. A "rst candidate [148] for the NaSch model 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 for the sharp transition at c from the free-#owing dynamical phase to the
congested 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 c; this probability is p for v "1 and decreases with increasing c.
The next obvious choice would be [149] the density of nearest-neighbour pairs in the stationary state 1 1 2 * nn , (79) m" H H> ¹¸ R H where, as de"ned earlier, n "0 for an empty cell and n "1 for a cell occupied by a vehicle H H (irrespective of its velocity). Because of the step 2 of the updating rule (deceleration due to other
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Fig. 25. The density dependence of the order parameter m in (a) the deterministic limit p"0 of the NaSch model (v "1, 2) and (b) the NaSch model with v "2, p"0.5.
vehicles) m gives the space}time-averaged density of those vehicles with velocity 0 which had to brake due to the next vehicle ahead. Fig. 25(a) shows that m vanishes at c if p"0. On the other hand, if pO0, m does not vanish
even for c(c although m becomes rather small at small densities (see Fig. 25(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 c'0 and, hence, inadequate to serve as an order parameter [150]. To slow down to v !2 a vehicle must be hindered by one randomly braking
vehicle in front. Similarly, to reach a speed v !3 a vehicle must "nd 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 cL. Therefore, the probability P (c) of "nding a vehicle with speed T v(v !1 is proportional to cT \\T and, hence, even for v"0, P (c) is, in general, nonzero for T
all cP0. 8.4.2. 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 [24,25]. Using lattice gas variables n , the equal-time density}density correlation function is de"ned by H 11 2 * n n !c . (80) G(r)" H H>P ¹¸ R H which measures the correlations in the density #uctuations that occur at the same time at two di!erent points in space separated by a distance r. Again it is very instructive to consider "rst the deterministic case p"0 (Fig. 26(a)). Since, as argued before, there are exactly v empty sites in front of each vehicle at c"c the correlation
function at c"c is given by
c(1!c) for,0 mod (v #1) ,
G(r)" !(c) else .
(81)
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Fig. 26. The density}density correlation function G(r) (a) the deterministic limit p"0 of the NaSch model (v "2) and
(b) the NaSch model with v "2, p"1/128.
For all v the correlation function for small nonzero p (Fig. 26(b)) has essentially the same
structure as that for p"0 (Fig. 26(a)) but the amplitude decays exponentially [149] for all c. In the general case of non-vanishing p, the asymptotic behaviour (rPR) of the correlation length m can be obtained analytically [143] only for v "1. It turns out that, for given p, m is maximum at
c"1/2"c and that m(c"1/2)Jp\ . Thus, for v "1, m diverges only for p"0 but remains
"nite for all nonzero p. For v '1 the trend of variation of m with c (Fig. 27(a)) in the vicinity of
c is the same as that for v "1 [149]. Moreover, for v '1, the maximum value of the
correlation length, m plotted against p (Fig. 27(b)), is also consistent with the corresponding
trend of variation for v "1. Thus, the correlation function G(r) gives a strong indication that the
NaSch model exhibits a second order phase transition, at c"c, only for p"0 but this transition
is smeared out if pO0. This noise-induced smearing of the phase transition in the NaSch model is very similar to the smearing of critical phenomena by "nite-size e!ects. 8.4.3. Distribution of lifetimes of jams Another quantity which should be able to give information about the nature of the transition from free-#ow to the jammed regime is the distribution of lifetimes of jams. Following Nagel [151] each vehicle which has a velocity less than v before the randomization step will be considered
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Fig. 27. The dependence of (a) m(c, p) on c for four di!erent values of p (v "2) and (b) m on p (v "2, 3).
Fig. 28. Lifetime distribution in the NaSch model for v "5 and p"0.5 and various densities below and above
c +0.085.
jammed. This de"nition is motivated by the cruise-control limit (see Section 9.1.1) where it is more natural than in the NaSch model. One expects, however, that the long-time behaviour of the lifetime distribution is independent of the exact de"nition of a jam. The short-time behaviour, on the other hand, might di!er strongly, e.g. for `compact jamsa where a jam is de"ned as a series of consecutive standing vehicles without any empty cells in between. Fig. 28 shows the results of Monte Carlo simulations for the lifetime distribution in the NaSch model for di!erent densities near the transition region, c+c "0.085$0.005 (for v "5,
p"0.5), where c is the density where the #ow is maximal. The most interesting feature of the
lifetime distribution is the existence of a cuto! near q "10000. It has been shown [151] that this cuto! is neither a "nite-size nor a "nite-time e!ect. For times smaller than q a scaling regime exists where the distribution decays algebraically. 8.4.4. Dissolution of a megajam Gerwinski and Krug [150] tried to "nd an intuitive criterion which allows the distinction of free-#ow and jammed phases. It is based on the investigation of jam dissolution times. Starting
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from a megajam con"guration, i.e., a block of N consecutive cells occupied by vehicles with the remaining ¸!N cells being empty, they determined the time until the jam has dissolved completely. A simple estimate gives the density at which the lifetime is expected to become in"nite. Suppose that the jam dissolves with velocity v . Since the "rst vehicle move freely with average velocity ( v "v !p it will reach the end of the jam at the same time as the dissolution wave if the $
condition (¸!N)/v "N/v is satis"ed. The corresponding density is then given by $ ( cH"v /(v #v )"v /(v #v !p) . (82) ( ( $ ( (
For v "1 vehicles accelerate immediately to v . In this case one has v "q"1!p. For
( higher velocities, q"1!p is only an upper bound for v . Inserting v "1!p into (82) one ( ( therefore obtains an upper bound for the density cH. Taking into account interactions between vehicles in the out#ow region of the jam, one can derive an e!ective acceleration rate q , and thus the jam dissolution velocity v "q , as a function of p [150]. ( Computer simulations show a sharp increase of the lifetime near the density cH. It becomes `in"nitea, i.e. the jam does not dissolve within the measurement time, only at a higher density cH which is considerably larger than the density c of maximum #ow. At intermediate densities
cH(c(cH the jam does not dissolve during the "rst lap, but later due to #uctuations of the two ends of the jam. During this time other jams have usually formed. All results found in [150] are consistent with the measurements of the lifetime distributions presented in the previous point. 8.4.5. Relaxation time 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 "rst by Nagel and Herrmann [113]. They found a maximum q of the relaxation time at the density c"1/(v #1) which diverges in
the thermodynamic limit ¸PR as q J¸. For "nite p the behaviour of the relaxation time is
more complicated. For technical reasons CsaH nyi and KerteH sz [152] made no direct measurements of the relaxation time, but used the following approach: Starting from a random con"guration of cars with velocity v "0 the average velocity v (t) is measured at each time step t. For tPR the H system reaches a stationary state with average velocity 1v 2. The relaxation time is characterized by the parameter [152]
q"
[min+vH(t), 1v 2,!1v (t)2] dt .
(83)
vH(t) denotes the average velocity in the acceleration phase tP0 for low vehicle density oP0. In this regime, due to the absence of interactions between the vehicles, one has vH(t)"(1!p)t. Thus, the relaxation time is obtained by summing up the deviations of the average velocity 1v (t)2 from the values of a system with one single vehicle which can move without interactions with other cars (oP0). Note that for a purely exponentially decaying quantity v(t)"v #C exp(!t/q ) the de"nition (83) is proportional to q , i.e., the standard de"nition of the relaxation time. One "nds In [150] the same de"nition of a jam as in the previous point `Distribution of lifetimes of jamsa (see [151]) has been used.
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Fig. 29. Relaxation parameter near the transition density for v "5 and p"0.25.
a maximum of the relaxation parameter near, but below, (see Fig. 29) the density of maximum #ow for p"0.25 and v "5 (see [152]).
The relaxation time (83) shows interesting behaviour [152,139,149] which is di$cult to interpret in terms of critical point phenomena. One "nds a maximum of the relaxation parameter near, but below, the density of maximal #ow. This maximum value increases with system size, but the width of the transition region does not seem to shrink [139]. Two scenarios are possible: (1) The relaxation time converges to a large but "nite value for large system sizes beyond the present computer power; (2) The relaxation time diverges for ¸PR. Scenario (1) appears to be more plausible in view of the "nite lifetimes of jams discussed above. Complicated interactions between jams could in principle lead to a divergence. Keeping in mind the unusual scaling behaviour of the width, this should occur in a "nite interval, not at a special (critical) point. The interpretation of the parameter q as a relaxation time can be problematic. This can be seen clearly for c'c , where q can become negative [149]. Here it is possible that during relaxation the system can temporarily reach states with a higher average velocity than in the stationary state. This overreaction can be divided into two phases for p'0. Within the "rst few time steps small clusters which occur in the initial con"guration vanish. The second phase is characterized by the growth of surviving jams. More and more cars get trapped into large jams and therefore the average #ow decreases to its stationary value. This decrease causes negative values of q at large densities. 8.4.6. Distribution of distance-headways In order to get information on the spatial organization of the vehicles, one can calculate the distance-headway distribution P (*x) by following either a site-oriented approach [153] or a car-oriented approach [146] if *x "x !x , i.e., if the number *x !1 of empty lattice sites H H> H H in front of the jth vehicle is identi"ed as the corresponding distance-headway. Stated precisely, P (k) is the conditional probability of "nding 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-"eld
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expression P ( j)"c(1!c)H (84) for the distance-headway distribution in the NaSch model with v "1 and the corresponding
Monte Carlo data [153] reveals the inadequacy of equation (84) at very short distances which indicates the existence of strong short-range correlations in the NaSch model that are neglected by the mean-"eld treatment. This is consistent with our earlier observation that there are particle-hole e!ective short-range attraction in the NaSch model with v "1. Again, this correlation disap pears when a random sequential updating is carried out! The exact distance-headway distribution in the NaSch model with v "1 is found to be [153,146]
H\ y y ( j"1, 2,2) , (85) 1! PA ( j)" (1!c) c(1!c)
where y"P (1, 0) is given by Eq. (76). For all v '1, at moderately high densities, P (*x) exhibits two peaks, in contrast to a single
peak in the distance-headway distributions for v "1 at all densities (Fig. 30); 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 free-#owing regions. At "rst sight, the simultaneous existence of free#owing and jammed regions may appear analogous to the coexistence of gaseous and liquid phases of matter in equilibrium. In fact, when the two-peaked structure of the distance-headway distribution was "rst observed [153,115], it was erroneously interpreted as a manifestation of the coexistence of two dynamical phases, namely, the free-#owing phase and the jammed phase. But, later works [154] established that the analogy between the coexistence of free-#owing and jammed regions in the NaSch model and the coexistence of the gas and liquid phases of matter cannot be pushed too far because the analogue of the gas}liquid interfacial tension is zero in the NaSch model. Thus, one cannot conclude that the NaSch model exhibits a "rst order dynamical phase transition.
Fig. 30. Distributions of distance-headways in the NaSch model for p"0.5 and di!erent densities for (a) v "1 and (b)
v "5. n denotes the number of empty cells in front of a vehicle and is related to the distance-headway by n"*x!1.
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8.4.7. 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 de"nition, such jams are compact). Similarly, when there are k lattice sites between two 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 2-cluster approximation or COMF [143,153,155,156]. The expressions are exact in the case v "1 with periodic boundary conditions. For higher velocities the results are only approxi mative. In COMF the probability C to "nd a jam of length k is given by I C!-+$"(1!P )PI\ , (86) I whereas in the 2-cluster approach one "nds 1 C\ " P(0"1)P(1"1)I\P(1"1)P(1"0) (k52) , I N ( 1 T P(0"v)P(v"0) . (87) C\AJ" N ( T For the n-cluster approximation similar expressions can be derived. Both distributions (86) and (87) decay exponentially for large jam sizes. COMF always predicts a monotonous distribution with C!-+$5C!-+$. In contrast, the jam size distribution in the I I> n-cluster approximation can in principle exhibit a maximum at small jam sizes 14k4n. 8.4.8. Distribution of time-headways Since #ux is equal to the inverse of the average time-headway, much more detailed information is contained in the full distribution of the time-headway than in the fundamental diagram. The time-headway distribution contains information on the temporal organization of the vehicles. Suppose, P (t ) is the probability that the following vehicle takes time t to reach the detector,
moving from its initial position where it was located when the leading vehicle just left the detector site. Suppose, after reaching the detector site, the following vehicle waits there for q!t time steps, either because of the presence of another vehicle in front of it or because of its own random braking; the probability for this event is denoted by Q(q!t "t ). The distribution P (q), of the time headway q, can be obtained from [154,157] O\ P (q)" P (t )Q(q!t "t ) . (88)
R Substituting the expressions for P (t ) and Q(q!t "t ) for v "1 in (88) we, "nally, get
[154,157]
qy qy R\ qy R\ qy # 1! 1! P(t)" d!y c d c!y
!
qy qy # pR\!q(t!1)pR\ , c!y d!y
(89)
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Fig. 31. Time-headway distribution in the NaSch model with (a) v "1 and (b) v "5.
where, q"1!p, d"1!c and, for the given c and p, y can be obtained from Eq. (77). The expression is plotted in Fig. 31(a) for a few typical values of c for a given p. A few typical time-headway distributions in the NaSch model for v '1, obtained through computer simula tion, are shown in Fig. 31(b). 8.4.9. Temporal correlations In order to probe the spatio-temporal correlations in the #uctuations of the occupation of the cells, one can study the space}time correlation function 11 * 2 n (t)n (t#q)!c (90) C(r, q)" G H>P ¹¸ H R which, by de"nition, vanishes in the absence of any correlation. In [158] three di!erent regimes have been distinguished. Free-yow (0(o4o ): `Minijamsa occur which resolve immediately. The correlation function shows anticorrelations around propagating peak. Jamming (o (o4o ): Free-#ow and jamming coexist, i.e., jams with a "nite lifetime and vehicles moving with v occur. This coexistence is re#ected in the behaviour of the correlation
function which exhibits a double-peak structure. Superjamming (o (o41): The whole system is congested. Jamming waves are connected and form an in"nite wave. As a consequence, the propagating peak in the correlation function has disappeared. Neubert et al. [159] 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 which will be introducted in later sections. 8.4.10. Structure factor Structure factors are known to give valuable information about driven systems [14]. For the NaSch model the static structure factor 1 S(k)" ¸
* n e IH H H
(91)
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has been investigated in [160]. Again n denotes the occupation number of cell j. Note that S(k) is H related to the Fourier transform of the density}density correlation function (80). For all densities, S(k) exhibits a maximum at k +0.72 which corresponds to the characteristic wavelength j "2p/k of the density #uctuations in the free-#ow regime. For v '1 one "nds
k (v #1)"const. In [161] these investigations have been extended to the dynamical structure factor in velocityspace,
1 v (t)e IL\SR , (92) S (k, u)" L T N¹ L R with k"2pm /N, u"2pm /¹, where N is the number of vehicles and m and m are integers. v (t) I S I S L is the velocity of the nth vehicle at time t. Compared to the dynamical structure factor in real space, (92) has the advantage that the free-#ow regime only contributes white noise, S (k, u)" "const. Therefore it is easier to study T U jamming properties. It is found in [161] that S (k, u) exhibits one ridge with negative slope, T corresponding to backward moving jams. One "nds that the velocity of the jams is a function of the randomization parameter p only. It is independent of the density c and the maximal velocity v [161]. This is consistent with results from a direct study of the autocorrelation function
&k\A of the structure [159]. Above a transition density, an algebraic behavior S (k, u)" T S IT
factor is found. This has been interpreted as an indication of critical behaviour in [161]. However, due to the di$culties involved in the calculation of (92) only relatively short times ¹42048 have been considered in [161]. This is much smaller than the cuto! found in the lifetime of jams (see the discussion above) and lies well in the region where an algebraic decay is found. Therefore the results for the dynamical structure factor (92) 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 cuto!, times ¹'10 would have to be considered. 8.5. Exact solution of the NaSch model with v "1 and open boundary conditions
The analytical methods presented in Section 8.3 are well suited for the investigation of translationally invariant stationary states which are achieved by imposing periodic boundary conditions. For both practical and theoretical reasons sometimes di!erent boundary conditions are preferable. Imagine a situation where a multilane road is reduced to one lane, e.g. due to road construction. Such a bottleneck can be modeled by using a NaSch model with open boundaries. The multilane part of the road acts as a particle reservoir. If the "rst cell of the one-lane part is empty a car is inserted here with probability a. At the other end a car is removed from the last cell with probability b (see Fig. 32). These boundary conditions break the translational invariance and in general one can expect stationary states with a nontrivial density pro"le 1q 2. From a more H theoretical point of view such models have been studied intensively as prototypes of systems exhibiting so-called boundary-induced phase transitions [162,163]. In contrast to what one expects Measurements of the jam dissolution speed in [150], however, show a decrease with increasing v and saturation
for large v .
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Fig. 32. De"nition of the ASEP with open boundary conditions. Fig. 33. Phase diagram of the ASEP for parallel dynamics. The inserts show typical density pro"les.
from experience with equilibrium systems, one-dimensional driven nonequilibrium systems can exhibit phase transitions, even when the interactions are short-ranged, just by `slightlya changing the boundary conditions. The probability distribution characterizing the steady state of the TASEP with parallel dynamics (i.e., NaSch model with v "1) and open boundary conditions has been obtained recently in
[164,165] using generalizations of techniques based on the matrix product ansatz (MPA) (see Appendix F for a more technical introduction). By varying the boundary rates a and b one obtains a surprisingly rich phase diagram (see Fig. 33) which is qualitatively the same for all types of dynamics. Three phases can be distinguished by the functional dependence of the current through the system on the system parameters. In the low-density phase A (a(b, a (p)) the current is independent of b. Here the current is limited by the rate a which then dominates the behaviour of the system. In the high-density phase B (b(a, b (p)) the behaviour is dominated by the output rate b and the current is independent of a. In the maximum current phase C (a'a (p) and b'b (p)) the limiting factor for the current is the bulk rate q"1!p. Here the current becomes independent of both a and b. High- and low-density phase can be subdivided into two phases AI, AII and BI and BII, respectively. These subphases can be distinguished by the asymptotic behaviour of the density pro"les at the boundaries. The transitions between the phases can be characterized by the behaviour of two correlation lengths m and m which only depend on p and a or b. These lengths can be obtained explicitly ? @ from the exact solution. Apart from m and m also a third length m\""m\!m\" plays an ? @ ? @ important role.
Note that conventionally the hopping rate in the ASEP is denoted as p. Since in the NaSch model p is the braking probability the hopping rate in the ASEP (for v "1) becomes q"1!p.
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The transition from AII (BII) to C is continuous with diverging correlation length m (m ). The ? @ transition from the high- to the low-density phase is of "rst order. Here both m and m are "nite, ? @ but m diverges. On the transition line one "nds a linear density pro"le created by the di!usion of a domain wall between a low-density region at the left end of the chain and a high-density region at the right end. For the case of parallel dynamics, i.e., the NaSch model with v "1, the currents in the three
phases are given by [164}166] a(q!a) , J " q!a
b(q!b) 1 J " , J " (1!(p) . ! 2 q!b
(93)
The corresponding bulk densities are a(1!a) q!b 1 c " , c " , c " . ! q!a q!b 2
(94)
The phase boundaries are determined by the critical rates (95) a (p)"b (p)"1!(p . In Fig. 33 also the special line (1!a)(1!b)"p is indicated. Here the density pro"le is #at (i.e., constant). On this line the exact solution can be obtained by the 2-cluster approach of Section 8.3.3 [166]. Since it goes through all three phase these results are su$cient to obtain exact analytic expressions, e.g., for the currents, once the structure of the phase diagram is established (e.g., by Monte Carlo simulations). The stationary state of the ASEP can also be obtained for other types of updates (see Appendix A), e.g., random-sequential [167,168], ordered-sequential [166,169,170] and sublattice-parallel update [171}173,166]. One "nds that the phase diagram has the same basic structure for all updates [166]. The functional dependence of the currents, density pro"les, etc., on the model parameters di!ers, however. For the important case of random-sequential updating (93)}(95) have to be replaced by [167,168] q J "qa(1!a), J "qb(1!b), J " , ! 4
(96)
1 o "a, o "1!b, o " , ! 2
(97)
q a (p)"b (p)" . 2
(98)
Results for other updates can be found in [166]. For a discussion of the calculation of di!usion constants and shock pro"les we refer to the reviews [22,174] and references therein. In the maximum current phase no real bulk density can be de"ned due to the algebraic behaviour of the density pro"le. c is therefore just 1q 2. ! *
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In [162] the behaviour of the ASEP for b"1 was explained by postulating a maximal-current principle. According to this principle, independent of the details of the dynamics, the system tries to maximize the stationary current J: (99) J" max J(c) AZ A\
Here J(c) is the fundamental diagram (for periodic boundary conditions) and c is the density at \ the left (input) boundary, i.e. c "a in the case described above. \ In [175] a nice physical picture has been developed which explains the structure of the phase diagram not only qualitatively, but also (at least partially) quantitatively. It is determined by the dynamics of domain walls. In nonequilibrium systems, a domain wall is an object connecting two possible stationary states. The notion of domain walls in the ASEP can be illustrated in the limit a¸;1 and b¸;1 of small boundary rates. At late times there will be a low-density region at the left end of the chain and a high-density region at the right end, with a domain wall in between. Schematically, this state can be depicted as 000011111. For general values of the rates the wall not be sharp in general, but spread over a few lattice sites. For late times the dynamics of the system can then be interpreted in terms of the motion of the domain wall. A particle entering the system leads moves the wall one cell to left, and a particle leaving the system moves it one cell to the right. Therefore the domain wall performs a biased random walk with drift velocity v "b!a and di!usion coe$cient D"(a#b)/2. For a(b the " domain wall moves to the right until it reaches the end of the system which is thereafter in the low-density stationary state. For a'b the wall moves to the left until it reaches the left end and the system goes into the high-density stationary state. In the case a"b there is no net drift in the position of the wall. It #uctuates with its rms displacement increasing with time as (Dt) , i.e., it can be anywhere in the system resulting in a linear density pro"le. In order to understand the case of general a and b one has to introduce a second kind of domain wall separating a maximum current phase from the high-density phase. Since the maximal possible #ow for periodic boundary conditions is J "1/4 (for p"0 and random-sequential update) the
dynamics for a"1/2 is dominated by the overfeeding at the left boundary. The injection rate could support a current larger than 1/4, but in the bulk it cannot exceed this value. Therefore at the left boundary a maximum current state is formed. If the particles are not extracted fast enough at the right boundary a high-density region will develop there. These two regions are separated by a new kind of domain wall, the maximum current/high-density domain wall. Schematically it can be represented as mmmm1111. Again this domain wall performs a biased random walk. In order to obtain more quantitative predictions one goes to a coarse-grained picture. Then it is useful to replace the boundary rates a and b by particle reservoirs with densities c and c . The \ > continuity equation Rc/Rt#RJ/Rx"0 in the continuum limit has traveling wave solutions of the form c(x!v t) with the domain wall velocity " v "(J !J )/(c !c ) (100) " > \ > \ which can be obtained by integration over the chain. A somewhat related approach has been used to obtain an approximate solution for the special case of parallel dynamics with deterministic bulk dynamics (p"0) [176].
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For the low-density/high-density domain wall one has c "1!b, J "J(c )"b(1!b) > > > and c "a, J "J(c )"a(1!a). This gives indeed v "b!a which should be valid for \ \ \ " a, b(1/2. For the maximum current/high-density domain wall c takes the value c "1/2 \ \ so that J "1/4 and thus v "b!1/2. \ " The arguments described above can be generalized to any process where the fundamental diagram J(c) of the periodic system has only one maximum at a density cH. For all currents J(J(cH) there exist two corresponding densities c and c with J(c )"J"J(c ). For funda mental diagrams with more than one maximum, more than two densities might exist for a given current J. This implies the existence of a larger number of domain wall types. The phase diagram of the open system than exhibits a larger number of phases [177]. The maximal-current principle (99) for the TASEP with b"1 is generalized to the extremal-current principle [177,15]
max > \ J(c) for c 'c , \ > AZ A A
(101) min \ > J(c) for c (c . \ > AZ A A
Since the above phenomenological picture does not depend on the microscopic details of the dynamics, it is plausible that the phase diagrams for di!erent updates are qualitatively the same. Boundary-induced phase transitions have recently been observed [178] in measurements on a German motorway [41]. One "nds a "rst-order nonequilibrium phase transition between a free-#ow and a congested phase. This transition is induced by the interplay between density waves induced by an on-ramp and a shock wave moving on the motorway [178]. J"
9. Generalizations and extensions of the NaSch model As stated earlier, the NaSch model is a minimal model. The "rst obvious possible generalization would be to replace the acceleration stage of updating rule (U1) to v Pmin(v #a , v ) , (U1 ) L L L where a , acceleration assigned to the nth vehicle, need not be unity and, in general, may depend on L n. In the following subsections we consider more nontrivial generalizations and extensions of the NaSch model. 9.1. Single-lane highways In the next few subsubsections we shall demonstrate the rich variety of tra$c phenomena that can be observed by appropriate modi"cations of the random braking. We have earlier mentioned in the context of empirical results that tra$c #ow exhibits metastability and the related hysteresis e!ects. Such phenomena have been observed in continuum formulations of `microscopica models, i.e., in coupled-map lattice models [115]. However, the NaSch model is too simple to account for these phenomena. We now brie#y describe a few generalizations of the NaSch model, each of which We assume p"0 and random-sequential dynamics.
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Fig. 34. Metastability and hysteresis.
is based on modi"cations of the braking rules of the original NaSch model; one common feature of all of these generalized models is that they show metastability and hysteresis. Before we begin our discussions on speci"c generalized versions of the NaSch model, which exhibit metastability, we make some general remarks. In the schematic stationary fundamental diagram of Fig. 34, the low-density branch corresponds to homogeneous free-#ow states, while the high-density branch corresponds to con"gurations, where jammed states are present. Obviously, at densities c (c(c , the #ow depends non-uniquely on the global density. In order to establish the existence of meta-stable states one can follow two basic strategies. In the "rst 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. On the other hand, by adding vehicles to a free #owing state (low densities), one 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 di!erent 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 c'c the homogeneous initialization leads to a free-#ow state, while the mega-jam initialization leads to the jammed high-density states. 9.1.1. Cruise-control limit and self-organized criticality In the cruise-control limit of the NaSch model [179] vehicles moving with their desired velocity v are not subject to noise. This is exactly the e!ect of a cruise-control which automatically keeps
the velocity constant at a desired value. In this model the acceleration, deceleration (due to other vehicles) and movement stages of updating are identical to those in the general case of the NaSch model; however, the randomization step is applied only to vehicles which have a velocity v(v
after step 2 of the update rule. We can express this more formally by recasting the randomization stage of the update rules in the NaSch model as follows: v Pmax(0, v!1) L
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with probability
p"
p T p
if v"v if v(v
, ,
(102)
where v is the velocity of the vehicle at the end of step 2 of the update rule, i.e., after deceleration due to blocking by other vehicles. In the original formulation of the NaSch model p "p. On the T other hand, the cruise-control limit corresponds to p P0, pO0. T For p ;1, at su$ciently low densities, all the vehicles move deterministically with the velocity T v ; this deterministic motion is, however, interrupted by small perturbations at a vanishingly
small rate. Consequently, the system gets enough time to relax back to the state corresponding to the deterministic algorithm before it is perturbed again. This e!ectively separates completely the time scales for perturbing the system and the response of the system. First, let us consider the periodic boundary conditions which is easier to treat than the open boundary conditions. In this model, a sharp transition from the `free-#owinga dynamical phase to the `congesteda phase takes place at a critical concentration c (p, v ) which depends on p as well H
as v and, for all pO0, c (p, v ) is smaller than c"1/(v #1). For a given v , c (p)
H
H increases with decreasing p and, in the deterministic limit pP0, c (0, v )Pc. H
In this model a jam is de"ned to consist of vehicles all of which have their instantaneous velocities smaller than v . For all c(c , jams present in the initial con"guration eventually
H disappear and in the jam-free stationary state every vehicle moves with the velocity v . Therefore,
in the density regime c(c the #ux increases linearly with density following J"cv , just like H
that in the deterministic limit p"0 of the NaSch model (Fig. 35). But, unlike the deterministic limit p"0, the cruise-control limit of the NaSch model exhibits metastability in the interval c (c(c. In this context, the metastability means that, in the interval c (c(c, on H
H
appropriate initialization, the system can reach apparently steady states where no jam appears and where the #uxes are higher than J(c ); but, perturbations of such a `metastablea state creates H long-lived jams thereby reducing the #ux to a level consistent with the stable branch of the fundamental diagram. At all c'c jams present in the initial con"guration never disappear H
Fig. 35. Fundamental diagram in the cruise-control limit of the NaSch model for v "5 and p "0, p"0.5.
T
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completely and, in this density regime, the stable steady-state is a mixture of laminar #ow regions and jams. The long-lived jams lower the #ux beyond c and the #ux decreases linearly with density H (Fig. 35). Let us assume that at densities slightly above c , only one jam of length ¸ containing H
N vehicles exists in the system. Then, because of the periodic boundary conditions, the total
number of vehicles N is conserved and, hence, N"c
¸ #c (¸!¸ ) , (103)
where c "N /¸ and c "(N!N )/(¸!¸ ) are the densities of the vehicles in the
jam and in the out#ow region, respectively. Dividing both sides of (103) by ¸ we get c"c
¸ ¸
#c 1!
¸ ¸
.
(104)
Since in the cruise-control limit of the NaSch model ¸ must vanish as cPc , we conclude [179]
H that we must have c "c , i.e., the average density in the outyow region of a jam is equal to the H critical density c . H In order to study the tra$c at the critical point of the cruise-control limit, Nagel and Paczuski [179] used a special boundary condition which enables the system to select automatically the state of maximum throughput, i.e., the system exhibits self-organized criticality. This special boundary condition consists of an in"nite jam from !R to 0 (i.e., at the left boundary) while the right boundary is open. Vehicles emerge from the in"nite jam in a jerky way, before attaining the velocity v . Far away from the in"nite jam all vehicles move with the same velocity v . In order to show
that the state selected this way is `criticala [179] we perturb a vehicle, far downstream from the in"nite jam, slightly by reducing its velocity from v to v !1. This particular vehicle initiates
a chain reaction and gives rise to a jam if the following vehicle is su$ciently close to it although it itself accelerates and, eventually, attains v . This phantom jam has a time-dependent size n(t),
measured by the number of vehicles n in this jam at time t and it has a lifetime q . The statistics of these features of the phantom jam can be obtained by repeating the computer experiment su$ciently large number of times; sometimes the phantom jam created is small and has a short lifetime and sometimes it is large and has quite long lifetime. Interestingly, the characteristic quantities like, for example, the distributions of the sizes of the jams, lifetimes of the jams, etc. do, indeed, exhibit power-laws which are hall mark of the self-organized criticality [26,27]. E.g. the branching behaviour of the jams gives rise to intermittent dynamics with a 1/f power law spectrum [179]. 1/f noise in real tra$c has been discovered by Musha and Higuchi [180]. They recorded transit times of vehicles passing underneath a bridge. The corresponding power spectral density of the #ow #uctuations shows 1/f behaviour at low frequencies. The exponents associated with the various power laws in the cruise-control limit of the NaSch model can be calculated analytically, at least for v "1, by utilizing a formal relation with
one-dimensional unbiased random walk. If v "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!r !r )P(n; t)#r P(n!1; t)#r P(n#1; t) ,
(105)
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where the phenomenological parameters r and r are the rates of incoming and outgoing vehicles. Of course, r depends on the density of the vehicles behind the jam. For large n and t, taking the continuum limit of Eq. (105) and expanding we get RP/Rt"(r
RP r #r RP . !r ) # Rn 2 Rn
(106)
If r 'r the jams would grow for ever. On the other hand, the jams would shrink, and eventually disappear, if r (r . If r "r , the "rst term on the right-hand side of Eq. (106) vanishes and the resulting equation governing the time evolution of P(n; t) is identical to that of the probability of "nding, at time t, an unbiased one-dimensional random walker at a distance n which was initially at the origin. Thus, when r "r , the jams exhibit large (`criticala) #uctuations which can be characterized by critical exponents. Using this formal mapping onto unbiased random walk, we "nd (a) that the mean size of jam at time t corresponds to the mean displacement of the random walker from the origin after time interval t and (b) that the lifetime of a jam corresponds to the time taken by the random walker to return to the origin for the "rst time. Hence, using the well-known results from the theory of random walks [181,182], we get n(t)Jt and P(q )Jq\ . (107) It turns out that the power-law exhibited by the size of the jams, the distributions of the lifetimes, etc., are not restricted merely to the special case v "1 of the cruise-control limit but is also
shown by the corresponding computer simulation data also for arbitrary v . The power-law
distributions of P(q ) in the cruise-control limit of the NaSch model is in sharp contrast with the exponential distribution observed in the NaSch model [151]. Thus, in the cruise-control limit of the NaSch model, the large jams are fractal [183] in the sense that there are smaller sub-jams inside larger jams, ad in"nitum. In other words, in between sub-jams, there are holes of all sizes. 9.1.2. Slow-to-start rules, metastability and hysteresis The slow-to-start rules can lead not only to metastability and, consequently, hysteresis, but also to phase separated states at high densities, as we now show. Takayasu}Takayasu slow-to-start rule. Takayasu and Takayasu (TT) [184] were the "rst to suggest a CA model with a slow-to-start rule. Here, we investigate the generalization, as suggested in [185], of the original slow-to-start rule. According to this generalized version, a standing vehicle (i.e., a vehicle with the instantaneous velocity v"0) with exactly one empty cell in front accelerates with probability q "1!p , while all other vehicles accelerate deterministically. The other steps of R R the update rule (U2-U4) of the NaSch model are kept unchanged. As in the case of the NaSch model, it is instructive to consider "rst the deterministic limits of the TT model [184,186]. The TT model reduces to the NaSch model in the limit p "0. What happens R in the other deterministic limit, namely, p "1? In the latter deterministic limit, a stopped vehicle R can move only if there are at least two empty cells in front [184]. Obviously, completely blocked states exist for densities c50.5, where the number of empty cells in front of each vehicle is smaller than two. However, in the region 0.54c:0.66 the number of blocked con"gurations is very small compared to the total number of con"gurations and states with a "nite #ow exist. Precisely at c"0.5, there are only two blocked states and the time to reach these states diverges exponentially with the system size.
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The fundamental diagram for the TT model with v "1 has been derived analytically by
carrying out (approximate) 2-cluster calculations in the site-oriented approach [185]. But, the fundamental diagrams of the TT model for all v '1 have been obtained so far only numerically
by carrying out computer simulations (see Fig. 36). Comparing these fundamental diagrams with the corresponding ones for the NaSch model (p "0), we "nd the following e!ects of the TT slow-to-start rule: (i) for a given density c, the R #ux J(c) is smaller in the TT model as compared to that in the NaSch model; (ii) the particle-hole symmetry is not exhibited by the TT model for any v (not even for v "1) and (iii) the TT
model exhibits metastability and hysteresis which are absent in the NaSch model. Note that the mechanism for meta-stability in the case p "1 is di!erent from that for the metastability observed R for 0(p (1. R Because of the slow-to-start rules, the separations between the vehicles coming out of a jam are larger than those between the vehicles coming out a jam in the NaSch model. Since the density far downstream is smaller than the density of maximum #ow, the vehicles can propagate freely in the low density regions of the lattice where spontaneous formation of jams is highly unlikely, if the parameter p is su$ciently small. Therefore, the phase-separated steady-states at high global densities consist of a macroscopic jam and a macroscopic free-#ow regime both of which coexist simultaneously (Fig. 37).
Fig. 36. The fundamental diagram in the TT model (v "5, p"0.01, p "0.75, ¸"10 000) obtained using two
R di!erent initial conditions, namely, a completely jammed state (jam) and a homogeneous state (hom) and averaging over 100 000 sweeps through the lattice. Fig. 37. Typical space}time diagram of the TT model with v "5 and c"0.20, p"0.01 and p "0.75. Each
R horizontal row of dots represents the instantaneous positions of the vehicles moving towards right while the successive rows of dots represent the positions of the same vehicles at the successive time steps.
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The BJH model of slow-to-start rule. Benjamin, Johnson and Hui (BJH) [187] modi"ed the updating rules of the NaSch model by introducing an extra step where their `slow-to-starta rule is implemented; this slow-to-start rule is di!erent from that introduced by TT [184]. According to the BJH `slow-to-starta rule, the vehicles which had to brake due to the next vehicle ahead will move on the next opportunity only with probability 1!p . The steps of the update rules can be stated as follows: Step 1: Acceleration. v Pmin(v #1, v ). L L
Step 2: Slow-to-start rule: If -ag"1, then v P0 with probability p . L Step 3: Blockage (due to other vehicles). v Pmin(v , d !1) and, then, -ag"1 if v "0, else L L L L -ag"0. Step 4: Randomization. v Pmax(v !1, 0) with probability p. L L Step 5: Vehicle movement. x Px #v . L L L Here -ag is a label distinguishing vehicles which have to obey the slow-to-start rule ( -ag"1) from those which do not have to ( -ag"0). Obviously, for p "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 (Fig. 38) clearly shows the existence of metastable states and, therefore, expected to exhibit hysteresis e!ects [155]. But, in the special case of v "1, for which approximate analytical calculations can be carried out [185], no meta-stable
states exist.
Fig. 38. The fundamental diagram in the BJH model (v "5, p"0.01, p "0.75) obtained using two di!erent initial
conditions, namely, a completely jammed state (jam) and a homogeneous state (hom). Fig. 39. Typical space}time diagram of the BJH model with v "5 and c"0.20, p"0.01 and p "0.75. Each
horizontal row of dots represents the instantaneous positions of the vehicles moving towards right while the successive rows of dots represent the positions of the same vehicles at the successive time steps.
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Fig. 40. Distribution of gaps between jams in the BJH and the NaSch model for v "5, p"0.05, p "0.5 and density
c"0.5.
Since for all v '1 in the BJH model, just as in the TT model, the out#ow from a jam is smaller
than the maximal #ow, the phase-separated steady-states at high global densities consist of a macroscopic jam and a macroscopic free-#ow regime both of which coexist simultaneously (Fig. 39) [155]. However, the macroscopic jam is not compact. The typical size of the macroscopic free-#ow regime can be estimated by measuring the distribution of the gaps between the successive jams [155]. A peak occurs in this distribution for headways of the order of the system size (see the inset of the right part of Fig. 40). The position of the peak indicates the typical size of the macroscopic free-#ow regime. The NaSch model with a velocity-dependent slow-to-start rule. Although the NaSch model does not exhibit metastable states and hysteresis, a simple generalization exists which is able to reproduce these e!ects. It is the so-called velocity-dependent-randomization (VDR) model [188]. 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 8) are supplemented by a new rule, Step 0: Determination of the randomization parameter. The randomization parameter used in step 3 for the nth vehicle is given by p"p(v (t)). L This new step has to be carried out before the acceleration step 1. The randomization parameter used in step 3 depends on the velocity v (t) of the nth vehicle after the previous timestep. In order to L implement a simple slow-to-start rule one chooses [188]
p(v)"
p p
for v"0 , for v'0 ,
(108)
with p 'p. This means that vehicles which have been standing in the previous timestep have a higher probability p of braking in the randomization step than moving vehicles. The rules of the VDR model can be recast in a form similar to those of the BJH model. We de"ne a label -ag which distinguishes between vehicles which have to obey the slow-to-start rule ( -ag"1) from those which do not have to ( -ag"0). -ag"1 if v "0 at the beginning of a time L
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step, else -ag"0. Explicitly, the update rules are as follows: Step 1: Acceleration. v Pmin(v #1, v ). L L
Step 2: Blockage (due to other vehicles). v Pmin(v , d !1), L L L Step 3: Randomization. v Pmax(v !1, 0) with probability p if -ag"1. else, v Pmax(v !1, 0) L L L L with probability p. Step 4: Vehicle movement. x Px #v . L L L Let us compare this VDR model with the cruise-control limit of the NaSch model. The vehicles with velocity v"v (at the end of step 2) are treated deterministically in the cruise-control limit
whereas in the VDR model velocities of all those with the velocity v'0 ( just before step 3) are updated stochastically, but using di!erent values of the braking parameter. Typical fundamental diagrams look like the one shown in Fig. 41 where, over a certain interval of c, J(c) 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. 42) 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 qualitative similar to those observed in the high-density regimes of the TT and BJH models but di!ers drastically from those found in the NaSch model. It is instructive to compare the fundamental diagram of the VDR model with those of the corresponding NaSch models. We now present a simple derivation of the fundamental diagram of the VDR model on the basis of heuristic arguments utilizing the observed structures of the steady states. For small densities c;1 there are no slow vehicles in the VDR model since interactions between vehicles are extremely rare. In this regime every vehicle can move with the free-#ow
Fig. 41. The fundamental diagram in the NaSch model with a velocity-dependent slow-to-start rule (v "5,
p "0.75, p"1/64) obtained using two di!erent initial conditions, namely, a completely jammed state ( jam) and a homogeneous state (hom). Fig. 42. Typical space}time diagram of the VDR model with v "5 and c"0.20, p"0.01 and p "0.75. Each
horizontal row of dots represents the instantaneous positions of the vehicles moving towards right while the successive rows of dots represent the positions of the same vehicles at the successive time steps.
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velocity v "(1!p)v #p(v !1)"v !p and, therefore, the #ux is given by
J (c)"c(v !p)
(109)
which is identical to the NaSch model with randomization p. On the other hand, for densities close to c"1, the vehicles are likely to have velocities v"0 or v"1 only and, therefore, the random braking is dominated by p , rather than p, while the #ow is determined by the movement of the holes. Hence, for large densities, i.e., 1!c;1, the #ow is given by J(c)+(1!p )(1!c) which corresponds to the NaSch model with randomization p . This expression for #ux 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 v "v !p in the free-#ow region. Neglecting interactions between
vehicles in the free-#owing region (which is justi"ed because of the corresponding low density), the average distance of two consecutive vehicles in the free-#ow region is given by *x"c\"¹ v #1 where the average waiting time ¹ of the "rst vehicle at the head of the megajam is given by ¹ "1/(1!p ). In other words, the density in the free-#ow regime c is determined by the average waiting time ¹ and v . Now suppose that N and N are the number of ( $ vehicles in the megajam and free-#owing regions, respectively. Using the normalization ¸"N #N *x we "nd that for the density c"(N #N )/¸, the #ux J (c) is given by ( $ $ ( J (c)"(N /¸)(v !p) and, hence, $
J (c)"(1!p )(1!c) .
(110)
Obviously, c is precisely the lower branching density c , because for densities below c the jam-length is zero. It should be noted that the heuristic arguments presented above remain valid for p
In the case v "1, vehicles can stop spontaneously, even in the free-#ow regime and these
vehicles might initiate a jam. This is the basic reason why hysteresis is usually not observed for v "1.
Analogous to the BJH model phase separation can be directly identi"ed using the results of the jam-gap distribution. Fig. 43 shows that the size of the free-#ow regime is proportional to the system size. The results for the slow-to-start models discussed above have been obtained by computer simulations of periodic systems of "nite length. It was shown that the fundamental diagram which is sketched in Fig. 34 is generic for all models under consideration. Now it is self-evident to ask what kind of stationary states are realized in the thermodynamic limit ¸PR. The simulation results indicate that *c"c !c decreases with larger system sizes and is expected to vanish for ¸PR, i.e., the jammed branch is stable in that limit. This is readily understood if one analyses the typical con"gurations which lead to an emerging jam or, vice versa, 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-#ow state where the gap between the vehicles is not larger than v . Obviously the probability to "nd large platoons of vehicles driving with small spatial
headways increases with the system size (for "xed density). In addition to that 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 #uctuates due to the stochastic movement and acceleration of the vehicles.
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Fig. 43. Peaks of the jam-gap distribution at macroscopic distances.
Jams can dissolve if the amplitude of these #uctuations are of the order of the length of the jam, which is impossible in the thermodynamic limit. Therefore, the non-unique behaviour of the fundamental diagram is only observable if "nite system sizes are considered or if the vehicles move deterministically in the free #ow regime. Nevertheless, the results discussed above are highly relevant for practical purposes, because the hysteresis e!ects have been observed at realistic system sizes (e.g. ¸"10 000 corresponds to a highway of length 75 km). 9.1.3. Flow-optimization and meta-stable states Hysteresis e!ects and meta-stable 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 #ow, if the homogeneous state is stabilized by controlling the density so that it never exceeds c . This strategy was followed in minimizing frequent jams in the Lincoln- and the Holland-Tunnels in New York. Before the tra$c lights were installed at the entrance of the tunnels jams used to form spontaneously within the tunnel because (a) the vehicle density used to be su$ciently high and (b) the drivers used to drive more carefully inside the tunnel thereby giving rise to stronger #uctuations which caused the jams. But, the tra$c lights installed at the entrance of the tunnels do not allow the density to exceed c and, consequently, jams are not formed spontaneously by the decay of any metastable high-density state. One can mimic the situation of Lincoln and Holland-tunnels within the framework of the CA models in the following way [188}190]. The tunnel is considered as part of the road, where the braking probabilities pR, pR are higher compared to the remaining part of the lattice (p, p , see Fig. 44). Therefore, if one allows for an uncontrolled in#ow of the vehicles, jams typically appear inside the `tunnela and the system capacity is governed by pR . The situation di!ers drastically if tra$c lights are implemented [188}190]. As shown in Fig. 45, a considerable increase of the maximum capacity can be achieved for an optimal combination of the red-/green-signal periods. The gain of capacity obtained for the optimal intervals of the signal is of the same order as for the realistic examples [191].
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Fig. 44. A schematic representation of a tunnel on a highway. Larger values of the braking parameters are used for the right part, which mimics the tunnel.
Fig. 45. Average #ow for di!erent combinations of red-/green-signal periods for a system of length ¸"1000 and v "5. Inside the tunnel pR"0.15, pR "0.60 and outside p"0.01, p "0.20 has been used.
9.2. Multi-lane highways For a realistic description of tra$c on highways the idealized single-lane models must be generalized to develop CA models of multi-lane tra$c; the main ingredient required for this generalization being the lane-changing rules. Several attempts have been made so far in this direction [192}199]. The lane changing rules for two-lane tra$c can be symmetric or asymmetric with respect to the lanes. Similarly, if there are two (or more) di!erent types of vehicles (say, cars and trucks) with two di!erent v , the lane-changing rule can be symmetric or asymmetric with
respect to the vehicles. In general, the update in the two-lane models is divided into two sub-steps: in one sub-step, the vehicles may change lanes in parallel following the lane-changing rules and in the other sub-step each vehicle may move forward e!ectively as in the single-lane NaSch model. Drivers must "nd some incentive in changing the lane. Two obvious incentives are (a) the situation on the other lane is more convenient for driving, and (b) the need to make a turn in near future. Two general prerequisites have to be ful"lled in order to initiate a lane change: "rst, there must be an incentive and second, the safety rules must be ful"lled [200]. Lane changing rules according to this scheme have been introduced by Rickert et al. [193]. They suggested that vehicles are allowed to
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change the lane if the following four criteria are satis"ed: (C1) (C2) (C3) (C4)
gap(i)(l, gap (i)'l , M M gap (i)'l , M M rand()(p . Here gap(i) and gap (i) are the gaps in front of vehicle i on the own lane and the other lane, M respectively. gap (i) is the gap on the other lane to the next vehicle behind. l, l , l and p are M M M parameters specifying the rule and rand() is a random number in the interval [0, 1]. The "rst rule C1 represents the incentive criterion, i.e. if the gap gap(i) in front of the vehicle is not su$ciently large vehicles want to change the lane. Typical choices of the parameter l are given by l"min(v#1, v ). 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 C2 it is checked if the situation on the other lane is indeed more convenient. This motivates the choice l"l . The third rule M C3 avoids too small distances to following vehicles on the other lane. Rickert and coworkers suggested l "v . It is also important to perform lane changing stochastically. Even if M
the incentive and safety criteria are ful"lled a lane change is performed only with probability p (C4). This avoids, at least partially, so called ping-pong lane changes, i.e. multiple lane changes of vehicles in consecutive timesteps. 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. [196] (see Fig. 46). The lane changing rules for two-lane tra$c can be symmetric or asymmetric with respect to the lanes [193]. If symmetric lane changing rules are applied the rules do not depend on the direction of the lane changing maneuver. In contrast also asymmetric lane changing rules have been considered. Lane changing rules can be asymmetric in two ways. First it is possible that it is preferred to drive on the right lane at low densities. This behaviour can be implemented simply by leaving out the "rst rule for a change from the left to the right lane. Second it is also possible that it is even forbidden to overtake a vehicle on the right lane, e.g. on german highways. Then the single lane dynamics on the right lane depends on the con"guration on the left lane. These examples show the #exibility of the CA approaches. Moreover the simulations also show that the details of the lane changing rules may lead to considerable changes of the model results [194,196]. In multi-lane tra$c it is of particular interest to investigate systems with di!erent types of vehicles. For CA models this has been done "rst by Chowdhury et al. [195], who simulated a periodic two-lane system with slow and fast vehicles, i.e. vehicles with di!erent v . The
simulation results have been shown that already for small densities the fast vehicles take on the
The gap on the other lane is de"ned in the same way as the gap on the own lane by imagining that the vehicle occupies the site parallel to its current position. This artifact of the parallel update was already pointed out by Nagatani [192], who simulated a two-lane system with v "1.
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Fig. 46. The table gives an overview over di!erent choices of the lane changing rules discussed in the literature. The numbers of the references correspond to those in [196]. The "rst column shows different incentive criteria and the second column denotes the corresponding minimal gap on the other lane. In general the incentive criteria can be di!erent for a change from left to right (¸PR) and (RP¸) (from [196]).
average the free-#ow velocity of the slow vehicles, even if only a small fraction of slow vehicles have been considered. Analogous results have been obtained by Helbing and Huberman [212] who used a di!erent CA model for the in-lane update (see Section 11.3 for the de"nition of the model). In addition to that Nagel et al. [196] have been shown that for a suitable choice of the lane changing rules and di!erent types of vehicles even the phenomenon of `lane inversiona which has been
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observed at german highways can be reproduced. The results discussed so far show the strong in#uence of slow vehicles in multi-lane systems. They "t fairly well the empirical results, which show an alignment of the speeds on di!erent lanes and of di!erent types of vehicles. Nevertheless recent simulation results of Knospe et al. [197] indicate that the in#uence of slow vehicles seems to be overestimated by the multi-lane 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 #ow. In order to weaken the e!ect of slow vehicles they suggested to consider anticipation e!ects, i.e. the driver estimates the velocity of the vehicle in the next timestep [197]. Another interesting quantity to look at is the frequency of lane changes at di!erent densities. Here the simulation results show that close to the density of maximal #ow 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 su$ciently large gaps exist. For larger values of the braking noise large density #uctuations are observable. Therefore the local minimum of the lane-changing frequency is not found for larger values of p. In general, the simulation results show that some generic multi-lane e!ects can be pointed out. First of all the maximal performance of multi-lane systems is slightly increased compared to corresponding single-lane network. In addition, slow vehicles lead to an alignment of velocities of di!erent type of vehicles already at low densities which is con"rmed by empirical observation. This e!ect is quite robust for di!erent choices of the CA model as well as for di!erent lane changing rules. It can be weakened most e$ciently if anticipation e!ects are applied. The details of the lane changing rules, however, may have strong in#uence on the lane usage characteristics. 9.3. Bidirectional trazc Simon and Gutowitz [201] 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 su$ciently low, a pass will be attempted. This means that at high global densities the lanes are e!ectively decoupled since only very few passes will occur. In principle, three types of jams can occur on a bidirectional road: (1) spontaneous jamming and start-and-stop waves on one of the lanes; (2) jams caused by drivers who try to pass but cannot return to their home lane since there is not enough space and (3) `super-jamsa when an adjacent pair of drivers tries to pass simultaneously. These super jams halt tra$c 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 tra$c described in the previous Section 9.2. First, the situation on the own lane is examined. If the motion is hindered by another vehicle (moving in the same direction), a pass is attempted. This will only be initiated if the safety criteria are satis"ed: (1) the gap on the other lane has to be su$ciently large, and (2) the 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 #ow the left lane is even higher frequented.
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number of vehicles to be passed has to be small. Even if these criteria are satis"ed a lane change occurs only with probability p . After this lane changing step the vehicles move forward similar to the dynamics of the NaSch model. There are, however, important di!erences. 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 tra$c decelerate deterministically by one unit. This rule prevents the occurrence of a super jam. The results of [201] show the expected behaviour, namely that passing makes tra$c more #uid. Start-stop waves are surpressed if the density is not too large. The improvement of the #ow on one lane compared to the the one-lane model depends on the density of vehicles on the other lane. It is maximal for very small densities (cP0) on the passing lane. If the density on the other lane is small (c(0.25) the #ow may be lower than in the one-lane model since passing oncoming vehicles create an additional hindrance. For large densities on at least on of the lanes there is little di!erence between the one- and two-lane models. Lee et al. [202] have proposed a toy model for bidirectional tra$c based on a multispecies generalization of the ASEP. Here no passing is allowed. Instead oncoming tra$c 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 update and v "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/b. 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 c when this cell is empty and by c/b if it is occupied. For c(1 the unin#uenced 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 b can be interpreted as a kind of road narrowness. For b"1 vehicles are not slowed down by oncoming tra$c. This corresponds to a highway with divider. The case bP0 corresponds to a narrow road being completely blocked by the oncoming tra$c. The behaviour of the model with only one truck is rather similar to that of the NaSch model with quenched disorder (see Section 10). For b'b 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 2-species variant of the ASEP. Using the matrix-product Ansatz (see Appendix F) many steady-state properties for the single-truck case can be obtained exactly. Two phase 5 can be distinguished: A low-density phase for cb(1 and a jammed phase for cb'1 where c is the density of vehicles on lane 1. In contrast to the case of a "xed defect site (see Section 10.3) only one critical density c "1/b exists since the particle-hole symmetry is broken. Generalizations of this model to other updates and higher velocities can be found in [203].
10. E4ects of quenched disorder on tra7c 10.1. Randomness in the braking probability of drivers and Bose}Einstein-like condensation We have seen how modi"cations of the random braking probability or the rule(s) for random braking in the NaSch model can give rise to a rich variety of physical phenomena, e.g.,
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self-organized criticality, metastability and hysteresis, etc. Now we consider the e!ects of quenched randomness in the random braking probability p, i.e., we study the e!ects of assigning randomly di!erent time-independent braking probabilities p to di!erent drivers i in the NaSch model. Such G `quencheda (i.e., time-independent) randomness in the random braking in the NaSch model can lead to exotic phenomena [204,197] which are reminiscent of `Bose-Einstein-like condensationa in the TASEP where particle-hopping rates are quenched random variables [205,206]. Various aspects of these phenomena have been thoroughly reviewed by Krug [207] and, therefore, we'll restrict our discussion to only the essential points. Let us "rst consider the special case of the NaSch model with v "1. As explained earlier, this
model reduces to the TASEP if the parallel updating is replaced by random sequential updating scheme. If the same hopping probability q is assigned to every particle except one for which the hopping probability is q (q, then the single `impuritya particle is the slowest moving one. The faster particles can be allowed to overtake the slow one at a nonzero rate [174,208]; 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 tra$c behind the slow vehicles (e.g., trucks). Here we are interested in a more general situation of quenched `disordera in the form of a distribution of intrinsic hopping probabilities of the vehicles in the system rather than that of the single `defecta particle. In such situations random initial conditions can lead to the formation of platoons if (a) slow particles are su$ciently rare and (b) if the density of vehicles is su$ciently low. 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, for example, binary alloys [28]. Suppose, m(t) is the typical platoon size at time t. Starting from a homogeneous spatial distribution of the vehicles, m(t) can be monitored as a function of time t to "nd out the law of `growtha of the size of the platoons. Before describing the e!ects of the quenched randomness in the hopping probabilities on the steady states of the TASEP and the NaSch model, we consider an even simpler model of platoon formation [209,210] which was developed using the language of aggregation phenomena. In this model an initial velocity v is assigned to each vehicle j, drawn randomly from a continuous H 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 m(t) increases inde"nitely according to the power law m(t)&tL> L> ,
(111)
where the exponent n characterizes the behaviour of f (v) in the vicinity of the minimal velocity v ,
i.e., f (v)&A(v!v )L as vPv with some positive constant A. An attempt has been made to
develop a coarse-grained description of this phenomenon [211]. It has been shown [205,206] that if quenched random hopping probabilities are assigned to each particle in the TASEP, there are small gaps between particles in the high-density congested phase but in the inhomogeneous low-density phase there is a macroscopically large empty region in front of the slowest particle (i.e., the particle with smallest hopping probability) behind which a platoon is formed. The phase transition from the low-density inhomogeneous phase (which consists of
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a macroscopic free region and a platoon) to the high-density congested phase is, in many respects, analogous to the Bose}Einstein transition. In order to see this analogy, let us imagine that the empty sites are bosons and the state of a boson is determined by which particle it is immediately in front of. In the language of the ideal Bose gas, in the high-density phase the bosons are thinly spread over all the states. On the other hand, in the low-density phase there is a "nite fraction of the empty sites are condensed in front of the slowest particle in such `Bose}Einstein-like condenseda state. The steady-state velocity of the particles is the analogue 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 vehicles. The Bose}Einstein-like-condensation in the TASEP with quenched random hopping probabilities of the individual particles survives when the random sequential updating is replaced by parallel updating [206]. 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 non-interacting quantum particles. The qualitative features of the dynamical phases and phase transitions observed in the NaSch model with random braking probabilities, for v "1 as well as for larger v , are very similar to
those described above for the TASEP with random hopping probabilities [204]. Typical snapshots of the system at three di!erent stages of evolution from a random initial state are shown in Fig. 47. The typical size of the platoons m(t) can be computed directly [204] by computing the correlation function (80) and identifying the separation r"R of the "rst zero-crossing of this correlation as m(t). Following this procedure, it has been observed that m(t) follows the power law (111) when the distribution of the random braking probabilities is given by P(p)"2L(n#1)(!p)L. 10.2. Random v
The two important parameters of the NaSch model are p and v . In the preceding subsection
we have seen the e!ects of randomizing p assigning the same v to all the vehicles. In this
subsection, on the other hand, we investigate the e!ects of randomizing v , assigning a non random constant p to every driver. The simplest possible model to investigate the e!ects of quenched randomness in v is that
considered by Ben-Naim et al. [209] which was discussed to motivate the phenomenon of platoon formation in the preceding subsection. In order to model tra$c consisting of two di!erent types of vehicles, say, for example, cars and trucks, of which a fraction f are intrinsically fast (say, cars) while the remaining fraction 1!f are intrinsically slow (say, trucks), Chowdhury et al. [195] assigned a higher v (e.g., v "5) to
a fraction f of vehicles chosen randomly while the remaining fraction 1!f were assigned a lower v (e.g., v "3). As the density of the vehicles increases, the vehicles with higher
v "nd it more di$cult to change lane in order to pass a vehicle with lower v ahead of it in the
same lane. This leads to the formation of `coherent moving blocksa of vehicles each of which is led by a vehicle of lower v [212]. Two main causes of tra$c accidents, namely, di!erences in vehicles
speeds and lane changes, are reduced considerably in this state thereby making this state of tra$c much safer. It is worth mentioning that even a small number of slow vehicles in 2-lane models, where overtaking is possible, can have a drastic e!ect. For details we refer to [197] and the discussion in Section 9.2.
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Fig. 47. Typical space}time diagram of the NaSch model with v "2 and c"0.1 and random braking probabilities.
Each horizontal row of dots represents the instantaneous positions of the vehicles moving towards right while the successive rows of dots represent the positions of the same vehicles at the successive timesteps.
10.3. Randomly placed bottlenecks on the roads and the maximum yux principle So far we have investigated the e!ects of two di!erent types of quenched randomness both of which were associated with the vehicles (i.e., particles). We now consider the e!ects of yet another type of quenched randomness which is associated with the road (i.e., lattice). In order to anticipate the e!ects of such randomness associated with the highway, let us begin with the simplest possible caricature of tra$c with a `point defecta [213]: a single `impuritya (or `defecta) site in the deterministic limit p"0 of the NaSch model with v "1. In this model,
vehicles move forward, in parallel, by one lattice spacing if the corresponding site in front is empty; each vehicle takes ¹ ('1) timesteps to cross the `impuritya site but only one time step to cross a normal site when the next site is empty. The impurity sites acts like a blockage for all ¹ '1. As explained in Section 8.2.1, in the absence of the impurity, J"c for 0(c41/2 and J"1!c for 1/2(c41. Note that, if the impurity is present, 1/¹ vehicle passes through the impurity site per unit time. Therefore, the bottleneck created by the impurity introduces an upper cut-o! of the #ux, viz., 1/¹ . Obviously, J"c(1/¹ so long as c(c "1/¹ . Similarly,
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J"1!c(1/¹ for c'c where c "1!c . In the density interval c (c(1!c , the bottleneck at the impurity is the #ow-limiting factor and, hence, in this regime, J"1/¹ is independent of c. Thus, in the simple caricature of tra$c under consideration one would expect the #ux to vary with density following the relation
c
J" 1/¹ 1!c where
if 0(c4c , if c (c4c , if c (c41 ,
(112)
1 (*t) , c " and c " (113) 1#(*t) 1#(*t) and ¹ "1#(*t) such that (*t) "0 for the normal sites but (*t) '0 for the impurity site. The fundamental diagram, obtained numerically through computer simulation of the NaSch model with a single defect and non-zero p (Fig. 48) is in qualitative agreement with those of the fundamental diagram (112). The qualitative features of the fundamental diagram in Fig. 48 are also similar to those of the TASEP with a single defect [214] where the hopping probability q is smaller than that at all the normal sites. Eq. (112) also indicates that the larger is (*t) the lower is the maximum #ux 1/[1#(*t) ] and the wider is the interval c 4c4c over which the #ux remains constant. What makes the problem of a single `point defecta nontrivial is that, over the interval c 4c4c of the density of the vehicles, where the J is maximum and independent of c, the localized blockage has global e!ects whereby the tra$c exhibits macroscopic phase segregation into high-density and low-density regions. Evidence for such macroscopic phase segregation can be obtained directly from the density pro"les (see Fig. 49). Fig. 49 implies that so long as c(c the
Fig. 48. Fundamental diagram of the NaSch model with v "1 and a blockage site located at the site 1. The hopping
probability in the bulk is given by q"0.75 and at the defect by q"0.25.
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Fig. 49. Density pro"les for di!erent values of c in the NaSch model with v "1 and a blockage site located at the site
1. The hopping probability in the bulk is given by q"0.75 and at the blockage site by q"0.25.
particles will not pile up but a local increase of density will compensate for the reduced local velocity at the blockage so that the #ux around the blockage is identical to that far from it. However, if the global density exceeds c , 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 c . We now develop a semi-phenomenological theory for the NaSch model with v "1,
non-zero p and a single `impuritya site assuming the steady state to be phase-segregated, as demonstrated by computer simulation (Fig. 49). 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 #ux in the phase-segregated regime. Our calculations are based on arguments similar to those suggested originally by Janowsky and Lebowitz [214] in the context of TASEP with a single defect. Using Eq. (78), the #ux in the high- and low-density regions, far from their interface, are given by J "(1!(1!4qc (1!c )) and Jl "(1!(1!4qcl (1!cl )) and that across the defect bond is given by J K(1!(1!4q c (1!cl )). Since, in the steady state, the #ux is same across the entire system, we must have qc (1!c )"qcl (1!cl ) and, hence, (114) c "cl or c "1!cl . The condition c "cl is satis"ed by the uniform density pro"les whereas the condition c "1!cl is satis"ed by the phase-segregated density pro"le (see Fig. 49). Moreover, using the condition J "J "Jl we get c (1!c )"cl (1!cl )Krc (1!cl ) , (115)
A microscopic approach for deterministic dynamics can be found in [176].
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where r"q /q(1 may be interpreted as the `transmission probabilitya or `permeabilitya of the blockage. From (approximate) Eqs. (115) we get p 1 " c K r#1 p#p and, hence,
r p and cl K " r#1 p#p
1 4qr J" 1! 1! . 2 (1#r)
(116)
(117)
The estimate (117) is in good agreement with the numerical data (Fig. 48) obtained from computer simulation [190]. Moreover, the estimates cl and c are also in good agreement with cl and c , respectively, in Fig. 49. Note that cl and c depend only on r and are independent of c. Moreover, the estimates (116) of cl and c are in excellent agreement with c and c , respectively, in Fig. 48. At "rst 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 demand that c¸"c h#cl "c h#cl (¸!h) , (118) where h and l"¸!h are the lengths of the high-density and low-density regions, respectively. Thus, h c!cl c(1#r)!r " " . (119) ¸ c !cl 1!r Eq. (119) shows that h/¸P0 as cPcl and h/¸P1 as cPc . Therefore, keeping r "xed as the density is increased beyond c "cl , the densities of the two regions remain "xed but the high-density region grows thicker at the cost of the length of the low-density region as more and more vehicles pile up and, eventually, at c"c "c the low-density region occupies a vanishingly small fraction of the total length of the system signaling the disappearance of the phase segregation. Interestingly, recasting the expressions for c and cl as cl "1/[1#(*t) ] and c "(*t) / [1#(*t) ], where (*t) "1/r, we "nd close formal analogies with c and c , respectively, in Eq. (113) [215]. SchuK tz [216] considered a TASEP with sublattice-parallel update (see Appendix A) where the motion of the particles is deterministic (i.e., q"1) everywhere except at a defect site where they move with the probability q (1 (i.e., r"q (1). Exact solution is possible through a mapping on a 6-vertex model. Later, a solution using the matrix-product ansatz was presented in [217]. Except for minor di!erences, the qualitative features of the results do not di!er from the corresponding approximate results obtained for q O1 [214]. Qualitatively similar phase segregation phenomena have also been observed in a related model [202]. The qualitative features of the fundamental diagram do not change signi"cantly if the `pointlikea defect (or, impurity) is replaced by an `extendeda defect [190], i.e., a few consecutive defect sites. However, with increasing length of the defect, the maximum value of the #ux decreases monotonically and approaches the maximum #ow of the homogeneous system where the hopping probability associated with each of the bonds is identical to that associated with the defects in our
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model (Fig. 50). From Figs. 50 and 51 we conclude that the monotonic decrease of the #ow with increasing length of the extended defects, leads to a larger di!erence c !cl between the densities of the high-density and the low-density regions of the phase-segregated steady state. Next, instead of a single point-like or extended `defecta, let us consider the more general case of quenched `disordera in the NaSch model with v "1 where the quenched random hopping
probabilities q "1!p are chosen independently from some probability distribution P(q), H H> H H> for the hopping from the cell j to the cell j#1 ( j"1, 2,2, ¸). 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 q and a given vehicle hops across di!erent bonds, in general, with di!erent G G> probabilities assigned to these bonds as it moves forward with time. A similar generalization of the TASEP has also been studied [137]. We shall refer to this model as disordered TASEP (i.e., DTASEP). Suppose, q are chosen from the binary distribution P(q "q )"f, P(q "q)"1!f , H H> H H>
(120)
i.e., a fraction f of the bonds have a permeability r(1 while the remaining fraction 1!f have unit permeability. A mean-"eld theory has been developed [137] (see Appendix G for details) for computing the fundamental diagram of the DTASEP. The #ux in this model has interesting symmetry properties under the operations of `charge conjugationa (which interchanges particles and holes), `paritya (which interchanges forward and backward hopping rates on each bond and `time reversala (which reverses the direction of the current) [137,218]. The quenched disorder in these `disordereda models can be viewed as `point-like impuritiesa distributed randomly over the lattice. But, the qualitative features of the fundamental diagram of
Fig. 50. Fundamental diagram for the NaSch model with v "1 and defects of di!erent length ¸ . Again p"0.75 and
p "0.25 is chosen. For comparison the fundamental diagram of the `fasta (p"0.75) and `slowa homogeneous systems are shown. Fig. 51. Density pro"les for di!erent lengths of the defect in the NaSch model with v "1. Again p"0.75 and 0.25 are
used and the average density is given by c"0.50. The defect extended over the "rst ¸ sites of the system.
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DTASEP are similar to those observed for a single point-like defect and those for a single extended defect. Although the random distribution of the point-like impurities leads to a `rougha density pro"le for all densities, in an intermediate regime of density, phase-segregated steady sates with macroscopic high- and low-density regions have been identi"ed. What is the underlying mechanism for the `macroscopica phase segregation in all the models DTASEP [137] ? Let us denote the stretches of bonds with permeability 1 by X and the stretches of bonds with permeability r by >. The two parabolas in Fig. 52 are the two steady-state fundamental diagrams for the two pure reference systems consisting of all X and all >, respectively. Since the #ux must be spatially constant in the steady-state, the possible densities are given by the four intersections of the line J"J with the two parabolas. If the average density is less (greater) than 1/2 then the two possible densities are c and c (c and c ). The variation of density between c and c (or c and c ) in the X and > 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 cK(1!f )c (J )#fc (J ) where f is given by Eq. (120). However, as the density increases the #ux also increases till it attains the maximum allowed #ux of the pure system consisting of all > (this happens at a global density smaller than 1/2). What happens when the density increases further? According to the `maximum current principlea [162], no further increase of the #ux is possible and the excess density is taken care of by increasing the density in some of the X stretches from c to c (or, vice versa if c'1/2). This conversion takes place adjacent to the largest stretch of > bonds where the density also changes from c to c (or, vice versa if c'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 di!erent mean densities- one with lower densities c , c in the X and > stretches and the other with the higher densities c , c in the X and > stretches. It is not di$cult to generalize the DTASEP to disordered NaSch (DNaSch) model by replacing the random sequential updating by parallel updating. However, we face subtle conceptual
Fig. 52. Origin of phase separation in DTASEP. Shown are the fundamental diagrams of two reference non-disordered TASEPs.
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di$culties in extending DNaSch model further to arbitrary v ; if the position of a vehicle at time
t#1 is decided by q at its current position at time t, it may be forced to move v sites downstream by hopping over sites of even smaller q if v'1. Some alternative parameterization of the defect (or, disordered) sites in the NaSch model for arbitrary v have also been suggested. In [219,190] localized defects have been investigated where
the randomization parameter p is larger than in the rest of the system. Csahok and Vicsek [220] B have considered the blockages as sites with a `permeabilitya smaller than unity whereas the permeability of all the other sites is unity. This e!ectively reduces v while the vehicle is at
a blockage. On the other hand, Emmerich and Rank [221,154] 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., v (t#1)"v (t)/2 is the nth vehicle is located L L within the hindrance region. Some e!ects of static hindrances on vehicular tra$c have also been investigated following alternative approaches, e.g., car-following theory [222]. From a practical point of view, ramps have e!ects very similar to those of a static defect. For the NaSch model this has been investigated in [223].
11. Other CA models of highway tra7c All the CA models of highway tra$c described so far are basically generalizations of the minimal CA model proposed originally by Nagel and Schreckenberg [20]. We now describe a few other alternative minimal CA models and the interesting features of the corresponding results. 11.1. Fukui}Ishibashi model The update rules of the Fukui and Ishibashi (FI) [120] model of single-lane highway tra$c are as follows: If v or more sites in front of the nth vehicle is empty at the time step t, then it has a probability
1!p to move forward by v sites and a probability p to move forward by v !1 sites in the
time step t#1. However, if only d sites (d(v ) in front of the nth vehicle are empty at time t then
it moves by d sites in the next time step. Since no site can be occupied simultaneously by more than one vehicle, a vehicle must not move forward in the time step t#1 if the site immediately in front of it is occupied by a vehicle at the time step t. The model becomes deterministic in both the limits p"0 and 1. The FI model di!ers from the NaSch model in two respects: (a) the increase of speed of the vehicles is not necessarily gradual and (b) the stochastic delay applies only to high-speed vehicles. The FI model, obviously, reduces to the NaSch model if v "1. A site-oriented mean-"eld theory
[224] and a car-oriented mean-"eld theory [225,226] for the FI model have been developed for arbitrary v and p. Note that the FI model is equivalent to the deterministic CA rule 184 (in the
notation of Wolfram [18]) in the limit v "1, p"0. Generalizations of the deterministic limit
p"0 of the FI model have also been proposed [227,228]. According to the classi"cation of Section 7.3, the FI model belongs to class I, i.e., the highacceleration limit where no spontaneous jamming exists. In [156] an alternative high-acceleration variant has been proposed. Here only the acceleration step (;1) of the NaSch model is changed to
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vPv , i.e., all vehicles accelerate immediately to the maximal possible velocity. 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 behaviour of this variant is therefore similar to that of the NaSch model, e.g. one "nds spontaneous jam formation. 11.2. Galilei-invariant vehicle}vehicle interaction and metastability In the NaSch model it is postulated that the gap between a pair of successive vehicles is adjusted according to the velocity of the leading vehicle alone. In contrast, often in real tra$c, drivers tend to adjust the gap in front taking into account the diwerence between the velocity of their own vehicle and that of the leading vehicle. The latter aspect of real tra$c is captured by a recent model developed by Werth, Froese and Wolf (WFW) [229,230]. If both the following vehicle and the leading vehicle move with constant acceleration b, then a collision between the two can be avoided provided gap#l(v )5v q #l(v ) , (121) *4 $4 P $4 where v and v are the velocities of the leading vehicle and the following vehicle, respectively, *4 $4 q is the reaction time of the following vehicle and l(v)"v/2b (122) is the distance covered by a vehicle with initial velocity v before it comes to a stop by moving with a constant deceleration b. Using Eq. (122) the condition (121) for avoiding collision can be written as gap5v q #(v /b)(v !v ) , (123) $4 $4 *4 where v "(v #v )/2 if the average velocity of the pair of vehicles under consideration. $4 *4 Therefore, a suzcient condition for avoiding a collision is
v q for v 4v , $4 *4 (124) gap5 $4 v v q # (v !v ) for v 'v . $4 *4 $4 *4 $4 b In the limiting case bPR the su$cient condition (124) reduces to gap5v q which is identical $4 to the form of vehicle}vehicle interaction in the NaSch model if one chooses q as the unit of time. In the opposite limit q"0, the su$cient condition (124) reduces to
0 for v 4v , $4 *4 (125) gap5 v
(v !v ) for v 'v . *4 $4 *4 b $4 Since this type of vehicle}vehicle interaction involves the di!erence of the velocities v }v , it is $4 *4 clearly invariant under a Galilean transformation and, hence, the name. The vehicle}vehicle interactions in real tra$c may be somewhere in between the two limiting cases of NaSch model and the Galilei-invariant model. Suppose the indices n!1 and n label the leading vehicle and the following vehicle, respectively, of a pair. The update rules suggested by WFW [230] for implementing the Galilei-invariant
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vehicle}vehicle interaction are as follows: Step 1: Acceleration, v"min(v #1, v ) . L L
Step 2: Deceleration (due to other vehicles), v"min(v, d !1#v ). L L L L\ Step 3: Randomization N v" max(v!1, 0) with probability p . L L Step 4: Deceleration (due to other vehicles), v"min(v, d !1#v ) . L L L L\ Step 5: Vehicle movement, x "x #v, v "v . L L L L L Thus, the rule for deceleration (due to other vehicles) is applied twice. Step 4 makes sure collisions are avoided. Since also the new velocity v of the preceding car enters, this step cannot be L\ performed in parallel for all cars. Instead it is performed sequentially, but the "nal con"guration is independent of the starting point of this sequential updating. Step 4 has then to be applied twice in order to determine all velocities v consistently. L The rules as given above de"ne the retarded version of the Galilei-invariant model. In the non-retarded version, in step 2 v is replaced by the new velocity v . To determine v L\ L\ L consistently for all cars, step 2 has then to be iterated v !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 cruise-control nor slow-to-start rules. The mechanism leading to the existence of metastable states is di!erent from the models with slow-to-start rules (see Section 9.1.2). The out#ow from jams is the same as in the NaSch model since it is independent of the interaction between vehicles. However, due to the inclusion of anticipation e!ects (i.e., the driver knows the velocity of the preceding vehicle) the free-#ow is less sensitive to #uctuations. 11.3. CA versions of the optimal-velocity model The tra$c jams appear spontaneously in both the OV models and the CA models. However, in the OV models spontaneous formation of the jams are caused by the non-linearity of the dynamical equations whereas in the CA models it is triggered primarily by the stochasticity of the update `rulesa. The mechanism for the spontaneous formation of jams in real tra$c may be a combination of these two. 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. Thus, unlike the CA
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models like the NaSch model, the vehicles in the OV models get an opportunity to avoid crash without any need to exert unphysically large deceleration. In fact, collision of vehicles may take place in naive discretization of the dynamical equations for the OV models unless special care is taken in the discretization process (see below). In the following we present several CA model analogues of the OV model that have been proposed by di!erent authors. In principle, the NaSch model is also an OV model, but with a linear OV function, v(d)"min[d!1, v ]. The "rst attempt to generalize this relation is due to
Emmerich and Rank [231]. The update rules of their model are as follows: Step 1: Find largest gap. Find the vehicle with the largest gap to the next vehicle ahead. Step 2: Acceleration, v Pmin(v , v #1) . L
L Step 3: Deceleration due to other vehicles, v PM L L B \ TL
if d !14v , L
i.e., a vehicle with velocity j and i empty cells in of it (i.e., a gap d "i#1) reduces its velocity to L M (04i, j4v ). G H
Step 4: Randomization, v Pmax(v !1, 0) with probability p . L L Step 5: Vehicle movement, x Px #v . L L L Step 6: Next vehicle. Repeat steps 2}5 for the next vehicle behind, i.e., proceed in the direction opposite to the motion of the vehicles. For the NaSch model with v "5 the matrix M is given by
G H
0 0 0 0 0 0 0 1 1 1 1 1
0 1 2 2 2 2 . M , 1" G H 0 1 2 3 3 3
(126)
0 1 2 3 4 4 0 1 2 3 4 5
A general matrix M has to satisfy certain conditions (e.g. M 4min(i, j) and M 4M for G H G H G H G I j4k) to guarantee, e.g. the absence of collisions in the model. In order to model the fact that faster vehicles keep a relatively larger headway to the preceding vehicle, Emmerich and Rank suggested
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the following matrix:
0 0 0 0 0 0 0 1 1 1 1 1
0 1 2 2 2 2 M#0" . G H 0 1 2 2 3 3
(127)
0 1 2 3 3 4 0 1 2 3 4 4
Using a parallel update scheme, the model shows unrealistic behaviour in the free-#ow regime, especially in the deterministic limit p;1. Here the fundamental diagram is non-monotonic [232], as can be seen from a simple example for p"0. At density c"1/7, the stationary state is of the form 525252 where numbers denote the velocity of vehicles and &.' an empty cell. At density c"1/6, on the other hand, the stationary state is 424242 . Comparing the corresponding #ows, one "nds J(c"1/7)"5/7'J(c"1/6)"2/3. At density c"1/5, the stationary state is 42.42.42 . with #ow J(c"1/5)"4/5 which is again larger than the #ow at c"1/6 and corresponds to the maximal possible #ow. This kind of behaviour persists even in the presence of randomness (p'0) [232]. In order to circumvent this problem, Emmerich and Rank had to introduce a special kind of ordered-sequential update, where "rst 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. Emmerich and Rank also investigated more general rules where even for gaps larger than v the velocity of the vehicles is reduced to a value v(v .
Later, a similar model has been proposed by Helbing and Schreckenberg [233]. It is closer to the spirit of the original optimal-velocity model (see Section 6.2). Step 1: Vehicle movement, x Px #v (t) . L L L Step 2: Acceleration, v (t#1)"v (t)#W j[< (d (t))!v (t)] X, L L L L Step 3: Randomization,
1 with probability p (if v (t#1)'0) , L v (t#1)"v (t#1)! L L 0 otherwise . Here W yX denotes the #oor function, i.e., the largest integer i4y. In [233] various optimal velocity functions < (d) have been used in order to "t experimental data. The simplest, but unrealistic, choice was < (d)"min(d, v ) where d is the distance-headway. The parameter j corresponds
formally to the sensitivity parameter in the OV model where it determines the timescale of relaxation towards the stationary fundamental diagram. Such an interpretation is not possible for discrete time models. Here the main e!ect of the parameter j is a rescaling of the OV function. The naive discretization of the OV function produces some undesirable features of the model, e.g. the #ow corresponding to the OV function is non-monotonic in the free-#ow region. Furthermore,
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one "nds } for certain initial conditions } a breakdown of the #ow at a "nite density c (1. This H indicates a lack of robustness of model against small modi"cations of the rules. For j(1 the model is not intrinsically collision-free [234], in contrast to most other models discussed in this review. Problems occur, e.g. when fast vehicles approach the end of a jam. For the < (d)"min(d, v ) and the `realistica choice j"0.77 [233] collisions can occur for v 55
[234]. For j'1, on the other hand, a backward motion of vehicles is possible. For a given OV function it is possible to derive conditions for the parameter j which ensure the absence of collisions and backward motion [234]. Nagatani [235] has suggested a CA model which combines the OV idea with the TASEP. It is the discrete analogue of the simpli"ed OV model (51) presented in Section 6.2. Here the nth vehicle moves ahead with probability v (t) where v (t) is interpreted as velocity. This velocity is obtained by L L integration of the OV function
a for *x (t)5*x , L (128) xK (t)" L !a for *x (t)(*x , L where a'0, *x (t)"x (t)!x (t) and *x is a safety distance. Furthermore, the velocity is L L> L restricted to the interval 04v (t)4v 41. L
11.4. CA from ultra-discretization Several CA which can be interpreted as tra$c models have been derived using the so-called ultra-discretization method (UDM) [236]. This approach allows to establish a direct connection between certain di!erential equations and CA. The problem in the derivation of CA from di!erential equations lies in the discretization of the &state' (or dependent) variable. E.g. in a numerical treatment, only space and time variables are discretized. The basic procedure of the UDM is as follows: (1) start from a non-linear wave equation, e.g. the KdV equation or Burgers' equation; (2) discretize space- and time-variables in a standard way to obtain the discrete analogue of the wave equation which is still continuous in the state variable u (t); (3) the discrete analogue is now ultra-discretized. De"ning ; (t)"e ln(u (t)) (where H H H e depends on the discretization *x and *t of space and time) one can use the identity lim e log(e C#e C#2)"max[A, B,2] to derive the CA analogue of the non-linear wave C equation. By applying the UDM to Burgers' equation v "2vv #v one obtains the (¸#1)-state, R V VV deterministic CA [237,238], n (t#1)"n (t)#min[M, n (t), ¸!n (t)]!min[M, n (t), ¸!n (t)] . (129) H H H\ H H H> n (t) is the occupation number of cell j at time t. In contrast to most other CA discussed in this H review multiple occupations of cells are allowed. The maximum number of particles which can occupy the same cell is given by ¸, i.e. 04n (t)4¸. The model de"ned by (129) might therefore H be interpreted as a simple model for a highway with ¸ lanes where the e!ects of lane changes are completely neglected. In [238] it has also been suggested to interpret n (t)/¸ for large ¸ as H a coarse-grained density. The parameter M denotes the maximum number of vehicles that can move from cell j to cell j#1.
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For M5¸"1 the model reduces to the rule-184 CA, i.e., the NaSch model with v "1 and
p"0. In the case ¸42M the fundamental diagram looks similar to that of rule-184 CA, but it is °enerate' in the sense that qualitative di!erent stationary states with the same #ow exist. For ¸'2M the fundamental diagram resembles that of rule-184 CA with a blockage site (see Section 10.3, especially Fig. 48). In the region M/¸4c4(¸!M)/¸ the #ow takes the constant value M/¸, i.e., the parameter M can be interpreted as a #ow limiter. In [238] also a generalization of (129) to higher velocities v '1 has been suggested. For
v "2 the generalized update rule is given by
n (t#1)"n (t)#a (t)!a (t)# min[b (t)!a (t), ¸!n (t)!a (t)] H H H\ H H\ H\ H H\ ! min[b (t)!a (t), ¸!n (t)!a (t)] . (130) H H H> H\ Here b (t)"min[n (t), ¸!n (t)] is the maximum number of vehicles at site j that can move and H H H> a (t)"min[n (t), ¸!n (t), ¸!n (t)] is the number of vehicles that move two cells forward. H H H> H> The idea behind this dynamics is that "rst the vehicles try to move two cells forward. This is only possible if the two cells ahead are not fully occupied, i.e., ¸!n (t)'0 and ¸!n (t)'0. H> H> Then min[b (t)!a (t), ¸!n (t)!a (t)] vehicles move forward one cell. H H H> H\ The model de"ned by (130) can be considered as a generalization of the Fukui}Ishibashi model (see Section 11.1) to which it reduces for ¸"1. Note that the #ow limiter M has been dropped in the extended model. The fundamental diagram of the model (130) has a structure similar to that shown in Fig. 34, i.e., states of high #ow exist. Due to the higher velocity the degeneracy found in the model (129) is lifted. In the case ¸"2, the c"1/2 states 211112 and 220202 are degenerate in the simple model. In the v "2 case, however, in the state 211112 the vehicles can increase their velocity,
while in the state 220202 they cannot due to the presence of the fully occupied cells `2a. The high-#ow states are unstable against local perturbations. Another CA model obtained by ultra-discretization of the modi"ed KdV-equation has been suggested in [239]. It is a second-order CA since the con"guration at time t#1 does not only depend on the con"guration at time t, but also on the previous one at time t!1. The update rule for the position x (t) of vehicle j is given by H x (t#2)"x (t)#*x (t)#max[0, *x (t#1)!M]!max[0, *x (t#1)!¸] , (131) H H H H H where *x (t)"x (t)!x (t) is the gap and M and ¸ are constants. H H> H Although this model is deterministic it exhibits start-stop waves similar to those found in the NaSch model [239]. It seems that a second-order deterministic CA might produce e!ects similar to those of noise in a "rst-order stochastic CA. One should note, however, that the rules allow vehicles to move backwards. Other models obtained using the UDM have been discussed in [240]. 12. Cellular automata models of city tra7c and road networks 12.1. Biham}Middleton}Levine model and its generalizations In the BML model [21], each of the sites of a square lattice represent the crossing of a east}west street and a north}south street. All the streets parallel to the x( -direction of a Cartesian coordinate
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system are assumed to allow only single-lane east-bound tra$c while all those parallel to the y( -direction allow only single-lane north-bound tra$c. Let us represent the east-bound (northbound) vehicles by an arrow pointing towards east (north). In the initial state of the system, vehicles are randomly distributed among the streets. The states of east-bound vehicles are updated in parallel at every odd discrete time step whereas those of the north-bound vehicles are updated in parallel at every even discrete time step following a rule which is a simple extension of the TASEP: a vehicle moves forward by one lattice spacing if and only if the site in front is empty, otherwise the vehicle does not move at that time step. Thus, the BML model is also a driven lattice gas model where each of the sites can be in one of the three possible states: either empty or occupied by an arrow ! or P. Note that the parallel update rules of the BML model is fully deterministic and, therefore, it may also be regarded as a deterministic CA. The randomness arises in this model only from the random initial conditions [241]. Suppose, N and N are the numbers of the east-bound and north-bound vehicles, t respectively, in the initial state of the system. If periodic boundary conditions are imposed in all directions, the number of vehicles in every street is conserved since no turning of the vehicles are allowed by the updating rules. In a "nite ¸;¸ system the densities of the east-bound and north-bound vehicles are given by c "N /¸ and c "N /¸, respectively, while the global t t density of the vehicles is c"c #c . t Computer simulations of the BML model with periodic boundary conditions demonstrate that a xrst order phase transition takes place at a "nite non-vanishing density c , where the average H velocity of the vehicles vanishes discontinuously signaling complete jamming; this jamming arises from the mutual blocking (`grid-lockinga) of the #ows of east-bound and north-bound tra$c at various di!erent crossings (see [242] for the corresponding results of the BML model with open boundary conditions). At concentrations just above c , in the jammed phase, all the vehicles together form a single H cluster which is stretched along the diagonal connecting the south}west to the north}east of the system. In other words, the lowest-density jammed con"gurations consist of a single diagonal band where the P and ! occupy nearest-neighbour 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 o!-diagonal branches and the in"nite cluster of the jammed vehicles looks more and more random. Thus, in general, a typical in"nite cluster of the jammed vehicles consists of a `backbonea and `dangling vehiclesa which are the analogs of the `backbonea and the `dangling endsa of the in"nite percolation clusters in the usual site/bond percolation [243]. However, in contrast to the in"nite percolation cluster in the usual random site/bond percolation, the in"nite 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 in"nite cluster of jammed vehicles in the BML model at c'c [244,245]. The distribution of the H waiting times of the vehicles at the signals (i.e., at the lattice sites) has also been investigated through computer simulations [246,247]. 12.1.1. Poor man's mean-xeld estimates for the BML model If one is not interested in detailed information on the `structurea of the dynamical phases, one can get a mean-"eld estimate of c by carrying out a back-of-the-envelope calculation. Suppose, H c and c denote the average densities while v and v denote the average speeds of the east-bound t t
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and north-bound vehicles, respectively. In order to estimate c one has to take into account H interaction of the east-bound (north-bound) vehicles not only with the north-bound (east-bound) vehicles [248] but also, in a self-consistent manner, with other east-bound (north-bound) vehicles [249]. Following the arguments of Appendix I, one can show that in the symmetric case c "c "c, a self-consistency equation for the speed is t c c 1 !4c v "v "v" 1# # 1# (132) t 2 2 2
for c(c . The critical density c is determined by the condition that at c5c the equation (132) H H H does not give a real solution. Hence we get c"c "6!(32K0.343 which, in spite of the H approximations made, is surprisingly close to the corresponding numerical estimate obtained from computer simulation [21]. Moreover, the mean-"eld estimate c K0.343 is also consistent with H the more rigorous result [251,253] that c is strictly less than 1/2. The BML model in three H dimension, although not relevant for vehicular tra$c, has also been studied [254]. 12.1.2. Mean-xeld theory of the BML model Recall that the occupation variables n(i; t) in the NaSch model describe the state of occupation of the sites i (i"1, 2,2) by the vehicles on the one-dimensional highway. A scheme for a truly microscopic analysis of the BML model begins [255] by introducing the corresponding generalized occupation variables n (x, y; t) and n (x, y; t), which describe the state of occupation of the sites t (x, y) by the north-bound and east-bound vehicles, respectively, on the two-dimensional streetnetwork. The analogs of Eqs. (64) and (65) (see Appendix I for details) have analogous physical interpretations. As usual, in the naive SOMF approximation one neglects the correlations between the occupations of di!erent sites [255]. Since none of the sites is allowed to be occupied by more than one particle at a time, Pauli operators may be used to develop an analytical theory of vehicular tra$c [256] but one should keep in mind that the particles representing the vehicles are purely classical and the system does not have any quantum mechanical characteristics. 12.1.3. Generalizations and extensions of the BML model The BML model has been generalized and extended to take into account several realistic features of tra$c in cities. Asymmetric distribution of the vehicles: Suppose the vehicles are distributed asymmetrically among the east-bound and north-bound streets [248], i.e., c Oc . For convenience, let us write t c "cf and c "c(1!f ) where f is the fraction of the vehicles moving towards north. Clearly t ? ? ? f "1/2 corresponds to the symmetric case c "c "c/2. On the other hand, f "0 ( f "1) ? t ? ? correspond to the extreme asymmetric case where all the vehicles are east-bound (north-bound). Obviously, the absence of grid-locking in the extreme limits f "0 and f "1 rules out the ? ? possibility of BML-like complete jamming transition, i.e., c "1 for both f "0 and f "1. H ? ? Moreover, c decreases with decreasing asymmetry in the distribution of the vehicles; c is the H H Ishibashi and Fukui [250] claimed that complete jamming can occur in the BML model only for c"1. However, a plausible #aw in their arguments was pointed out by Chau et al. [251]. In [252] it has been argued that c J¸\ , H i.e., c "0 in the thermodynamic limit. H
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smallest for f "1/2, i.e., symmetric distribution of the vehicles. These results can be presented ? graphically by plotting the curve c ( f ) in the phase diagram on the c}f plane [248]. H ? ? Unequal maximum velocities: In the BML model both east-bound and north-bound vehicles can move by a maximum of one lattice spacing at a time and, therefore, the average speeds of both types of vehicles can never exceed unity. On the other hand, recall that in the Fukui}Ishibashi model [120] of highway tra$c vehicles can move up to a maximum of M lattice sites at a time and, hence, have average velocities larger than unity. Incorporating similar high-speed vehicles Fukui et al. [257] generalized the BML model where the east-bound vehicles are allowed tom ove by M sites at a single time step while the north-bound vehicles can move by only one lattice spacing. Overpasses or two-level crossings: The BML model has been extended to take into account the e!ects of overpasses (or two-level crossings) [258]. A fraction f of the lattice sites in the BML model are randomly identi"ed as overpasses each of which can accommodate up to a maximum of two vehicles simultaneously. The overpasses weaken the grid-locking in the BML model. Therefore, c is expected to increase with increasing f . Besides, c is expected to be unity if f "1. H H Naturally, we address the question: does jamming disappear (i.e., c is unity) only at f "1 or for H even smaller values of f ? In order to answer this question we extend the self-consistent mean-"eld arguments, which led to Eq. (132), incorporating the e!ects of the overpasses thereby getting the generalized self-consistency equation [259]
1!f 1 c# v "v "v" 1# t 2 2
1!f c !4(1!f )c 1# 2
(133)
in the symmetric case c "c "c. Eq. (133) reduces to the equation (132) in the limit f "0. It t predicts that, if f O0, a moving phase exists in the BML model with overpasses for vehicle densities c4c "(6!(32)/(1!f ) and that the jammed phase disappears altogether (i.e., H c becomes unity) for f 51!0.343"0.657, an underestimate when compared with the correH sponding computer simulations. These results can be presented graphically by drawing the curve c ( f ) in the mean-"eld phase diagram on the c!f plane not only for the symmetric distribution H (i.e., for f"1/2) but also for asymmetric distributions of the vehicles among the east-bound and north-bound streets [259]. Faulty trazc lights: The e!ects of faulty tra$c lights have been modeled by generalizing the BML as follows [260]: a fraction f of the sites are identi"ed randomly as the locations of the faulty tra$c lights. Both the north-bound vehicles currently south of the faulty tra$c lights and the east-bound vehicles currently west of a faulty tra$c light are allowed to hop onto the empty crossings where the faulty tra$c lights are located, irrespective of whether the corresponding time step of updating is odd or even. If an east-bound vehicle and a north-bound vehicle simultaneously attempt to enter the same crossing, where the faulty tra$c light is located, then only one of then is allowed to enter that crossing by selecting randomly. Since a north-bound (east-bound) vehicle will be able to move forward although only east-bound (north-bound) vehicles would have moved forward had f been zero, the average velocity of the vehicles is expected to increase with increasing f . However, with the increase of f there is an increasing likelihood that an east-bound (north-bound) vehicle would be blocked by a northbound (east-bound) vehicle at a faulty tra$c light. Thus, the increasing density of faulty tra$c lights increases the e!ect of grid-locking thereby decreasing c . H
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Static hindrances: The BML model has been extended to incorporate the e!ects of static hindrances or road blocks (e.g., vehicles crashed in tra$c accident), i.e., stagnant points [261,262]. A vehicle, which stays at a normal site for only one time step before attempting to move out of it, stays at a point-like blockage for ¹ time steps before attempting to move out of it. Obviously, the longer is ¹ , the lower is the corresponding c . The time-dependent phenomenon of spreading of H the jam from the blockage site during the approach of the system to its jammed steady-state con"gurations has also been investigated [261,263]. Stagnant street: Let us consider the e!ects of a stagnant street, where the local density c of the vehicles is initially higher than that in the other streets [264], on the tra$c #ow in the BML model. The stagnant street, e!ectively, acts like a `line-defecta, rather than a `point-defecta. However, in contrast to the static roadblocks, a stagnant street o!ers a time-dependent hindrance to the vehicles moving in the perpendicular streets. As intuitively expected, the jamming transition has been found to occur at a lower global density when the local density in the stagnant street is higher. Independent turning of the vehicles: Let us now generalize the BML model by assigning a trend or preferred direction of motion, = (x, y), to each vehicle n (n"1, 2,2) located at the site x, y. L According to this de"nition, the vehicle n, located at x, y jumps to the next site towards east with probability = (x, y) while 1!= (x, y) is the corresponding probability that it hops to the next site L L towards north [265]. In this generalized model vehicles can take a turn but the processes of turning from east-bound (north-bound) to north-bound (east-bound) streets is stochastic. For simplicity, suppose, N/2 vehicles are assigned = (x, y)"c while the remaining N/2 vehicles are assigned L = (x, y)"1!c where 04c41/2; this implies that N/2 vehicles move preferentially east-ward L whereas the remaining N/2 vehicles move preferentially towards north. In the limit c"0 no vehicle can turn and we recover the original BML model with deterministic update rules. The most dramatic e!ect of the stochastic turning is that the discontinuous jump D1v2 of the average velocity 1v2 decreases with increasing c and, eventually, the "rst order jamming transition ends at a critical point where D1v2 just vanishes. Jam-avoiding turn and drive: In the model of turning considered in Refs. [265] a vehicle turns stochastically independently of the other vehicles. In real tra$c, however, a vehicle is likely to turn if its forward movement is blocked by other vehicles ahead of it in the same street. Therefore, let us now consider a model [266] where an east-bound (north-bound) vehicle turns north (east) with probability p if blocked by another vehicle in front of it. Computer simulations of this model shows that c (p ) increases with increasing p . In a slightly di!erent model [267], on being H blocked by a vehicle in front, an east-bound (north-bound) vehicle hops with probability p to the H? next east-bound (north-bound) street towards north (east). A single north-bound street cutting across east-bound streets: Let us now consider a special situation where only one north-bound street exists which cuts all of the ¸ equispaced mutually parallel east-bound streets of length ¸ [268]. This situation can be modelled as a ¸;¸ square lattice and each cell on the north-bound street can be in one of the three allowed states, namely, either empty or occupied by a P or a !. But, in contrast to the BML model, there are two allowed states for each cell (outside the crossings) on the east-bound street; these can be either empty or occupied by only P. Since no grid-locking is possible with only one north-bound street, complete jamming occurs trivially in this case only if each of the cells either on the east-bound streets or on the north-bound street or on all the streets are occupied simultaneously by vehicles. Nevertheless, at any "nite non-vanishing density, the crossings of the east-bound and north-bound streets act like
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hindrances with "nite non-vanishing permeability for the #ow of the east-bound tra$c. Obviously, the higher is the density of the north-bound vehicles, the lower is the permeability and the stronger is the rate-limiting factor of the bottleneck. A comparison of this problem with that of highway tra$c in the presence of static hindrances [214,137] explains not only why the #ux along the east-bound streets exhibits a #at plateau over an intermediate density regime but also why the plateau appears at lower values of the #ux with increasing density of the north-bound vehicles [268]. A mean-"eld theory for the macroscopic phase segregation in this model has been developed by appropriately modifying that for the similar phenomenon in the one-dimensional models of highway tra$c in the presence of static hindrances. It is worth emphasizing here that in [268] each site on the east-bound streets has been interpreted as a cell, which can accommodate one vehicle at a time, rather than as a crossing of the east-bound street with a north-bound street. Conceptually, this is an extension of the BML model. Finally, we brie#y mention that in [269] the phase diagram of a system consisting of one east-bound and one north-bound street with one crossing has been investigated. Green-wave synchronization: Often the tra$c lights along the main streets in cities are synchronized to allow continuous #ow; this is usually referred to as `green-wavea synchronization. A green-wave (GW) model has been developed [270] by replacing the parallel updating scheme of the BML model by an updating scheme which is partly backward-ordered sequential (see Appendix A for a general explanation of this update scheme). At odd time steps, an east-bound vehicle moves by one lattice spacing if the target site was empty at the end of the previous time step or has become empty in the current time step (this is possible because of the backward-ordered sequential updating at every time-step). Similarly the positions of the north-bound vehicles are updated at every even time step. The main di!erence between the BML model and the GW model (see Fig. 53), arising from the di!erent updating schemes, is that in the GW model vehicles move as `convoysa (a cluster of vehicles with no empty cell between them) thereby mimicking the e!ects of green-wave synchronization of the tra$c lights in real tra$c. The jamming transition in the GW model has been investigated by a combination of a mean-"eld argument and numerical input from computer simulations [270].
Fig. 53. Comparison of the update procedure in the BML model (left) and the GW model (right) (from [270]).
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More realistic description of streets and junctions: At "rst sight the BML model may appear very unrealistic because the vehicles seem to hop from one crossing to the next. However, it may not appear so unrealistic if each unit of discrete time interval in the BML model is interpreted as the time for which the tra$c lights remain green (or red) before switching red (or green) simultaneously in a synchronized manner, and over that time scale each vehicle, which faces a green signal, gets an opportunity to move from jth crossing to the ( j#1)th (or, more generally [257], 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 identi"ed as the crossings. Brunnet and Goncalves [271] considered a modi"ed version where, instead, the vehicles are located on the bonds and, therefore, never block the #ow of vehicles in the transverse direction. Consequently, in this version of the CA model of city tra$c jams of only xnite sizes can form and these jams have xnite lifetime after which they disappear while new jams may appear elsewhere in the system; an in"nitely long-lived jam spanning the entire system is possible only in the trivial limit c"1. In contrast, Horiguchi and Sakakibara [272] generalized the BML by replacing each of the bonds connecting the nearest-neighbour lattice sites by a bond decorated with an extra lattice site in between. In [273] a generalization to s extra lattice sites between crossings has been presented. However, the vehicles are still allowed to hop forward by only one lattice spacing in the model of Horiguchi and Sakakibara. Moreover, generalizing the rules for turning of the vehicles in Ref. [265], Horiguchi and Sakakibara also allowed probabilistic turning of the vehicles in their model. The model exhibits a transition from the #owing phase to a completely jammed phase. The streets in the original BML model were assumed to allow only one-way tra$c. This restriction was relaxed in a more realistic model proposed by Freund and PoK schel [274] which allows both-way tra$c on all the streets. Thus, each east}west (north}south) street is implicitly assumed to consist of two lanes one of which allows east-bound (north-bound) tra$c while the other allows west-bound (south-bound) tra$c. 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-neighbour crossings (i.e., from north, south, east and west) and queued up at the crossing under consideration. So, in this extended version of the BML model, each site can accommodate at most 4Q particles if each of the four queues of vehicles associated with it is allowed to grow to a maximum length Q. In the model proposed by Freund and PoK schel [274], initially, each of the vehicles is assigned a site selected at random, the queue to which it belongs (i.e., whether it is approaching the crossing from north, south, east or west) and the desired direction (i.e., left, right or straight) for its intended motion at the next time step. At each discrete time step a vehicle is allowed to move forward in its desired direction of motion by one lattice spacing provided (a) it is at the front of the queue in its present location and (b) there are fewer than Q vehicles queued up at the next crossing in the same desired direction of motion. Once a vehicle moves to the next crossing it "nds a place at the end of the corresponding queue at the new crossing while the vehicles in the queue is left behind are moved `closer to the crossinga by one position (by mere relabelling as no physical movement of the vehicles in the queue takes place explicitly). Various reasonable choices for the rule which determines the desired direction of each vehicle at every time step have also been considered. The "nite space of the streets in between successive crossings do not appear explicitly in the extension of the BML model suggested by Freund and PoK schel [274] although it is more realistic
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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. [275] have developed a more realistic CA model of city tra$c where the stretches of the streets in between successive crossings appear explicitly. In this model also each of the streets consist of two lanes which allow oppositely directed tra$c. The rule for implementing the motion of the vehicles at the crossing is formulated assuming a rotary to be located at each crossing. Depending on the details of the rules to be followed by the vehicles at the rotary, the system can exhibit a variety of phenomena. For example, the #ow can be metastable at all densities if each of the vehicles on the rotary is required to stop till the destination cell becomes available for occupation [275]. Moreover, the bottleneck created by the vehicles on the rotaries at the junctions can lead to a plateau in the fundamental diagram which is analogous to that caused by a static hindrances on a highway [214,137]. 12.2. Marriage of the NaSch model and the BML model; a `unixed a CA model of city trazc If one wants to develop a more detailed `"ne-graineda description of city tra$c than that provided by the BML model then one must "rst decorate each bond [268,272] with D!1 (D'1) sites to represent D!1 cells in between each pair of successive crossings [268,275] thereby modeling each segment of the streets in between successive crossings in the same manner in which the entire highway is modelled in the NaSch model. Then, one can follow the prescriptions of the NaSch model for describing the positions, speeds and accelerations of the vehicles [275,276] as well as for taking into account the interactions among the vehicles moving along the same street. Moreover, one should #ip the color of the signal periodically at regular interval of ¹ (¹<1) 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 tra$c in cities has, indeed, been proposed very recently [277,278] where the rules of updating have been formulated in such a way that, (a) a vehicle approaching a crossing can keep moving, even when the signal is red, until it reaches a site immediately in front of which there is either a halting vehicle or a crossing; and (b) no grid-locking would occur in the absence of random braking. Let us model the network of the streets as a N;N square lattice. For simplicity, let us assume that the streets parallel to x( and y( axes allow only single-lane east-bound and north-bound tra$c, 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 ¸"N;D. Each of these cells can be either empty or occupied by at most one single vehicle at a time. Because of these cells, the network of the streets can be viewed as a decorated lattice. However, unlike the BML model [21], which corresponds to D"1, and the model of Horiguchi and Sakakibara [272], which corresponds to D"2, D((¸) in this model is to be treated as a parameter. Note that D introduces a new length scale into the problem. 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 interval ¹ and then, simultaneously, all the signals turn red for the east-bound vehicles (and, green for the north-bound
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vehicles) for the next ¹ timesteps. Clearly, the parameter ¹ introduces a new time scale into the problem. Thus, in contrast to the BML model, the forward movement of the individual vehicles over smaller distances during shorter time intervals are described explicitly in this `uni"eda model. If no turning of the vehicles is 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 v #1 integer values v"0, 1,2, v .
Suppose, v is the speed of the nth vehicle at time t while moving either towards east or towards L north. At each discrete time step tPt#1, the arrangement of N vehicles is updated in parallel according to the following `rulesa: Step 1: Acceleration v Pmin(v #1, v ) . L L
Step 2: Deceleration (due to other vehicles or signal). Suppose, d is the gap in between the nth vehicle and the vehicle in front of it, and s is the L L distance between the same nth vehicle and the closest signal in front of it (see Fig. 54). Case I: The signal is red for the nth vehicle under consideration: v Pmin(v , d !1, s !1). L L L L Case II: The signal is green for the nth vehicle under consideration: Suppose, q is the number of the remaining time steps before the signal turns red. Now there are two possibilities in this case: (i) When d 4s , then v Pmin(v , d !1). The motivation for this choice comes from the fact L L L L L that, when d 4s , the hindrance e!ect comes from the leading vehicle. L L (ii) When d 's , then v Pmin(v , d !1) if min(v , d !1);q's ; else v Pmin(v , s !1). L L L L L L L L L L L The motivation for this choice comes from the fact that, when d 's , the speed of the nth vehicle L L 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. Step 3: Randomization. v Pmax(v !1, 0) with probability p (04p41); p, the random decelerL L ation probability, is identical for all the vehicles and does not change during the updating. Step 4: Vehicle movement. For the east-bound vehicles, x Px #v while for the north-bound L L L vehicles, y Py #v . L L L The rule in case II of step 2 can be simpli"ed without changing the overall behaviour of the model [278]: Case II : If the signal turns to red in the next timestep (q"1): v Pmin(v , s !1, d !1) L L L L
Fig. 54. A segment of a east}west street, where the nth and the (n#1)th vehicles are located, is shown schematically together with one of the nearest signals.
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else v Pmin(v , d !1) . L L L These rules are not merely a combination of the rules proposed by Biham et al. [21] and those introduced by Nagel and Schreckenberg [20] but also involve some modi"cations. For example, unlike all the 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 vehicle or a crossing. Moreover, 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) tra$c when the signal is red for the east-bound (north-bound) vehicles. Initially, we put N and N vehicles at random positions on the east-bound and north t bound streets, respectively. We update the positions and velocities of the vehicles in parallel following the rules mentioned above. After the initial transients die down, at every time step t, we compute the average velocities 1v (t)2 and 1v (t)2 of the east-bound and north-bound vehicles, V W respectively. A phase transition from the `free-#owinga dynamical phase to the completely `jammeda phase takes place in this model at a vehicle density c (D). The dependence on the dynamical parameters p, H v and ¹ is not clear yet [278]. The data obtained so far from computer simulations do not
conclusively rule out the possibility that the transition density only depends on the structure of the underlying lattice, similar to the percolation transition [243], and is independent of the dynamical parameters. The intrinsic stochasticity of the dynamics, which triggers the onset of jamming, is similar to that in the NaSch model, while the phenomenon of complete jamming through self-organization as well as the "nal jammed con"gurations (see Fig. 55) are similar to those in the BML model. The variations of 1v 2 and 1v 2 with time (see Fig. 56) as well as with c, D, ¹ and p in V W the #owing phase are certainly more realistic that in the BML model [277]. The `uni"eda model has been formulated intentionally to keep it as simple as possible and at the same time capture some of interesting features of the NaSch model as well as the BML model. We believe that this model can be generalized (i) to allow tra$c #ow both ways on each street which may consist of more than one lane, (ii) to make more realistic rules for the right-of-the-way at the crossings and turning of the vehicles, (iii) to implement di!erent types of synchronization or staggering of tra$c lights [276] (including green-wave), etc. 12.3. Practical applications of the models of vehicular trazc; on-line simulation of trazc networks A large fraction of the available resources are spent by the governments, particularly in the industrialized developed countries, to construct more highways and other infrastructural facilities related to transportation. The car-following models, the coupled-map lattice models as well as the CA models have been used for computer simulation with a hope to utilize the results for on-line tra$c control. For planning and design of the transportation network [19], for example, in a metropolitan area [279}281], one needs much more than just micro-simulation of how vehicles move on an idealized linear or square lattice under a speci"ed 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.
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Fig. 55. A typical jammed con"guration of the vehicles (N"5, D"8). The east-bound and north-bound vehicles are represented by the symbols P and !, respectively. Fig. 56. Time-dependence of average speeds of vehicles. The symbols #, ;, * and 䊐 correspond, respectively, to 1v 2, V 1v 2, and the fractions of vehicles with instantaneous speed v"0, f and f , respectively. The common parameters are W V W v "5, p"0.1, D"100, ¹"100 and c"0.1. The continuous line has been obtained from heuristic arguments given
in [277].
Micro-simulations are carried out for all possible di!erent routes to execute these plans; the results give informations on the e$ciency of the di!erent routes and these informations are utilized in the designing of the transportation network [19]. Some socio-economic questions as well as questions on the environmental impacts of the planned transportation infrastructure also need to be addressed during such planning and design. For a thorough discussion of these aspects we refer to the recent review by Nagel et al. [19].
13. Some related systems, models and phenomena In this section we will brie#y present some stochastic models and phenomena which are somehow related to the main topic of this review. For some models the relation to tra$c #ow is rather obvious, e.g. there are toy models which share some features with tra$c models, but which can be solved exactly. We also point out similarities in the description of other phenomena, e.g. granular #ow or surface growth. Similarities also exist to systems from solid state physics, namely ionic conductors. This point will not discussed here, instead we refer to Refs. [282,283]. Finally we like to mention that there are some resemblances between the CA models of vehicular tra$c and the CA models of driven di!usive Frenkel}Kontorova-type systems [284].
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13.1. Generalizations of the TASEP The TASEP is probably one of the best studied models in non-equilibrium physics. Using powerful methods like MPA or Bethe ansatz recently it became possible to calculate not only simple expectation values in the stationary states, but also more complicated quantities like di!usion constants or the large-deviation function. For a more detailed discussion and a list of references we refer to the recent reviews on the TASEP [22,174] (see also [15]). Several variants of the TASEP have been proposed. Most of those preserve the exact solvability. In the following we will discuss brie#y some of the variants and generalizations of the TASEP discussed so far in the literature. Partially asymmetric exclusion process. An obvious generalization of the TASEP is to allow hopping processes in both directions [285]. Here only results for the random-sequential update exist, since for parallel dynamics ambiguities in the updating appear when two particles attempt to hop onto the same site. One "nds that the phase diagram looks essentially like that of the TASEP with three di!erent phases (see Section 8.5). Recently it has been shown [286], however, that the high- and low-density phases can be divided into three subphases (AI, AII, AIII and BI, BII, BIII) instead of two in the TASEP case. Again the phase boundaries of these subphases are determined by the behaviour of the density pro"les and the corresponding correlation lengths. Multispecies models I. Karimipour and collaborators [287] have developed a multispecies generalization of the TASEP which retains the solvability by MPA. It is similar to the disordered model discussed in Section 10.1 where each particle is characterized by a hopping rate (also called `velocitya in this context) v ( j"1,2, N ), i.e., there are N di!erent `speciesa of particles. In H contrast to the models with quenched disorder discussed in Section 10, however, overtaking of particles is possible, i.e., the ordering of the particles is not "xed. More speci"cally the dynamics of the model is de"ned as follows: A particle is chosen at random (random-sequential update). If the particle is of species j and the cell to its right is empty, it hops there with rate v . If the cell is occupied by a particle of species l and v (v , then they interchange H J H there positions with rate v !v . This means that a fast particle can overtake a slower one with H J a rate proportional to their velocity di!erence. Multispecies models II. In several papers multispecies generalizations of the ASEP have been suggested which exhibit phase separation and spontaneous symmetry breaking. Arndt et al. [288] considered a system of positive and negative charged particles di!using on a ring in opposite directions. Positive particles move to an empty right neighbour and negative particles move to an empty left neighbour with the same rate j. Furthermore, positive and negative particles on neighbouring sites can exchange their positions. The process !#P#! occurs with rate 1, and the inverse process #!P!# with a di!erent rate q. For equal densities of positive and negative particles the system exhibits three phases. For q(1 in the thermodynamic limit, the system organizes itself into con"gurations consisting of blocks of the type 0002###2*2. The dynamics out of theses states is extremely slow. Translational invariance is broken and the current vanishes. This phase is called &pure phase'. For 1(q(q the system is in the &mixed phase' which consists of two coexisting phase, the dense phase and the #uid phase. The dense phase where no vacancies exist covers a macroscopic region which shrinks to 0 for qPq . It is remarkable that the current for 1(q(q takes the value J"(q!1)/4 independent of the total density and the rate j. For q'q the #uid phase extends through the whole system. There is no charge separation and density pro"les are uniform.
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Similar results have been found by Evans et al. [289] in a slightly di!erent model. The dynamics of their 3-species model is given by O AB P QBA,
O BC P QCB,
O CA P QAC ,
(134)
i.e. the rates are cyclic in A, B and C and the numbers N , N and N of particles of each species is ! conserved. In the case N "N "N the dynamics satis"es the detailed balance condition with respect to ! a Hamiltonian with long-range asymmetric interactions. Stationary states are of the form A2AB2BC2C and exhibit phase separation, i.e., for large separations r the two-point function satis"es lim [1A A 2!1A 21A 2]'0. * P P 13.2. Surface growth, KPZ equation and Bethe Ansatz In Section 8.1.2 we have explained how the NaSch model with v "1 can be mapped onto
a stochastic surface growth model. This connection can be employed to calculate several properties of the noisy Burgers and KPZ equation exactly. Gwa and Spohn [290] used the Bethe ansatz (see e.g. [291]) to determine the spectrum of the stochastic Hamiltionian (see Appendix F) 1 * (135) H"! [r ) r !1#ie(pVpW !pWpV )] H H> H H> H H> 4 H corresponding to the ASEP with random-sequential updating and periodic boundary conditions. r "(pV, pW, pX) are the standard Pauli matrices at site j and e is the asymmetry of the hopping rates, H H H H q "(1#e) and q "(1!e). The `ground statea of H has eigenvalue 0 and is ¸-fold degenerate. For a "xed number N of up-spins every con"guration has equal weight in the ground state. In order to determine the dynamical scaling exponent of the noisy Burgers and KPZ equations, Gwa and Spohn investigated the "nite-size behaviour of the energy gap of H. Since H is non-Hermitian its spectrum is complex. The "rst excited state is then de"ned as the eigenvalue with the smallest (positive) real part E . In [290] it was shown that E J¸\ for e"1 and N"¸/2. This implies that the dynamical exponent z for the stationary correlations of the KPZ equation is given by z"3/2. The dynamical exponent relates temporal and spatial scaling behaviour on large scale. Generalizations and related results can be found in [292]. By an extension of the Bethe ansatz method of Gwa and Spohn, Derrida and Lebowitz [293] calculated the large deviation function (LDF) of the time-averaged current of the TASEP. The LDF is related to the total displacement >(t), i.e., the total number of hops to the right minus the total number of hops to the left between time 0 and time t. In the corresponding growth model >(t)
One species may be interpreted as a vacant site. I.e., a "xed number N of particles.
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is the total number of particles deposited until time t. The LDF is then de"ned as
1 >(t) f (y)"lim ln Prob "v #y (136) t t R where v "lim 1>(t)/t2"ec(1!c)¸/(¸!1) is the mean current for a ring of "nite size ¸ and R density c"N/¸. The results of [293] have been extended and generalized in [294]. Apart from the treatment of "nite systems, the BA can also be used to solve the master equation for an in"nite system with a "nite number of particles [295]. This allows e.g. to study the collective di!usion of two single particles. 13.3. Protein synthesis You must have noticed in the earlier sections that some of the models of tra$c are non-trivial generalizations or extensions of the TASEP, the simplest of the driven-dissipative systems which are of current interest in non-equilibrium statistical mechanics [14,15,17]. Some similarities between these systems and a dynamical model of protein synthesis have been pointed out [296,15]. In a simpli"ed picture of the mechanism of biopolymerisation ribosomes read the genetic information encoded in triplets of base pairs. They attach to one end of a messenger-RNA and then move along the chain after adding a monomer to a biopolymer attached to the ribosome. The type of monomer added depends on the genetic information read by the ribosome. When the ribosome reaches the other end of the m-RNA the biopolymer is fully synthesized and the ribosome is released. MacDonald et al. [297] have described the kinetics of this process using an ASEP-type model. The m-RNA is represented by a chain of ¸ sites where each site corresponds to one triplet of base pairs. The ribosome is given by a hard-core particle covering r neighbouring sites (r+20}30) which moves forward by one site with rate q. At the beginning of the chain particles are added with rate aq and at the end they are released with rate bq. In the idealized case r"1 this is exactly the TASEP of Sections 8.1.1 and 8.5. The relevant case for the experiments is a"b(1/2. The exact solution of the TASEP allowed for an explanation of many aspects of the experiments [296]. 13.4. Granular yow Another quasi-one-dimensional driven-dissipative system, which is also receiving wide attention of physicists in recent years, is the granular material #owing through a pipe [3,4]. Since the fascinating phenomena (e.g. size segregation, convection, standing waves, localized excitations) found in granular materials have been subject of several excellent reviews [298] we discuss only brie#y the similarities between the clustering of vehicles on a highway and particle}particle (and particle-cluster) aggregation process [299,300,209]. Obviously both highway tra$c and granular #ow through a pipe may be described as quasione-dimensional systems consisting of discrete elements (vehicles, grains). The dynamics of these elements is determined by an intricate interplay between a driving force (driver, gravitation) and dissipation (braking, inelastic scattering processes). These similarities already show that both systems can be described by similar approaches. One important di!erence between tra$c #ow
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and granular #ow exist, however. In granular #ow density waves can move both in and against the direction of the #ow whereas in tra$c #ow they only move backwards. An important success of a description of granular #ow using the optimal velocity model (Section 6.2) is the explanation of the experimentally observed f \ -behaviour of the power spectrum [301]. 13.5. The bus route model The bus route model (BRM) [302] has been formulated as a one-dimensional lattice with periodic boundary conditions. The sites represent cells, each of which may be thought of as a bus stop and are labeled by an index i (i"1, 2,2, ¸) [302]. Two binary variables p and q are assigned G G to each cell i: (i) If the cell i is occupied by a bus then p "1; otherwise p "0. (ii) If cell i has G G passengers waiting for a bus then q "1; otherwise q "0. Since a cell cannot have simultaneously G G a bus and waiting passengers, let us impose the condition that a cell cannot have both p "1 and G q "1 simultaneously. Each bus is assumed to hop from one stop to the next. G Next, let us specify the update rules (see Fig. 57): a cell i is picked up at random. Then, (i) if p "0 G and q "0 (i.e., cell i contains neither a bus nor waiting passengers), then qP1 with probability j, G where j is the probability of arrival of passenger(s) at the bus stop. (ii) If p "1 (i.e., there is a bus at G the cell i) and p "0, then the hopping rate k of the bus is de"ned as follows: (a) if q "0, then G> G> k"a but (b) if q "1, then k"b , where a is the hopping rate of a bus onto a stop which has @ G> @ @ no waiting passengers and b is the hopping rate onto a stop with waiting passenger(s). Generally, @ b (a , which re#ects the fact that a bus has to slow down when it has to pick up passengers. We @ @ can set a "1 without loss of generality. When a bus hops onto a stop i with waiting passengers @ q is reset to zero as the bus takes all the passengers. Note that the density of buses c"N/¸ in G 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 #uctuation, 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, because during the period of delay more passengers would be waiting for it and, as a result, it would get further delayed. On the other hand, 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 grow further and the steady state in a "nite 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
Fig. 57. The bus-route model (from [302]).
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models with slow impurities. On the basis of heuristic arguments and mean-"eld approximation it has been argued [302] that this model exhibits a true phase transition from an inhomogeneous low-density phase to a homogeneous (but congested) high-density phase only in the limit jP0. Finally, we mention that the BRM with parallel dynamics has been recently been studied in [303] where also its connection with the NaSch model has been elucidated. 13.6. Mobile directional impurities We have considered the e!ects of random hopping probabilities, assigned either to the lattice sites [214,137,190] or to the particles [204}206] in the TASEP and in the NaSch-type models of vehicular tra$c, on the nature of the corresponding steady states as well as their approach to the steady states starting from random initial conditions. Toroczkai and Zia [304] solved analytically a model with one `mobile directional impuritya; this model is also an extension of the ASEP. In this model, N particles, labeled by integers 1 to N (from left to right), occupy the sites of a one-dimensional lattice of length N#1 where periodic boundary conditions are applied. Thus, there is a single `holea (i.e., empty site) in this model. The shifting of the hole from the site in between the particles n and n#1 to the site in between particles n#1 and n#2 is described by the statement `hole jump from position n to n#1a. In the absence of any impurity, the hole at position n can exchange position with either the particle on its left (with probability = ) L\ L or the particle on its right (with probability = ). These probabilities are arbitrary and L> L direction-dependent (i.e., in general, = O= ) but time-independent. Note that the hopL\ L L> L ping probabilities of the hole is determined by the particles (rather than the lattice sites) in front and behind it. The general case, where = O= and the probabilities = as well as L\ L L> L L\ L = for di!erent n are chosen randomly, is referred to as the random asymmetric case. As the hole L> L wanders, the string of particles also shifts as a whole. However, this system is translationally invariant in the sense that the jump rate of any particular particle}hole pair is independent of its location on the lattice. The `directional impuritya is introduced by identifying a speci"c bond (whose position is "xed with respect to the lattice) as a defect bond such that the time-independent rates of particle}hole exchanges across it are "xed at, say, q and q irrespective of the particle}hole pair involved. In other words, when the hole is in between the particles n and n#1, the hopping probability to the right (left) is always = (= ), except when the defect bond is involved. This model can be L> L L\ L represented as in Fig. 58 where the defect bond is shown as a kink; the motivation for such a kink came from an earlier model [305] of gel electrophoresis [306].
Fig. 58. (a) The translationally invariant Toroczkai}Zia model without any defect bond. (b) The translational invariance is broken by the defect bond (represented by the kink denoted by the dashed line).
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More speci"cally, suppose, the probabilities for the exchange of the hole with a particle across the defect bond are q if the hole moved upward and q if the hole moved downward, independent of the particle involved. In the so-called pure limit, we have, for all n, = "= and L> L = "= . In this limit, the hole can also be regarded as a particle undergoing biased di!usion L\ L (if = O= ) everywhere except across a speci"c defect bond. If we make the further assumption that no backward motion of any particle is allowed, then this model reduces to the model of TASEP with a single blockage [214]. 13.7. Computer networks Inspired by the recent success of the methods of statistical mechanics outside the traditional domain of physics, tools of statistical mechanics have also been applied to analyze fundamental properties of information tra$c on the international network of computers (Internet) [307}310]. 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 "nite bu!er 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 which give rise to the information congestion on the internet. Measuring the #uctuations in the round-trip time taken by a message on the internet (using the ping command of the UNIX operating system), 1/f-like power spectrum has been observed [307,308]. In the square lattice model of a computer network developed by Ohira and Sawatari [310] information packets are generated at the sites on the boundary at a rate j 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 j of the packets, Ohira and Sawatari [310] observed a transition from a low-congestion phase to a high-congestion phase at a nonzero "nite value j . H 14. Summary and conclusion In this section we summarize our conclusions regarding the current status of understanding of the statistical physics of vehicular tra$c. We also speculate on the future trends of research in this area. As we stated in the beginning, one of the main aims of basic research on vehicular tra$c, from the point of view of statistical physics, is to understand the nature of the steady states of the system. We have summarized the empirical evidences available at present in support of the occurrence of
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three distinct dynamical phases, namely, those corresponding to free-yow, synchronized yow and stop-and-go tra$c. Our critical review of the theoretical works has made it clear that, at present, the physical mechanisms at the `microscopica level, which give rise to the synchronized tra$c, are not as well understood as those responsible for the free-#ow and stop-and-go tra$c. There are, however, strong indications that for a complete theory, which would account for all these three phases, one must take into account not only the vehicles on a given stretch of the highway but also on the on- and o!-ramps. We have explored the possibility of transitions from one dynamical phase to another in the NaSch model (and TASEP) with periodic boundary conditions. Moreover, we have also presented the generic phase diagram of the TASEP with open boundary conditions and explained the notion of boundary-induced phase transitions in such one-dimensional driven-di!usive lattice gases which are far from equilibrium. Furthermore, we have found that while some models exhibit "rst-order phase transitions, some others exhibit second-order phase transitions and the signatures of `criticalitya while in some rare situations, e.g., in the cruise-control limit of the NaSch model, the system is found to exhibit even `self-organizeda criticality. In the beginning, we stated that one of the aims of basic research on vehicular tra$c is to understand the nature of the dynamical #uctuations around the steady states. The time-dependent correlations functions and the distributions of the relaxation times have been computed for the NaSch model and some other models but the general questions of the validity of dynamic scaling and dynamic universality classes have not been addressed. Another aim of the statistical mechanical approach to vehicular tra$c, as we stated also in the beginning, is to investigate how the system evolves from initial states which are far from the corresponding steady state. The phenomenon of `coarseninga of the platoons of vehicles during evolution from random initial states have been studied in some models. But, the questions of `universalitya, if any, of the growth exponents have not been addressed so far. Metastable states have been observed in several CA models. But, to our knowledge, the mechanisms of spontaneous decay of such states (analogue of homogeneous nucleation) has not been investigated so far. Besides, to our knowledge, so far it has not been possible to develop any powerful analytical technique for calculating the dynamical properties of the tra$c models. While stating the aims of basic research on vehicular tra$c, we also mentioned the need to understand the e!ects of quenched disorder on the steady-states as well as on the dynamical properties of the systems. We have seen that the randomization of the hopping probability of the vehicles can lead to some exotic platooning phenomena which are close analogs of the `Bose}Einstein-condensationa. An alternative prescription for introducing quenched disorder into the tra$c models is to install random bottlenecks on the road and assign a time-independent hopping probability (or, equivalently, a `permeabilitya) for hopping across bonds in such locations. It has been found that such localized bottlenecks can lead to global phase-segregation. The NaSch model is the most extensively studied minimal CA model of vehicular tra$c on idealized single-lane highways. We have explained the conceptual framework, and illustrated the use of the mathematical formalism, of the cluster-theoretic analytical calculations for the NaSch model. This formalism, which yields exact results for the NaSch model with v "1, gives quite accurate
estimates of various quantities of interest as long as v is not too large. It would be desirable to
develope a new formalism to carry out exact analytical calculations for higher velocities too. Here also the limit v "R is interesting since it shows a rather peculiar behaviour.
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In addition to the detailed discussions on the NaSch model and its various generalizations, we have also mentioned brie#y some other alternative CA models of single-lane highway tra$c, e.g., the VDR model, the Fukui}Ishibashi model, the Werth}Froese}Wolf model, etc., so that one can appreciate the ongoing e!orts to formulate the most satisfactory minimal model. From the point of view of practical applications, modelling vehicular tra$c on multi-lane highways are more relevant than that on idealized single-lane highways which are, nevertheless, interesting from the point of fundamental understanding of truly non-equilibrium phenomena in driven-di!usive lattice gases. At present, there are several di!erent alternative prescriptions for formulating the CA rules for lane-changing of the vehicles on multi-lane highways. But, in order to pick out the most appropriate one from among these CA models theorists would require input from careful further observations of the phenomenon of lane-changing on real multi-lane highways. Empirical observations may also indicate modi"cations or extensions of the CA rules necessary for more realistic modelling of the multi-lane tra$c. The generalizations of the CA models of tra$c on idealized single-lane highways to those on multi-lane highways may be regarded as extensions of one-dimensional model chains to onedimensional strips. The BML model of vehicular tra$c in cities may be regarded as a further generalizations of these models from one-dimensional chains to two-dimensional lattices, or further, to decorated lattices. A few di!erent CA rules have been considered so far for taking into account the e!ects of the tra$c lights at the crossings of streets in such idealized street networks. We have emphasized the intrinsic di!erences between percolation clusters and the cluster of jammed vehicles in the BML (and similar) models in spite of some apparent similarities between them. We have focused attention mainly on the progress made in the recent years using `particlehoppinga models, formulated in terms of cellular automata, and compared these with several other similar systems. Although this may be a slightly biased overview (as all reviews usually are) of the theory of vehicular tra$c, we have also discussed the main ideas behind all the major approaches including the #uid-dynamical, gas-kinetic and car-following theories of vehicular tra$c. At present the relationships between di!erent approaches of modelling have not explored in great detail. It would be very useful if the phenomenological parameters of the macroscopic theories can be estimated by utilizing the mathematical formulae relating these with those of the `microscopica models. It is now quite clear that, in order to make signi"cant further progress, we not only need more realistic models and better techniques of calculation but we also need more detailed and accurate empirical data from real tra$c on highways as well as more careful re-analysis of the existing data in the light of recently developed concepts. So far as the observation of the real tra$c is concerned, a lot can be learnt from a systematic analysis of aereal pictures or video photographs. Alternatively, as the second best choice, a series of counting loops along the highway can give more insight, by providing detailed information on, for example, time-headways (#ux), velocity and local density. If you are a critical thinker (or a pragmatist) you may ask: `armed with the theoretical tools at our disposal now, can we predict the occurrence of a tra$c jam at a speci"c place on a given highway (or street) at a particular instant of timea? This question sounds similar to questions often asked in the context of some other interdisciplinary topics of current research in the area of complex systems, e.g., `can we predict an earthquakea, or, `can we predict a stock market crasha? Of course, we know that, at present, the best we can hope for is to predict (if at all possible)
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probabilities of occurrences of all these phenomena. But, we must admit that, we have a long way to go before we come even close to this goal. Nevertheless, we hope you have enjoyed the fascinating twists and turns of the way we have covered so far. Our endeveour will be more successful if your interest has been stimulated by the intellectual challenges posed by the open problems and if you are willing to uncover the current mysteries as well as anticipating new surprises that may lie ahead. We are just at the beginning of a long road! The volume of vehicular tra$c in the past several years has rapidly outstripped the capacities of the nation's highways. It has become increasingly necessary to understand the dynamics of tra$c #ow and obtain a mathematical description of the process } H. Greenberg (1959)
Acknowledgements It is our pleasure to thank P. Arndt, R. Barlovic, M. Barma, J.G. Brankov, E. Brockfeld, G. Diedrich, B. EisenblaK tter, J. Esser, K. Ghosh, N. Ito, J. KerteH sz, K. Klauck, W. Knospe, S. Krauss, J. Krug, D. Ktitarev, A. Majumdar, K. Nagel, L. Neubert, A. Pasupathy, V. Popkov, V.B. Priezzhev, N. Rajewsky, M. Schreckenberg, G. SchuK tz, S. Sinha, D. Stau!er, R.B. Stinchcombe, Y. Sugiyama, P. Wagner, D.E. Wolf and J. Zittartz for enjoyable collaborations the results of some of which have been reviewed here, for useful discussions and for critical comments as well as suggestions on a preliminary draft of this review. One of us (DC) acknowledges warm hospitality of ICTP, Trieste during the preparation of this manuscript. This work is supported by SFB341 KoK ln-Aachen-JuK lich.
Appendix A. De5nition of update orders A dynamical model is not fully de"ned just by its local transition rules. In addition one has to specify the order in which the rules are applied to the di!erent particles, i.e., the update ordering (sometimes also called &dynamics'). This is an essential part of the de"nition of the model since the transient and even the stationary state may di!er dramatically [311,312]. For the NaSch model one uses a parallel update scheme [140] where the rules are applied to all particles (i.e., vehicles) at the same time. This kind of ordering is sometimes also called synchronous updating. Among the various types of asynchronous update schemes most frequently the so-called randomsequential update is used. Here one picks one particle at random and applies the transition rules to it. Then one makes another random choice (which can also be the same particle again) and so on. This update is sometimes called continuous since it can be described by a master equation in continuous time. Apart from the parallel update there are other updates which are discrete in time. We just mention the ordered-sequential updates. Here one starts by applying the transition rules to one particle. After that the rules are applied to the other particles in a "xed order, e.g. one might continue with the next particle ahead of the "rst one ( forward-ordered ) or the next particle behind it (backward-ordered ). We would like to point out that, in principle, one has to distinguish two
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di!erent types of ordered-sequential updates which one could name site-ordered-sequential and particle-ordered-sequential, respectively. In contrast to the particle-ordered-sequential update described above, in the site-ordered-sequential update the rules are applied to all sites consecutively. This might have a strong e!ect, since a particle might move to next cell ahead which then is updated next (for the forward-particle-ordered-sequential update). Then this particle might move again and so on. This is di!erent from the particle-ordered-sequential case where a particle at most moves once during a sweep through the lattice. As an example consider the extreme case of the NaSch mode with v "1 where only one particle is present which moves with probability q"1 to
an empty cell in front. This particle will move through the whole lattice during one sweep! By looking at a lattice with two particles, one can already see that the two di!erent updates might introduce rather di!erent correlations. Starting with particles separated by d empty sites, in the site-orderedsequential update 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 stay always d or d!1 sites apart. For general values of q the situation is similar. There are several other updates which can be de"ned. We refer to the literature (see e.g. [166,311}315]) for a comparison of di!erent update procedures. The parallel update usually produces the strongest correlations and is used for tra$c simulations [140]. Note that the forwardparticle-ordered-sequential update is almost identical to the parallel update. In the case of periodic boundary conditions a di!erence only occurs during the update of the last particle. In the forwardordered case the particle in front of it (i.e. the "rst particle) might already have moved since it has been updated earlier. Although this di!erence appears to be minor it can have a large e!ect. The di!erence between parallel and forward-particle-ordered update can be viewed as a dynamical defect.
Appendix B. TASEP This simple model of driven systems of interacting particles is one of the most exhaustively studied prototype models in non-equilibrium statistical mechanics [14}16,316,317]. This model can be divided into four classes on the basis of the boundary conditions and the update scheme for the implementation of the dynamics. In this appendix we consider the TASEP with only randomsequential dynamics. Let us consider the TASEP with periodic boundary conditions and random-sequential dynamics. Since only two states, namely empty and occupied, are allowed for each site we can use a two-state variable n to denote the state of the ith site where n "0 if the ith site is empty and G G n "1 if the ith site is occupied. For any given initial con"guration +n (0),, we can write the G G equations governing the time evolution of 1n (t)2 (and all the correlation functions) by taking into G account all the processes during the elementary time interval dt. It is not di$cult to establish that
n (t) with probability 1!2dt , G n (t#dt)" n (t)#n (t)!n (t)n (t) with probability dt , G\ G G\ G G n (t)n (t) with probability dt , G G> dn /dt"n (1!n )!n (1!n ) G G\ G G G>
(B.1)
(B.2)
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which, upon averaging over the history between times 0 and t, leads to the equation d1n 2/dt"1n 2!1n 2!1n n 2#1n n 2 G G\ G G\ G G G>
(B.3)
for 1n 2, the average occupation of the ith site. Note that the equation for 1n 2 involves two-site G G correlations. Similarly, it is straightforward to see that the equations for the two-site correlations involve three-site correlations and so on. Thus, the problem is an intrinsically N-body problem! The probability distribution for this system in the steady state is given by [22] P (+n ,)"N!(¸!N)!/¸! , U G
(B.4)
where ¸ and N refer to the total number of sites and the total number of particles, respectively. From this distribution it follows that 1n 2"N/¸ and 1v2"(¸!N)/(¸!1) which lead to G 1n 2"c and 1v2"(1!c) in the thermodynamic limit. G The stationary state of the TASEP with open boundary conditions and random-sequential dynamics has been determined exactly using the so-called matrix product ansatz (MPA) (see Appendix F for a more technical introduction) in [167,168] using recursion relations. This solution has been generalized to di!erent types of discrete dynamics in [172,173,166]. The solution for parallel dynamics was obtained recently in [164] and [165] using generalizations of the MPA technique.
Appendix C. Naive site-oriented mean-5eld treatment of the NaSch model Suppose, c (i, t), Probability that there is a vehicle with speed v (v"0, 1, 2,2, v ) at the site T
i at the time step t. Then, obviously, c(i, t)" T c (i, t), Probability that the site i is occupied H H t) is the corresponding probability that the site by a vehicle at the time step t and d(i, t)"1!c(i, i is empty at the time step t. Using the de"nition T J(c, p)" vc . T T for the #ux J(c, p) one can get the mean-"eld fundamental diagram for the given p provided one can get c in the mean-"eld approximation. T Step I. Acceleration stage (tPt ): c (i, t )"0 ,
(C.1)
c (i, t )"c (i, t), (0(v(v ) T T\
(C.2)
c (i, t )"c (i, t)#c (i, t) T T \ T
(C.3)
Step II. Deceleration stage (t Pt ): T c (i, t )"c (i, t )#c(i#1, t ) c (i, t ) , T T
(C.4)
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T T c (i, t )"c(i#v#1, t ) d(i#j, t ) c (i, t ) T TY H TYT> T # c (i, t ) d(i#j, t ) (0(v(v ) , T
H T c (i, t )" d(i#j, t )c (i, t ) . T T H Step III. Randomization stage (t Pt ): c (i, t )"c (i, t )#pc (i, t ) , c (i, t )"qc (i, t )#pc (i, t ) (0(v(v ) , T T T>
c (i, t )"qc (i, t ) . T T Step IV. Movement stage (t Pt#1): c (i, t#1)"c (i!v, t ) (04v4v ) . T T
In the special case v "1, Eqs. (C.1)}(C.10) get simpli"ed and, hence, we get
c (i, t#1)"c(i, t)c(i#1, t)#pc(i, t)d(i#1, t) , c (i, t#1)"qc(i!1, t)d(i, t) . Similarly, in the case of v "2, one gets
c (i, t#1)"[c(i, t)#pd(i, t)]c (i, t)#[1#pd(i, t)]c(i, t)[c (i, t)#c (i, t)] , c (i, t#1)"d(i, t)qc (i, t)#d(i, t)[qc(i, t)#pd(i, t)][c (i, t)#c (i, t)] , c (i, t#1)"qd(i, t)[c (i, t)#c (i, t)] .
315
(C.5) (C.6)
(C.7) (C.8) (C.9)
(C.10)
(C.11) (C.12)
(C.13) (C.14) (C.15)
Appendix D. Paradisical mean-5eld theory For v "1 the question, whether a state is a GoE state or not, can be decided locally by
investigating just nearest-neighbour con"gurations. By analysing the update rules one "nds that all states containing the local con"gurations (0, 1) or (1, 1), i.e., con"gurations 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 con"gurations. The momentary velocity gives the number of cells that the car moved in the previous timestep. In both con"gurations the "rst car moved one cell. Therefore it is immediately clear that (0, 1) is a GoE state since otherwise there would have been a doubly occupied cell before the last timestep. The con"guration (1, 1) is also not possible since both cars must have occupied neighbouring cells before the last timestep too. Therefore, according to step 2, the second car could not move. The MFT equations (64) and (65) have to be modi"ed to take into account the existence of GoE states. In general, one has to follow the procedure outlined in Appendix C. A quicker way to derive
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the paradisical mean-"eld (pMF) equations is to analyse the MF equations (64) and (65). In (64) the contribution c(i; t)c(i#1; t) appears. Since 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 modi"ed to c(i; t)c (i#1; t) in pMFT. All other contributions are left unchanged compared to MFT. Due to this modi"cation and the corresponding reduction of the con"guration space the normalization c #c "c is no longer satis"ed automatically. Therefore a normalization constant N has to be introduced. The "nal equations for a homogeneous stationary state are than given by c "N(c #pd)c , c "Nqcd , with the normalization
(D.1) (D.2)
N"1/(c #d) . (D.3) Since c #c "c we have only one independent variable for "xed density c, e.g. c . Solving (D.1), (D.2) for c we obtain (D.4) c "(1!(1!4q(1!c)c) . The #ow is given by f (c)"c and we recover the exact solution for the case v "1.
In the case v "2 more GoE states exist. In order to identify these it is helpful to note that the
rule steps 1}4 imply d (t)"d (t!1)#v (t)!v (t) and therefore H H H> H d (t)5v (t)!v (t) , (D.5) H H> H v (t)4d (t!1) . (D.6) H H The second inequality (D.6) is a consequence of step 2. In the following we list the elementary GoE states, i.e., the local con"gurations which are dynamically forbidden (cars move from left to right): (0, 1), (0, 2), (1, 2), (0, 䢇, 2) ,
(D.7)
(1, 1), (2, 1), (2, 2), (1, 䢇, 2), (2, 䢇, 2) ,
(D.8)
(0, 䢇,
(D.9)
䢇,
2) .
The numbers give the velocity of a vehicle in an occupied cell and 䢇 denotes an empty cell. The elementary GoE states in (D.7) violate the inequality (D.5), and the con"gurations in (D.8) violate (D.6). The state in (D.9) is a second order GoE state. Going one step back in time leads to a "rst order GoE state since (0, 䢇, 䢇, 2) must have evolved from (0, v) (with v"1 or v"2). Again we can derive the pMF equations by modifying the method for the derivation of the MFT. Taking into account only the "rst order GoE states (D.7) and (D.8) one obtains the following pMF equations: c "N[c c#pd(c #c c)] , c "N[pd(c #c )#qd(c #c c)] , c "Nqd(c #c ) .
(D.10) (D.11) (D.12)
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The normalization N ensures c #c #c "c and is given explicitly by 1 1 " . (D.13) N" c #d(1!c ) c #dc #dc These equations have been analysed in [142]. After expressing c through c and c by c "(1/2d)(c #d!((c #d)!4qd(c!c )) . (D.14) Inserting this result into (D.10) we obtain a cubic equation which determines c in terms of the parameters c and p [142]. Results for di!erent values of p are shown in Fig. 19. These results are only slightly modi"ed when also the second order GoE state is taken into account [142].
Appendix E. Equations of car-oriented theory of NaSch model and COMF approximation In terms of p, q, g and g (t)"1!g(t), the equations describing the time evolution of the probabilities P (t) for the NaSch model with v "1 are given by L
P (t#1)"g (t)[P (t)#qP (t)] , (E.1) P (t#1)"g(t)P (t)#[qg(t)#pg (t)]P (t)#qg (t)P (t) , (E.2) P (t#1)"pg(t)P (t)#[qg(t)#pg (t)]P (t)#qg (t)P (t) (n52) . (E.3) L L\ L L> It is worth mentioning here that, for the NaSch model with v "1, the 2-cluster probabilities
P (p , p ) of the SOMF theory are related to the probabilities P of the COMF theory through G H L P (1, 1)"cP , (E.4) P (1, 0)"c(1!c)P , (E.5) P (E.6) P (0, 0)"(1!c) L> (n51) . P L Appendix F. The matrix-product Ansatz for stochastic systems For the stochastic systems considered here the time-evolution of the probability P(s, t) to "nd the system in the con"guration s"(q ,2,q ) is determined by the master equation. For random * sequential dynamics it has the form RP(s, t)/Rt" w(s Ps)P(s , t)! w(sPs )P(s, t) , s
(F.1)
s
with transition rates w(s Ps) from state s to state s. Eq. (F.1) can be rewritten in the form of a SchroK dinger equation in imaginary time [318], (R/Rt)"P(t)2"!H"P(t)2 ,
(F.2)
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with the state vector "P(t)2" s P(s, t)"s2. The vectors "s2""q ,2, q 2 corresponding to the * con"gurations s form an orthonormal basis of the con"guration space. The stochastic Hamiltonian H is de"ned through its matrix elements (F.3) 1s"H"s 2"!w(s Ps), 1s"H"s2" w(sPs ) (sOs ) . s s $ The stationary state of the stochastic process corresponds to the eigenvector "P 2 of H with eigenvalue 0. For discrete-time dynamics the master equation takes the form P(s, t#1)" =(s Ps)P(s , t) ,
(F.4)
s
where =(s Ps)"w(s Ps) ) *t are transition probabilities. This can be rewritten as "P(t#1)2"T"P(t)2 .
(F.5)
Here the stationary state corresponds to the eigenvector "P 2 of the transfer matrix T with eigenvalue 1. A very powerful method for the determination of stationary solutions of the master equation is the so-called matrix-product ansatz (MPA). For a system with open boundaries the weights P(s) in the stationary state can be written in the form
* 1 = [q D#(1!q )E] < . P(q ,2, q )" H H * Z * H For periodic boundary condition the MPA takes the form
(F.6)
* P(q ,2, q )"Tr [q D#(1!q )E] . (F.7) H H * H For simplicity, we have assumed a two-state system where e.g. q "0 corresponds to an empty cell H j and q "1 to an occupied cell. Z is a normalization constant that can be calculated as H * Z "1="C*"<2. In (F.6), (F.7) E and D are matrices and 1=" and "<2 are vectors characterizing * the boundary conditions. The explicit form of these quantities has to be determined from the condition that (F.6) or (F.7) solves the master equation. This leads in general to a algebraic relations between the matrices E and D and the boundary vectors 1=" and "<2. Once one these have been determined one has a simple recipe for determining P(q ,2, q ): First, translate the * con"guration q ,2, q into a product of matrices by identifying each empty cell (q "0) with * H a factor E and each occupied cell (q "1) with D. In that way the con"guration 0110012 H corresponds to the product EDDEED2"EDED2. The weight of the con"guration is then just the matrix element with the vectors 1w" and "v2. A simple example is the ASEP discussed in Section 8.1.1. Here these quantities have to satisfy p DE"D#E ,
(F.8)
a1="E"1=" ,
(F.9)
bD"<2""<2 .
(F.10)
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319
If one is able to "nd explicit representations for this algebra one can determine in principle all expectation values in the stationary state exactly. For (F.8)}(F.10) one can show that all representations are in"nite-dimensional [167]. Only on the line a#b"p one-dimensional representations (with E, D and 1=", "<2 being real numbers) are possible. At this point it might appear that the MPA works only in very special cases. However, it can be shown that the stationary state of one-dimensional stochastic processes is generically of matrixproduct form [319}321]. Even if it is not straightforward to "nd general representations of the resulting algebras, one can at least search systematically for "nite-dimensional representations on special lines in the parameter space of the model. Furthermore, since the mathematical structure of the stationary state is known it is sometimes possible to derive rather general results. As an example in [320] interesting relations between expectation values for ordered-sequential and sublatticeparallel dynamics have been derived. For a more detailed description of the MPA for di!erent types of dynamics and its relation with the MPA technique for quantum-mechanical spin systems [322] we refer to [166]. A review of the treatment of the ASEP using the MPA is given in [22]. The MPA has also been extended to treat the full dynamics, not only the stationary state [323]. Using time-dependent matrices D(t) and E(t) one obtains the Bethe ansatz equations for the corresponding stochastic Hamiltonian [323}325].
Appendix G. Two schemes for solving the mean-5eld approximation of the DTASEP A mean-"eld approximation scheme for this model has also been developed [137]. The timeaveraged steady-state current J in the bond ( j, j#1) is given by J "q 1n (1!n )2. H H> H H> H H> H H> In the mean-"eld approximation, 1n (1!n )2"1n 21(1!n )2 and, hence, H H> H H> J"J "q c (1!c ) , (G.1) H H> H H> H H> where c "1n 2. In order to calculate the steady-state #ux J as a function of the mean density c of H H the particles, Tripathy and Barma [137] used two di!erent iteration schemes based on Eq. (G.1). (i) Constant-current iteration scheme: In this scheme, for a given system length ¸ and a "xed #ux J"J , one starts with some value of c and, computes all the other c ( j'1) using Eq. (G.1), i.e., H J , j"1, 2,2, ¸ (G.2) c "1! H> q c H H> H together with the periodic boundary condition c "c . If the iteration converges, i.e., one gets all H>* H the site densities in the physically acceptable range [0, 1], one accepts the average of these "nal site densities to be the global mean density of the particles corresponding to the #ux J . (ii) Constant density iteration scheme: In this scheme, for a given system length ¸ and "xed global average density c, one begins by assigning the site densities 04c (0)41 to the lattice sites subject to the global H
This can be seen easily from (F.8)}(F.10). We have to mention here that these results up to now do not include the case of parallel dynamics.
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constraint (1/¸) c (0)"c. Then, the site densities are updated in parallel according to H H c (t#1)"c (t)#J (t)!J (t), j"1, 2,2, ¸ (G.3) H H H\ H H H> which follows from Eq. (G.1). It is straightforward to verify that the this iteration scheme keeps the average global density c unchanged at every step of updating and hence the name. After su$cient number of iterations the set of densities converge to a set +c , and the #ux on each bond converge H to the steady-state #ux J . Appendix H. Self-consistent equations for *x and *y in the mean-5eld approximation of the BML model Suppose, v and v denote the average speeds of east-bound and north-bound vehicles, respecV W tively. Then, on the average, an east-bound vehicle spends a time 1/v at a site whereas a northV bound vehicle spends a time 1/v at a site. The north-bound vehicles lead to a reduction of the W speed of the east-bound vehicles by n /v . Moreover, because of the hindrance of the east-bound W W vehicles by other east-bound vehicles ahead of it there will be further reduction of the speed of the east-bound vehicles by n [(1/v )!1]. Furthermore, if the density of the overpasses is f , then V V n 1 !1 . (H.1) v "1!(1!f ) W #n V v V v W V Similarly, the corresponding equation for v is given by W n 1 !1 . (H.2) v "1!(1!f ) V #n W v W v V W In the special case n "n "n/2 both Eqs. (H.1) and (H.2) reduce to the form V W 1 1 (H.3) v"1!(1!f ) ! n , v 2
where v "v "v. The solution of the quadratic equation (H.3) for v is V W
1!f 1 n# v" 1# 2 2
1!f n !4(1!f )n . 1# 2
(H.4)
Appendix I. Derivation of the equations in the microscopic theory of the BML model By de"nition, n (x, y; t) (n (x, y : t)) is unity if the site (x, y) is occupied at time t by a north-bound t (east-bound) vehicle and zero if the site (x, y) is not occupied by a north-bound (east-bound) vehicle. Normalization requires n (x, y; t)#n (x, y; t)"1!n (x, y; t) where the two-state variable t n (x, y; t) is unity if, at time t, the site (x, y) is empty and zero if the site (x, y) is not empty. In order to describe the state of the signals at time t, one also de"nes a two-state variable S(t): S(t)"1(0) if the signal is green (red) for the vehicles under consideration. The space average of n (x, y; t) and n (x, y; t) are c (t) and c (t), respectively. Besides, the time average of S(t) is 1/2. t t
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The updating rules of the BML model lead to the equations n (x, y; t#1)"n (x, y; t)[n (x#1, y; t)#n (x#1, y; t)]S(t) , t # n (x!1, y; t)[1!n (x, y; t)!n (x, y; t)]S(t)#n (x, y; t)[1!S(t)] . t
(I.1)
The "rst term on the right-hand side of Eq. (I.1) describes the situation when the east-bound vehicle, which was at the site (x, y) at time t, "nds a green signal but cannot move because the next site towards east is occupied by another vehicle. The second term on the right-hand side of (I.1) corresponds to the situation where the east-bound vehicle, which was at the site (x!1, y) at time t, "nds a green signal and moves to the next site towards east, which was empty. The last term on the right-hand side of Eq. (I.1) arises from the possibility that the east-bound vehicle, which was at (x, y) at time t, could not move because of a red signal, irrespective of the state of occupation of the next site towards east. Following the similar arguments, one can also write down the corresponding equation for the north-bound vehicles.
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CONTENTS VOLUME 329 D. Sorokin. Superbranes and superembeddings
1
S. Boccaletti, C. Grebogi, Y.-C. Lai, H. Mancini, D. Maza. The control of chaos: theory and applications
103
D. Chowdhury, L. Santen, A. Schadschneider. Statistical physics of vehicular tra$c and some related systems
199
