Lectures on Dynamics of Stochastic Systems

Lectures on Dynamics of Stochastic Systems

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Lectures on Dynamics of Stochastic Systems

Edited by

Valery I. Klyatskin Translated from Russian by

A. Vinogradov

AMSTERDAM • BOSTON • HEIDELBERG • LONDON • NEW YORK OXFORD • PARIS SAN DIEGO • SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO

Elsevier 32 Jamestown Road London NW1 7BY 30 Corporate Drive, Suite 400, Burlington, MA 01803, USA First edition 2011 c 2011 Elsevier Inc. All rights reserved Copyright No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangement with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress ISBN: 978-0-12-384966-3 For information on all Elsevier publications visit our website at www.elsevierdirect.com Typeset by: diacriTech, India This book has been manufactured using Print On Demand technology. Each copy is produced to order and is limited to black ink. The online version of this book will show color figures where appropriate.

Contents

Preface Introduction

ix xi

Part I

1

1

Dynamical Description of Stochastic Systems

Examples, Basic Problems, Peculiar Features of Solutions 1.1

Ordinary Differential Equations: Initial-Value Problems 1.1.1 Particles Under the Random Velocity Field 1.1.2 Particles Under Random Forces 1.1.3 The Hopping Phenomenon 1.1.4 Systems with Blow-Up Singularities 1.1.5 Oscillator with Randomly Varying Frequency (Stochastic Parametric Resonance) 1.2 Boundary-Value Problems for Linear Ordinary Differential Equations (Plane Waves in Layered Media) 1.3 Partial Differential Equations 1.3.1 Linear First-Order Partial Differential Equations 1.3.2 Quasilinear and Nonlinear First-Order Partial Differential Equations 1.3.3 Parabolic Equation of Quasioptics (Waves in Randomly Inhomogeneous Media) 1.3.4 Navier–Stokes Equation: Random Forces in Hydrodynamic Theory of Turbulence Problem

2

3

Solution Dependence on Problem Type, Medium Parameters, and Initial Data

3 3 3 8 10 18 19 20 24 24 33 39 42 50 53

2.1

Functional Representation of Problem Solution 2.1.1 Variational (Functional) Derivatives 2.1.2 Principle of Dynamic Causality 2.2 Solution Dependence on Problem’s Parameters 2.2.1 Solution Dependence on Initial Data 2.2.2 Imbedding Method for Boundary-Value Problems Problems

53 53 59 60 60 62 65

Indicator Function and Liouville Equation

69

3.1 3.2

69 72 72

Ordinary Differential Equations First-Order Partial Differential Equations 3.2.1 Linear Equations

vi

Contents

3.2.2 Quasilinear Equations 3.2.3 General-Form Nonlinear Equations 3.3 Higher-Order Partial Differential Equations 3.3.1 Parabolic Equation of Quasi-Optics 3.3.2 Random Forces in Hydrodynamic Theory of Turbulence Problems

Part II 4

5

6

7

Statistical Description of Stochastic Systems

77 79 80 80 83 85

87

Random Quantities, Processes, and Fields

89

4.1 4.2

89 95 95

Random Quantities and their Characteristics Random Processes, Fields, and their Characteristics 4.2.1 General Remarks 4.2.2 Statistical Topography of Random Processes and Fields 4.2.3 Gaussian Random Process 4.2.4 Gaussian Vector Random Field 4.2.5 Logarithmically Normal Random Process 4.2.6 Discontinuous Random Processes 4.3 Markovian Processes 4.3.1 General Properties 4.3.2 Characteristic Functional of the Markovian Process Problems

99 102 105 108 110 115 115 117 119

Correlation Splitting

123

5.1 5.2 5.3 5.4 5.5

General Remarks Gaussian Process Poisson’s Process Telegrapher’s Random Process Delta-Correlated Random Processes 5.5.1 Asymptotic Meaning of Delta-Correlated Processes and Fields Problems

123 125 127 128 130

General Approaches to Analyzing Stochastic Systems

141

6.1 6.2

Ordinary Differential Equations Completely Solvable Stochastic Dynamic Systems 6.2.1 Ordinary Differential Equations 6.2.2 Partial Differential Equations 6.3 Delta-Correlated Fields and Processes 6.3.1 One-Dimensional Nonlinear Differential Equation 6.3.2 Linear Operator Equation Problems

141 144 144 158 160 162 165 166

Stochastic Equations with the Markovian Fluctuations of Parameters

183

7.1

184

Telegrapher’s Processes

133 135

Contents

8

9

10

7.2 Gaussian Markovian Processes Problems

187 188

Approximations of Gaussian Random Field Delta-Correlated in Time

191

8.1 8.2 8.3

The Fokker–Planck Equation Transition Probability Distributions The Simplest Markovian Random Processes 8.3.1 Wiener Random Process 8.3.2 Wiener Random Process with Shear 8.3.3 Logarithmic-Normal Random Process 8.4 Applicability Range of the Fokker–Planck Equation 8.4.1 Langevin Equation 8.5 Causal Integral Equations 8.6 Diffusion Approximation Problems

191 194 196 197 197 200 211 211 215 218 220

Methods for Solving and Analyzing the Fokker–Planck Equation

229

9.1 9.2

229 230 231 232 234

Integral Transformations Steady-State Solutions of the Fokker–Planck Equation 9.2.1 One-Dimensional Nonlinear Differential Equation 9.2.2 Hamiltonian Systems 9.2.3 Systems of Hydrodynamic Type 9.3 Boundary-Value Problems for the Fokker–Planck Equation (Hopping Phenomenon) 9.4 Method of Fast Oscillation Averaging Problems

242 245 247

Some Other Approximate Approaches to the Problems of Statistical Hydrodynamics

253

10.1 Quasi-Elastic Properties of Isotropic and Stationary Noncompressible Turbulent Media 10.2 Sound Radiation by Vortex Motions 10.2.1 Sound Radiation by Vortex Lines 10.2.2 Sound Radiation by Vortex Rings

254 258 260 263

Part III Examples of Coherent Phenomena in Stochastic Dynamic Systems 11

vii

269

Passive Tracer Clustering and Diffusion in Random Hydrodynamic and Magnetohydrodynamic Flows

271

11.1 General Remarks 11.2 Particle Diffusion in Random Velocity Field 11.2.1 One-Point Statistical Characteristics 11.2.2 Two-Point Statistical Characteristics 11.3 Probabilistic Description of Density Field in Random Velocity Field

271 276 276 281 284

viii

Contents

11.4 Probabilistic Description of Magnetic Field and Magnetic Energy in Random Velocity Field 11.5 Integral One-Point Statistical Characteristics of Passive Vector Fields 11.5.1 Spatial Correlation Function of Density Field 11.5.2 Spatial Correlation Tensor of Density Field Gradient and Dissipation of Density Field Variance 11.5.3 Spatial Correlation Function of Magnetic Field 11.5.4 On the Magnetic Field Helicity 11.5.5 On the Magnetic Field Dissipation Problems 12

13

291 298 299 302 310 313 315 319

Wave Localization in Randomly Layered Media

325

12.1 General Remarks 12.1.1 Wave Incidence on an Inhomogeneous Layer 12.1.2 Source Inside an Inhomogeneous Layer 12.2 Statistics of Scattered Field at Layer Boundaries 12.2.1 Reflection and Transmission Coefficients 12.2.2 Source Inside the Layer of a Medium 12.2.3 Statistical Energy Localization 12.3 Statistical Theory of Radiative Transfer 12.3.1 Normal Wave Incidence on the Layer of Random Media 12.3.2 Plane Wave Source Located in Random Medium 12.4 Numerical Simulation Problems

325 325 327 330 330 337 338 339 340 347 350 352

Caustic Structure of Wavefield in Random Media

355

13.1 Input Stochastic Equations and Their Implications 13.2 Wavefield Amplitude–Phase Fluctuations. Rytov’s Smooth Perturbation Method 13.2.1 Random Phase Screen (1x  x) 13.2.2 Continuous Medium (1x = x) 13.3 Method of Path Integral 13.3.1 Asymptotic Analysis of Plane Wave Intensity Fluctuations 13.4 Elements of Statistical Topography of Random Intensity Field 13.4.1 Weak Intensity Fluctuations 13.4.2 Strong Intensity Fluctuations Problems

355

References

361 365 366 367 371 381 383 386 388 393

Preface

I keep six honest serving-men (They taught me all I knew); Their names are What and Why and When And How and Where and Who. R. Kipling

This book is a revised and more comprehensive re-edition of my book Dynamics of Stochastic Systems, Elsevier, Amsterdam, 2005. Writing this book, I sourced from the series of lectures that I gave to scientific associates at the Institute of Calculus Mathematics, Russian Academy of Sciences. In the book, I use the functional approach to uniformly formulate general methods of statistical description and analysis of dynamic systems. These are described in terms of different types of equations with fluctuating parameters, such as ordinary differential equations, partial differential equations, boundary-value problems, and integral equations. Asymptotic methods of analyzing stochastic dynamic systems – the deltacorrelated random process (field) approximation and the diffusion approximation – are also considered. General ideas are illustrated using examples of coherent phenomena in stochastic dynamic systems, such as clustering of particles and passive tracer in random velocity field, dynamic localization of plane waves in randomly layered media and caustic structure of wavefield in random media. Working at this edition, I tried to take into account opinions and wishes of readers about both the style of the text and the choice of specific problems. Various mistakes and misprints have been corrected. The book is aimed at scientists dealing with stochastic dynamic systems in different areas, such as acoustics, hydrodynamics, magnetohydrodynamics, radiophysics, theoretical and mathematical physics, and applied mathematics; it may also be useful for senior and postgraduate students. The book consists of three parts. The first part is, in essence, an introductory text. It raises a few typical physical problems to discuss their solutions obtained under random perturbations of parameters affecting system behavior. More detailed formulations of these problems and relevant statistical analysis can be found in other parts of the book. The second part is devoted to the general theory of statistical analysis of dynamic systems with fluctuating parameters described by differential and integral equations. This theory is illustrated by analyzing specific dynamic systems. In addition, this part

x

Preface

considers asymptotic methods of dynamic system statistical analysis, such as the deltacorrelated random process (field) approximation and diffusion approximation. The third part deals with the analysis of specific physical problems associated with coherent phenomena. These are clustering and diffusion of particles and passive tracer (density and magnetic fields) in a random velocity field, dynamic localization of plane waves propagating in layered random media and caustic structure of wavefield in random multidimensional media. These phenomena are described by ordinary differential equations and partial differential equations. The material is supplemented with sections concerning dynamical and statistical descriptions of simplest hydrodynamic-type systems, the relationship between conventional methods (based on the Lyapunov stability analysis of stochastic dynamic systems), methods of statistical topography and statistical description of magnetic field generation in the random velocity field (stochastic [turbulent] dynamo). Each lecture is appended with problems for the reader to solve on his or her own, which will be a good training for independent investigations. V. I. Klyatskin Moscow, Russia

Introduction

Different areas of physics pose statistical problems in ever greater numbers. Apart from issues traditionally obtained in statistical physics, many applications call for including fluctuation effects. While fluctuations may stem from different sources (such as thermal noise, instability, and turbulence), methods used to treat them are very similar. In many cases, the statistical nature of fluctuations may be deemed known (either from physical considerations or from problem formulation) and the physical processes may be modeled by differential, integro-differential or integral equations. We will consider a statistical theory of dynamic and wave systems with fluctuating parameters. These systems can be described by ordinary differential equations, partial differential equations, integro-differential equations and integral equations. A popular way to solve such systems is by obtaining a closed system of equations for statistical characteristics, to study their solutions as comprehensively as possible. We note that wave problems are often boundary-value problems. When this is the case, one may resort to the imbedding method to reformulate the equations at hand to initial-value problems, thus considerably simplifying the statistical analysis. The purpose of this book is to demonstrate how different physical problems described by stochastic equations can be solved on the basis of a general approach. In stochastic problems with fluctuating parameters, the variables are functions. It would be natural therefore to resort to functional methods for their analysis. We will use a functional method devised by Novikov [1] for Gaussian fluctuations of parameters in a turbulence theory and developed by the author of this book [2] for the general case of dynamic systems and fluctuating parameters of an arbitrary nature. However, only a few dynamic systems lend themselves to analysis yielding solutions in a general form. It proved to be more efficient to use an asymptotic method where the statistical characteristics of dynamic problem solutions are expanded in powers of a small parameter. This is essentially a ratio of the random impact’s correlation time to the time of observation or to another characteristic time scale of the problem (in some cases, these may be spatial rather than temporal scales). This method is essentially a generalization of the theory of Brownian motion. It is termed the deltacorrelated random process (field) approximation. For dynamic systems described by ordinary differential stochastic equations with Gaussian fluctuations of parameters, this method leads to a Markovian problem solving model, and the respective equation for transition probability density has the form of the Fokker–Planck equation. In this book, we will consider in depth the methods of analysis available for this equation and its boundary conditions. We will analyze solutions and validity conditions by way of integral transformations. In more complicated problems described by partial differential equations, this method leads to a

xii

Introduction

generalized equation of the Fokker–Planck type in which variables are the derivatives of the solution’s characteristic functional. For dynamic problems with non-Gaussian fluctuations of parameters, this method also yields Markovian type solutions. Under the circumstances, the probability density of respective dynamic stochastic equations satisfies a closed operator equation. In physical investigations, Fokker–Planck and similar equations are usually set up from rule-of-thumb considerations, and dynamic equations are invoked only to calculate the coefficients of these equations. This approach is inconsistent, generally speaking. Indeed, the statistical problem is completely defined by dynamic equations and assumptions on the statistics of random impacts. For example, the Fokker–Planck equation must be a logical sequence of the dynamic equations and some assumptions on the character of random impacts. It is clear that not all problems lend themselves for reduction to a Fokker–Planck equation. The functional approach allows one to derive a Fokker–Planck equation from the problem’s dynamic equation along with its applicability conditions. In terms of formal mathematics, our approach corresponds to that of R.L. Stratonovich (see, e.g., [3]). For a certain class of Markovian random process (telegrapher’s processes, Gaussian process and the like), the developed functional approach also yields closed equations for the solution probability density with allowance for a finite correlation time of random interactions. For processes with Gaussian fluctuations of parameters, one may construct a better physical approximation than the delta-correlated random process (field) approximation – the diffusion approximation that allows for finiteness of correlation time radius. In this approximation, the solution is Markovian and its applicability condition has transparent physical meaning, namely, the statistical effects should be small within the correlation time of fluctuating parameters. This book treats these issues in depth from a general standpoint and for some specific physical applications. Recently, the interest of both theoreticians and experimenters has been attracted to the relation of the behavior of average statistical characteristics of a problem solution with the behavior of the solution in certain happenings (realizations). This is especially important for geophysical problems related to the atmosphere and ocean where, generally speaking, a respective averaging ensemble is absent and experimenters, as a rule, deal with individual observations. Seeking solutions to dynamic problems for these specific realizations of medium parameters is almost hopeless due to the extreme mathematical complexity of these problems. At the same time, researchers are interested in the main characteristics of these phenomena without much need to know specific details. Therefore, the idea of using a well-developed approach to random processes and fields based on ensemble averages rather than separate observations proved to be very fruitful. By way of example, almost all physical problems of the atmosphere and ocean to some extent are treated by statistical analysis. Randomness in medium parameters gives rise to a stochastic behavior of physical fields. Individual samples of scalar two-dimensional fields ρ(R, t), R = (x, y), say, recall a rough mountainous terrain with randomly scattered peaks, troughs, ridges and

Introduction

xiii

4 2

2 1

0

0

−2 140

140 100

100 60

−1

3 140 100

100 60

60

140

120

120

100

100

80

80

60

60 80

4

140

−2

140

60

3 2 1

100 (a)

120

140

60

80

100 (b)

120

2 1

60

140

Figure 0.1 Realizations of the fields governed by (a) Gaussian and (b) lognormal distribution and the corresponding topographic level lines. The bold curves in the bottom patterns show level lines corresponding to levels 0 (a) and 1 (b).

saddles. Figure 0.1 shows examples of realizations of two random fields characterized by different statistical structures. Common methods of statistical averaging (computing mean-type averages –
 hρ (R, t)i, space-time correlation function – ρ (R, t) ρ R0 , t0 etc., where h· · · i implies averaging over an ensemble of random parameter samples) smooth the qualitative features of specific samples. Frequently, these statistical characteristics have nothing in common with the behavior of specific samples, and at first glance may even seem to be at variance with them. For example, the statistical averaging over all observations makes the field of average concentration of a passive tracer in a random velocity field ever more smooth, whereas each realization sample tends to be more irregular in space due to the mixture of areas with substantially different concentrations. Thus, these types of statistical average usually characterize ‘global’ space–time dimensions of the area with stochastic processes, but give no details about the process behavior inside the area. For this case, details heavily depend on the velocity field pattern, specifically, on whether it is divergent or solenoidal. Thus, the first case will show with total probability that clusters will be formed, i.e. compact areas of enhanced concentration of tracer surrounded by vast areas of low-concentration tracer. In the

xiv

Introduction

circumstances, all statistical moments of the distance between the particles will grow with time exponentially; that is, on average, a statistical recession of particles will take place [4]. In a similar way, in the case of waves propagating in random media, an exponential spread of the rays will take place on average; but simultaneously, with total probability, caustics will form at finite distances. One more example to illustrate this point is the dynamic localization of plane waves in layered randomly inhomogeneous media. In this phenomenon, the wave field intensity exponentially decays inward to the medium with the probability equal to unity when the wave is incident on the half-space of such a medium, while all statistical moments increase exponentially with distance from the boundary of the medium [5]. These physical processes and phenomena occurring with the probability equal to unity will be referred to as coherent processes and phenomena [6]. This type of statistical coherence may be viewed as some organization of the complex dynamic system, and retrieval of its statistically stable characteristics is similar to the concept of coherence as self-organization of multicomponent systems that evolve from the random interactions of their elements [7]. In the general case, it is rather difficult to say whether or not the phenomenon occurs with the probability equal to unity. However, for a number of applications amenable to treatment with the simple models of fluctuating parameters, this may be handled by analytical means. In other cases, one may verify this by performing numerical modeling experiments or analyzing experimental findings. The complete statistic (say, the whole body of all n-point space-time moment functions), would undoubtedly contain all the information about the investigated dynamic system. In practice, however, one may succeed only in studying the simplest statistical characteristics associated mainly with simultaneous and one-point probability distributions. It would be reasonable to ask how with these statistics on hand one would look into the quantitative and qualitative behavior of some system happenings? This question is answered by methods of statistical topography [8]. These methods were highlighted by Ziman [9], who seems to have coined this term. Statistical topography yields a different philosophy of statistical analysis of dynamic stochastic systems, which may prove useful for experimenters planning a statistical processing of experimental data. These issues are treated in depth in this book. More details about the material of this book and more exhaustive references can be found in the textbook quoted in reference [2] and recent reviews [6, 10–16].

Part I Dynamical Description of Stochastic Systems

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Lecture 1 Examples, Basic Problems, Peculiar Features of Solutions

In this chapter, we consider several dynamic systems described by differential equations of different types and discuss the features in the behaviors of solutions to these equations under random disturbances of parameters. Here, we content ourselves with the problems in the simplest formulation. More complete formulations will be discussed below, in the sections dealing with statistical analysis of the corresponding systems.

1.1

Ordinary Differential Equations: Initial-Value Problems

1.1.1

Particles Under the Random Velocity Field

In the simplest case, a particle under random velocity field is described by the system of ordinary differential equations of the first order d r(t) = U(r, t), dt

r(t0 ) = r0 ,

(1.1)

where U(r, t) = u0 (r, t) + u(r, t), u0 (r, t) is the deterministic component of the velocity field (mean flow), and u(r, t) is the random component. In the general case, field u(r, t) can have both nondivergent (solenoidal, for which div u(r, t) = 0) and divergent (for which div u(r, t) 6= 0) components. We dwell on stochastic features of the solution to problem (1.1) for a system of particles in the absence of mean flow (u0 (r, t) = 0). From Eq. (1.1) formally follows that every particle moves independently of other particles. However, if random field u(r, t) has a finite spatial correlation radius lcor , particles spaced by a distance shorter than lcor appear in the common zone of infection of random field u(r, t) and the behavior of such a system can show new collective features. For steady velocity field u(r, t) ≡ u(r), Eq. (1.1) reduces to d r(t) = u(r), dt

r(0) = r0 .

Lectures on Dynamics of Stochastic Systems. DOI: 10.1016/B978-0-12-384966-3.00001-5 c 2011 Elsevier Inc. All rights reserved. Copyright

(1.2)

4

Lectures on Dynamics of Stochastic Systems

This equation clearly shows that steady points er (at which u(er ) = 0) remain the fixed points. Depending on whether these points are stable or unstable, they will attract or repel nearby particles. In view of randomness of function u(r), points er are random too. It is expected that the similar behavior will be also characteristic of the general case of space-time random velocity field u(r,t). If some points er remain stable during sufficiently long time, then clusters of particles (i.e., compact regions with elevated particle concentration, which occur merely in rarefied zones) must arise around these points in separate realizations of random field u(r, t). On the contrary, if the stability of these points alternates with instability sufficiently rapidly and particles have no time for significant rearrangement, no clusters of particles will occur. Simulations [17, 18] show that the behavior of a system of particles essentially depends on whether the random field of velocities is nondivergent or divergent. By way of example, Fig. 1.1a shows a schematic of the evolution of the two-dimensional system of particles uniformly distributed within the circle for a particular realization of the nondivergent steady field u(r). Here, we use the dimensionless time related to statistical parameters of field u(r). In this case, the area of surface patch within the contour remains intact and particles relatively uniformly fill the region within the deformed contour. The only feature consists in the fractal-type irregularity of the deformed contour. This phenomenon – called chaotic advection – is under active study now (see, e.g., [19]). On the contrary, in the case of the divergent velocity field u(r), particles uniformly distributed in the square at the initial instant will form clusters during the temporal evolution. Results simulated for this case are shown in Fig. 1.1b. We emphasize that the formation of clusters is purely a kinematic effect. This feature of particle dynamics disappears on averaging, over an ensemble of realizations of random velocity field. To demonstrate the process of particle clustering, we consider the simplest problem [11], in which random velocity field u(r, t) has the form u(r, t) = v(t)f (kr),

(1.3)

where v(t) is the random vector process and f (kr) = sin 2(kr)

(1.4)

is the deterministic function of one variable. This form of function f (kr) corresponds to the first term of the expansion in harmonic components and is commonly used in numerical simulations [17, 18]. In this case, Eq. (1.1) can be written in the form d r(t) = v(t) sin 2(kr), dt

r(0) = r0 .

In the context of this model, motions of a particle along vector k and in the plane perpendicular to vector k are independent and can be separated. If we direct the x-axis

Examples, Basic Problems, Peculiar Features of Solutions

−10

5

10

10

5

5

−5

5

−5

10

−10

−10 t=0 10

t=2 10 5

5 −5

10

−5

−5

−10

5

5

−10

10

−5

5

10

−5

−5 −10 t=3

−10 t=4 (a)

t = 0.5

t =0

t =1

t = 2.0 (b)

Figure 1.1 Diffusion of a system of particles described by Eqs. (1.2) numerically simulated for (a) solenoidal and (b) nondivergent random steady velocity field u(r).

6

Lectures on Dynamics of Stochastic Systems

along vector k, then the equations assume the form d x(t) = vx (t) sin(2kx), dt d R(t) = vR (t) sin(2kx), dt

x(0) = x0 , (1.5) R(0) = R0 .

The solution of the first equation in system (1.5) is x(t) =

h i 1 arctan eT(t) tan(kx0 ) , k

(1.6)

where Zt T(t) = 2k

dτ vx (τ ).

(1.7)

0

Taking into account the equalities following from Eq. (1.6) sin(2kx) =

sin(2kx0 ) e−T(t) cos2 (kx0 ) + eT(t) sin2 (kx0 )

, (1.8)

cos(2kx) =

1 − e2T(t) tan2 (kx0 ) , 1 + e2T(t) tan2 (kx0 )

we can rewrite the second equation in Eqs. (1.5) in the form sin(2kx0 )vR (t) d R(t|r0 ) = . −T(t) dt e cos2 (kx0 ) + eT(t) sin2 (kx0 ) As a result, we have R(t|r0 ) = R0 +

Zt dτ 0

sin(2kx0 )vR (τ ) . −T(τ ) e cos2 (kx0 ) + eT(τ ) sin2 (kx0 )

(1.9)

Consequently, if the initial particle position x0 is such that π kx0 = n , 2

(1.10)

where n = 0, ±1, . . . , then the particle will be the fixed particle and r(t) ≡ r0 . Equalities (1.10) define planes in the general case and points in the one-dimensional case. They correspond to zeros of the field of velocities. Stability of these points depends on the sign of function v(t), and this sign changes during the evolution process. It can be expected that particles will gather around these points if vx (t) 6= 0, which just corresponds to clustering of particles.

Examples, Basic Problems, Peculiar Features of Solutions

7

x 1.5

T (t ) 10

1.0 0 0.5

−10 −20 0

10

20 30 (a)

40 t

0

10

20 (b)

15 (c)

16

17 t

30

40 t

x 1.5

1.0

0.5

13

14

Figure 1.2 (a) Segment of a realization of random process T(t) obtained by numerically integrating Eq. (1.7) for a realization of random process vx (t); (b), (c) x-coordinates simulated with this segment for four particles versus time.

In the case of a nondivergent velocity field, vx (t) = 0 and, consequently, T(t) ≡ 0; as a result, we have x(t|x0 ) ≡ x0 ,

R(t|r0 ) = R0 + sin 2(kx0 )

Zt

dτ vR (τ ),

0

which means that no clustering occurs. Figure 1.2a shows a fragment of the realization of random process T(t) obtained by numerical integration of Eq. (1.7) for a realization of random process vx (t); we used this fragment for simulating the temporal evolution of coordinates of four particles π i (i = 1, 2, 3, 4) (see x(t), x ∈ (0, π/2) initially located at coordinates x0 (i) = 25 Fig. 1.2b). Figure 1.2b shows that particles form a cluster in the vicinity of point x = 0 at the dimensionless time t ≈ 4. Further, at time t ≈ 16 the initial cluster disappears and new one appears in the vicinity of point x = π/2. At moment t ≈ 40, the cluster appears again in the vicinity of point x = 0, and so on. In this process, particles in clusters remember their past history and significantly diverge during intermediate temporal segments (see Fig. 1.2c). Thus, we see in this example that the cluster does not move from one region to another; instead, it first collapses and then a new cluster appears. Moreover, the

8

Lectures on Dynamics of Stochastic Systems

lifetime of clusters significantly exceeds the duration of intermediate segments. It seems that this feature is characteristic of the specific model of velocity field and follows from steadiness of points (1.10). As regards the particle diffusion along the y-direction, no cluster occurs there. Note that such clustering in a system of particles was found, to all appearance for the first time, in papers [20, 21] as a result of simulating the so-called Eole experiment with the use of the simplest equations of atmospheric dynamics. In this global experiment, 500 constant-density balloons were launched in Argentina in 1970–1971; these balloons traveled at a height of about 12 km and spread along the whole of the southern hemisphere. Figure 1.3 shows the balloon distribution over the southern hemisphere for day 105 from the beginning of this process simulation [20]; this distribution clearly shows that balloons are concentrated in groups, which just corresponds to clustering.

1.1.2

Particles Under Random Forces

The system of equations (1.1) describes also the behavior of a particle under the field of random external forces f (r, t). In the simplest case, the behavior of a particle in the presence of linear friction is described by the differential equation of the second order (Newton equation) d d2 r(t) = −λ r(t) + f (r, t), dt dt2 r(0) = r0 ,

(1.11)

d r(0) = v0 , dt

or the systems of differential equation of the first order d r(t) = v(t), dt

d v(t) = −λv(t) + f (r, t), dt

r(0) = r0 ,

(1.12)

v(0) = v0 .

Results of numerical simulations of stochastic system (1.12) can be found in Refs [22, 23]. Stability of the system was studied in these papers by analyzing Lyapunov’s characteristic parameters. The behavior of a particle under the deterministic potential field in the presence of linear friction and random forces is described by the system of equations d r(t) = v(t), dt

d ∂U(r, t) v(t) = −λv(t) − + f (r, t), dt ∂r r(0) = r0 ,

(1.13)

v(0) = v0 .

which is the simplest example of Hamiltonian systems with linear friction. If friction and external forces are absent and function U is independent of time, U(r, t) = U(r),

Examples, Basic Problems, Peculiar Features of Solutions

9

Figure 1.3 Balloon distribution in the atmosphere for day 105 from the beginning of process simulation.

the system has an integral of motion d E(t) = const, dt

E(t) =

v2 + U(r) 2

expressing energy conservation. In statistical problems, equations of type (1.12), (1.13) are widely used to describe the Brownian motion of particles. In the general case, Hamiltonian systems are described by the system of equations d ∂H(r, v, t) r(t) = , dt ∂v r(0) = r0 ,

d ∂H(r, v, t) v(t) = − , dt ∂r

(1.14)

v(0) = v0 ,

where H(r, v, t) = H(r(t), v(t), t) is the Hamiltonian function. In the case of conservative Hamiltonian systems, function H(r, v, t) has no explicit dependence on time, H(r, v, t) = H(r, v), and the system has the integral of motion H(r, v) = const.

10

Lectures on Dynamics of Stochastic Systems

1.1.3

The Hopping Phenomenon

Now, we dwell on another stochastic aspect related to dynamic equations of type (1.1); namely, we consider the hopping phenomenon caused by random fluctuations. Consider the one-dimensional nonlinear equation   d x(t) = x 1 − x2 + f (t), dt

x(0) = x0 ,

(1.15)

where f (t) is the random function of time. In the absence of randomness ( f (t) ≡ 0), the solution of Eq. (1.15) has two stable steady states x = ±1 and one instable state x = 0. Depending on the initial-value, solution of Eq. (1.15) arrives at one of the stable states. However, in the presence of small random disturbances f (t), dynamic system (1.15) will first approach the vicinity of one of the stable states and then, after the lapse of certain time, it will be transferred into the vicinity of another stable state. Note that Eq. (1.15) corresponds to limit process λ → ∞ in the equation   d2 d dU(x) x(t) + λ x(t) − λ + f (t) = 0, dt dx dt2 that is known as the Duffing equation and is the special case of the one-dimensional Hamiltonian system (1.13) d x(t) = v(t), dt

  d dU(x) v(t) = −λ v(t) − − f (t) dt dx

(1.16)

with the potential function U(x) =

x4 x2 − . 2 4

In other words, Eq. (1.15) corresponds to great friction coefficients λ. Statistical description of this problem will be considered in Sect. 8.4.1, page 211. Additionally, we note that, in the context of statistical description, reduction of the Hamiltonian system (1.16) to the ‘short-cut equation’ is called the Kramers problem. It is clear that the similar behavior can occur in more complicated situations.

Hydrodynamic-Type Nonlinear Systems An important problem of studying large-scale processes in the atmosphere considered as a single physical system consists in revealing the mechanism of energy exchange between different ‘degrees of freedom’. The analysis of such nonlinear processes on the base of simple models described by a small number of parameters (degrees of freedom) is recently of great attention. In this connection, A.M. Obukhov (see, for example [24]) introduced the concept of hydrodynamic-type systems (HTS). These systems have a finite number of parameters v1, . . . , vn , but the general features of the dynamic equations governing system motions coincide with those characteristic of the

Examples, Basic Problems, Peculiar Features of Solutions

11

hydrodynamic equations of perfect incompressible liquid, including quadratic nonlinearity, energy conservation, and regularity (phase volume invariance during system motions). The general description of HTS is given in Sect. 8.3.3, page 200. Here, we dwell only on the dynamic description of simplest systems. The simplest system of this type (S3 ) is equivalent to the Euler equations in the dynamics of solids; it describes the known problem on liquid motions in an ellipsoidal cavity [24]. Any finite-dimensional approximation of hydrodynamic equations also belongs to the class of HTS if it possesses the above features. To model the cascade mechanism of energy transformation in a turbulent flow, Obukhov [25] suggested a multistage HTS. Each stage of this system consists of identical-scale triplets; at every next stage, the number of triplets is doubled and the scale is decreased in geometrical progression with ratio Q & 1. As a result, this model describes interactions between the motions of different scales. The first stage consists of a singe triplet whose unstable mode v01 is excited by an external force f0 (t) applied to the system (Fig. 1.4a). The stable modes of this triplet v1,1 and v1,2 are the unstable modes of two triplets of the second stage; their stable modes v2,1 , v2,2 , v2,3 , and v2,4 are, in turn, the unstable modes of four triplets of the third stage; and so on (Fig. 1.4b). It should be noted however that physical processes described in terms of macroscopic equations occur in actuality against the background of processes characterized by shorter scales (noises). Among these processes are, for example, the molecular noises (in the context of macroscopic hydrodynamics), microturbulence (against the large-scale motions), and the effect of truncated (small-scale) terms in the finite-dimensional approximation of hydrodynamic equations. The effect of these small-scale noises should be analyzed in statistical terms. Such a consideration can be performed in terms of macroscopic variables. With this goal, one must include external random forces with certain statistical characteristics in the corresponding macroscopic equations. The models considered here require additionally the inclusion of dissipative terms in the equations of motion to ensure energy outflow to smaller-scale modes. Accordingly, the simplest hydrodynamic models that allow simulating actual processes are the HTS with random forces and linear friction. An important problem which appeared, for example, in the theory of climate consists in the determination of a possibility of significantly different circulation processes f0(t)

f0(t) v1,0 (t)

v1,0 (t) v1,1(t)

v1,2(t)

v1,1(t)

v1,2(t)

v2,4(t)

v2,1(t) v2,2(t) (a)

Figure 1.4

v2,3(t) (b)

Diagrams of (a) three- and (b) seven-mode HTS.

12

Lectures on Dynamics of Stochastic Systems

which occurred under the same distribution of incoming heat, i.e., the problem of the existence of different motion regimes of a given hydrodynamic system under the same ‘external conditions’. A quite natural phenomenon that should be considered in this context is the ‘hopping’ phenomenon that consists in switching between such regimes of motion. Characteristic of these regimes is the fact that the duration of switching is small in comparison with the lifetime of the corresponding regimes. It is expedient to study these problems using the above simple models. The corresponding systems of quadratically nonlinear ordinary differential equations may generally have several stable regimes, certain parameters describing external conditions being identical. The hopping events are caused by variations of these conditions in time and by the effect of random noises. If the system has no stable regimes, its behavior can appear extremely difficult and require statistical description as it is the case in the Lorentz model [26]. In the case of availability of indifferent equilibrium states, the system may allow quasisteady-state regimes of motion. Switching between such regimes can be governed by the dynamic structure of the system, and system’s behavior can appear ‘stochastic’ in this case, too. In what follows, we study these hopping phenomena with the use of the simplest hydrodynamic models.

Dynamics of a Triplet (Gyroscope) Consider first the case of a single stage, i.e., the triplet in the regime of forced motion (Fig. 1.4a). With this goal, we set force f0 (t) = f0 = const and assume the availability of dissipative forces acting on the stable modes of the triplet. The corresponding equations of motion have the form   d v1,0 (t) = µ v21,1 (t) − v21,2 (t) + f0 , dt d v1,1 (t) = −µv1,0 (t)v1,1 (t) − λv1,1 (t), dt

(1.17)

d v1,2 (t) = µv1,0 (t)v1,2 (t) − λv1,2 (t). dt If f0 > 0, component v1,1 (t) vanishes with time, so that the motion of the triplet (1.17) is described in the dimensionless variables r x=

µ λ v1,0 − √ , f0 µf0

r y=

µ v1,2 , f0

τ=

p µf0 t,

(1.18)

by the two-mode system d x(t) = −y2 (t) + 1, dt

d y(t) = x(t)y(t). dt

(1.19)

Examples, Basic Problems, Peculiar Features of Solutions

13

This system has the integral of motion H1 = x2 (t) + y2 (t) − 2 ln y(t). The change of variables p(t) = x(t), q(t) = ln y(t) reduces the system to the Hamiltonian form ∂ H(p, q) d p(t) = − , dt ∂q

d ∂ H(p, q) q(t) = , dt ∂p

p2 (t) 1 2q(t) + e − q(t). 2 2 Thus, the behavior of a system with friction (1.17) under the action of a constant external force is described in terms of the Hamiltonian system. Stationary points (0, 1) and (0, −1) of system (1.19) are the centers. If H1 −1  1, the period T1 of motion along closed trajectories around each of these singular points is determined by the asymptotic formula (it is assumed that the sign of y(t) remains intact during this motion)   √ H1 − 1 T1 ≈ 2π 1 + . 12 with the Hamiltonian H(p, q) =

In the opposite limiting case H1  1 (the trajectories are significantly distant from the mentioned centers), we obtain 1 T1 ≈ √ [2H1 + ln H1 ]. H1 Supplement now the dynamic system (1.17) with the linear friction acting on component v1,0 (t):   d v1,0 (t) = µ v21,1 (t) − v21,2 (t) − λv1,0 (t) + f0 , dt d (1.20) v1,1 (t) = −µv1,0 (t)v1,1 (t) − λv1,1 (t), dt d v1,2 (t) = µv1,0 (t)v1,2 (t) − λv1,2 (t). dt Introducing again the dimensionless variables t → t/λ,

v1,0 (t) →

λ v0 (t), µ

v1,2 (t) →

λ v1 (t), µ

v1,1 (t) →

λ v2 (t), µ

we arrive at the system of equations d v0 (t) = v22 (t) − v21 (t) − v0 (t) + R, dt d v1 (t) = v0 (t)v1 (t) − v1 (t), dt d v2 (t) = −v0 (t)v2 (t) − v2 (t), dt where quantity R =

µf0 is the analog of the Reynolds number. λ2

(1.21)

14

Lectures on Dynamics of Stochastic Systems

Dynamic system (1.21) has steady-state solutions that depend now on parameter R, and R = Rcr = 1 is the critical value. For R < 1, the system has the stable steady-state solution v1 = v2 = 0,

v0 = R.

For R > 1, this solution becomes unstable with respect to small disturbances of parameters, and the steady-state regimes v0 = 1,

√ v1 = ± R − 1,

v2 = 0,

(1.22)

become available. Here, we have an element of randomness because component v1 can be either positive or negative, depending on the amplitude of small disturbance. Assume now that all components of the triplet are acted on by random forces. This means that system (1.21) is replaced with the system of equations d v0 (t) = v22 (t) − v21 (t) − v0 (t) + R + f0 (t), dt d v1 (t) = v0 (t)v1 (t) − v1 (t) + f1 (t), dt d v2 (t) = −v0 (t)v2 (t) − v2 (t) + f2 (t). dt

(1.23)

This system describes the motion of a triplet (gyroscope) with the linear isotropic friction, which is driven by the force acting on the instable mode and having both regular (R) and random (f (t)) components. Such a situation occurs, for example, for a liquid moving in the ellipsoidal cavity. For R > 1, dynamic system (1.23) under the action of random disturbances will first reach the vicinity of one of the stable states (1.22), and then, after the lapse of certain time, it will be hoppingly set into the vicinity of the other stable state. Figure 1.5

4 3 5 2 1 0

2

4

6

8

10

12

14

t

−1 −2 − 5

Figure 1.5 Hopping phenomenon simulated from system (1.23) for R = 6 and σ = 0.1 (the solid and dashed lines show components v0 (t) and v1 (t), respectively).

Examples, Basic Problems, Peculiar Features of Solutions

15

shows the results of simulations of this phenomenon for R = 6 and different realizations of random force f (t), whose components were simulated as the Gaussian random processes. (See Sect. 8.3.3, page 200 for the statistical description of this problem.) Thus, within the framework of the dynamics of the first stage, hopping phenomena can occur only due to the effect of external random forces acting on all modes.

Hopping Between Quasisteady-State Regimes The simplest two-stage system can be represented in the form d v1,0 (t) = v21,1 (t) − v21,2 (t) + 1, dt   d v1,1 (t) = −v1,0 (t)v1,1 (t) + Q v22,1 (t) − v22,2 (t) , dt   d v1,2 (t) = −v1,0 (t)v1,2 (t) + Q v22,3 (t) − v22,4 (t) , dt d v2,1 (t) = −Qv1,1 (t)v2,1 (t), dt

d v2,2 (t) = Qv1,1 (t)v2,2 (t), dt

d v2,3 (t) = −Qv1,2 (t)v2,3 (t), dt

d v2,4 (t) = Qv1,2 (t)v2,4 (t), dt

(1.24)

where we used the dimensionless variables similar to (1.18). Only the components v1,0 (t), v1,2 (t), v2,3 (t) and v2,4 (t) survive for f0 > 0 (see Fig. 1.6). These components satisfy the system of equations d v1,0 (t) = −v21,2 (t) + 1, dt   d v1,2 (t) = −v1,0 (t)v1,2 (t) + Q v22,3 (t) − v22,4 (t) , dt d v2,3 (t) = −Qv1,2 (t)v2,3 (t), dt

d v2,4 (t) = Qv1,2 (t)v2,4 (t), dt

f0(t) v1,0 (t) v1,2(t) v2,3(t)

Figure 1.6

v2,4(t)

Diagram of the excited seven-mode HTS.

16

Lectures on Dynamics of Stochastic Systems

that have the integral of motion v2,3 (t)v2,4 (t) = I = const. Introducing notation x(t) = v1,0 (t),

y(t) = v1,2 (t),

z(t) =

v2,4 (t) , v2,3 (t)

we arrive at the system of three equations d x(t) = −y2 (t) + 1, dt   d 1 y(t) = x(t)y(t) + QI − z(t) , dt z(t)

(1.25)

d z(t) = y(t)z(t), dt that describes the behavior of the seven-mode model (1.24). Inclusion of the second stage significantly changes the dynamics of the first stage. Indeed, the initial-values of components v2,3 and v2,4 being arbitrarily small, they nevertheless determine certain value of the integral of motion I, and, by virtue of Eqs. (1.25), variable y will repeatedly change the sign. Consider the case of small values of constant I in more detail. Figure 1.7 shows the numerical √solution of Eqs. (1.25) with the initial conditions x = 0.05, y = 1, and z = 1 at Q = 8 and I = 10−20 . As may be seen, two types of motion are characteristic of system (1.25) for small values of constant I; namely, ‘fast’ motions occur in a small vicinity of either closed trajectory of the Hamiltonian system (1.19) near the plane z = 1, and relatively rare hopping events occur due to the changes of sign 1 of variable y at z ∼ I or z ∼ − . Every such hopping event significantly changes the I parameters of fast motion trajectories of system (1.25), so that the motion of the system acquires the features characteristic of the dynamic systems with strange attractors (see, e.g., [26]). To describe ‘slow’ motions, we introduce variables X and Y by the formulas x(t) = X(t)x1 (t),

y(t) = Y(t)y1 (t),

where (x1 (t), y1 (t)) is a solution to Eqs. (1.19). According to Eqs. (1.25), X(t), Y(t), and z(t) satisfy the equations  i d 1 h 2 X(t) = X (t) − Y 2 (t) y21 (t) + 1 − X(t) , dt x1 (t)   d QI 1 Y(t) = (X(t) − 1) Y(t)x1 (t) + − z(t) , dt y1 (t) z(t) d z(t) = 2QY(t)y1 (t)z(t). dt

Examples, Basic Problems, Peculiar Features of Solutions

17

y log z 20

y 2

2 1

10

1 0.5

0

0

−10

−1

−20

−2

10

50

100

t

−1

−0.5

0

0.5

1

x

−0.5

(a)

(b)

Figure 1.7 (a) Time-dependent behavior of components of system (1.25) (curve 1 for y and curve 2 for log z) and (b) projection of the phase trajectory of system (1.25) on plane (x, y).

√ Averaging these equations in slow varying quantities over the period1 T1 = 2π, we obtain   d d 1 d X(t) = 0, Y(t) = QI − Z(t) , Z(t) = 2QY(t)Z(t), (1.26) dt dt Z(t) dt where Z = e z(t). The system (1.26) has the integral of motion   1 H2 = Y 2 (t) + I + Z(t) , Z(t)

(1.27)

so that we can use the corresponding variables to rewrite it in the Hamiltonian form, as it was done for the system (1.19). The motion of system (1.26) along closed trajectories around steady point (0, 1) is characterized by the half-period T2 (the time between hopping events) given by the formula 2 T2 = Q

ZZ2 Z1

1

s √  2−1 dZ 1 2 r  , s √  = r  K 2−1 √ r + r 1 Q I r + r2 − 1 Z H2 − I +Z Z

This expression for T1 appears to be quite adequate for the solution shown in Fig. 1.7. On closed trajectories of system (1.19) near which this solution passes, the values of Hamiltonian do not exceed 2. Because these trajectories are located in small vicinities of critical points (0, 1) and (0, −1) of this system, we ^ performed averaging assuming that e y1 (t) = 1 and 1/y 1 (t) = 1.

18

Lectures on Dynamics of Stochastic Systems

log Z

Y

20

2

10

1

0

0

−10

−1

−20

−2

2

1

10

50

t

Figure 1.8 Time-dependent behavior of components of system (1.26) (curve 1 for Y and curve 2 for log Z ).

where Z1 and Z2 are the roots of Eq. (1.27) at Y = 0, r = 2H2 /I, and K(z) is the complete elliptic integral of the first kind. For small I, we have   1 4H2 . T2 ≈ √ ln I Q H2

(1.28)

Figure 1.8 shows the numerical solution of Eqs. (1.26) with the initial conditions Y = 1, Z = 1 (they correspond to the initial conditions used earlier for solving Eqs. (1.25)), constants Q and I also being coincident with those used for solving Eqs. (1.25). The comparison of curves in Figs. 1.7 and 1.8 shows that system (1.26) satisfactorily describes hopping events in system (1.25). The values of halfperiod T2 determined by Eq. (1.28) and obtained from the numerical integration of Eqs. (1.25) are 33.54 and 33.51, respectively. We note that system (1.25) has an additional characteristic time T3 ∼ 1/Q, whose meaning is the duration of the hopping event.

1.1.4

Systems with Blow-Up Singularities

The simplest stochastic system showing singular behavior in time is described by the equation commonly used in the statistical theory of waves, d x(t) = −λx2 (t) + f (t), dt

x(0) = x0 ,

where f (t) is the random function of time.

λ > 0,

(1.29)

Examples, Basic Problems, Peculiar Features of Solutions

19

x(t) x0 0

Figure 1.9

t

Typical realization of the solution to Eq. (1.29).

In the absence of randomness ( f (t) = 0), the solution to Eq. (1.29) has the form x(t) =

1 , λ (t − t0 )

t0 = −

1 . λx0

For x0 > 0, we have t0 < 0, and solution x(t) monotonically tends to zero with increasing time. On the contrary, for x0 < 0, solution x(t) reaches −∞ within a finite time t0 = −1/λx0 , which means that the solution becomes singular and shows the blow-up behavior. In this case, random force f (t) has insignificant effect on the behavior of the system. The effect becomes significant only for positive parameter x0 . Here, the solution, slightly fluctuating, decreases with time as long as it remains positive. On reaching sufficiently small value x(t), the impact of force f (t) can cause the solution to hop into the region of negative values of x, where it reaches the value of −∞ within a certain finite time. Thus, in the stochastic case, the solution to problem (1.29) shows the blow-up behavior for arbitrary values of parameter x0 and always reaches −∞ within a finite time t0 . Figure 1.9 schematically shows the temporal realization of the solution x(t) to problem (1.29) for t > t0 ; its behavior resembles a quasi-periodic structure.

1.1.5

Oscillator with Randomly Varying Frequency (Stochastic Parametric Resonance)

In the above stochastic examples, we considered the effect of additive random impacts (forces) on the behavior of systems. The simplest nontrivial system with multiplicative (parametric) impact can be illustrated by the example of stochastic parametric resonance. Such a system is described by the second-order equation d2 x(t) + ω02 [1 + z(t)]x(t) = 0, dt2 (1.30) x(0) = x0 ,

d x(0) = v0 , dt

20

Lectures on Dynamics of Stochastic Systems

TLe ik(L0−x)

L0

T1e ik(L0−x)

e ik(L−x)

ε(x) L

x

ε (x)

T2e − ik(L−x)

x0 L

L0

x

RLe − ik(L−x) (a)

Figure 1.10 layer.

(b)

(a) Plane wave incident on the medium layer and (b) source inside the medium

where z(t) is the random function of time. This equation is characteristic of almost all fields of physics. It is physically obvious that dynamic system (1.30) is capable of parametric excitation, because random process z(t) has harmonic components of all frequencies, including frequencies 2ω0 /n (n = 1, 2, . . .) that exactly correspond to the frequencies of parametric resonance in the system with periodic function z(t), as, for example, in the case of the Mathieu equation.

1.2

Boundary-Value Problems for Linear Ordinary Differential Equations (Plane Waves in Layered Media)

In the previous section, we considered several dynamic systems described by a system of ordinary differential equations with given initial-values. Now, we consider the simplest linear boundary-value problem, namely, the steady one-dimensional wave problem. Assume the layer of inhomogeneous medium occupies part of space L0 < x < L, and let the unit-amplitude plane wave u0 (x) = e−ik(x−L) is incident on this layer from the region x > L (Fig. 1.10a). The wavefield satisfies the Helmholtz equation, d2 u(x) + k2 (x)u(x) = 0, dx2

(1.31)

where k2 (x) = k2 [1 + ε(x)], and function ε(x) describes medium inhomogeneities. We assume that ε(x) = 0, i.e., k(x) = k outside the layer; inside the layer, we set ε(x) = ε1 (x) + iγ , where the real part ε1 (x) is responsible for wave scattering in the medium and the imaginary part γ  1 describes absorption of the wave in the medium. In region x > L, the wavefield has the structure u(x) = e−ik(x−L) + RL eik(x−L) ,

Examples, Basic Problems, Peculiar Features of Solutions

21

where RL is the complex reflection coefficient. In region x < L0 , the structure of the wavefield is u(x) = TL eik(L0 −x) , where TL is the complex transmission coefficient. Boundary conditions for Eq. (1.31) are the continuity conditions for the field and the field derivative at layer boundaries; they can be written as follows i du(x) i du(x) u(L) + = 0. (1.32) = 2, u(L ) − 0 k dx x=L k dx x=L0 Thus, the wavefield in the layer of an inhomogeneous medium is described by the boundary-value problems (1.31), (1.32). Dynamic equation (1.31) coincides in form with Eq. (1.30). Note that the problem under consideration assumes that function ε(x) is discontinuous at layer boundaries. We will call the boundary-value problem (1.31), (1.32) the unmatched boundary-value problem. In such problems, wave scattering is caused not only by medium inhomogeneities, but also by discontinuities of function ε(x) at layer boundaries. If medium parameters (function ε1 (x)) are specified in the statistical form, then solving the stochastic problem (1.31), (1.32) consists in obtaining statistical characteristics of the reflection and transmission coefficients, which are related to the wavefield values at layer boundaries by the relationships RL = u(L) − 1,

TL = u(L0 ),

and the wavefield intensity I(x) = |u(x)|2 inside the inhomogeneous medium. Determination of these characteristics constitutes the subject of the statistical theory of radiative transfer. Note that, for x < L, from (1.31) follows the equality kγ I(x) =

d S(x), dx

where energy-flux density S(x) is determined by the relationship   i d ∗ d ∗ S(x) = u(x) u (x) − u (x) u(x) . 2k dx dx By virtue of boundary conditions, we have S(L) = 1 − |RL |2 and S(L0 ) = |TL |2 . For non-absorptive media (γ = 0), conservation of energy-flux density is expressed by the equality |RL |2 + |TL |2 = 1.

(1.33)

22

Lectures on Dynamics of Stochastic Systems

2

1

0

2.5

5

Δx

Figure 1.11 Dynamic localization phenomenon simulated for two realizations of medium inhomogeneities.

Consider some features characteristic of solutions to the stochastic boundary-value problem (1.31), (1.32). On the assumption that medium inhomogeneities are absent (ε1 (x) = 0) and absorption γ is sufficiently small, the intensity of the wavefield in the medium slowly decays with distance according to the exponential law I(x) = |u(x)|2 = e−kγ (L−x) .

(1.34)

Figure 1.11 shows two realizations of the intensity of a wave in a sufficiently thick layer of medium. These realizations were simulated for two realizations of medium inhomogeneities. The difference between them consists in the fact that the corresponding functions ε1 (x) have different signs in the middle of the layer at a distance of the wavelength. This offers a possibility of estimating the effect of a small medium mismatch on the solution of the boundary problem. Omitting the detailed description of problem parameters, we mention only that this figure clearly shows the prominent tendency of a sharp exponential decay (accompanied by significant spikes toward both higher and nearly zero-valued intensity values), which is caused by multiple reflections of the wave in the chaotically inhomogeneous random medium (the phenomenon of dynamic localization). Recall that absorption is small (γ  1), so that it cannot significantly affect the dynamic localization. The imbedding method offers a possibility of reformulating boundary-value problem (1.31), (1.32) to the dynamic initial-value problem with respect to parameter L (this parameter is the geometrical position of the layer right-hand boundary) by considering the solution to the boundary-value problem as a function of parameter L (see the next lecture). On such reformulation, the reflection coefficient RL satisfies the Riccati equation d ik RL = 2ikRL + ε(L) (1 + RL )2 , dL 2

RL0 = 0,

(1.35)

Examples, Basic Problems, Peculiar Features of Solutions

23

and the wavefield in the medium layer u(x) ≡ u(x; L) satisfies the linear equation ik ∂ u(x; L) = iku(x; L) + ε(L) (1 + RL ) u(x; L), ∂L 2 u(x; x) = 1 + Rx ,

(1.36)

The equation for the reflection coefficient squared modulus WL = |RL |2 for absent attenuation (i.e., at γ = 0) follows from Eq. (1.35)
d ik WL = − ε1 (L) RL − R∗L (1 − WL ) , dL 2

WL0 = 0.

(1.37)

Note that condition WL0 = 1 will be the initial condition to Eq. (1.37) in the case of totally reflecting boundary at L0 . In this case, the wave incident on the layer of a non-absorptive medium (γ = 0) is totally reflected from the layer, i.e., WL = 1. In the general case of arbitrarily reflecting boundary L0 , the steady-state (independent of L) solution WL = 1 corresponding to the total reflection of incident wave formally exists for a half-space (L0 → −∞) filled with non-absorptive random medium, too. This solution, as it will be shown later, is actually realized in the statistical problem with a probability equal to unity. If, in contrast to the above problem, we assume that function k(x) is continuous at boundary x = L, i.e., if we assume that the wave number in the free half-space x > L is equal to k(L), then boundary conditions (1.32) of problem (1.31) will be replaced with the conditions i du(x) i du(x) u(L) + = 0. (1.38) = 2, u(L0 ) − k(L) dx x=L k(L0 ) dx x=L0 We will call the boundary-value problem (1.31), (1.38) the matched boundary-value problem. The field of a point source located in the layer of random medium is described by the similar boundary-value problem for Green’s function of the Helmholtz equation: d2 G(x; x0 ) + k2 [1 + ε(x)]G(x; x0 ) = 2ikδ(x − x0 ), dx2 i dG(x; x0 ) i dG(x; x0 ) G(L; x0 ) + = 0, G(L ; x ) − 0 0 k dx k dx x=L

= 0.

x=L0

Outside the layer, the solution has the form of outgoing waves (Fig. 1.10b) G(x; x0 ) = T1 eik(x−L)

(x ≥ L),

G(x; x0 ) = T2 e−ik(x−L0 )

(x ≤ L0 ).

Note that, for the source located at the layer boundary x0 = L, this problem coincides with the boundary-value problem (1.31), (1.32) on the wave incident on the layer, which yields G(x; L) = u(x; L).

24

Lectures on Dynamics of Stochastic Systems

1.3

Partial Differential Equations

Consider now several dynamic systems (dynamic fields) described by partial differential equations.

1.3.1

Linear First-Order Partial Differential Equations

Diffusion of Density Field Under Random Velocity Field In the context of linear first-order partial differential equations, the simplest problems concern the equation of continuity for the concentration of a conservative tracer and the equation of transfer of a nonconservative passive tracer by random velocity field U(r, t):   ∂ ∂ + U(r, t) ρ(r, t) = 0, ρ(r, 0) = ρ0 (r), (1.39) ∂t ∂r   ∂ ∂ + U(r, t) q(r, t) = 0, q(r, 0) = q0 (r). (1.40) ∂t ∂r The conservative tracer is a tracer whose total mass remains intact Z Z M0 = drρ(r, t) = drρ0 (r)

(1.41)

We can use the method of characteristics to solve the linear first-order partial differential equations (1.39), (1.40). Introducing characteristic curves (particles) d r(t) = U(r, t), dt

r(0) = r0 ,

(1.42)

we can write these equations in the form d ∂U(r, t) ρ(t) = − ρ(t), ρ(0) = ρ0 (r0 ), dt ∂r d q(t) = 0, q(0) = q0 (r0 ). dt

(1.43)

This formulation of the problem corresponds to the Lagrangian description, while the initial dynamic equations (1.39), (1.40) correspond to the Eulerian description. Here, we introduced the characteristic vector parameter r0 in the system of equations (1.42), (1.43). With this parameter, Eq. (1.42) coincides with Eq. (1.1) that describes particle dynamics under random velocity field. The solution of system of equations (1.42), (1.43) depends on initial-value r0 , r(t) = r(t|r0 ),

ρ(t) = ρ(t|r0 ),

which we will isolate in the argument list by the vertical bar.

(1.44)

Examples, Basic Problems, Peculiar Features of Solutions

25

The first equality in Eq. (1.44) can be considered as the algebraic equation in characteristic parameter; the solution of this equation r0 = r0 (r, t) exists because divergence j(t|r0 ) = det k∂ri (t|r0 )/∂r0k k is different from zero. Consequently, we can write the solution of the initial equation (1.39) in the form ρ(r, t) = ρ(t|r0 (r, t)) =

Z

dr0 ρ(t|r0 )j(t|r0 )δ (r(t|r0 ) − r) .

Integrating this expression over r, we obtain, in view of Eq. (1.41), the relationship between functions ρ(t|r0 ) and j(t|r0 ) ρ(t|r0 ) =

ρ0 (r0 ) , j(t|r0 )

(1.45)

and, consequently, the density field can be rewritten in the form of equality ρ(r, t) =

Z

dr0 ρ(t|r0 )j(t|r0 )δ (r(t|r0 ) − r) =

Z

dr0 ρ0 (r0 )δ (r(t|r0 ) − r) (1.46)

that ascertains the relationship between the Lagrangian and Eulerian characteristics. For the position of the Lagrangian particle, the delta-function appeared in the righthand side of this equality is the indicator function (see Lecture 3). For a nondivergent velocity field (div U(r, t) = 0), both particle divergence and particle density are conserved, i.e., j(t|r0 ) = 1,

ρ(t|r0 ) = ρ0 (r0 ),

q(t|r0 ) = q0 (r0 ).

Consider now the stochastic features of solutions to problem (1.39). A convenient way of analyzing dynamics of a random field consists in using topographic concepts. Indeed, in the case of the nondivergent velocity field, temporal evolution of the contour of constant concentration ρ = const coincides with the dynamics of particles under this velocity field and, consequently, coincides with the dynamics shown in Fig. 1.1a. In this case, the area within the contour remains constant and, as it is seen from Fig. 1.1a, the pattern becomes highly indented, which is manifested in gradient sharpening and the appearance of contour dynamics for progressively shorter scales. In the other limiting case of a divergent velocity field, the area within the contour tends to zero, and the concentration field condenses in clusters. One can find examples simulated for this case in papers [17, 18]. These features of particle dynamics disappear on averaging over an ensemble of realizations. Cluster formation in the Eulerian description can be traced using the random velocity field of form (1.3), (1.4), page 4. If vx (t) 6= 0, then concentration field in

26

Lectures on Dynamics of Stochastic Systems

Lagrangian description ρ(t|r0 ) in the particular case of uniform (independent of r) initial distribution ρ0 (r) = ρ0 can be described by the following equation d ρ(t) = −2kvx (t) cos(2kx)ρ(t), dt

ρ(0) = ρ0 ,

which can be rewritten, by virtue of Eq. (1.6), page 6, in the form 1 − e2T(t) tan2 (kx0 ) d ρ(t|r0 ) = −2kvx (t) ρ(t|r0 ), dt 1 + e2T(t) tan2 (kx0 )

(1.47)

where function T(t) is given by Eq. (1.7), page 6. Integrating Eq. (1.47), we obtain the Lagrangian representation of the velocity field in the framework of the model under consideration i h ρ(t|x0 )/ρ0 = e−T(t) cos2 (kx0 ) + eT(t) sin2 (kx0 ) . Eliminating characteristic parameter x0 with the use of equalities sin2 (kx0 ) = cos (kx0 ) =

e−T(t) sin2 (kx(t)) eT(t) cos2 (kx(t)) + e−T(t) sin2 (kx(t)) eT(t) cos2 (kx(t))

2

eT(t) cos2 (kx(t)) + e−T(t) sin2 (kx(t))

, (1.48) ,

following from Eq. (1.6), page 6, we pass to the Eulerial description ρ(r, t)/ρ0 =

1 eT(t) cos2 (kx) + e−T(t) sin2 (kx)

.

(1.49)

Expression (1.49) shows that the density field is low everywhere excluding the π neighborhoods of points kx = n , where ρ(x, t)/ρ0 = e±T(t) and is sufficiently high 2 if random factor T(t) has appropriate sign. Thus, in the problem under consideration, the cluster structure of the density field in the Eulerian description is formed in the neighborhoods of points kx = n

π 2

(n = 0, ±1, ±2, . . .).

Note that Eulerian density field (1.49) averaged over spatial variables is independent of random factor T(t), ρ(x, t)/ρ0 = 1, and the average square of the density mainly grows with time (ρ(x, t)/ρ0 )2 =

 1  T(t) e + e−T(t) . 2

Examples, Basic Problems, Peculiar Features of Solutions

27

1000

10000

1000 ρ /ρ0 + 1

ρ /ρ0 + 1

100 100 t =1 t =4

10

t =2 t =5

10

t =3

t =6 t =9

t =7 t = 10

t=8

1

1 0

0.4

0.8 x

1.2

1.6

0

0.4

0.8 x

1.2

1.6

(a)

(a) 10000 10000 ρ /ρ0 + 1

ρ /ρ0 + 1

1000

100 t = 16 t = 19

10

t = 17 t = 20

1000 100

t = 18

10

1

t = 21 t = 24

t = 22 t = 25

t = 23

0.4

0.8 x

1.2

1 0

0.4

0.8 x (c)

Figure 1.12

1.2

1.6

0

1.6

(d)

Space-time evolution of the Eulerian density field given by Eq. (1.49).

Figure 1.12 shows the Eulerian concentration field 1 + ρ(r, t)/ρ0 and its spacetime evolution calculated by Eq. (1.49) in the dimensionless space-time variables (the density field is added with a unity to avoid the difficulties of dealing with nearly zerovalued concentrations in the logarithmic scale). This figure shows successive patterns of concentration field rearrangement toward narrow neighborhoods of points x ≈ 0 and x ≈ π/2, i.e., the formation of clusters, in which relative density is as high as 103 − 104 , while relative density is practically zero in the whole other space. Note that the realization of the density field passes through the initial homogeneous state at the instants t such that T(t) = 0. As is seen from figures, the lifetimes of such clusters coincide on the order of magnitude with the time of cluster formation. This model provides an insight into the difference between the diffusion processes in divergent and nondivergent velocity fields. In nondivergent (incompressible) velocity fields, particles (and, consequently, density field) have no time for attracting to stable centers of attraction during the lifetime of these centers, and particles slightly fluctuate relative to their initial location. On the contrary, in the divergent (compressible)

28

Lectures on Dynamics of Stochastic Systems

velocity field, lifetime of stable centers of attraction is sufficient for particles to attract to them, because the speed of attraction increases exponentially, which is clearly seen from Eq. (1.49). From the above description, it becomes obvious that dynamic equation (1.39) considered as a model equation describing actual physical phenomena can be used only on finite temporal intervals. A more complete analysis assumes the consideration of the field of tracer concentration gradient p(r, t) = ∇ρ(r, t) that satisfies the equation (repeating indices assume summation) 

 ∂ ∂Uk (r, t) ∂ 2 Uk (r, t) ∂ + U(r, t) pi (r, t) = −pk (r, t) − ρ(r, t) , ∂t ∂r ∂ri ∂ri ∂rk p(r, 0) = p0 (r) = ∇ρ0 (r).

(1.50)

In addition, one should also include the effect of the dynamic molecular diffusion (with the dynamic diffusion coefficient µρ ) that smooths the mentioned sharpen of the gradient; this effect is described by the linear second-order partial differential equation   ∂ ∂ + U(r, t) ρ(r, t) = µρ 1ρ(r, t), ρ(r, 0) = ρ0 (r). (1.51) ∂t ∂r

Diffusion of Magnetic Field in a Random Velocity Field The diffusion of such passive fields as the scalar density (particle concentration) field and the magnetic field is an important problem of the theory of turbulence in magnetohydrodynamics. Here, the basic stochastic equations are the continuity equation (1.39) for density field ρ(r, t) (see previous section) and the induction equation for nondivergent magnetic field H(r, t) (div H(r, t) = 0) [27] ∂ H(r, t) = curl [U(r, t) × H(r, t)] , ∂t

H(r, 0) = H0 (r).

(1.52)

In Eq. (1.52), U(r, t) is the hydrodynamic velocity field and pseudovector C = A × B is the vector product of vectors A and B, Ci = εijk Aj Bk , where εijk is the pseudotensor such that εijk = 0 if indices i, j, and k are not all different and εijk = 1 or εijk = −1, if indices i, j, and k are all different and form cyclic or anticyclic sequence (see, e.g., [28]). The operator curl C(r, t) = [∇ × C(r, t)], is called the vortex of field C(r, t).

curl C(r, t)|i = εijk

∂ Ck (r, t) ∂rj

Examples, Basic Problems, Peculiar Features of Solutions

29

In magnetohydrodynamics, velocity field U(r, t) is generally described by the Navier–Stokes equation complemented with the density of extraneous electromagnetic forces f (r, t) =

1 [curl H(r, t) × H(r, t)]. 4π

Nevertheless we, as earlier, will consider velocity field U(r, t) as random field whose statistical parameters are given. The product of two pseudotensors is a tensor, and, in the case of pseudotensors ε, we have the following equality εilm εjpq = δij δlp δmq + δip δlq δmj + δiq δlj δmp − δij δlq δmp − δip δlj δmq − δiq δlp δmj . (1.53) Setting j = m (repeated indices assume summation), we obtain εilm εmpq = (d − 2)(δip δlq − δiq δlp ),

(1.54)

where d is the dimension of space, so that the above convolution becomes zero in the two-dimensional case. Thus, the double vector product is given by the formula [C × [A × B]]i = εilm εmpq Cl Ap Bq = Cq Ai Bq − Cp Ap Bi .

(1.55)

If fields C, A, and B are the conventional vector fields, Eq. (1.55) assumes the form [C × [A × B]] = (C · B) A − (C · A) B.

(1.56)

∂ , Eq. (1.55) results in the expression ∂r     ∂ ∂ curl [A(r) × B(r)] = · B(r) A(r) − · A(r) B(r). (1.57) ∂r ∂r

In the case operator vector field C = ∇ =

Note that, if vector field A is an operator in Eq. (1.55), A = ∇ = [C(r) × curl B(r)] = Cq (r)

∂ , then we have ∂r

  ∂ ∂ Bq (r) − C(r) · B(r) ∂r ∂r

and, in particular,   1 ∂ 2 ∂ [B(r) × curl B(r)] = B (r) − B(r) · B(r). 2 ∂r ∂r

(1.58)

30

Lectures on Dynamics of Stochastic Systems

Using Eq. (1.57), we can rewrite Eq. (1.52) in the form     ∂ ∂ ∂ + u(r, t) H(r, t) = H(r, t) · u(r, t), ∂t ∂r ∂r H(r, 0) = H0 (r).

(1.59)

R Dynamic system (1.59) is a conservative system, and magnetic field flux dr H(r, t) remains constant during evolution. We are interested in the evolution of magnetic field in space and time from given smooth initial distributions and, in particular, simply homogeneous ones, H0 (r) = H0 . Clearly, at the initial evolutionary stages of the process, the effects of dynamic diffusion are insignificant, and Eq. (1.59) describes namely this case. Further stages of evolution require consideration of the effects of molecular diffusion; these effects are described by the equation     ∂ ∂ ∂ + u(r, t) H(r, t) = H(r, t) · u(r, t) + µH 1H(r, t), (1.60) ∂t ∂r ∂r where µH is the dynamic diffusion coefficient for the magnetic field. Note that, similarly to the case of tracer density [16], velocity filed model (1.3), (1.4), page 4, allows obtaining the magnetic field in an explicit form if molecular diffusion can be neglected. With the use of this model, the induction equation for homogeneous initial condition (1.59) assumes the form   ∂ ∂ H(x, t) + vx (t) sin 2(kx) ∂t ∂x = 2k cos 2(kx) [v(t)Hx (x, t) − vx (t)H(x, t)], H(x, 0) = H0 , from which follows that the x-component of the magnetic field remains constant (Hx (x, t) = Hx0 ), and the existence of this component Hx0 causes the appearance of an additional source of magnetic field in the transverse (y, z)-plane   ∂ ∂ + vx (t) sin 2(kx) H⊥ (r, t) ∂t ∂x = 2k cos 2(kx) [v⊥ (t)Hx0 − vx (t)H⊥ (x, t)], H⊥ (x, 0) = H⊥0 . (1.61) Equation (1.61) is a partial differential equation, and we can solve it using the method of characteristics (the Lagrangian description). The characteristics satisfy the equations d x(t|x0 ) = vx (t) sin 2(kx(t|x0 )), dt d H⊥ (t|x0 ) = 2k cos 2(kx|x0 ) [v⊥ (t)Hx0 − vx (t)H⊥ (t|x0 )], dt x(0|x0 ) = x0 , H⊥ (0|x0 ) = H⊥0 ,

(1.62)

Examples, Basic Problems, Peculiar Features of Solutions

31

where the vertical bar separates the dependence on characteristic parameter x0 . The first equation in Eqs. (1.62) describes particle diffusion, and its solution has the form Eq. (1.6) h i 1 arctan eT(t) tan(kx0 ) , k

x(t|x0 ) =

where function T(t) is given by Eq. (1.7), page 6. The solution of the equation in the magnetic field has the form h i H⊥ (t|x0 ) = e−T(t) cos2 (kx0 ) + eT(t) sin2 (kx0 ) H⊥0 h i + 2k e−T(t) cos2 (kx0 ) + eT(t) sin2 (kx0 ) h i Zt e−T(τ ) cos2 (kx0 ) − eT(τ ) sin2 (kx0 ) × dτ h i2 vx (τ )v⊥ (τ )Hx0 . −T(τ ) cos2 (kx ) + eT(τ ) sin2 (kx ) e 0 0 0 Eliminating now characteristic parameter x0 with the use of Eqs. (1.48), page 26, we pass to the Eulerian description H⊥ (x, t) =

ρ(x, t) H⊥0 ρ0 Zt + 2kHx0 0

h i eT(t)−T(τ ) cos2 (kx) − e−T(t)+T(τ ) sin2 (kx) dτ h i2 vx (τ )v⊥ (τ ), eT(t)−T(τ ) cos2 (kx) + e−T(t)+T(τ ) sin2 (kx) (1.63)

where the density of passive tracer ρ(x, t) is described by Eq. (1.49). Making now the change of integration variables t − τ = λ in Eq. (1.63), we can rewrite it as

H⊥ (x, t) =

ρ(x, t) H⊥0 + 2kHx0 ρ0

Zt 0

h i eT(t)−T(τ ) cos2 (kx) − e−T(t)+T(τ ) sin2 (kx) dλ h i2 eT(t)−T(τ ) cos2 (kx) + e−T(t)+T(τ ) sin2 (kx)

× vx (t − λ)v⊥ (t − λ), where

T(t) − T(τ ) =

Zt τ

dξ vx (ξ ) =

Zt−τ 0

dηvx (t − η) =

Zλ 0

dηvx (t − η).

32

Lectures on Dynamics of Stochastic Systems

Hence, dealing with the one-time statistical characteristics of magnetic field, we can replace vx (t − λ) with vx (λ) in view of stationarity of the velocity field (see Section 4.2.1, page 95) and rewrite Eq. (1.63) in a statistically equivalent form, H⊥ (x, t) =

ρ(x, t) H⊥0 ρ0 h i Zt eT(τ ) cos2 (kx) − e−T(τ ) sin2 (kx) + 2kHx0 dτ h i2 vx (τ )v⊥ (τ ). (1.64) 2 −T(τ ) T(τ ) 2 e cos (kx) + e sin (kx) 0

The first term describes magnetic field clustering like the density field clustering if H⊥0 6= 0. The second term describes the generation of a magnetic field in the transverse (y, z)-plane due to the presence of an initial field Hx0 . At H⊥0 = 0, this term, proportional to the square of the random velocity field, determines the situation. Like the density field, the structure of this field is also clustered. Figures 1.13, 1.14 present the calculated space–time evolution of a realization of energy of the magnetic field generated in the transverse plane, E(x, t) = H2⊥ (x, t), in dimensionless variables (see Problem 8.8, page 226) at H⊥0 = 0 for the same realization of the random process T(t) as that presented previously in Fig. 1.2a. First of all, we note that the total energy of the generated magnetic field concentrated in the segment [0, π/2] increases rapidly with time (Fig. 1.13a). A general space–time structure of the magnetic energy clustering is shown in Fig. 1.13b. This structure was calculated in the following way. The coordinates of points xi are plotted along the x-axis and the time is plotted along the t-axis (at 0.1 steps). The points are marked as white squares (unseen) if they contain less than 1% of the energy available in the entire layer at current t and as black squares if they contain more than 1% of the energy available in the entire layer at the time in question. There are a total of 40,000 points (100 steps in x and 400 steps in time).

20 18 log E(t )

16 14

1,5

12 1 x

10 8

0,5

6 xxx

0

4 0

10

20 t (a)

30

40

0

10

20 t

30

40

(b)

Figure 1.13 (a) Temporal evolution of the total magnetic energy in segment [0, π/2] and (b) cluster structure in the (x, t)-plane.

Examples, Basic Problems, Peculiar Features of Solutions

33

100

40

log E(x, t )

E(x, t) E (t)

60

100

80

20 0 0

0.5

1

1.5

12 11 10 9 8 7 6 5 4 3 0

0.5

1

x

x

(a)

(b)

1.5

Figure 1.14 Dynamics of disappearance of a cluster at point 0 and appearance of a cluster at point π/2. The circles, triangles, and squares mark the curves corresponding to time instants t = 10.4, 10.8, and 11.8, respectively.

A more detailed pattern of the evolution of clusters with time is presented in Fig. 1.14. Figure 1.14a shows the percentage ratio of the generated magnetic energy field contained in a cluster to the total energy in the layer at the time under consideration; Fig. 1.14b shows the dynamics of the flow of magnetic energy perturbations from one boundary of the region to the other.

1.3.2

Quasilinear and Nonlinear First-Order Partial Differential Equations

Consider now the simplest quasilinear equation for scalar quantity q(r, t), which we write in the form 

 ∂ ∂ + U(t, q) q(r, t) = Q (t, q), ∂t ∂r

q(r, 0) = q0 (r),

(1.65)

where we assume for simplicity that functions U(t, q) and Q(t, q) have no explicit dependence on spatial variable r. Supplement Eq. (1.65) with the equation for the gradient p(r, t) = ∇q(r, t), which follows from Eq. (1.65), and the equation of continuity for conserved quantity I(r, t): 

 ∂ ∂ {U(t, q)p(r, t)} ∂Q(t, q) ∂ + U(t, q) p(r, t) + p(r, t) = p(r, t), ∂t ∂r ∂q ∂q (1.66)

∂ ∂ {U(t, q)I(r, t)} = 0. I(r, t) + ∂t ∂r

34

Lectures on Dynamics of Stochastic Systems

From Eqs. (1.66) it follows that Z Z drI(r, t) = drI0 (r).

(1.67)

In terms of characteristic curves determined by the system of ordinary differential equations, Eqs. (1.65) and (1.66) can be written in the form d d r(t) = U (t, q), q(t) = Q (t, q) , r (0) = r0 , q (0) = q0 (r0 ) , dt dt ∂ {U (t, q) p(t)} ∂Q(t, q) ∂q0 (r0 ) d p(t) = − p(t) + p(t), p (0) = , (1.68) dt ∂q ∂q ∂r0 ∂ {U (t, q) p(t)} d I(t) = − I(t), I (0) = I0 (r0 ) . dt ∂q Thus, the Lagrangian description considers the system (1.68) as the initial-value problem. In this description, the two first equations form a closed system that determines characteristic curves. Expressing now characteristic parameter r0 in terms of t and r, one can write the solution to Eqs. (1.65) and (1.66) in the Eulerian description as Z q (r, t) = dr0 q (t|r0 ) j (t|r0 ) δ (r (t|r0 ) − r), Z (1.69) I (r, t) = dr0 I (t|r0 ) j (t|r0 ) δ (r (t|r0 ) − r). The feature of the transition from the Lagrangian description (1.68) to the Eulerian description (1.69) consists in the general appearance of ambiguities, which results in discontinuous solutions. These ambiguities are related to the fact that the divergence– Jacobian



∂

ri (t|r0 ) j (t|r0 ) = det

∂r0k – can vanish at certain moments. Quantities I(t|r0 ) and j(t|r0 ) are not independent. Indeed, integrating I (r, t) in Eq. (1.69) over r and taking into account Eq. (1.67), we see that there is the evolution integral j (t|r0 ) =

I0 (r0 ) , I (t|r0 )

(1.70)

from which follows that zero-valued divergence j(t|r0 ) is accompanied by the infinite value of conservative quantity I(t|r0 ). It is obvious that all these results can be easily extended to the case in which functions U(r, t, q) and Q(r, t, q) explicitly depend on spatial variable r and Eq. (1.65) itself is the vector equation.

Examples, Basic Problems, Peculiar Features of Solutions

35

As a particular physical example, we consider the equation for the velocity field V(r, t) of low-inertia particles moving in the hydrodynamic flow whose velocity field is u(r, t) (see, e.g., [29]) 

 ∂ ∂ + V(r, t) V(r, t) = −λ [V(r, t) − u(r, t)]. ∂t ∂r

(1.71)

We will assume this equation the phenomenological equation. In the general case, the solution to Eq. (1.71) can be non-unique, it can have discontinuities, etc. However, in the case of asymptotically small inertia property of particles (parameter λ → ∞), which is of our concern here, the solution will be unique during reasonable temporal intervals. Note that, in the right-hand side of Eq. (1.71), term F(r, t) = λV(r, t) linear in the velocity field V(r, t) is, according to the known Stokes formula, the resistance force acting on a slowly moving particle. If we approximate the particle by a sphere of radius a, parameter λ will be λ = 6π aη/mp , where η is the dynamic viscosity coefficient and mp is the mass of the particle (see, e.g., [30]). From Eq. (1.71) follows that velocity field V(r, t) is the divergent field (div V(r, t) 6= 0) even if hydrodynamic flow u(r, t) is the nondivergent field (div u(r, t) = 0). As a consequence, particle number density n(r, t) in nondivergent hydrodynamic flows, which satisfies the linear equation of continuity 

 ∂ ∂ + V(r, t) n(r, t) = 0, ∂t ∂r

n(r, 0) = n0 (r),

(1.72)

similar to Eq. (1.39), shows the cluster behavior [31]. For large parameters λ → ∞ (inertia-less particles), we have V(r, t) ≈ u(r, t),

(1.73)

and particle number density n(r, t) in nondivergent hydrodynamic flows shows no cluster behavior. The first-order partial differential equation (1.71) (the Eulerian description) is equivalent to the system of ordinary differential characteristic equations (the Lagrangian description) d r(t) = V (r(t), t), r(0) = r0 , dt d V(t) = −λ [V(t) − u (r(t), t)] , dt

(1.74) V(0) = V 0 (r0 ),

that describe diffusion of a particle under random external force and linear friction and coincide with Eq. (1.12). Conventional statistical description usually assumes that fluctuations of the hydrodynamic velocity field are sufficiently small. For this reason, we can linearize the

36

Lectures on Dynamics of Stochastic Systems

system of equations (1.74) and rewrite it in the form (for simplicity, we assume that the mean flow is absent and use the zero-valued initial conditions) d d r(t) = v(t), v(t) = −λ [v(t) − f (t)], dt dt r(0) = 0, v(0) = 0,

(1.75)

the stochastic solution to which has the form v(t) = λ

Zt

−λ(t−τ )

dτ e

f (τ ),

Zt r(t) =

0

h i dτ 1 − e−λ(t−τ ) f (τ ).

0

Note that the closed linear equation of the first order in velocity v(t) is called the Langevin equation. Now, we turn back to Eq. (1.71). Setting λ = 0, we arrive at the equation 

 ∂ ∂ + V(r, t) V(r, t) = 0, ∂t ∂r

V(r, 0) = V 0 (r),

(1.76)

called the Riemann equation. It describes free propagation of the nonlinear Riemann wave. The solution to this equation obviously satisfies the transcendental equation V(r, t) = V 0 (r − tV(r, t)) . In the special case of the one-dimensional equation 

 ∂ ∂ + q(x, t) q(x, t) = 0, ∂t ∂x

q(x, 0) = q0 (x),

(1.77)

the solution can be drawn in both implicit and explicit form. This equation coincides with Eq. (1.65) at G(t, q) = 0, U(t, q) = q(x, t). The method of characteristics applied to Eq. (1.77) gives q(t|x0 ) = q0 (x0 ),

x(t|x0 ) = x0 + tq0 (x0 ),

(1.78)

so that the solution of Eq. (1.77) can be written in the form of the transcendental equation q(x, t) = q0 (x − tq(x, t)), from which follows an expression for spatial derivative p(x, t) =

q00 (x0 ) ∂ q(x, t) = , ∂x 1 + tq00 (x0 )

(1.79)

Examples, Basic Problems, Peculiar Features of Solutions

37

where x0 = x − tq(x, t) and q00 (x0 ) =

d q0 (x0 ). dx0

The function p(x, t) by itself satisfies here the equation 

 ∂ ∂ + q(x, t) p(x, t) = −p2 (x, t), ∂t ∂x

p(x, 0) = p0 (x) = q00 (x).

(1.80)

For completeness, we give the equation of continuity for the density field ρ(x, t) 

 ∂ ∂ + q(x, t) ρ(x, t) = −p(x, t)ρ(x, t), ∂t ∂x

ρ(x, 0) = ρ0 (x),

(1.81)

and its logarithm χ(x, t) = ln ρ(x, t) 

 ∂ ∂ + q(x, t) χ(x, t) = −p(x, t), ∂t ∂x

χ(x, 0) = χ0 (x),

(1.82)

which are related to the Riemann equation (1.77). The solution to Eq. (1.77) has the form ρ(x, t) =

ρ0 (x0 ) ρ0 (x − tq(x, t)) = . 1 + tp0 (x0 ) 1 + tp0 (x − tq(x, t))

(1.83)

∂ q(x, t) and solution of Eq. (1.77) becomes discon∂x tinuous. For times prior to t0 , the solution is unique and representable in the form of a quadrature. To show this fact, we calculate the variational derivative (for variational derivative definitions and the corresponding operation rules, see Lecture 2) If q00 (x0 ) < 0, then derivative

1 δq (x, t) = δ (x − tq (x, t) − x0 ). δq0 (x0 ) 1 + tq00 (x0 ) Because q(x, t) = q0 (x0 ) and x − tq0 (x0 ) = x0 , the argument of delta function vanishes at x = F(x0 , t) = x0 + tq0 (x0 ). Consequently, we have δq (x, t) 1 = δ(x − F(x0 , t)) = δq0 (x0 ) 2π

Z∞

dk eik(x−x0 )−iktq0 (x0 ) .

−∞

We can consider this equality as the functional equation in variable q0 (x0 ). Then, integrating this equation with the initial value q (x, t)|q0 (x0 )=0 = 0,

38

Lectures on Dynamics of Stochastic Systems

in the functional space, we obtain the solution of the Riemann equation in the form of the quadrature (see e.g., [32]) i q (x, t) = 2πt

Z∞ −∞

dk k

Z∞

h i dξ eik(x−x0 ) e−iktq0 (ξ ) − 1 .

−∞

The mentioned ambiguity can be eliminated by considering the Burgers equation ∂ ∂2 ∂ q(x, t) + q(x, t) q(x, t) = µ 2 q(x, t), ∂t ∂x ∂x

q(x, 0) = q0 (x),

(it includes the molecular viscosity and also can be solved in quadratures) followed by the limit process µ → 0. In the general case, a nonlinear scalar first-order partial differential equation can be written in the form ∂ q(r, t) + H (r, t, q, p) = 0, ∂t

q (r, 0) = q0 (r),

(1.84)

where p(r, t) = ∇q(r, t). In terms of the Lagrangian description, this equation can be rewritten in the form of the system of characteristic equations: ∂ d r(t|r0 ) = H (r, t, q, p), r(0|r0 ) = r0 ; dt ∂p   d ∂ ∂ p(t|r0 ) = − +p H (r, t, q, p), p(0|r0 ) = p0 (r0 ); dt ∂r ∂q   ∂ d q(t|r0 ) = p − 1 H (r, t, q, p), q(0|r0 ) = q0 (r0 ). dt ∂p

(1.85)

Now, we supplement Eq. (1.84) with the equation for the conservative quantity I(r, t)   ∂ ∂H (r, t, q, p) ∂ I(r, t) + I(r, t) = 0, I (r, 0) = I0 (r). (1.86) ∂t ∂r ∂p From Eq. (1.86) follows that Z Z drI(r, t) = drI0 (r).

(1.87)

Then, in the Lagrangian description, the corresponding quantity satisfies the equation ∂ 2 H (r, t, q, p) d I(t|r0 ) = − I(r, t), dt ∂r∂p

I (0|r0 ) = I0 (r0 ) ,

Examples, Basic Problems, Peculiar Features of Solutions

39

so that the solution to Eq. (1.86) has the form I (r, t) = I (t|r0 (t, r)) =

Z

dr0 I (t|r0 ) j (t|r0 ) δ (r (t|r0 ) − r),

(1.88)

where j (t|r0 ) = det ∂ri (t|r0 ) /∂r0j is the divergence (Jacobian). Quantities I(t|r0 ) and j(t|r0 ) are related to each other. Indeed, substituting Eq. (1.88) for I(r, t) in Eq. (1.87), we see that there is the evolution integral j (t|r0 ) =

I0 (r0 ) , I (t|r0 )

and Eq. (1.88) assumes the form I (r, t) =

1.3.3

Z

dr0 I0 (r0 ) δ (r (t|r0 ) − r).

Parabolic Equation of Quasioptics (Waves in Randomly Inhomogeneous Media)

We will describe propagation of a monochromatic wave U(t, x, R) = e−iωt u(x, R) in the media with large-scale three-dimensional inhomogeneities responsible for smallangle scattering on the base of the complex parabolic equation of quasioptics, ∂ i ik u(x, R) = 1R u(x, R) + ε(x, R)u(x, R), ∂x 2k 2

u(0, R) = u0 (R),

(1.89)

where k = ω/c is the wave number, c is the velocity of wave propagation, x-axis is directed along the initial direction of wave propagation, vecor R denotes the coordinates in the transverse plane, 1R = ∂ 2 /∂R2 , and function ε(x, R) is the deviation of the refractive index (or dielectric permittivity) from unity. This equation was successfully used in many problems on wave propagation in Earth’s atmosphere and oceans [33, 34]. Introducing the amplitude–phase representation of the wavefield in Eq. (1.89) by the formula u(x, R) = A(x, R)eiS(x,R) , we can write the equation for the wavefield intensity I(x, R) = u(x, R)u∗ (x, R) = |u(x, R)|2

40

Lectures on Dynamics of Stochastic Systems

in the form ∂ 1 I(x, R) + ∇ R {∇ R S(x, R)I(x, R)} = 0, ∂x k

I(0, R) = I0 (R).

(1.90)

From this equation follows that the power of a wave in plane x = const is conserved in the general case of arbitrary incident wave beam: Z Z E0 = I(x, R)dR = I0 (R)dR. Equation (1.90) coincides in form with Eq. (1.39). Consequently, we can treat it as the transport equation for conservative tracer in the potential velocity field. However, this tracer can be considered the passive tracer only in the geometrical optics approximation, in which case the phase of the wave S(x, R), the transverse gradient of the phase p(x, R) =

1 ∇ R S(x, R), k

and the matrix of the phase second derivatives uij (x, R) =

1 ∂2 S(x, R) k ∂Ri ∂Rj

characterizing the curvature of the phase front S(x, R) = const satisfy the closed system of equations k k ∂ S(x, R) + p2 (x, R) = ε(x, R), ∂x 2 2   ∂ 1 + p(x, R)∇ R p(x, R) = ∇ R ε(x, R), ∂x 2   1 ∂2 ∂ + p(x, R)∇ R uij (x, R) + uik (x, R)ukj (x, R) = ε(x, R). ∂x 2 ∂Ri ∂Rj

(1.91)

In the general case, i.e., with the inclusion of diffraction effects, this tracer is the active tracer. According to the material of Sect. 1.3.1, realizations of intensity must show the cluster behavior, which manifests itself in the appearance of caustic structures. An example demonstrating the appearance of the wavefield caustic structures is given in Fig. 1.15, which is a fragment of the photo on the back of the cover – the flyleaf – of book [33] that shows the transverse section of a laser beam in turbulent medium (laboratory measurements). A photo of the pool in Fig. 1.16 also shows the prominent caustic structure of the wavefield on the pool bottom. Such structures appear due to light refraction and reflection by rough water surface, which corresponds to scattering by the so-called phase screen.

Examples, Basic Problems, Peculiar Features of Solutions

Figure 1.15

Transverse section of a laser beam in turbulent medium.

Figure 1.16

Caustics in a pool.

41

Note that Eq. (1.89) with parameter x concidered as time coincides in form with the Schr¨odinger equation. In a similar way, the nonlinear parabolic equation describing self-action of a harmonic wave field in multidimensional random media, i ik ∂ u(x, R) = 1R u(x, R) + ε(x, R; I(x, R))u(x, R), ∂x 2k 2

u(0, R) = u0 (R),

coincides in form with the nonlinear Schr¨odinger equation. Consequently, clustering of wave field energy must occur in this case too, because Eq. (1.90) is formally independent of the shape of function ε(x, R; I(x, R)).

42

Lectures on Dynamics of Stochastic Systems

Consider now geometrical optics approximation (1.91) for parabolic equation (1.89). In this approximation, the equation for the wave phase is the Hamilton–Jacobi equation and the equation for the transverse gradient of the phase (1.91) is the closed quasilinear first-order partial differential equation, and we can solve it by the method of characteristics. Equations for the characteristic curves (rays) have the form d R(x) = p(x), dx

d 1 p(x) = ∇ R ε(x, R), dx 2

(1.92)

and the wavefield intensity and matrix of the phase second derivatives along the characteristic curves will satisfy the equations d I(x) = −I(x)uii (x), dx d 1 ∂2 uij (x) + uik (x)ukj (x) = ε(x, R). dx 2 ∂Ri ∂Rj

(1.93)

Equations (1.92) coincide in appearance with the equations for a particle under random external forces in the absence of friction (1.12) and form the system of the Hamilton equations. In the two-dimensional case (R = y), Eqs. (1.92), (1.93) become significantly simpler and assume the form d d 1 ∂ y(x) = p(x), p(x) = ε(x, y), dx dx 2 ∂y d d 1 ∂2 I(x) = −I(x)u(x), u(x) + u2 (x) = ε(x, y). dx dx 2 ∂y2

(1.94)

The last equation for u(x) in (1.94) is similar to Eq. (1.29) whose solution shows the singular behavior. The only difference between these equations consists in the random term that has now a more complicated structure. Nevertheless, it is quite clear that solutions to stochastic problem (1.94) will show the blow-up behavior; namely, function u(x) will reach minus infinity and intensity will reach plus infinity at a finite distance. Such a behavior of a wavefield in randomly inhomogeneous media corresponds to random focusing, i.e., to the formation of caustics, which means the appearance of points of multivaluedness (and discontinuity) in the solutions of quasilinear equation (1.91) for the transverse gradient of the wavefield phase.

1.3.4

Navier–Stokes Equation: Random Forces in Hydrodynamic Theory of Turbulence

Consider now the turbulent motion model that assumes the presence of external forces f (r, t) acting on the liquid. Such a model is evidently only imaginary, because there are no actual analogues for these forces. However, assuming that forces f (r, t) on average ensure an appreciable energy income only to large-scale velocity components, we can expect that, within the concepts of the theory of local isotropic turbulence, the

Examples, Basic Problems, Peculiar Features of Solutions

43

imaginary nature of field f (r, t) will only slightly affect statistical properties of smallscale turbulent components [35]. Consequently, this model is quite appropriate for describing small-scale properties of turbulence. Motion of an incompressible liquid under external forces is governed by the Navier–Stokes equation 

 1 ∂ ∂ ∂ + u(r, t) u(r, t) = − p(r, t) + ν1u(r, t) + f (r, t), ∂t ∂r ρ0 ∂r

(1.95)

div f (r, t) = 0.

div u(r, t) = 0,

Here, ρ0 is the density of the liquid, ν is the kinematic viscosity, and pressure field p(x, t) is expressed in terms of the velocity field at the same instant by the relationship Z

1−1 r, r0

p(r, t) = −ρ0



∂ 2 ui (r0 , t)uj (r0 , t) 0 dr , ∂ri0 ∂rj0

(1.96)

where 1−1 (r, r0 ) is the integral operator inverse to the Laplace operator (repeated indices assume summation). Note that the linearized equation (1.95) ∂ u(r, t) = ν1u(r, t) + f (r, t), ∂t

u(r, 0) = 0,

(1.97)

describes the problem on turbulence degeneration for t → ∞. The solution to this problem can be drawn in an explicit form: Zt u(r, t) =

ν(t−τ )1

dτ e

f (r, τ ) =

0

=

=

ντ 1

Z

dτ e

dr0 δ(r − r0 )f (r0 , t − τ )

0

1 (2π)3 1 (2π)3 Z

=

Zt

dr

0

Zt

dτ eντ 1

Z

dr0

Z

dq eiq(r−r ) f (r0 , t − τ ) 0

0

Zt

Z dτ

dr0

Z

2

0

dq e−ντ q eiqr f (r − r0 , t − τ )

0

Zt 0

  r02 dτ exp − f (r − r0 , t − τ ). 4ντ (4πντ )3/2

(1.98)

The linear equation (1.97) is an extension of the Langevin equation in velocity (1.75), page 36 to random fields.

44

Lectures on Dynamics of Stochastic Systems

Neglecting the effect of viscosity and external random forces in Eq. (1.95), we arrive at the equation 1 ∂ u(r, t) + (u(r, t) · ∇) u(r, t) = − ∇p(r, t) ∂t ρ

(1.99)

that describes dynamics of perfect liquid and is called the Euler equation. Using Eq. (1.58), page 29, this equation can be rewritten in terms of the velocity field vortex ω = curl u(r, t) as ∂ ω(r, t) = curl [u(r, t) × ω(r, t)] , ∂t

ω(r, 0) = ω0 (r),

(1.100)

or 

   ∂ ∂ ∂ + u(r, t) ω(r, t) = ω(r, t) · ω(r, t), ∂t ∂r ∂r

ω(r, 0) = ω0 (r).

(1.101)

These equations coincide with Eqs. (1.52) and (1.59), pages 28 and 30, and appeared in the problem on the diffusion of magnetic field. However, this coincidence is only formal, because different boundary conditions to these problems results in drastically different behaviors of fields ω(r, t) and H(r, t) as functions of the velocity field u(r, t). If we substitute Eq. (1.96) in Eq. (1.95) to exclude the pressure field, then we obtain in the three-dimensional case that the Fourier transform of the velocity field with respect to spatial coordinates Z Z 1 b ui (k, t) = drui (r, t)e−ikr , ui (r, t) = dkb ui (k, t)eikr , (2π)3 (b u∗i (k, t) = b ui (−k, t)) satisfies the nonlinear integro-differential equation Z Z i ∂ αβ b ui (k, t) + dk1 dk2 3i (k1 , k2 , k)b uα (k1 , t)b uβ (k2 , t) ∂t 2 − νk2b ui (k, t) = b fi (k, t),

(1.102)

where αβ

3i (k1 , k2 , k) =

1 (2π)

1ij (k) = δij −

 3

kα 1iβ (k) + kβ 1iα (k) δ(k1 + k2 − k),

ki kj k2

(i, α, β = 1, 2, 3),

and b f (k, t) is the spatial Fourier harmonics of external forces, Z Z 1 −ikr b f (k, t) = dr f (r, t)e , f (r, t) = dkb f (k, t)eikr . (2π)3 A specific feature of the three-dimensional hydrodynamic motions consists in the fact that the absence of external forces and viscosity-driven effects is sufficient for energy conservation.

Examples, Basic Problems, Peculiar Features of Solutions

45

It appears convenient to describe the stationary turbulence in terms of the spacetime Fourier harmonics of the velocity field b ui (K) =

Z∞

Z dx

dtui (x, t)e−i(kx+ωt) ,

−∞

1 ui (x, t) = (2π)4

Z∞

Z dk

dωb ui (K)ei(kx+ωt) ,

−∞

where K is the four-dimensional wave vector {k, ω} and field b ui∗ (K) = b ui (−K) because field ui (r, t) is real. In this case, we obtain the equation for component b ui (K) by performing the Fourier transformation of Eq. (1.102) with respect to time: Z Z i αβ 2 4 (iω + νk )b ui (K) + d K 1 d4 K 2 3i (K 1 , K 2 , K)b uα (K 1 )b uβ (K) = b fi (K) , 2 (1.103) where αβ

3i (K 1 , K 2 , K) =

1 αβ 3 (k1 , k2 , k) δ(ω1 + ω2 − ω), 2π i

and b fi (K) are the space-time Fourier harmonics of external forces. The obtained Eq. (1.103) is now the integral (and not integro-differential) nonlinear equation.

Plane Motion Under the Action of a Periodic Force Consider now the two-dimensional motion of an incompressible viscous liquid U(r, t) = {u(r, t), v(r, t)} in plane r = {x, y} under the action of a spatially periodic force directed along the x-axis fx (r, t) = γ sin py (γ > 0). Such a flow is usually called the Kolmogorov flow (stream). The corresponding motion is described by the system of equations ∂u ∂u2 ∂uv 1 ∂P + + =− + ν1u + γ sin py, ∂t ∂x ∂y ρ ∂x ∂v ∂uv ∂v2 1 ∂P + + =− + ν1v, ∂t ∂x ∂y ρ ∂y ∂u ∂v + = 0, ∂x ∂y

(1.104)

where P(r, t) is the pressure, ρ is the density, and ν is the kinematic viscosity. The system of the Navier–Stokes and continuity equations (1.104) has the steadystate solution that corresponds to the laminar flow at constant pressure along the x-axis and has the following form us-s (r) =

γ sin py, νp2

vs-s (r) = 0,

Ps-s (r) = const.

(1.105)

46

Lectures on Dynamics of Stochastic Systems

Introducing scales of length p−1 , velocity p−2 υ −1 γ , and time pνγ −1 and using dimensionless variables, we reduce system (1.104) to the form ∂uv ∂P 1 1 ∂u ∂u2 + + =− + 1u + sin y, ∂t ∂x ∂y ∂x R R ∂v ∂uv ∂v2 ∂P 1 + + =− + 1v, ∂t ∂x ∂y ∂y R ∂u ∂v + = 0, ∂x ∂y where R =

γ ν 2 p3

(1.106)

is the Reynolds number. In these variables, the steady-state solution

has the form

us-s (r) = sin y,

vs-s (r) = 0,

Ps-s (r) = const.

Introducing flow function ψ(r, t) by the relationship u(r, t) =

∂ ψ(r, t), ∂y

v(r, t) = −

∂ ψ(r, t), ∂x

we obtain that it satisfies the equation 

 ∂ 1 ∂ψ ∂1ψ ∂ψ ∂1ψ 1 − 1ψ − + = cos y, ∂t R ∂x ∂y ∂y ∂x R

(1.107)

and ψs-s (r) = − cos y. It was shown [43, 44] that, in the linear problem formulation, the steady-state solution (1.105) corresponding to the laminar flow is unstable with respect to small disturbances for certain values of parameter R. These disturbances rapidly increase in time getting the energy from the flow (1.105); this causes the Reynolds stresses described by the nonlinear terms in Eq. (1.107) to increase, which results in decreasing the amplitude of laminar flow until certain new steady-state flow (called usually the secondary flow) is formed. We represent the hydrodynamic fields in the form u(r, t) = U(y, t) + e u(r, t), v(r, t) = e v(r, t), e(r, t). P(r, t) = P0 + e P(r, t), ψ(r, t) = 9(y, t) + ψ Here, U(y, t) is the new profile of the steady-state flow to be determined together with the Reynolds stresses and the tilde denotes the corresponding ultimate disturbances. Abiding by the cited works, we will consider disturbances harmonic in variable x with

Examples, Basic Problems, Peculiar Features of Solutions

47

wavelength 2π/α (α > 0). The new flow profile U(y, t) is the result of averaging with respect to x over distances of about a wavelength. One can easily see that, for α ≥ 1, the laminar flow (1.105) is unique and stable for all R [44] and instability can appear only for disturbances with α < 1. According to the linear stability theory, we will consider first the nonlinear interaction only between the first harmonic of disturbances and the mean flow and neglect the generation of higher harmonics and their mutual interactions and interaction with the mean flow. We represent all disturbances in the form e ϕ (r, t) = ϕ (1) (y, t)eiαx + ϕ (−1) (y, t)e−iαx ,
e(r, t) , e ϕ (r, t) = e u(r, t), e v(r, t), e P(r, t), ψ where quantity ϕ (−1) (y, t) is the complex conjugated of ϕ (1) (y, t). Then, using this representation in system (1.106) and eliminating quantities e P(r, t) and e u(r, t), we obtain the system of equations in mean flow U(y, t) and disturbances v(1) (y, t) [24, 25]: ! 2 (1) 2 (−1) i 1 ∂ 2U 1 ∂ (−1) ∂ v (1) ∂ v U+ v − v = + sin y, 2 2 2 ∂t α R ∂y R ∂y ∂y (1.108)     2 ∂ 1 (1) (1) (1) ∂ U = 0. − 1v + iα U1v − v ∂t R ∂y2 The second equation in system (1.108) is the known Orr–Sommerfeld equation. A similar system can be derived for the flow function. To examine the stability of the laminar regime (1.105), we set U(y) = sin y in the second equation of system (1.108) to obtain   ∂ 1 − 1v(1) (y, t) + iα sin y[1 + 1]v(1) (y, t) = 0. ∂t R

(1.109)

Representing disturbances v(1) (y, t) in the form v(1) (y, t) =

∞ X

σ t+iny v(1) , n e

(1.110)

n=−∞

and substituting this representation in Eq. (1.109), we arrive at the recurrent system in (1) quantities vn   h i 2 2 α 2 + n2 (1) (1) α + n2 σ + vn + vn−1 α 2 − 1 + (n − 1)2 α R h i (1) − vn+1 α 2 − 1 + (n + 1)2 = 0, n = −∞, . . . , +∞. (1.111)

48

Lectures on Dynamics of Stochastic Systems

The analysis of system (1.111) showed [43, 44] that, under certain restrictions on wave number α and Reynolds number R, positive values of parameter σ can exist, i.e., the solutions are unstable. The corresponding dispersion equation in σ has the √ form of an infinite continued fraction, and the critical Reynolds number is Rcr = 2 for α → 0. In other words, long-wave disturbances along the applied force appear to be most unstable. For this reason, we can consider parameter α as a small parameter of the problem at hand and integrate the Orr–Sommerfeld equation asymptotically. We will not dwell on details of this solution. Note only that the components of eigenvec (1) tor vn of problem (1.111) have different orders of magnitude in parameter α. For  (1) example, all components of vector vn with n = ±2, ±3, . . . will have an order of α 4 at least. As a result, we can confine ourselves to the most significant harmonics with n = 0, ±1, which, in essence, is equivalent to the Galerkin method with the trigonometric coordinate functions. In this case, U(y, t) = U(t) sin y, and the equation in v(1) assumes the form 

 1 ∂ − 1v(1) (y, t) + iαU(t) sin y[1 + 1]v(1) (y, t) = 0. ∂t R

Substituting the expansion v(1) (y, t) =

1 X

iny v(1) n (t)e ,

n=−1

in Eqs. (1.108), we obtain that functions U(t),

(1)

z− (t) =

(1)

(1)

z+ (t) = v1 (t) + v−1 (t),

(1)

(1)

z0 (t) = v0 (t),

v1 (t) − v−1 (t) 2

,

satisfy the system of equations [24, 45] 

 d 1 1 4 + U(t) = − z0 (t)z− (t), dt R R α   2 α d + z0 (t) = αU(t)z− (t), dt R     d 1 α d 1 + z− (t) = U(t)z0 (t), + z+ (t) = 0. dt R 2 dt R

(1.112)

Examples, Basic Problems, Peculiar Features of Solutions

49

Equation in quantity z+ (t) is independent of other equations; as a consequence, the corresponding disturbances can only decay with time. Three remaining equations form the simplest three-component hydrodynamic-type system (see Sect. 1.1.3, page 10). As was mentioned earlier, this system is equivalent to the dynamic system describing the motion of a gyroscope with anisotropic friction under the action of an external moment of force relative to the unstable axis. An analysis of system (1.112) shows √ √ that, for R < Rcr = 2, it yields the laminar regime with U = 1, zi = 0. For R > 2, this regime becomes unstable, and new regime (the secondary flow) is formed that corresponds to the mean flow profile and steady-state Reynolds stresses √ √ 4 R− 2 2 , z0 z− = U= , z+ = 0, R α R2 √ h i2 α (1) R− 2 (1) (1) , v1 = √ v0 , α  1, v0 = √ 2 2 2R 2

R≥

√

2.

Turning back to the dimensional quantities, we obtain √ U(y) = 2υp sin py,

√ R− 2 he cos py. ue vi = − p R γ

(1.113)

Note that the amplitude of the steady-state mean flow is independent of the ampli(1) tude of exciting force. Moreover, quantity v0 can be both positive and negative, depending on the signs of the amplitudes of small initial disturbances. Flow function of the steady-state flow has the form √ i 2 2 (1) h√ ψ1 (x, y) = − cos y − v0 2α sin y cos αx + sin αx . R α Figure 1.17 shows the current lines α cos y +

√ 2α sin y cos αx + sin αx = C

√ (1) of flow (1.113) at R = 2Rcr = 2 2 (v0 > 0). In addition, Fig. 1.17 shows schematically the profile of the mean flow. As distinct from the laminar solution, systems of spatially periodic vortices appear here, and the tilt of longer axes of these vortices is determined by the sign of the derivative of the mean flow profile with respect to y. Flow (1.113) was derived under the assumption that the nonlinear interactions between different harmonics of the disturbance are insignificant in comparison with their interactions with the mean flow. This assumption will hold if flow (1.113) is, in turn, stable with respect to small disturbances. The corresponding stability analysis can be carried out by the standard procedure, i.e., by linearizing the equation for flow function (1.107) relative to flow (1.113) [45]. The analysis shows that flow (1.113) is stable

50

Lectures on Dynamics of Stochastic Systems

y

y

π

π

C = −1

x

π 2

− π 2

−π C = 1 + α

0

C = 1−α

αx

−π C = −1

2√ 3α

Figure 1.17 (1) (v0 > 0).

π

− π 2

C=0

−π

π 2

0

2√ 2α 4√ α

C=0 − 3π 2

2√ 3α

C=0

2√ 2α 4√ α

2√ 3α

√ Mean velocity and current lines of the secondary flow at R = 2Rcr = 2 2

if we restrict ourselves to the harmonics of the types same as in solution (1.113). However, the solution appears to be unstable with respect to small-scale disturbances. In this case, the nonlinear interaction of infinitely small disturbances governs the motion along with the interaction of the disturbances with the mean flow. Moreover, we cannot here content ourselves with a finite number of harmonics in x-coordinate and need to consider the infinite series. As regards the harmonics in y-coordinate, we, as earlier, can limit the consideration to the harmonics with n = 0, ±1.

Problem Problem 1.1 Derive the system of equations which takes into account the whole infinite series of harmonics in x-coordinate and extends the system of gyroscopic-type equations (1.112) to the infinite-dimensional case. Solution Represent the flow function in the form

ψ(x, y, t) = ψ−1 (x, t)e−iy + ψ0 (x, t) + ψ1 (x, t)eiy (ψ1∗ (x, t) = ψ−1 (x, t)),

(1.114)

where ψi (x, t) are the periodic functions in x-coordinate with a period 2π/α. Substituting Eq. (1.114) in Eq. (1.107), neglecting the terms of order α 3 in

Examples, Basic Problems, Peculiar Features of Solutions

interactions of harmonics and the terms of order α 2 in the dissipative terms of harmonics ψ±1 , and introducing new functions ψ+ (x, t) =

ψ−1 + ψ1 , 2

ψ− (x, t) =

ψ−1 − ψ1 , 2i

we obtain the system of equations  1 1 ∂ψ0 ∂ + ψ+ − ψ− =− , ∂t R ∂x 2R   ∂ 1 ∂ψ0 + ψ− + ψ+ = 0, ∂t R ∂x     ∂ 1 ∂2 ∂ψ+ ∂ψ− ψ + 2 ψ − − ψ = 0. 0 − + ∂t R ∂x2 ∂x ∂x 

Its characteristic feature consists in the absence of steady-state solutions periodic in x-coordinate (except the solution corresponding to the laminar flow).

51

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Lecture 2 Solution Dependence on Problem Type, Medium Parameters, and Initial Data

Below, we consider a number of dynamic systems described by both ordinary and partial differential equations. Many applications concerning research of statistical characteristics of the solutions to these equations require the knowledge of solution dependence (generally, in the functional form) on the medium parameters appearing in the equation as coefficients and initial values. Some properties appear common to all such dependencies, and two of them are of special interest in the context of statistical descriptions. We illustrate these dependencies by the example of the simplest problem, namely, the system of ordinary differential equations (1.1) that describes particle dynamics under random velocity field and which we reformulate in the form of the nonlinear integral equation Zt r(t) = r0 +

dτ U(r(τ ), τ ).

(2.1)

t0

The solution to Eq. (2.1) functionally depends on vector field U(r0 , τ ) and initial values r0 , t0 .

2.1

Functional Representation of Problem Solution

2.1.1

Variational (Functional) Derivatives

Recall first the general definition of a functional. One says that a functional is given if a rule is fixed that associates a number to every function from a certain function family. Below, we give some examples of functionals:

(a) F[ϕ(τ )] =

Zt2

dτ a(τ )ϕ(τ ),

t1

Lectures on Dynamics of Stochastic Systems. DOI: 10.1016/B978-0-12-384966-3.00002-7 c 2011 Elsevier Inc. All rights reserved. Copyright

54

Lectures on Dynamics of Stochastic Systems

ϕ (τ) ϕ (τ) + δϕ(τ)

ϕ (τ)

t−

Figure 2.1

Δτ 2

t

t + Δτ 2

τ

To definition of variational derivative.

where a(t) is the given (fixed) function and limits t1 and t2 can be both finite and infinite. This is the linear functional. (b) F[ϕ(τ )] =

Zt2 Zt2

dτ1 dτ2 B(τ1 , τ2 )ϕ(τ1 )ϕ(τ2 ),

t1 t1

where B(t1 , t2 ) is the given (fixed) function. This is the quadratic functional. (c) F[ϕ(τ )] = f (8[ϕ(τ )]) , where f (x) is the given function and quantity 8[ϕ(τ )] is the functional. Estimate the difference between the values of a functional calculated for functions 1τ 1τ ϕ(τ ) and ϕ(τ ) + δϕ(τ ) for t − <τ
(2.2)

1τ

is called the variational (or functional) derivative (see, e.g., [35]). δF[ϕ(τ )] δF[ϕ(τ )] For short, we will use notation instead of . δϕ(t) δϕ(t)dt Note that, if we use function δϕ(τ ) = αδ(τ ), where δ(τ ) is the Dirac delta function, then Eq. (2.2) can be represented in the form of the ordinary derivative δF[ϕ(τ )] d = lim F[ϕ(τ ) + αδ(τ − t)]. α→0 dα δϕ(t)

Solution Dependence on Problem Type, Medium Parameters, and Initial Data

55

The variational derivative of functional F[ϕ(τ )] is again the functional of ϕ(τ ), which depends additionally on point t as a parameter. As a result, this variational derivative will have two types of derivatives; one can differentiate it in the ordinary sense with respect to parameter t and in the functional sense with respect to ϕ(τ ) at point τ = t0 , thus obtaining the second variational derivative of the initial functional   δ δF[ϕ(τ )] δ 2 F[ϕ(τ )] = . δϕ(t0 )δϕ(t) δϕ(t0 ) δϕ(t) The second variational derivative will now be the functional of ϕ(τ ) dependent on two points t and t0 , and so forth. Determine the variational derivatives of functionals (a), (b), and (c). In case (a), we have t+ 1τ 2

δF[ϕ(τ )] = F[ϕ(τ ) + δϕ(τ )] − F[ϕ(τ )] =

Z

dτ a(τ )δϕ(τ ).

t− 1τ 2

If function a(t) is continuous on segment 1τ , then, by the average theorem, Z δF[ϕ(τ )] = a(t0 ) dτ δϕ(τ ), 1τ

where point

t0

  1τ 1τ belongs to segment t − ,t + . Consequently, 2 2

δF[ϕ(τ )] = lim a(t0 ) = a(t). 1τ →0 δϕ(t)

(2.3)

In case (b), we obtain similarly δF[ϕ(τ )] = δϕ(t)

Zt2

dτ [B(τ,t)+B(t, τ )] ϕ(τ )

(t1 < t < t2 ).

t1

Note that function B(τ1 , τ2 ) can always be assumed a symmetric function of its arguments here. In case (c), we have ∂f (8[ϕ(τ )]) δ8[ϕ(τ )] + · · · ∂8 ∂f (8[ϕ(τ )]) = F[ϕ(τ )] + δ8[ϕ(τ )] + · · · ∂8

F[ϕ(τ ) + δϕ(τ )] = f (8[ϕ(τ )]) +

and, consequently, δ ∂f (8[ϕ(τ )]) δ f (8[ϕ(τ )]) = 8[ϕ(τ )]. δϕ(t) ∂8 δϕ(t)

(2.4)

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Lectures on Dynamics of Stochastic Systems

Consider now functional 8[ϕ(τ )] = F1 [ϕ(τ )]F2 [ϕ(τ )]. We have δ8[ϕ(τ )] = F 1 [ϕ(τ ) + δϕ(τ )]F 2 [ϕ(τ ) + δϕ(τ )]−F 1 [ϕ(τ )]F 2 [ϕ(τ )] = F1 [ϕ(τ )]δF2 [ϕ(τ )] + F2 [ϕ(τ )]δF1 [ϕ(τ )] and, consequently, δ δ δ F1 [ϕ(τ )]F2 [ϕ(τ )] = F1 [ϕ(τ )] F2 [ϕ(τ )] + F2 [ϕ(τ )] F1 [ϕ(τ )]. δϕ(t) δϕ(t) δϕ(t) (2.5) We can define the expression for the variational derivative of functional ϕ(τ0 ) with respect to function ϕ(t) by the formal relationship δϕ(τ0 ) = δ(τ0 − t). δϕ(t)

(2.6)

Formula (2.6) can be proved, for example, by considering the linear functional of the form

1

F[ϕ(τ )] = √ 2πσ

Z∞

  (τ − τ0 )2 dτ ϕ(τ ) exp − . 2σ 2

(2.7)

−∞

According to Eq. (2.3), the variational derivative of this functional has the form ( ) 1 δ (t − τ0 )2 F[ϕ(τ )] = √ exp − . δϕ(t) 2σ 2 2πσ

(2.8)

Performing now formal limit process σ → 0 in Eq. (2.7)) and (2.8), we obtain the desired formula (2.6). Moreover, ∂F[ϕ(τ )] δϕ(τ ) ∂F[ϕ(τ )] δF[ϕ(τ )] = = δ(τ − t). δϕ(t) ∂ϕ(τ ) δϕ(t) ∂ϕ(τ )

Solution Dependence on Problem Type, Medium Parameters, and Initial Data

57

Formula (2.6) is very convenient for functional differentiation of functionals explicitly dependent on ϕ(τ ). Indeed, for the quadratic functional (b), we have

δ δϕ(t)

Zt2 Zt2

dτ1 dτ2 B(τ1 , τ2 )ϕ(τ1 )ϕ(τ2 )

t1 t1

(2.5)

Zt2 Zt2

=

dτ1 dτ2 B(τ1 , τ2 )

t1 t1 (2.6)

Zt2

=



δϕ(τ1 ) δϕ(τ2 ) ϕ(τ2 ) + ϕ(τ1 ) δϕ(t) δϕ(t)

dτ [B(t, τ )+B(τ, t)] ϕ(τ )



(t1 < t < t2 ).

t1

Consider the functional

F[ϕ(τ )] =

Zt2

  dϕ(τ ) dτ L τ, ϕ(τ ), dτ

t1

as another example. In this case, δ F[ϕ(τ )] δϕ(t)       dϕ(τ ) dϕ(τ ) t ∂L τ, ϕ(τ ), Z2 ∂L τ, ϕ(τ ),  d  dτ dτ (2.4)  δϕ(τ ) = dτ  +  ∂ϕ(τ ) ∂ ϕ(τ ˙ ) dτ  δϕ(t) t1

(2.6)

Zt2

=

t1

     dϕ(τ ) dϕ(τ ) ∂L τ, ϕ(τ ), ∂L τ, ϕ(τ ),  d  dτ dτ  δ(τ − t) + dτ   ∂ϕ(τ ) ∂ ϕ(τ ˙ ) dτ  

    d ∂ ∂ dϕ(t) = − + L t, ϕ(t), , dt ∂ ϕ(t) ˙ ∂ϕ(t) dt

where ϕ(t) ˙ =

d ϕ(t) if point t belongs to interval (t1 , t2 ). dt

58

Lectures on Dynamics of Stochastic Systems

Just as a function can be expanded in the Taylor series, a functional F[ϕ(τ ) + η(τ )] can be expanded in the functional Taylor series in function η(τ ) for η(τ ) ∼ 0 F[ϕ(τ ) + η(τ )] = F[ϕ(τ )] +

Z∞ dt

−∞

+

δF[ϕ(τ )] η(t) δϕ(t)

Z∞ Z∞

1 2!

dt1 dt2 −∞ −∞

δ 2 F[ϕ(τ )] η(t1 )η(t2 ) + · · · . δϕ(t1 )δϕ(t2 )

(2.9)

Note that the operator expression Z∞ 1+

dtη(t)

−∞

δ 1 + δϕ(t) 2! Z∞

=1+ −∞

Z∞ Z∞

dt1 dt2 η(t1 )η(t2 )

−∞ −∞

δ2 + ··· δϕ(t1 )δϕ(t2 )

 ∞ 2 Z 1  δ  δ + dtη(t) + ··· dtη(t) δϕ(t) 2! δϕ(t)

(2.10)

−∞

can be written shortly as the operator

exp

 ∞ Z 

−∞

 δ  , dtη(t) δϕ(t) 

(2.11)

whose action should be treated precisely in the sense of expansion (2.10). Using this operator, we can rewrite Eq. (2.9) in the form R∞

F[ϕ(τ ) + η(τ )] = e−∞

δ dtη(t) δϕ(t)

F[ϕ(τ )],

(2.12)

which enables us to interpret operator (2.11) as the functional shift operator. Consider now functional F[t; ϕ(τ )] dependent on parameter t. We can differentiate this functional with respect to t and determine its variational derivative with respect to ϕ(t0 ), as well. One can easily see that these operations commute, i.e., the equality ∂ δF[t; ϕ(τ )] δ ∂F[t; ϕ(τ )] = 0 ∂t δϕ(t ) δϕ(t0 ) ∂t

(2.13)

holds. If the domain of τ is independent of t, the validity of Eq. (2.13) is obvious. Otherwise, for example, for functionals F[t; ϕ(τ )] with 0 ≤ τ ≤ t, the validity of Eq. (2.13) can be checked on by expanding functional F[t; ϕ(τ )] in the functional Taylor series.

Solution Dependence on Problem Type, Medium Parameters, and Initial Data

2.1.2

59

Principle of Dynamic Causality

Vary Eq. (2.1) with respect to field U (r, t). Assuming that the initial position r0 is independent of field U, we obtain the equation linear in variational derivative (the linear variational differential equation)



δri (t) = δij δ r − r t0 θ t0 − t0 θ t − t0 + δUj (r, t0 )

Zt dτ t0

∂Ui (r (τ ) , τ ) δrk (τ ) , ∂rk δUj (r, t0 ) (2.14)


where δ r − r0 is the Dirac delta function, and θ (z) is the Heaviside step function. From Eq. (2.14) follows that δri (t) =0 δUj (r, t0 )

for t0 > t or t0 < t0 ,

(2.15)

which
means that solution r(t)
to the dynamic problem (2.1) as a functional of field U r, t0 depends on U r, t0 only
for t0 < t0 < t. Consequently, function r(t) will remain unchanged if field U r, t0 varies outside the interval (t0 , t), i.e., for t0 < t0 or t0 > t. We will call condition (2.15) the dynamic causality condition. Taking this condition into account, we can rewrite Eq. (2.14) in the form



δri (t) = δij δ r − r t0 θ t0 − t0 θ t − t0 δUj (r, t0 ) Zt ∂Ui (r (τ ) , τ ) δrk (τ ) + dτ . ∂rk δUj (r, t0 )

(2.16)

t0

As a consequence, limit t → t0 + 0 yields the equality

δri (t) = δij δ r − r t0 . 0 δUj (r, t ) t=t0 +0

(2.17)

Integral equation (2.16) in variational derivative is obviously equivalent to the linear differential equation with the initial value     δri (t) ∂Ui (r (t) , t) δrk (t) ∂ = , ∂t δUj (r, t0 ) ∂rk δUj (r, t0 ) (2.18) δri (t) δU (r, t0 ) j



= δij δ r − r t0 .

t=t0

The dynamic causality condition is the general property of problems described by differential equations with initial values. The boundary-value problems possess no

60

Lectures on Dynamics of Stochastic Systems

such property. Indeed, in the case of problem (1.31), (1.32), page 20–21 that describes propagation of a plane wave in a layer of inhomogeneous medium, wavefield u(x) at point x and reflection and transmission coefficients depend functionally on function ε(x) for all x of layer (L0 , L). However, using the imbedding method, we can convert this problem into the initial-value problem with respect to an auxiliary parameter L and make use the causality property in terms of the equations of the imbedding method.

2.2

Solution Dependence on Problem’s Parameters

2.2.1

Solution Dependence on Initial Data

Here, we will use the vertical bar symbol to isolate the dependence of solution r(t) to Eq. (2.1) on the initial parameters r0 and t0 : r (t) = r (t|r0 , t0 ) , r0 = r (t0 |r0 , t0 ) . Let us differentiate Eq. (2.1) with respect to parameters r0k and t0 . As a result, we ∂ ∂ ri (t|r0 , t0 ) and quantity ri (t|r0 , t0 ) obtain linear equations for Jacobi’s matrix ∂r0k ∂t0 ∂ri (t|r0 , t0 ) = δik + ∂r0k

Zt dτ t0

∂Ui (r (τ ) , τ ) ∂rj (τ |r0 , t0 ) , ∂rj ∂r0k

∂ri (t|r0 , t0 ) = −Ui (r0 (t0 ), t0 ) + ∂t0

(2.19)

Zt dτ t0

∂Ui (r (τ ) , τ ) ∂rj (τ |r0 , t0 ) . ∂rj ∂t0

Multiplying now the first of these equations by Uk (r0 (t) , t), summing over index k, adding the result to the second equation, and introducing the vector function Fi (t|r0 , t0 ) =



∂ ∂ + U (r0 , t0 ) ∂t0 ∂r0



ri (t|r0 , t0 ) ,

we obtain that this function satisfies the linear homogeneous equation Fi (t|r0 , t0 ) =

Zt dτ t0

∂Ui (r (τ ) , τ ) Fk (τ |r0 , t0 ) . ∂rk

(2.20)

Differentiating this equation with respect to time, we arrive at the ordinary differential equation ∂ ∂Ui (r (t) , t) Fi (t|r0 , t0 ) = Fk (t|r0 , t0 ) ∂t ∂rk

Solution Dependence on Problem Type, Medium Parameters, and Initial Data

61

with the initial condition Fi (t0 |r0 , t0 ) = 0 at t = t0 , which follows from Eq. (2.20); as a consequence, we have Fi (t|r0 , t0 ) ≡ 0. Therefore, we obtain the equality   ∂ ∂ + U (r0 , t0 ) ri (t|r0 , t0 ) = 0, (2.21) ∂t0 ∂r0 which can be considered as the linear partial differential equation with the derivatives with respect to variables r0 , t0 and the initial value at t0 = t r (t|r0 , t) = r0 .

(2.22)

The variable t appears now in problem (2.21), (2.22) as a parameter. Equation (2.21) is solved using the time direction inverse to that used in solving problem (1.1), pages 50–51; for this reason, we will call it the backward equation. Equation (2.21) with the initial condition (2.22) can be rewritten as the integral equation Zt

r(t|r0 , t0 ) = r0 +

t0

  ∂ r(t|r0 , τ ). dτ U (r0 , τ ) ∂r0

(2.23)

Varying now Eq. (2.23) with respect to function Uj (r0 , t0 ), we obtain the integral equation 0 δri (t|r0 , t0 ) 0 0 0 ∂ri (t|r0 , t ) = δ(r − r )θ(t − t )θ(t − t ) 0 0 δUj (r0 , t0 ) ∂rj0   Zt ∂ δri (t|r0 , τ ) + dτ U (r0 , τ ) , ∂r0 δUj (r0 , t0 )

(2.24)

t0

from which follows that δri (t|r0 , t0 ) = 0, δUj (r0 , t0 )

if t0 > t

or t0 < t0 ,

which means that function r(t|r0 , t0 ) also possesses the property of dynamic causality with respect to parameter t0 (2.15) – which is quite natural – and Eq. (2.24) can be rewritten in the form (for t0 < t0 < t) ∂ri (t|r0 , t0 ) δri (t|r0 , t0 ) = δ(r0 − r0 ) + 0 0 δUj (r , t ) ∂rj0

Zt0 t0

  δri (t|r0 , τ ) ∂ dτ U (r0 , τ ) . ∂r0 δUj (r0 , t0 ) (2.25)

Setting now t0 → t0 + 0, we obtain the equality δri (t|r0 , t0 ) ∂ri (t|r0 , t0 ) = δ (r0 − r) . δUj (r, t0 ) t0 =t0 +0 ∂r0j

(2.26)

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Lectures on Dynamics of Stochastic Systems

2.2.2

Imbedding Method for Boundary-Value Problems

Consider first boundary-value problems formulated in terms of ordinary differential equations. The imbedding method (or invariant imbedding method, as it is usually called in mathematical literature) offers a possibility of reducing boundary-value problems at hand to the evolution-type initial-value problems possessing the property of dynamic causality with respect to an auxiliary parameter. The idea of this method was first suggested by V.A. Ambartsumyan (the so-called Ambartsumyan invariance principle) [36–38] for solving the equations of linear theory of radiative transfer. Further, mathematicians grasped this idea and used it to convert boundary-value (nonlinear, in the general case) problems into evolution-type initialvalue problems that are more convenient for simulations. Several monographs (see, e.g., [39–41]) deal with this method and consider both physical and computational aspects. Consider the dynamic system described in terms of the system of ordinary differential equations d x(t) = F (t, x(t)) , dt

(2.27)

defined on segment t ∈ [0, T] with the boundary conditions gx(0) + hx(T) = v,

(2.28)

where g and h are the constant matrixes. Dynamic problem (2.27), (2.28) possesses no dynamic causality property, which means that the solution x(t) to this problem at instant t functionally depends on external forces F (τ, x(τ )) for all 0 ≤ τ ≤ T. Moreover, even boundary values x(0) and x(T) are functionals of field F (τ, x(τ )). The absence of dynamic causality in problem (2.27), (2.28) prevents us from using the known statistical methods of analyzing statistical characteristics of the solution to Eq. (2.27) if external force functional F (t, x) is the random space- and time-domain field. Introducing the one-time probability density P(t; x) of the solution to Eq. (2.27), we can easily see that condition (2.28) is insufficient for determining the value of this probability at any point. The boundary condition imposes only certain functional restriction. Note that the solution to problem (2.27), (2.28) parametrically depends on T and v, i.e., x(t) = x(t; T, v). Abiding by paper [42], we introduce functions R(T, v) = x(T; T, v),

S(T, v) = x(0; T, v)

that describe the boundary values of the solution to Eq. (2.27). Differentiate Eq. (2.27) with respect to T and v. We obtain two linear equations in the corresponding derivatives d ∂xi (t; T, v) ∂Fi (t, x) ∂xl (t; T, v) = , dt ∂T ∂xl ∂T d ∂xi (t; T, v) ∂Fi (t, x) ∂xl (t; T, v) = . dt ∂vk ∂xl ∂vk

(2.29)

Solution Dependence on Problem Type, Medium Parameters, and Initial Data

63

These equations are identical in form; consequently, we can expect that their solutions are related by the linear expression ∂xi (t; T, v) ∂xi (t; T, v) = λk (T, v) , ∂T ∂vk

(2.30)

if vector quantity λ(T, v) is such that boundary conditions (2.28) are satisfied and the solution is unique. To determine vector quantity λ(T, v), we first set t = 0 in Eq. (2.30) and multiply the result by matrix g; then, we set t = T and multiply the result by matrix h; and, finally, we combine the obtained expressions. Taking into account Eq. (2.28), we obtain ∂x(0; T, v) ∂x(t; T, v) g +h = λ(T, v). ∂T ∂T t=T In view of the fact that ∂x(T; T, v) ∂x(t; T, v) ∂R(T, v) ∂x(t; T, v) = − = − F (T, R(T, v)) ∂T ∂T ∂t ∂T t=T t=T (with allowance for Eq. (2.27)), we obtain the desired expression for quantity λ(T, v), λ(T, v) = −hF (T, R(T, v)) .

(2.31)

Expression (2.30) with parameter λ(T, v) defined by Eq. (2.31), i.e., the expression ∂xi (t; T, v) ∂xi (t; T, v) = −hkl Fl (T, R(T, v)) , ∂T ∂vk

(2.32)

can be considered as the linear differential equation; one needs only to supplement it with the corresponding initial condition x(t; T, v)|T=t = R(t, v) assuming that function R(T, v) is known. The equation for this function can be obtained from the equality ∂x(t; T, v) ∂x(t; T, v) ∂R(T, v) = + . ∂T ∂t ∂T t=T t=T

(2.33)

The right-hand side of Eq. (2.33) is the sum of the right-hand sides of Eq. (2.27) and (2.30) at t = T. As a result, we obtain the closed nonlinear (quasilinear) equation ∂R(T, v) ∂R(T, v) = −hkl Fl (T, R(T, v)) + F (T, R(T, v)). ∂T ∂vk

(2.34)

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Lectures on Dynamics of Stochastic Systems

The initial condition for Eq. (2.34) follows from Eq. (2.28) for T → 0 R(T, v)|T=0 = (g + h)−1 v.

(2.35)

Setting now t = 0 in Eq. (2.29), we obtain for the secondary boundary quantity S(T, v) = x(0; T, v) the equation ∂S(T, v) ∂S(T, v) = −hkl Fl (T, R(T, v)) ∂T ∂vk

(2.36)

with the initial condition S(T, v)|T=0 = (g + h)−1 v following from Eq. (2.35). Thus, the problem reduces to the closed quasilinear equation (2.34) with initial value (2.35) and linear equation (2.30) whose coefficients and initial value are determined by the solution of Eq. (2.34). In the problem under consideration, input 0 and output T are symmetric. For this reason, one can solve it not only from T to 0, but also from 0 to T. In the latter case, functions R(T, v) and S(T, v) switch the places. An important point consists in the fact that, despite the initial problem (2.27) is nonlinear, Eq. (2.30) is the linear equation, because it is essentially the equation in variations. It is Eq. (2.34) that is responsible for nonlinearity. Note that the above technique of deriving imbedding equations for Eq. (2.27) can be easily extended to the boundary condition of the form [42] g (x(0)) + h (x(T)) +

ZT

dτ K (τ, x(τ )) = v,

0

where g (x), h (x) and K (T, x) are arbitrary given vector functions. If function F (t, x) is linear in x, Fi (t, x) = Aij (t)xj (t), then boundary-value problem (2.27), (2.28) assumes the simpler form d x(t) = A (t) x(t), gx(0) + hx(T) = v, dt and the solution of Eq. (2.30), (2.34) and (2.36) will be the function linear in v x(t; T, v) = X(t; T)v.

(2.37)

As a result, we arrive at the closed matrix Riccati equation for matrix R(T) = X(T; T) d R(T) = A(T)R(T) − R(T)hA(T)R(T), R(0) = (g + h)−1 . (2.38) dT As regards matrix X(t, T), it satisfies the linear matrix equation with the initial condition ∂ X(t; T) = −X(t; T)hA(T)R(T), ∂T

X(t; T)T=t = R(t).

(2.39)

Solution Dependence on Problem Type, Medium Parameters, and Initial Data

Problems Problem 2.1 Helmholtz equation with unmatched boundary. Derive the imbedding equations for the stationary wave boundary-value problem  2  d 2 + k (x) u(x) = 0, dx2     d d + ik1 u(x) = 0, − ik0 u(x) = −2ik0 . dx dx x=L0 x=L Reformulate this boundary-value problem as the initial-value in ∂ u(x; L) terms of functions u(x) = u(x; L) and v(x; L) = ∂x Instruction

d d u(x; L) = v(x; L), v(x; L) = −k2 (x)u(x; L), dx dx v(L0 ; L) + ik1 u(L0 ; L) = 0, v(L; L) − ik0 u(L; L) = −2ik0 .

Solution

  i i h 2 ∂ u(x; L) = ik0 + k (L) − k02 u(L; L) u(x; L), ∂L 2k0 u(x; L)|L=x = u(x; x), i d i h 2 u(L; L) = 2ik0 [u(L; L) − 1] + k (L) − k02 u(L; L)2 , dL 2k0 2k0 u(L0 ; L0 ) = . k0 + k1

Problem 2.2 Helmholtz equation with matched boundary. Derive the imbedding equations for the stationary wave boundary-value problem  d2 2 + k (x) u(x; L) = 0, dx2   d + ik1 u(x; L) = 0, dx x=L0





 d − ik(L) u(x; L) = −2ik(L). dx x=L

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Solution

  ∂ 1 k0 (L) [2 − u(L; L)] u(x; L), u(x; L) = ik(L) + ∂L 2 k(L) u(x; L)|L=x = u(x; x), 1 k0 (L) d [2 − u(L; L)] u(L; L), u(L; L) = ik(L) [u(L; L) − 1] + dL 2 k(L) 2k(L0 ) u(L0 ; L0 ) = . k(L0 ) + k1

Problem 2.3 Derive the imbedding equations for the boundary-value problem  d2 ρ 0 (x) d 2 + p [1 + ε(x)] u(x) = 0, − ρ(x) dx dx2     1 d 1 d + ip u(x) = 0, − ip u(x) = −2ip, ρ(x) dx ρ(x) dx x=L0 x=L



dρ(x) dx Solution The imbedding equations with respect to parameter L have the form where ρ 0 (x) =

∂ u(x; L) = {ipρ(L) + ϕ(L)u(L; L)} u(x; L), ∂L u(x; L)|L=x = u(x; x); d u(L; L) = 2ipρ(L) [u(L; L) − 1] + ϕ(L)u2 (L; L), dL u(L; L)|L=L0 = 1, where ϕ(x) =

i ip h 1 + ε(x) − ρ 2 (x) . 2ρ(x)

Remark The above boundary-value problem describes many physical processes,

such as acoustic waves in a medium with variable density and some types of electromagnetic waves. In these cases, function ε(x) describes inhomogeneities of the velocity of wave propagation (refractive index or dielectric permittivity) and function ρ(x) appears only in the case of acoustic waves and characterize medium density.

Solution Dependence on Problem Type, Medium Parameters, and Initial Data

Problem 2.4 Derive the imbedding equations for the matrix Helmholtz equation  d d2 + γ (x) + K(x) U(x) = 0, dx dx2     d d + B U(x) = D, +C U(x) = 0, dx dx x=L x=0



where γ (x), K(x), and U(x) are the variable matrixes, while B, C, and D are the constant matrixes. Solution The imbedding equations with respect to parameter L have the form

∂ U(x; L) = U(x; L)3(L), U(x; L)|L=x = U(x; x), ∂L n o d U(L; L) = D − BU(L; L) + U(L; L)D−1 BD + U(L; L)D−1 γ (L)D dL n o + U(L; L)D−1 K(L) − γ (L)B + B2 U(L; L),

U(0; 0) = (B − C)−1 D, where matrix 3(L) is given by the formula h i 3(L) = D−1 [γ (L) − B] D + D−1 K(L) + B2 − γ (L)B U(L; L).

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Lecture 3 Indicator Function and Liouville Equation

Modern apparatus of the theory of random processes is able of constructing closed descriptions of dynamic systems if these systems meet the condition of dynamic causality and are formulated in terms of linear partial differential equations or certain types of integral equations. One can use indicator functions to perform the transition from the initial, generally nonlinear system to the equivalent description in terms of the linear partial differential equations. However, this approach results in increasing the dimension of the space of variables. Consider such a transition in the dynamic systems described in the Lecture 1 as examples.

3.1

Ordinary Differential Equations

Let a stochastic problem is described by the system of equations (1.1), which we’ll rewrite in the form d r(t) = U (r(t), t), dt

r(t0 ) = r0 .

(3.1)

We introduce the scalar function ϕ(r, t) = δ(r(t) − r),

(3.2)

which is concentrated on the section of the random process r(t) by a given plane r(t) = r and is usually called the indicator function. Differentiating Eq. (3.2) with respect to time t, we obtain, using Eq. (3.1), the equality ∂ dr(t) ∂ ∂ ϕ(r, t) = − δ(r(t) − r) = −U (r(t), t) δ(r(t) − r) ∂t dt ∂r ∂r ∂ = − [U (r(t), t) δ(r(t) − r)] . ∂r Using then the probing property of the delta-function U (r(t), t) δ(r(t) − r) = U (r, t) δ(r(t) − r), Lectures on Dynamics of Stochastic Systems. DOI: 10.1016/B978-0-12-384966-3.00003-9 c 2011 Elsevier Inc. All rights reserved. Copyright

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we obtain the linear partial differential equation 

 ∂ ∂ + U(r, t) ϕ(r, t) = 0, ∂t ∂r

ϕ(r, t0 ) = δ(r0 − r),

(3.3)

equivalent to the initial system. This equation is called the Liouville equation. In terms of phase points moving in phase space {r}, this equation corresponds to the equation of continuity. The above derivation procedure of the Liouville equation clearly shows the necessity of distinguishing the function r(t) and the parameter r. In this connection, correct working with the delta-function assumes that the argument of function r(t) was explicitly written in all intermediate calculations (cf. the forms of Eqs. (3.1) and (1.1), page 3). Neglecting this rule usually results in errors. The transition from system (3.1) to Liouville equation (3.3) entails enlargement of the phase space (r, t), which, however, has a finite dimension. Note that Eq. (3.3) coincides in form with the equation of tracer transfer by the velocity field U(r, t) (1.39); the only difference consists in the initial values. Equation (3.3) can be rewritten in the form of the linear integral equation ϕ(r, t) = δ(r − r0 ) −

Zt dτ t0

∂ {U(r, τ )ϕ(r, τ )}. ∂r

Varying this equation with respect to function Ui (r0 , t0 ) with the use of Eq. (2.6), page 56, we obtain the integral equation δϕ(r, t) =− δUj (r0 , t0 )

Zt dτ t0

∂ ∂ri



 Zt   δUi (r, τ ) ∂ δϕ(r, τ ) ϕ(r, τ ) − dτ U(r, τ ) δUj (r0 , t0 ) ∂r δUj (r0 , t0 ) t0



∂  = −θ(t − t0 )θ(t0 − t0 ) δ r − r0 ϕ r, t0 ∂rj t   Z δϕ(r, τ ) ∂ − dτ U(r, τ ) , ∂r δUj (r0 , t0 ) t0

where θ(t) is, as earlier, the Heaviside step function. As a result, we have δϕ(r, t) ∼ θ(t − t0 )θ(t0 − t0 ), δUj (r0 , t0 ) and the variational derivative satisfies the condition δϕ(r, t) = 0, δUj (r0 , t0 )

if t0 < t0

or t0 > t,

(3.4)

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71

which expresses the condition of dynamic causality for the Liouville equation (3.3). For t0 < t0 < t, Eq. (3.4) can be written in the form

∂  δϕ(r, t) δ r − r0 ϕ r, t0 − =− 0 0 δUj (r , t ) ∂rj

Zt t0

  ∂ δϕ(r, τ ) dτ U(r, τ ) , ∂r δUj (r0 , t0 )

(3.5)

from which follows that
δϕ(r, t) ∂  =− δ r − r0 ϕ(r, t0 ) . 0 0 δUi (r , t ) t=t0 +0 ∂ri

(3.6)

Note that equality (3.6) can be derived immediately from Eq. (2.17), page 59. Indeed, by definition of function ϕ(r, t), variational derivative δϕ(r, t)/δUi (r0 , t0 ) has the form δ ∂ δri (t) δϕ(r, t) = δ(r(t) − r) = − δ(r(t) − r) . δUj (r0 , t0 ) δUj (r0 , t0 ) ∂ri δUj (r0 , t0 )

(3.7)

As a consequence, for t = t0 + 0, we have the equality δϕ(r, t) ∂ δri (t) 0 = − δ(r(t ) − r) δUj (r0 , t0 ) t=t0 +0 ∂ri δUj (r0 , t0 ) t=t0 +0

∂  ∂  =− δ(r(t0 ) − r)δ r0 − r = − ϕ(r, t0 )δ r0 − r . ∂rj ∂rj The solution to Eq. (3.1) and, consequently, function (3.2) depends on the initial values t0 , r0 . Indeed, function r(t) = r(t|r0 , t0 ) as a function of variables r0 and t0 satisfies the linear first-order partial differential equation (1.105). The equations of such type also allow the transition to the equations in the indicator function ϕ(r, t|r0 , t0 ) (see the next section); in the case under consideration, this equation is again the linear first-order partial differential equation including the derivatives with respect to variables r0 and t0 

∂ ∂ + U(r0 , t0 ) ∂t0 ∂r0



ϕ(r, t|r0 , t0 ) = 0,

ϕ(r, t|r0 , t) = δ(r0 − r).

(3.8)

Equation (3.8) can be called the backward Liouville equation. At the same time, as a consequence of the dynamic causality, the variational derivative of indicator function ϕ(r, t|r0 , t0 ), δϕ(r, t|r0 , t0 ) δ ∂ δri (t|r0 , t0 ) = δ(r(t|r0 , t0 ) − r) = − δ(r(t|r0 , t0 ) − r) , δUj (r0 , t0 ) δUj (r0 , t0 ) ∂ri δUj (r0 , t0 )

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can be expressed in view of Eq. (2.26), page 61, as follows ∂ δri (t|r0 , t0 ) δϕ(r, t|r0 , t0 ) = − δ(r(t|r0 , t0 ) − r) δUj (r0 , t0 ) t0 =t0 +0 ∂ri δUj (r0 , t0 ) t0 =t0 +0 = −δ(r0 − r0 )

∂ri (t|r0 , t0 ) ∂ϕ(r, t|r0 , t0 ) . ∂r0j ∂ri

(3.9)

On the other hand, ∂ϕ(r, t|r0 , t0 ) ∂ri (t|r0 , t0 ) ∂ϕ(r, t|r0 , t0 ) =− , ∂r0j ∂r0j ∂ri and Eq. (3.9) assumes the final closed form ∂ϕ(r, t|r0 , t0 ) δϕ(r, t|r0 , t0 ) = δ(r0 − r0 ) . δUj (r0 , t0 ) t0 =t0 +0 ∂r0j

3.2

(3.10)

First-Order Partial Differential Equations

If the initial-value problem is formulated in terms of partial differential equations, we always can convert it to the equivalent formulation in terms of the linear variational differential equation in the infinite-dimensional space (the Hopf equation). For some particular types of problems, this approach is simplified. Indeed, if the initial dynamic system satisfies the first-order partial differential equation (either linear as Eq. (1.39), page 24, or quasilinear as Eq. (1.65), page 33, or in the general case nonlinear as Eq. (1.84), page 38), then the phase space of the corresponding indicator function will be the finite-dimension space, which follows from the fact that first-order partial differential equations are equivalent to systems of ordinary (characteristic) differential equations. Consider these cases in more detail.

3.2.1

Linear Equations

Backward Liouville Equation First of all, we consider the derivation of the backward Liouville equation in indicator function ϕ(r, t|r0 , t0 ) = δ (r (t|r0 , t0 ) − r) considered as a function of parameters (r0 , t0 ). Differentiating function ϕ(r, t|r0 , t0 ) with respect to t0 and r0j and taking into account Eq. (2.21), page 61, we obtain the expressions ∂ϕ(r, t|r0 , t0 ) ∂ϕ(r, t|r0 , t0 ) ∂ri (t|r0 , t0 ) = U (r0 , t0 ) , ∂t0 ∂ri ∂r0 ∂ϕ(r, t|r0 , t0 ) ∂ϕ(r, t|r0 , t0 ) ∂ri (t|r0 , t0 ) =− . ∂r0 ∂ri ∂r0

(3.11)

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73

Taking scalar product of the second equation in Eq. (3.11) with U (r0 , t0 ) and summing the result with the first one, we arrive at the closed equation 

∂ ∂ + U (r0 , t0 ) ∂t0 ∂r0



ϕ(r, t|r0 , t0 ) = 0,

ϕ(r, t|r0 , t) = δ(r − r0 ),

(3.12)

which is just the backward Liouville equation given earlier without derivation. Rewrite Eq. (3.12) as the linear integral equation ϕ(r, t|r0 , t0 ) = δ(r − r0 ) +

Zt

dτ U(r0 , τ )

t0

∂ ϕ(r, t|r0 , τ ). ∂r0

According to this equation, variational derivative

(3.13)

δϕ(r, t|r0 , t0 ) satisfies the linear δUj (r0 , t0 )

integral equation 0 δϕ(r, t|r0 , t0 ) 0 0 0 ∂ϕ(r, t|r0 , t ) = δ(r − r )θ(t − t )θ(t − t ) 0 0 δUj (r0 , t0 ) ∂r0j Zt ∂ δϕ(r, t|r0 , τ ) + dτ U(r0 , τ ) , ∂r0 δUj (r0 , t0 ) t0

from which follows that δϕ(r, t|r0 , t0 ) ∼ θ(t0 − t0 )θ(t − t0 ), δUj (r0 , t0 ) i.e., the condition of dynamic causality holds. For t0 < t0 < t, this equation is simplified to the form 0 δϕ(r, t|r0 , t0 ) 0 ∂ϕ(r, t|r0 , t ) = δ(r − r ) + 0 δUj (r0 , t0 ) ∂r0j

Zt0

dτ U(r0 , τ )

t0

∂ δϕ(r, t|r0 , τ ) , ∂r0 δUj (r0 , t0 )

and we arrive, in the limit t0 = t0 + 0, at Eq. (3.10) derived earlier in a different way.

Density Field of Passive Tracer Under Random Velocity Field Consider the problem on tracer transfer by random velocity field in more details. The problem is formulated in terms of Eq. (1.39), page 24, that we rewrite in the form 

 ∂ ∂ ∂U(r, t) + U(r, t) ρ(r, t) + ρ(r, t) = 0, ∂t ∂r ∂r

ρ(r, 0) = ρ0 (r).

(3.14)

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To describe the density field in the Eulerian description, we introduce the indicator function ϕ(r, t; ρ) = δ(ρ(t, r) − ρ),

(3.15)

which is similar to function (3.2) and is localized on surface ρ(r, t) = ρ = const, in the three-dimensional case or on a contour in the two-dimensional case. An equation for this function can be easily obtained either immediately from Eq. (3.14), or from the Liouville equation in the Lagrangian description. Indeed, differentiating Eq. (3.15) with respect to time t and using dynamic equation (3.14) and probing property of the delta-function, we obtain the equation ∂U(r, t) ∂ ∂ρ(r, t) ∂ ∂ ϕ(r, t; ρ) = ρϕ(r, t; ρ) + U(r, t) ϕ(r, t; ρ). ∂t ∂r ∂ρ ∂r ∂ρ

(3.16)

However, this equation is unclosed because the right-hand side includes term ∂ρ(r, t)/∂r that cannot be explicitly expressed through ρ(r, t). On the other hand, differentiating function (3.15) with respect to r, we obtain the equality ∂ ∂ρ(r, t) ∂ ϕ(r, t; ρ) = − ϕ(r, t; ρ). ∂r ∂r ∂ρ

(3.17)

Eliminating now the last term in Eq. (3.16) with the use of (3.17), we obtain the closed Liouville equation in the Eulerian description 

 ∂ ∂ ∂U(r, t) ∂ [ρϕ(r, t; ρ)] , + U(r, t) ϕ(r, t; ρ) = ∂t ∂r ∂r ∂ρ ϕ(r, 0; ρ) = δ(ρ0 (r) − ρ).

(3.18)

Note that the Liouville equation (3.18) becomes the one-dimensional equation in the case of velocity field described by the simplest model Eq. (1.15), page 10, U(x, t) = v(t)f (x) (x is the dimensionless variable), 

 ∂ ∂ ∂f (x) ∂ [ρϕ(x, t; ρ)], + vx (t)f (x) ϕ(x, t; ρ) = vx (t) ∂t ∂x ∂x ∂ρ ϕ(x, 0; ρ) = δ(ρ0 (x) − ρ).

(3.19)

To obtain more complete description, we consider the extended indicator function including both density field ρ(r, t) and its spatial gradient p(r, t) = ∇ρ(r, t) ϕ(r, t; ρ, p) = δ (ρ(r, t) − ρ) δ (p(r, t) − p).

(3.20)

Indicator Function and Liouville Equation

75

Differentiating Eq. (3.20) with respect to time, we obtain the equality   ∂ ∂ρ(r, t) ∂ ∂pi (r, t) ∂ ϕ(r, t; ρ, p) = − + ϕ(r, t; ρ, p). ∂t ∂ρ ∂t ∂pi ∂t

(3.21)

Using now dynamic equation (3.14) for density and Eq. (1.50), page 28 for the density spatial gradient, we can rewrite Eq. (3.21) as the equation  ∂U(r, t) ρ + U(r, t)p ϕ(r, t; ρ, p) ∂r  ∂pi (r, t) ∂U(r, t) ∂ U(r, t) + pi + ∂pi ∂r ∂r  ∂Uk (r, t) ∂ 2 U(r, t) + pk ϕ(r, t; ρ, p), +ρ ∂ri ∂ri ∂r

∂ ∂ ϕ(r, t; ρ, p) = ∂t ∂ρ



(3.22)

which is unclosed because of the term ∂pi (r, t)/∂r in the right-hand side. Differentiating function (3.20) with respect to r, we obtain the equality   ∂ ∂ ∂pi (r, t) ∂ ϕ(r, t; ρ, p) = − p + ϕ(r, t; ρ, p). ∂r ∂ρ ∂r ∂pi

(3.23)

Now, multiplying Eq. (3.23) by U(r, t) and adding the result to Eq. (3.22), we obtain the closed Liouville equation for the extended indicator function 

 ∂ ∂ + U(r, t) ϕ(r, t; ρ, p) ∂t ∂r     ∂Uk (r, t) ∂ ∂ ∂ 2 U(r, t) ∂ ∂U(r, t) ∂ = ρ+ p + pk + ρ × ϕ(r, t; ρ, p) ∂ri ∂pi ∂r ∂ρ ∂p ∂ri ∂r ∂pi (3.24)

with the initial value
ϕ(r, 0; ρ, p) = δ (ρ0 (r) − ρ) δ p0 (r) − p . In the case of nondivergent velocity field (div U(r, t) = 0), the Liouville equation (3.24) is simplified to the form 

 ∂ ∂Uk (r, t) ∂ ∂ + U(r, t) ϕ(r, t; ρ, p) = pk ϕ(r, t; ρ, p), ∂t ∂r ∂ri ∂pi
ϕ(r, 0; ρ, p) = δ (ρ0 (r) − ρ) δ p0 (r) − p .

(3.25)

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Magnetic Field Diffusion Under Random Velocity Field Let us introduce the indicator function of magnetic field H(r, t), ϕ(r, t; H) = δ(H(r, t) − H). Differentiating this function with respect to time with the use of dynamic equation (1.59), page 30, and probing property of the delta function, we arrive at the equation ∂Hi (r, t) ∂ ∂ ϕ(r, t; H) = − ϕ(r, t; H) ∂t ∂t ∂Hi   ∂ui (r, t) ∂u(r, t) ∂ = − H(r, t) − Hi (r, t) ϕ(r, t; H) ∂r ∂r ∂Hi ∂Hi (r, t) ∂ + u(r, t) ϕ(r, t; H) ∂r ∂Hi   ∂ui (r, t) ∂ ∂u(r, t) H − Hi ϕ(r, t; H) =− ∂Hi ∂r ∂r ∂Hi (r, t) ∂ + u(r, t) ϕ(r, t; H), ∂r ∂Hi which is unclosed because of the last term. However, since ∂Hi (r, t) ∂ ∂ ϕ(r, t; H) = − δ(H(r, t) − H) ∂r ∂r ∂Hi and, hence, −u(r, t)

∂Hi (r, t) ∂ ∂ ϕ(r, t; H) = u(r, t) δ(H(r, t) − H), ∂r ∂r ∂Hi

we obtain the desired closed Liouville equation [16]     ∂ ∂ ∂ui (r, t) ∂u(r, t) ∂ + u(r, t) ϕ(r, t; H) = − H − Hi ϕ(r, t; H) ∂t ∂r ∂Hi ∂r ∂r (3.26) with the initial condition ϕ(r, 0; H) = δ(H0 (r) − H). Note that, in the case of velocity field described by the simplest model (1.15), page 10, u(x, t) = v(t)f (x) (x is the dimensionless variable), the Liouville equation (3.26) with homogeneous initial conditions becomes the equation    ∂ ∂ ∂ ϕ(x, t; H) = − f (x) + f 0 (x) 1 + H vx (t)ϕ(x, t; H) ∂t ∂x ∂H ∂ − f 0 (x)Hx0 vi (t) ϕ(x, t; H) (3.27) ∂Hi

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with the initial condition ϕ(x, 0; H) = δ(H0 − H).

3.2.2

Quasilinear Equations

Consider now the simplest quasilinear equation for scalar quantity q(r, t) (1.65), page 33, 

 ∂ ∂ + U(t, q) q(r, t) = Q (t, q), ∂t ∂r

q(r, 0) = q0 (r).

(3.28)

In this case, an attempt of deriving a closed equation for the indicator function ϕ(r, t; q) = δ(q(r, t) − q) on the analogy of the linear problem will fail. Here, we must supplement Eq. (3.28) with (1.66), page 33, for the gradient field p(r, t) = ∇q(r, t) 

 ∂ ∂ ∂ {U(t, q)p(r, t)} ∂Q(t, q) + U(t, q) p(r, t) + p(r, t) = p(r, t) ∂t ∂r ∂q ∂q

(3.29)

and consider the extended indicator function ϕ(r, t; q, p) = δ(q(r, t) − q)δ(p(r, t) − p).

(3.30)

Differentiating Eq. (3.30) with respect to time and using Eqs. (3.28) and (3.29), we obtain the equation   ∂ ∂q(r, t) ∂ ∂pk (r, t) ∂ ϕ(r, t; q, p) = − + ϕ(r, t; q, p) ∂t ∂q ∂t ∂pk ∂t   ∂ ∂ ∂pk (r, t) [ pU(t, q) − Q(t, q)] + = U(t, q) ϕ(r, t; q, p) ∂q ∂pk ∂r    ∂ ∂U(t, q) ∂Q(t, q) + pk p − ϕ(r, t; q, p), (3.31) ∂pk ∂q ∂q which is unclosed, however, because of the term ∂pk (r, t)/∂r in the right-hand side. Differentiating function (3.30) with respect to r, we obtain the equality   ∂ ∂ ∂pk (r, t) ∂ ϕ(r, t; q, p) = − p + ϕ(r, t; q, p), ∂r ∂q ∂r ∂pk from which follows that   ∂ ∂pk (r, t) ∂ ∂ ϕ(r, t; q, p) = − +p ϕ(r, t; q, p). ∂pk ∂r ∂r ∂q

(3.32)

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Consequently, Eq. (3.31) can be rewritten in the closed form     ∂ ∂U(t, q) ∂ ∂ + U(t, q) ϕ(r, t; q, p) = p − Q(t, q) ϕ(r, t; q, p) ∂t ∂r ∂q ∂q    ∂ ∂U(t, q) ∂Q(t, q) + pk p − ϕ(r, t; q, p), ∂pk ∂q ∂q (3.33) which is just the desired Liouville equation for the quasilinear equation (3.28) in the extended phase space {q, p} with the initial value ϕ(r, 0; q, p) = δ(q0 (r) − q)δ( p0 (r) − p). Note that the equation of continuity for conserved quantity I(r, t) ∂ ∂ {U(t, q)I(r, t)} = 0, I(r, t) + ∂t ∂r

I(r, 0) = I0 (r)

can be combined with Eqs. (3.28), (3.29). In this case, the indicator function has the form ϕ(r, t; q, p, I) = δ(q(r, t) − q)δ(p(r, t) − p)δ(I(r, t) − I) and derivation of the closed equation for this function appears possible in space {q, p, I}, which follows from the fact that, in the Lagrangian description, quantity inverse to I(r, t) coincides with the divergence. Note that the Liouville equation in indicator function ϕ(x, t; q, p, χ) = δ (q(x, t) − u) δ (p(x, t) − p) δ (χ(x, t) − χ) in the case of the one-dimensional Riemann equation (1.77), page 36, in function ∂q(x, t) q(x, t) complemented with the equations for the gradient field p(x, t) = − ∂x (1.80) and the logarithm of the density field χ(x, t) = ln ρ(x, t)− (1.82) assumes the form of the equation     ∂ ∂ ∂ 2 ∂ +q ϕ(x, t; q, p, χ) = p + p +p ϕ(x, t; q, p, χ) (3.34) ∂t ∂x ∂p ∂χ with the initial condition ϕ(x, 0; q, p, χ) = δ(q0 (x) − q)δ(p0 (x) − p)δ (χ0 (x) − χ). The solution to Eq. (3.34) can be represented in the form    ∂ ∂ ϕ(x, t; q, p, χ) = exp −t q − p e ϕ (x, t; q, p, χ), ∂x ∂χ

Indicator Function and Liouville Equation

79

where function e ϕ (x, t; q, p) satisfies the equation 

 ∂ 2 ∂ −p e ϕ (x, t; q, p, χ) = 3pe ϕ (x, t; q, p, χ) ∂t ∂p

(3.35)

with the initial condition ϕ(x, 0; q, p, χ) = δ(q0 (x) − q)δ( p0 (x) − p)δ (χ0 (x) − χ). Solving now Eq. (3.35) by the method of characteristics, we obtain the expression 1



p δ(q0 (x) − q)δ p0 (x) − e ϕ (x, t; q, p, χ) = 3 1 − pt (1 − pt)



× δ (χ0 (x) − ln (1 + tp0 (x)) − χ), that can be rewritten in the form p0 (x) −p e ϕ (x, t; q, p, χ) = [1 + tp0 (x)] δ(q0 (x) − q)δ 1 + tp0 (x) × δ (χ0 (x) − ln (1 + tp0 (x)) − χ). 

 (3.36)

Thus, we obtain the solution to Eq. (3.34) in the final form     ∂ p0 (x) [1 + tp0 (x)] δ(q0 (x) − q)δ ϕ(x, t; q, p, χ) = exp −qt −p ∂x 1 + tp0 (x) × δ (χ0 (x) − ln (1 + tp0 (x)) − χ), (3.37) in agreement with solutions (1.78), (1.79) and (1.83), pages 36 and 37.

3.2.3

General-Form Nonlinear Equations

Consider now the scalar nonlinear first-order partial differential equation in the general form (1.84), page 38, ∂ q(r, t) + H (r, t, q, p) = 0, ∂t

q (r, 0) = q0 (r),

(3.38)

where p(r, t) = ∂q(r, t)/∂r. In order to derive the closed Liouville equation in this case, we must supplement Eq. (3.38) with equations for vector p(r, t) and second derivative matrix Uik (r, t) = ∂ 2 q(r, t)/∂ri ∂rk . Introduce now the extended indicator function ϕ(r, t; q, p, U, I) = δ(q(r, t) − q)δ(p(r, t) − p)δ(U(r, t) − U)δ(I(r, t) − I), (3.39)

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where we included for generality an additional conserved variable I(r, t) satisfying the equation of continuity (1.86) ∂ ∂ I(r, t) + ∂t ∂r



 ∂H (r, t, q, p) I(r, t) = 0, ∂p

I (r, 0) = I0 (r).

(3.40)

Equations (3.38), (3.40) describe, for example, wave propagation in inhomogeneous media within the frames of the geometrical optics approximation of the parabolic equation of quasi-optics. Differentiating function (3.39) with respect to time and using dynamic equations for functions q(r, t), p(r, t), U(r, t) and I(r, t), we generally obtain an unclosed equation containing third-order derivatives of function q(r, t) with respect to spatial variable r. However, the combination 

 ∂H (r, t, q, p) ∂ ∂ + ϕ(r, t; q, p, U, I) ∂t ∂p ∂r

will not include the third-order derivatives; as a result, we obtain the closed Liouville equation in space {q, r, U, I}.

3.3

Higher-Order Partial Differential Equations

If the initial dynamic system includes higher-order derivatives (e.g., Laplace operator), derivation of a closed equation for the corresponding indicator function becomes impossible. In this case, only the variational differential equation (the Hopf equation) can be derived in the closed form for the functional whose average over an ensemble of realizations coincides with the characteristic functional of the solution to the corresponding dynamic equation. Consider such a transition using the partial differential equations considered in Lecture 1 as examples.

3.3.1

Parabolic Equation of Quasi-Optics

The first example concerns wave propagation in random medium within the frames of linear parabolic equation (1.89), page 39, ∂ i ik u(x, R) = 1R u(x, R) + ε(x, R)u(x, R), ∂x 2k 2

u(0, R) = u0 (R).

(3.41)

Equation (3.41) as the initial-value problem possesses the property of dynamic causality with respect to parameter x. Indeed, rewrite this equation in the integral form i u(x, R) = u0 (R) + 2k

Zx 0

ik dξ 1R u(ξ, R) + 2

Zx 0

dξ ε(ξ, R)u(ξ, R).

(3.42)

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81

Function u(x, R) is a functional of field ε(x, R), i.e. 
 u(x, R) = u x, R; ε e x, e R . Varying Eq. (3.42) with respect to function ε(x0 , R0 ), we obtain the integral equation δu(x, R) in variational derivative δε(x0 , R0 ) δu(x, R) ik = θ(x − x0 )θ(x0 )δ(R − R0 )u(x0 , R) 0 0 δε(x , R ) 2 Zx Zx ik δu(ξ, R) i δu(ξ, R) + dξ ε(ξ, R) + dξ 1R . 0 0 2 δε(x , R ) 2k δε(x0 , R0 ) 0

(3.43)

0

From this equation follows that δu(x, R) =0 δε(x0 , R0 )

if

x < x0

or

x0 < 0,

(3.44)

which just expresses the dynamic causality property. As a consequence of this fact, we can rewrite Eq. (3.43) in the form (for 0 < x0 < x) ik δu(x, R) = δ(R − R0 )u(x0 , R) δε(x0 , R0 ) 2 Zx Zx ik δu(ξ, R) i δu(ξ, R) + dξ ε(ξ, R) + dξ 1R , 0 0 2 δε(x , R ) 2k δε(x0 , R0 ) x0

(3.45)

x0

from which follows the equality δu(x, R) ik = δ(R − R0 )u(x, R). 0 δε(x − 0, R ) 2 Consider now the functional  Z    ϕ[x; v(R0 ), v∗ (R0 )] = exp i dR0 u(x, R0 )v(R0 ) + u∗ (x, R0 )v∗ (R0 ) ,

(3.46)

(3.47)

where wavefield u(x, R) satisfies Eq. (3.41) and u∗ (x, R) is the complex conjugated function. Differentiating (3.47) with respect to x and using dynamic equation (3.41) and its complex conjugate, we obtain the equality Z 1 ∂ 0 ∗ 0 ϕ[x; v(R ), v (R )] = − dR [v(R)1R u(x, R) ∂x 2k  −v∗ (R)1R u∗ (x, R) ϕ[x; v(R0 ), v∗ (R0 )] Z   k − dRε(x, R) v(R)u(x, R) − v∗ (R)u∗ (x, R) ϕ[x; v(R0 ), v∗ (R0 )], 2

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which can be written as the variational differential equation Z ik ∂ 0 ∗ 0 b (R)ϕ[x; v(R0 ), v∗ (R0 )] dRε(x, R)M ϕ[x; v(R ), v (R )] = ∂x 2   Z i δ δ + dR v(R)1R − v∗ (R)1R ∗ × ϕ[x; v(R0 ), v∗ (R0 )], 2k δv(R) δv (R)

(3.48)

with the initial value   Z   ϕ[0; v(R0 ), v∗ (R0 )] = exp i dR0 u0 (R0 )v(R0 ) + u∗0 (R0 )v∗ (R0 ) , and with the Hermitian operator b (R) = v(R) M

δ δ − v∗ (R) ∗ . δv(R) δv (R)

Equation (3.48) is equivalent to the input equation (3.41). Rewrite problem (3.48) with the initial condition in the integral form ϕ[x; v(e R), v∗ (e R)] = ϕ[0; v(e R), v∗ (e R)] Zx Z ik b (R)ϕ[ξ ; v(e + dξ dRε(ξ, R)M R), v∗ (e R)] 2 0

i + 2k

Zx

Z dξ

0

 δ δ ∗ dR v(R)1R − v (R)1R ∗ × ϕ[ξ ; v(e R), v∗ (e R)], δv(R) δv (R) 

and vary the resulting equation with respect to function ε(x0 , R0 ). Then we obtain that δϕ[x; v(e R), v∗ (e R)] satisfies the integral equation variational derivative 0 0 δε(x , R ) δϕ[x; v(e R), v∗ (e R)] ik b (R0 )ϕ[x0 ; v(e = θ (x − x0 )θ(x0 ) M R), v∗ (e R)] δε(x0 , R0 ) 2 Zx Z ik δϕ[ξ ; v(e R), v∗ (e R)] b (R) + dξ dRε(ξ, R)M 0 0 2 δε(x , R ) 0

i + 2k

Zx

Z dξ

0

 dR v(R)1R

 δ δ δϕ[ξ ; v(e R), v∗ (e R)] − v∗ (R)1R ∗ × . 0 0 δv(R) δv (R) δε(x , R )

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83

In view of the dynamic causality property, we can rewrite it in the form of the equation (for 0 < x0 < x) ik δϕ[x; v(e R), v∗ (e R)] b (R0 )ϕ[x0 ; v(e = M R), v∗ (e R)] δε(x0 , R0 ) 2 Zx Z ik δϕ[ξ ; v(e R), v∗ (e R)] b (R) + dξ dRε(ξ, R)M 0 0 2 δε(x , R ) x0

i + 2k

Zx

Z dξ

 dR v(R)1R

x0

 δϕ[ξ ; v(e R), v∗ (e R)] δ δ − v∗ (R)1R ∗ × , 0 0 δv(R) δv (R) δε(x , R )

from which follows the equality δ ik b (R)ϕ[x; v(e ϕ[x; v(e R), v∗ (e R)] = M R), v∗ (e R)]. δε(x − 0, R) 2

3.3.2

(3.49)

Random Forces in Hydrodynamic Theory of Turbulence

Consider now integro-differential equation (1.102), page 44, for the Fourier harmonics b u(k, t) of the solution to the Navier–Stokes equation (1.95) Z Z ∂ i αβ b ui (k, t) + dk1 dk2 3i (k1 , k2 , k)b uα (k1 , t)b uβ (k2 , t) ∂t 2 − νk2b ui (k, t) = b fi (k, t), (3.50) where αβ

1  kα 1iβ (k) + kβ 1iα (k) δ(k1 + k2 − k), (2π)3 ki kj 1ij (k) = δij − 2 (i, α, β = 1, 2, 3), k

3i (k1 , k2 , k) =

f (k, t) is the spatial Fourier harmonics of external forces. and b Function b ui (k, t) satisfies the principle of dynamic causality, which means that the variational derivative of function 
 b ui (k, t) = b ui k, t;b fα e k,e t
(considered as a functional of field b f (k, t)) with respect to field b f k0 , t0 vanishes for t0 > t and t0 < 0, δb ui (k, t)
= 0, δb fj k0 , t0

if

t0 > t

or

t0 < 0,

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Lectures on Dynamics of Stochastic Systems

and, moreover, δb ui (k, t)
= δij δ(k − k0 ). δb fj k0 , t − 0

(3.51)

Consider the functional  Z  ϕ[t; z] = ϕ[t; z(k0 )] = exp i dk0b u(k0 , t)z(k0 ) .

(3.52)

Differentiating this functional with respect to time t and using dynamic equation (3.50), we obtain the equality Z n o ∂ ϕ[t; z] = −i dkzi (k) νk2b ui (k, t) − b fi (k, t) ϕ[t; z] ∂t Z Z Z 1 αβ dkzi (k) dk1 dk2 3i (k1 , k2 , k)b uα (k1 , t)b uβ (k2 , t) ϕ[t; z], + 2 which can be rewritten in the functional space as the linear Hopf equation containing variational derivatives Z Z Z ∂ 1 δ 2 ϕ[t; z] αβ ϕ[t; z] = − dkzi (k) dk1 dk2 3i (k1 , k2 , k) ∂t 2 δzα (k1 ) δzβ (k2 )   Z δ − dkzi (k) νk2 − ib fi (k, t) ϕ[t; z], (3.53) δzi (k) with the given initial condition at t = 0. This means that Eq. (3.53) satisfies the condition of dynamic causality; a consequence of this condition is the equality δ δb f (k, t − 0)

ϕ[t; z] = iz(k)ϕ[t; z].

(3.54)

In a similar way, considering the space-time harmonics of the velocity field b ui (K), where K is the four-dimensional wave vector {k, ω} and b ui∗ (K) = b ui (−K) because field ui (r, t) is real, we obtain the nonlinear integral equation (1.103), page 45, Z Z i αβ (iω + νk2 )b ui (K) + d4 K 1 d4 K 2 3i (K 1 , K 2 , K)b uα (K 1 )b uβ (K) = b fi (K), 2 (3.55) where αβ

3i (K 1 , K 2 , K) =

1 αβ 3 (k1 , k2 , k) δ(ω1 + ω2 − ω), 2π i

and b fi (K) are the space-time Fourier harmonics of external forces. In this case, dealing with the functional  Z  ϕ[z] = ϕ[z(K 0 )] = exp i d4 K 0b u(K 0 )z(K 0 ) , (3.56)

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85

we derive the linear variational integro-differential equation of the form (iω + νk2 )

δϕ[z] = ib fi (K) ϕ[z] δzi (K) Z Z 1 δϕ[z] αβ − d4 K 1 d4 K 2 3i (K 1 , K 2 , K) . 2 δzα (K 1 ) δzβ (K 2 ) (3.57)

Problems Problem 3.1 Derive the Liouville equation for the Hamiltonian system of equations (1.14), page 9. Solution Indicator function ϕ(r, v, t) = δ(r(t) − r)δ(v(t) − v) satisfies the Liou-

ville equation (equation of continuity in the {r, v} space)   ∂ϕ(r, v, t) ∂ ∂H(r, v, t) ∂ ∂H(r, v, t) = − + ϕ(r, v, t) ∂t ∂rk ∂vk ∂vk ∂rk that can be rewritten in the form ∂ϕ(r, v, t) = {H(r, v, t), ϕ(r, v, t)}, ∂t where   ∂H ∂ϕ ∂H ∂ϕ {H, ϕ} = − ∂rk ∂vk ∂vk ∂rk

(3.58)

is the Poisson bracket of functions H(r, v, t) and ϕ(r, v, t).

Problem 3.2 Determine the relationship between the Eulerian and Lagrangian indicator functions for linear problem (3.14), page 73. Solution

ϕ(r, t; ρ) =

Z∞

Z dr0

dj jϕLag (r, ρ, j, t|r0 ).

0

Problem 3.3 Derive Eq. (3.18), page 74, from the dynamic system (1.42), (1.43), page 24, in the Lagrangian description.

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Part II Statistical Description of Stochastic Systems

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Lecture 4 Random Quantities, Processes, and Fields

Prior to consider statistical descriptions of the problems mentioned in Lecture 1, we discuss basic concepts of the theory of random quantities, processes, and fields.

4.1

Random Quantities and their Characteristics

The probability for a random quantity ξ to fall in interval −∞ < ξ < z is the monotonic function F(z) = P(−∞ < ξ < z) = hθ(z − ξ )iξ ,

F(∞) = 1,

(4.1)

where θ(z) =



1, 0,

if z > 0, if z < 0,

is the Heaviside step function and h· · · iξ denotes averaging over an ensemble of realizations of random quantity ξ . This function is called the probability distribution function or the integral distribution function. Definition (4.1) reflects the real-world procedure of finding the probability according to the rule n , P(−∞ < ξ < z) = lim N→∞ N where n is the integer equal to the number of realizations of event ξ < z in N independent trials. Consequently, the probability for a random quantity ξ to fall into interval z < ξ < z + dz, where dz is the infinitesimal increment, can be written in the form P(z < ξ < z + dz) = P(z)dz, where function P(z) called the probability density is represented by the formula d P(z) = P(−∞ < ξ < z) = hδ(z − ξ )iξ , (4.2) dz

Lectures on Dynamics of Stochastic Systems. DOI: 10.1016/B978-0-12-384966-3.00004-0 c 2011 Elsevier Inc. All rights reserved. Copyright

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where δ(z) is the Dirac delta function. In terms of probability density p(z), the integral distribution function is expressed by the formula Zz

F(z) = P(−∞ < ξ < z) =

dξ P(ξ ),

(4.3)

−∞

so that Z∞

P(z) > 0,

dzP(z) = 1. −∞

Multiplying Eq. (4.2) by arbitrary function f (z) and integrating over the whole domain of variable z, we express the mean value of the arbitrary function of random quantity in the following form Z∞

h f (ξ )iξ =

dzP(z)f (z).

(4.4)

−∞

The characteristic function defined by the equality

 8(v) = eivξ ξ =

Z∞

dzeivz P(z)

−∞

is a very important quantity that exhaustively describes all characteristics of random quantity ξ . The characteristic function being known, we can obtain the probability density (via the Fourier transform) Z∞

1 P(x) = 2π

dv8(v)e−ivx,

−∞

moments

 Mn = ξ n =

Z∞

n

dzp(z)z = −∞

cumulants (or semi-invariants)  Kn =

d idv

n

2(v)

v=0

,



d idv

n

8(v)

v=0

,

Random Quantities, Processes, and Fields

91

where 2(v) = ln 8(v), and other statistical characteristics. In terms of moments and cumulants of random quantity ξ , functions 2(v) and 8(v) are the Taylor series 8(v) =

∞ n X i Mn vn , n! n=0

2(v) =

∞ n X i Kn vn . n!

(4.5)

n=1

In the case of multidimensional random quantity ξ = {z1 , . . . , zn }, the exhaustive statistical description assumes the multidimensional characteristic function D E 8(v) = eivξ , ξ

v = {v1 , . . . , vn }.

(4.6)

The corresponding joined probability density for quantities ξ1 , . . . , ξn is the Fourier transform of characteristic function 8(v), i.e., Z 1 dv8(v)e−ivx , x = {x1 , . . . , xn }. (4.7) P(x) = (2π)n Substituting function 8(v) defined by Eq. (4.6) in Eq. (4.7) and integrating the result over v, we obtain the obvious equality P(x) = hδ(ξ − x)iξ = hδ(ξ1 − x1 ) . . . δ(ξn − xn )i

(4.8)

that can serve the definition of the probability density of random vector quantity ξ . In this case, the moments and cumulants of random quantity ξ are defined by the expressions ∂n ∂n Mi1 ,...,in = n 8(v) , Ki1 ,...,in = n 2(v) , i ∂vi1 . . . ∂vin i ∂vi1 . . . ∂vin v=0 v=0 where 2(v) = ln 8(v), and functions 2(v) and 8(v) are expressed in terms of moments Mi1 ,...,in and cumulants Ki1 ,...,in via the Taylor series 8(v) =

∞ n X i Mi ,...,i vi . . . vin , n! 1 n 1 n=0

2(v) =

∞ n X i Ki ,...,i vi . . . vin . n! 1 n 1

(4.9)

n=1

Note that, for quantities ξ assuming only discrete values ξi (i = 1, 2, . . .) with probabilities pi , formula (4.8) is replaced with its discrete analog

 Pk = δz,ξk , where δi,k is the Kronecker delta (δi,k = 1 for i = k and 0 otherwise). Consider now statistical average hξ f (ξ )iξ , where f (z) is arbitrary deterministic function such that the above average exists. We calculate this average using the procedure that will be widely used in what follows. Instead of f (ξ ), we consider function

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f (ξ + η), where η is arbitrary deterministic quantity. Expand function f (ξ + η) in the Taylor series in powers of ξ , i.e., represent it in the form   ∞ X 1 (n) d n f (ξ + η) = f (η)ξ = exp ξ f (η), n! dη n=0

where we introduced the shift operator with respect to η. Then we can write the equality    d hξ f (ξ + η)iξ = ξ exp ξ f (η) dη ξ    d ξ exp ξ      dη ξ d d   h f (ξ + η)iξ , exp ξ =  f (η) =  d dη ξ idη exp ξ dη ξ (4.10) where function  ξ eiξ v ξ d d ln 8(v) = 2(v), (v) = iξ v  = idv idv e ξ

and 8(v) is the characteristic function of random quantity ξ . Using now the Taylor series (4.5) for function 2(v), we obtain function (v) in the form of the series (v) =

∞ n X i Kn+1 vn . n!

(4.11)

n=0

Because variable η appears in the right-hand side of Eq. (4.10) only as the term of sum ξ + η, we can replace differentiation with respect to η with differentiation with respect to ξ (in so doing, operator (d/idξ ) should be introduced into averaging brackets) and set η = 0. As a result, we obtain the equality     d hξ f (ξ )iξ =  f (ξ ) , idξ ξ which can be rewritten, using expansion (4.11) for (v), as the series in cumulants Kn  n  ∞ X d f (ξ ) 1 hξ f (ξ )iξ = Kn+1 . n! dξ n ξ n=0

(4.12)

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93

Note that, setting f (ξ ) = ξ n−1 in Eq. (4.12), we obtain the recurrence formula that relates moments and cumulants of random quantity ξ in the form Mn =

n X k=1

(n − 1)! Kk Mn−k (k − 1)!(n − k)!

(M0 = 1,

n = 1, 2, . . .).

(4.13)

In a similar way, we can obtain the following operator expression for statistical average hg(ξ )f (ξ )iξ        d d d d hg(ξ + η1 )f (ξ + η2 )iξ = exp 2 −2 −2 + idη1 idη2 idη1 idη2 × hg(ξ + η1 )iξ h f (ξ + η2 )iξ . (4.14) In the particular case g(z) = eωz , where parameter ω assumes complex values too, we obtain the expression      

ωξ  d d 1 h f (ξ + η)iξ . ω+ −2 e f (ξ + η) ξ = exp 2 (4.15) i dη idη To illustrate practicability of the above formulas, we consider two types of random quantities ξ as examples. 1. Let ξ be the Gaussian random quantity with the probability density ) ( 1 z2 P(z) = √ exp − 2 . 2σ 2π σ Then, we have (

v2 σ 2 8(v) = exp − 2

) ,

2(v) = −

v2 σ 2 , 2

so that M1 = K1 = hξ i = 0,

D E M2 = K2 = σ 2 = ξ 2 ,

Kn>2 = 0.

In this case, the recurrence formula (4.13) assumes the form Mn = (n − 1)σ 2 Mn−2 ,

n = 2, . . . ,

(4.16)

from which follows that M2n+1 = 0,

M2n = (2n − 1)!!σ 2n .

For averages (4.12) and (4.15), we obtain the expressions ( )   D E E df (ξ ) ω2 σ 2 D ωξ hξ f (ξ )iξ = σ 2 , e f (ξ ) = exp f (ξ + ωσ 2 ) . ξ ξ dξ ξ 2

(4.17)

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Additional useful formulas can be derived from Eqs. (4.17); for example, D

e

ωξ

(

E ξ

ω2 σ 2 = exp 2

)

D

, ξe

ωξ

(

) ω2 σ 2 = ωσ exp , 2

E

2

ξ

and so on. If random quantity ξ has a nonzero mean value, hξ i 6= 0, the probability density is given by the expression ) (ξ − hξ i)2 . P(z) = √ exp − 2σ 2 2π σ 2 (

1

(4.18)

In this case, the characteristic function assumes the form (

) v2 σ 2 8(v) = exp iv hξ i − , 2

2(v) = iv hξ i −

v2 σ 2 , 2

and, consequently, moments and cumulants are given by the expressions M1 = K1 = hξ i, M2 = hξ i2 + σ 2 , K2 = σ 2 , Kn>2 = 0. The corresponding integral distribution function (4.3) assumes the form

F(z) = √

Zz

1 2π σ 2

1 = √ 2π

(

(ξ − hξ i)2 dξ exp − 2σ 2

)

−∞

z−hξ i σ

Z

(

y2 dy exp − 2

)

 = Pr

−∞

 z − hξ i , σ

(4.19)

where function 1 Pr (z) = √ 2π

Zz

(

x2 dx exp − 2

) (4.20)

−∞

is known as the probability integral. It is clear that Pr(∞) = 1

and

Pr(0) =

1 . 2

(4.21)

For negative z (z < 0), this function satisfies the relationship 1 Pr (−|z|) = √ 2π

Z∞ |z|

(

x2 dx exp − 2

) = 1 − Pr (|z|).

(4.22)

Random Quantities, Processes, and Fields

95

Asymptotic behavior of probability integral for z → ±∞ can be easily derived from Eqs. (4.20) and (4.22); namely,  ( )  z2 1    1 − z√2π exp − 2 , z → ∞, ) ( (4.23) Pr (z) ≈  1 z2   z → −∞. exp − √   2 |z| 2π If we deal with the random Gaussian vector whose components are ξi (hξi iξ = 0, i = 1, . . . , n), then the characteristic function is given by the equality   1 1 8(v) = exp − Bij vi vj , 2(v) = − Bij vi vj , 2 2

 where matrix Bij = ξi ξj and repeated indices assume summation. In this case, Eq. (4.17) is replaced with the equality   df (ξ ) hξ f (ξ )iξ = B . (4.24) dξ ξ 2. Let ξ ≡ n be the integer random quantity governed by Poisson distribution Pn =

n¯ n −¯n e , n!

where n¯ is the average value of quantity n. In this case, we have n  o   8(v) = exp n¯ eiv − 1 , 2(v) = n¯ eiv −1 . The recurrence formula (4.13) and Eq. (4.12) assume for this random quantity the forms Ml = n¯

l−1 X k=0

4.2

D E (l − 1)! Mk ≡ n¯ (n + 1)l−1 , hnf (n)i = n¯ hf (n + 1i. k!(l − 1 − k)!

(4.25)

Random Processes, Fields, and their Characteristics

4.2.1

General Remarks

If we deal with random function (random process) z(t), then all statistical characteristics of this function at any fixed instant t are exhaustively described in terms of the one-time probability density P(z, t) = hδ (z(t) − z)i, dependent parametrically on time t by the following relationship h f (z(t))i =

Z∞ −∞

dzf (z)P(z, t).

(4.26)

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The integral distribution function for this process, i.e., the probability of the event that process z(t) < Z at instant t, is calculated by the formula F(t, Z) = P(z(t) < Z) =

ZZ dzP(z, t), −∞

from which follows that F(t, Z) = hθ(Z − z(t))i,

F(t, ∞) = 1,

(4.27)

where θ(z) is the Heaviside step function equal to zero for z < 0 and unity for z > 0. Note that the singular Dirac delta function ϕ(z, t) = δ (z(t) − z) appeared in Eq. (4.26) in angle brackets of averaging is called the indicator function. Similar definitions hold for the two-time probability density P(z1 , t1 ; z2 , t2 ) = hϕ(z1 , t1 ; z2 , t2 )i, and for the general case of the n-time probability density P(z1 , t1 ; . . . ; zn , tn ) = hϕ(z1 , t1 ; . . . ; zn , tn )i, where ϕ(z1 , t1 ; . . . ; zn , tn ) = δ(z(t1 ) − z1 ) . . . δ(z(tn ) − zn ). is the n-time indicator function. Process z(t) is called stationary if all its statistical characteristics are invariant with respect to arbitrary temporal shift, i.e., if P(z1 , t1 + τ ; . . . ; zn , tn + τ ) = P(z1 , t1 ; . . . ; zn , tn ). In particular, the one-time probability density of stationary process is at all independent of time, and the correlation function depends only on the absolute value of difference of times, Bz (t1 , t2 ) = hz(t1 )z(t2 )i = Bz (|t1 − t2 |). Temporal scale τ0 characteristic of correlation function Bz (t) is called the temporal correlation radius of process z(t). We can determine this scale, say, by the equality Z∞ 0

D E hz(t + τ )z(t)i dτ = τ0 z2 (t) .

(4.28)

Random Quantities, Processes, and Fields

97

Note that the Fourier transform of stationary process correlation function 8z (ω) =

Z∞

dtBz (t)eiωt,

−∞

is called the temporal spectral function (or simply temporal spectrum). For random field f (x, t), the one- and n-time probability densities are defined similarly P(x, t; f ) = hϕ(x, t; f )i,

(4.29)

P(x1 , t1 , f1 ; . . . ; xn , tn , fn ) = hϕ(x1 , t1 , f1 ; . . . ; xn , tn , fn )i,

(4.30)

where the indicator functions are defined as follows: ϕ(x, t; f ) = δ( f (x, t) − f ), ϕ(x1 , t1 , f1 ; . . . ; xn , tn , fn ) = δ ( f (x1 , t1 ) − f1 ) . . . δ ( f (xn , tn ) − fn ).

(4.31)

For clarity, we use here variables x and t as spatial and temporal coordinates; however, in many physical problems, some preferred spatial coordinate can play the role of the temporal coordinate. Random field f (x, t) is called the spatially homogeneous field if all its statistical characteristics are invariant relative to spatial translations by arbitrary vector a, i.e., if P(x1 + a, t1 , f1 ; . . . ; xn + a, tn , fn ) = P(x1 , t1 , f1 ; . . . ; xn , tn , fn ). In this case, the one-point probability density P(x, t; f ) = P(t; f ) is independent of x, and the spatial correlation function Bf (x1 , t1 ; x2 , t2 ) depends on the difference x1 − x2 : Bf (x1 , t1 ; x2 , t2 ) = h f (x1 , t1 )f (x2 , t2 )i = Bf (x1 − x2 ; t1 , t2 ). If random field f (x, t) is additionally invariant with respect to rotation of all vectors xi by arbitrary angle, i.e., with respect to rotations of the reference system, then field f (x, t) is called the homogeneous isotropic random field. In this case, the correlation function depends on the length |x1 − x2 |: Bf (x1 , t1 ; x2 , t2 ) = hf ( x1 , t1 )f (x2 , t2 )i = Bf (|x1 − x2 |; t1 , t2 ). The corresponding Fourier transform of the correlation function with respect to the spatial variable defines the spatial spectral function (called also the angular spectrum) Z 8f (k, t) = dxBf (x, t)eikx ,

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and the Fourier transform of the correlation function of random field f (x, t) stationary in time and homogeneous in space defines the space–time spectrum 8f (k, ω) =

Z∞

Z dx

dtBf (x, t)ei(kx+ωt) .

−∞

In the case of isotropic random field f (x, t), the space–time spectrum appears to be isotropic in the k-space: 8f (k, ω) = 8f (k, ω). An exhaustive description of random function z(t) can be given in terms of the characteristic functional  ∞ + *  Z  8[v(τ )] = exp i dτ v(τ )z(τ ) ,   −∞

where v(t) is arbitrary (but sufficiently smooth) function. Functional 8[v(τ )] being known, one can determine such characteristics of random function z(t) as mean value hz(t)i, correlation function hz(t1 )z(t2 )i, n-time moment function hz(t1 ) . . . z(tn )i, etc. Indeed, expanding functional 8[v(τ )] in the functional Taylor series, we obtain the representation of characteristic functional in terms of the moment functions of process z(t): Z∞ ∞ n Z∞ X i dt1 . . . dtn Mn (t1 , . . . , tn )v(t1 ) . . . v(tn ), 8[v(τ )] = n! n=0 −∞ −∞ 1 δn 8[v(τ )] . Mn (t1 , . . . , tn ) = hz(t1 ) . . . z(tn )i = n i δv(t1 ) . . . δv(tn ) v=0 Consequently, the moment functions of random process z(t) are expressed in terms of variational derivatives of the characteristic functional. Represent functional 8[v(τ )] in the form 8[v(τ )] = exp{2[v(τ )]}. Functional 2[v(τ )] also can be expanded in the functional Taylor series Z∞ ∞ n Z∞ X i 2[v(τ )] = dt1 . . . dtn Kn (t1 , . . . , tn )v(t1 ) . . . v(tn ), n! n=1

−∞

−∞

where function 1 δn Kn (t1 , . . . , tn ) = n 2[v(τ )] i δv(t1 ) . . . δv(tn ) v=0 is called the n-th order cumulant function of random process z(t).

(4.32)

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99

The characteristic functional and the n-th order cumulant functions of scalar random field f (x, t) are defined similarly  + Z∞  Z   8[v(x0 , τ )] = exp i dx dtv(x,t)f (x, t) = exp 2[v(x0 , τ )] ,   −∞ δn 1 0 8[v(x , τ )] , Mn (x1 , t1 , . . . , xn , tn ) = n i δv(x1 , t1 ) . . . δv(xn , tn ) v=0 1 δn Kn (x1 , t1 , . . . , xn , tn ) = n 2[v(x0 , τ )] . i δv(x1 , t1 ) . . . δv(xn , tn ) v=0 *

In the case of vector random field f (x, t), we must assume that v(x,t) is the vector function. As we noted earlier, characteristic functionals ensure the exhaustive description of random processes and fields. However, even one-time probability densities provide certain information about temporal behavior and spatial structure of random processes for arbitrary long temporal intervals. The ideas of statistical topography of random processes and fields can assist in obtaining this information.

4.2.2

Statistical Topography of Random Processes and Fields

Random Processes We discuss first the concept of typical realization curve of random process z(t). This concept concerns the fundamental features of the behavior of a separate process realization as a whole for temporal intervals of arbitrary duration. Random Process Typical Realization Curve We will call the typical realization curve of random process z(t) the deterministic curve z∗ (t), which is the median of the integral distribution function (4.27) and is determined as the solution to the algebraic equation


F t, z∗ (t) =

zZ∗ (t)

dz P(z, t) =

1 . 2

(4.33)

−∞

The reason for this definition rests on the median property consisting in the fact that, for any temporal interval (t1 , t2 ), random process z(t) entwines about curve z∗ (t) in a way to force the identity of average times during which the inequalities z(t) > z∗ (t) and z(t) < z∗ (t) hold (Fig. 4.1):



 1 Tz(t)>z∗ (t) = Tz(t)
(4.34)

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Lectures on Dynamics of Stochastic Systems

z ∗(t ) z(t)

Δt 1 t1

Figure 4.1

Δt 2

Δt3 t2

t

To the definition of the typical realization curve of a random process.

Indeed, integrating Eq. (4.33) over temporal interval (t1 , t2 ), we obtain Zt2


1 dtF t, z∗ (t) = (t2 − t1 ). 2

(4.35)

t1

On the other hand, in view of the definition of integral distribution function (4.27), the integral in the right-hand side of Eq. (4.35) can be represented as Zt2


dtF t, z∗ (t) = hT(t1 , t2 )i,

(4.36)

t1

where T(t1 , t2 ) =

N P

1tk is the combined time during which the realization of process

1

z(t) is above curve z∗ (t) in interval (t1 , t2 ). Combining Eqs. (4.35) and (4.36), we obtain Eq. (4.34). Curve z∗ (t) can significantly differ from any particular realization of process z(t) and cannot describe possible magnitudes of spikes. Nevertheless, the definitional domain of typical realization curve z∗ (t) derived from the one-point probability density of random process z(t) is the whole temporal axis t ∈ (0, ∞). Consideration of specific random processes allows additional information to be obtained concerning the realization’s spikes relative to the typical realization curve. Statistics of Random Process Cross Points with a Line The one-time probability density (4.26) of random process z(t) is a result of averaging the singular indicator function over an ensemble of realizations of this process. This function is concentrated at the points at which process z(t) crosses line z = const. Because the cross points are determined as roots of the algebraic equation

z(tn ) = z (n = 0, 1, . . . , ∞),

Random Quantities, Processes, and Fields

101

we can rewrite the indicator function in the following form ϕ(z, t) =

n X k=1

1 δ(t − tk ), |p(tk )|

d z(t). dt The number of cross points by itself is obviously a random quantity described by the formula where p(t) =

Zt n(t, z) =

dτ |p(τ )|ϕ(τ ; z).

(4.37)

−∞

As a consequence, the average number of points where process z(t) crosses line z = const can be described in terms of the correlation between the process derivative with respect to time and the process indicator function, or in terms of joint one-time probad bility density of process z(t) and its derivative with respect to time z(t). dt In a similar way, we can determine certain elements of statistics related to some other special points (such as points of maxima or minima) of random process z(t).

Random Fields Similar to topography of mountain ranges, statistical topography studies the systems of contours (level lines in the two-dimensional case and surfaces of constant values in the three-dimensional case) specified by the equality f (r, t) = f = const. For analyzing a system of contours (in this section, we will deal for simplicity with the two-dimensional case and assume r = R), we introduce the singular indicator function (4.31) concentrated on these contours. The convenience of function (4.31) consists, in particular, in the fact that it allows simple expressions for quantities such as the total area of regions where f (R, t) > f (i.e., within level lines f (R, t) = f ) and the total mass of the field within these regions [4] S(t; f ) =

Z

M(t; f ) =

Z

θ( f (R, t) − f )dR =

Z∞

df 0

Z

dRϕ(t, R;f 0 ),

f

f (R, t)θ( f (R, t) − f )dR =

Z∞

0

f df

0

Z

dRϕ(t, R; f 0 ).

f

As we mentioned earlier, the mean value of indicator function (4.31) over an ensemble of realizations determines the one-time (in time) and one-point (in space) probability density P(R, t; f ) = hϕ(R, t; f )i = hδ (f (R, t)−f )i.

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Consequently, this probability density immediately determines ensemble-averaged values of the above expressions. If we include into consideration the spatial gradient p(R, t) = ∇f (R, t), we can obtain additional information on details of the structure of field f (R, t). For example, quantity Z I l(t; f ) = dR |p(R, t)| δ( f ( R, t) − f ) = dl, (4.38) is the total length of contours and extends formula (4.37) to random fields. The integrand in Eq. (4.38) is described in terms of the extended indicator function ϕ(R, t; f , p) = δ ( f (R, t) − f ) δ ( p(R, t)−p) ,

(4.39)

so that the average value of total length (4.38) is related to the joint one-time probability density of field f (R, t) and its gradient p(R, t), which is defined as the ensemble average of indicator function (4.39), i.e., as the function P(R, t; f , p) = hδ ( f (R, t) − f ) δ ( p(R, t) − p)i. Inclusion of second-order spatial derivatives into consideration allows us to estimate the total number of contours f (R, t) = f = const by the approximate formula (neglecting unclosed lines) Z 1 dRκ(t, R; f ) |p(R, t)| δ (f (R, t) − f ), N(t; f ) = Nin (t; f ) − Nout (t; f ) = 2π where Nin (t; f ) and Nout (t; f ) are the numbers of contours for which vector p is directed along internal and external normals, respectively, and κ(R, t; f ) is the curvature of the level line. Recall that, in the case of the spatially homogeneous field f (R, t), the corresponding probability densities P(R, t; f ) and P(R, t; f , p) are independent of R. In this case, if statistical averages of the above expressions (without integration over R) exist, they will characterize the corresponding specific (per unit area) values of these quantities. In this case, random field f (R, t) is statistically equivalent to the random process whose statistical characteristics coincide with the spatial one-point characteristics of field f (R, t). Consider now several examples of random processes.

4.2.3

Gaussian Random Process

We start the discussion with the continuous processes; namely, we consider the Gaussian random process z(t) with zero-valued mean (hz(t)i = 0) and correlation function B(t1 , t2 ) = hz(t1 )z(t2 )i. The corresponding characteristic functional assumes the

Random Quantities, Processes, and Fields

103

form     1 Z∞ Z∞ dt1 dt2 B(t1 , t2 )v(t1 )v(t2 ) . 8[v(τ )] = exp −   2

(4.40)

−∞ −∞

Only one cumulant function (the correlation function) K2 (t1 , t2 ) = B(t1 , t2 ), is different from zero for this process, so that 2[v(τ )] = −

1 2

Z∞ Z∞

dt1 dt2 B(t1 , t2 )v(t1 )v(t2 ).

(4.41)

−∞ −∞

Consider the nth-order variational derivative of functional 8[v(τ )]. It satisfies the following line of equalities: δ n−1 δ2[v(τ )] δn 8[v(τ )] = 8[v(τ )] δv(t1 ) . . . δv(tn ) δv(t2 ) . . . δv(tn ) δv(t1 ) =

δ 2 2[v(τ )] δ n−2 8[v(τ )] δv(t1 )δv(t2 ) δv(t3 ) . . . δv(tn ) +

δ2[v(τ )] δ8[v(τ )] δ n−2 . δv(t3 ) . . . δv(tn ) δv(t1 ) δv(t2 )

Setting now v = 0, we obtain that moment functions of the Gaussian process z(t) satisfy the recurrence formula Mn (t1 , . . . , tn ) =

n X

B(t1 , t2 )Mn−2 (t2 , . . . , tk−1 , tk+1 , . . . , tn ).

(4.42)

k=2

From this formula it follows that, for the Gaussian process with zero-valued mean, all moment functions of odd orders are identically equal to zero and the moment functions of even orders are represented as sums of terms which are the products of averages of all possible pairs z(ti )z(tk ). If we assume that function v(τ ) in Eq. (4.41) is different from zero only in interval 0 < τ < t, the characteristic functional  t + * Z 8[t; v(τ )] = exp i dτ z(τ )v(τ ) 0

  Zτ1  Zt  = exp − dτ1 dτ2 B(τ1 , τ2 )v(τ1 )v(τ2 ) ,   0

0

(4.43)

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becomes a function of time t and satisfies the ordinary differential equation d 8[t; v(τ )] = −v(t) dt

Zt

dt1 B(t, t1 )v(t1 )8[t; v(τ )],

8[0; v(τ )] = 1.

(4.44)

0

To obtain the one-time characteristic function of the Gaussian random process at instant t, we specify function v(τ ) in Eq. (4.40) in the form v(τ ) = vδ(τ − t). Then, we obtain D

8(v, t) = e

ivz(t)

E

Z∞ =

ivz

dz P(z, t)e

  1 2 2 = exp − σ (t)v , 2

(4.45)

−∞

where σ 2 (t) = B(t, t). Using the inverse Fourier transform of (4.45), we obtain the one-time probability density of the Gaussian random process 1 P(z, t) = 2π

Z∞

dv 8(v, t)e

−ivz

−∞

 z2 =p . exp − 2 2σ (t) 2πσ 2 (t) 1



(4.46)

Note that, in the case of stationary random process z(t), variance σ 2 (t) is independent of time t, i.e., σ 2 (t) = σ 2 = const. Density P(z, t) as a function of z is symmetric relative to point z = 0, P(z, t) = P(−z, t). If mean value of the Gaussian random process is different from zero, then we can consider process z(t) − hz(t)i to obtain instead of Eq. (4.46) the expression ( ) 1 (z − hz(t)i)2 P(z, t) = p exp − , (4.47) 2σ 2 (t) 2πσ 2 (t) and the corresponding integral distribution function assumes the form ( )   Zz 1 z − hz(t)i (z − hz(t)i)2 F(z, t) = p dz exp − = Pr , σ (t) 2σ 2 (t) 2πσ 2 (t) −∞

where probability integral Pr(z) is defined by Eq. (4.20), page 94. In view of Eq. (4.21), page 94, the typical realization curve (4.33) of the Gaussian random process z(t) in this case coincides with the mean value of process z(t) z∗ (t) = hz(t)i .

(4.48)

Random Quantities, Processes, and Fields

4.2.4

105

Gaussian Vector Random Field

Consider now the basic space–time statistical characteristics of the vector Gaussian random field u(r, t) with zero-valued mean and correlation tensor

 Bij (r, t; r0 , t0 ) = ui (r, t)uj (r0 , t0 ) . For definiteness, we will identify this vector field with the velocity field. In the general case, random field u(r, t) is assumed to be the divergent (div u(r, t) 6= 0) Gaussian field statistically homogeneous and possessing spherical symmetry (but not possessing reflection symmetry) in space and stationary in time with correlation and spectral tensors (τ = t − t1 ) Z

 Bij (r − r1 , τ ) = ui (r, t)uj (r1 , t1 ) = dk Eij (k, τ )eik(r−r1 ) , Z 1 dr Bij (r, τ )e−ikr , Eij (k, τ ) = (2π)d where d is the dimension of space. In view of the assumed symmetry conditions, correlation tensor Bij (r − r1 , τ ) has vector structure [28] (r − r1 → r) Bij (r, τ ) = Biso ij (r, τ ) + C(r, τ )εijk rk ,

(4.49)

where the isotropic portion of the correlation tensor is expressed as follows Biso ij (r, τ ) = A(r, τ )ri rj + B(r, τ )δij . Here, εijk is the pseudotensor described in Sect. 1.3.1, page 24. This pseudotensor is used, for example, to determine the velocity vortex field ω(r, t) = curl u(r, t) = [∇ × u(r, t)] ,

(4.50)

whose components have the form ωi (r, t) = εijk

∂uk (r, t) . ∂rj

Note that, in the two-dimensional case, vector ω(r, t) has a single component orthogonal to velocity field u(r, t); as a result, quantity u(r, t) · ω(r, t) called velocity field helicity vanishes, u(r, t) · ω(r, t) = 0.

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Expression (1.53), page 29, allows obtaining the following representations for the vortex field correlation and spectral tensors [28]:

 ∂ 2 Bll (r − r0 , t − t0 ) ωi (r, t)ωj (r0 , t0 ) = −δij 1r Bll (r − r0 , t − t0 ) + ∂ri ∂rj + 1r Bij (r − r0 , t − t0 ), ij (k, t − t0 ) = (δij k2 − ki kj )Ell (k, t − t0 ) − k2 Eij (k, t − t0 ). Convolving them with respect to indices i and j and setting r = r0 , we obtain the expressions for the correlation and spectral functions of the vortex field



 ωi (r, t)ωi (r, t0 ) = −1r Bll (0, t − t0 ) = − u(r, t)1r u(r, t0 ) , ii (k, t − t0 ) = k2 Eii (k, t − t0 ).

(4.51)

Note that the procedure similar to that used to calculate vortex field variance and spectrum allows additionally calculating average helicity of the velocity field χ(r, t) = u(r, t)ω(r, t) = u(r, t) · curl u(r, t), in the three-dimensional case in the absence of reflection symmetry. Indeed, the use of Eq. (1.54), page 29, results in the following expression for statistically averaged

quantity ωi (r, t)uj (r, t)

ωi (r, t)up (r , t) r=r0 0



 ∂uk (r, t) ∂ 0 = εijk up (r , t) = εijk Bkp (r, 0)r=0 ∂rj ∂r j r=r0 ∂ = lim εijk C(r, 0)εkpm rm = C(0, 0)εijm εmpj = 2C(0, 0)δip , r→0 ∂rj 

and, hence, the average helicity of the Gaussian random field in the three-dimensional case is hχ(r, t)i = 6C(0, 0).

(4.52)

The isotropic portion of correlation tensor corresponds to the spatial spectral tensor of the form p

Eij (k, τ ) = Eijs (k, τ ) + Eij (k, τ ), where the spectral components of the tensor of velocity field have the following structure   ki kj ki kj p s s Eij (k, τ ) = E (k, τ ) δij − 2 , Eij (k, τ ) = Ep (k, τ ) 2 . k k

Random Quantities, Processes, and Fields

107

Here, Es (k, τ ) and Ep (k, τ ) are the solenoidal and potential components of the spectral density of velocity field, respectively. Define now function Bij (r) as the integral of correlation function (4.49) over time, i.e., Z∞ Bij (r) = dτ Bij (r, τ ) = Biso ij (r) + C(r)εijk rk .

(4.53)

0

Then, Bij (0) = D0 δij and, hence, quantity Bii (0) = D0 d = τ0 σu2 =

Z

  dk (d − 1) Es (k) + Ep (k) ,

(4.54)

2

defines temporal correlation radius of the velocity field τ0 .s Here, σu p = Bii (0, 0) = 2 u (r, t) is the variance of the velocity field, and functions E (k) and E (k) are defined as follows

E (k) = s

Z∞

dτ E (k, τ ), s

E (k) = p

0

Z∞

dτ Ep (k, τ ).

(4.55)

0

The use of correlation function (4.53) appreciably simplifies calculations dealing with spatial derivatives of the velocity field. Indeed, we have ∂Bij (0) = C(0)εijk , ∂rk −

(4.56)

  ∂ 2 Bij (0) Ds (d + 1)δkl δij − δki δlj − δkj δli = ∂rk ∂rl d(d + 2)   Dp δkl δij + δki δlj + δkj δli , d(d + 2)

(4.57)


∂ 3 Bkp (0) = −2α εkpj δnm + εkpm δnj + εkpn δmj , ∂rn ∂rm ∂rj

(4.58)

+

and, consequently, ∂ 3 Bkp (0) = −2α(d + 2)εkpj , ∂r2 ∂rj

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where s

Z

D =

p

Z

D =

Z∞

1 dk k E (k) = d−1 2 s

Z∞

dk k E (k) = 2 p

0

dτ hω(r, t + τ )ω(r, t)i,

0

 ∂u(r, t + τ ) ∂u(r, t) dτ , ∂r ∂r 

(4.59)

C(r) = C(0) − αr , 2

and ω(r, t) is the velocity field vortex.

4.2.5

Logarithmically Normal Random Process

Note that the so-called logarithmically normal (lognormal) random process y(t) y(t) = ez(t) , whose logarithm is the Gaussian random process, is described by the one-time probability density P(y, t) of the form (  ) ln2 e−hz(t)i y 1 1 P(y, t) = P(z = ln y, t) = p exp − , y 2σ 2 (t) y 2πσ 2 (t) and, consequently, the corresponding integral distribution function assumes the form Zy F(y, t) = −∞

1

dy0 P(y0 , t) = p 2πσ 2 (t)

Zy −∞

  ln e−hz(t)i y

1

=p 2πσ 2 (t)

Z −∞

 dz exp −

(  ) ln2 e−hz(t)i y0 dy0 exp − y0 2σ 2 (t)

z2 2σ 2 (t)



 ! ln e−hz(t)i y = Pr , σ (t)

where probability integral Pr(z) is defined by Eq. (4.20), page 94. In view of Eq. (4.21), page 94, the right-hand side of this equality turns into 1/2 if e−hz(t)i y(t) = 1, which means that the typical realization curve y∗ (t) of the lognormal random process y(t) is given by the equality y∗ (t) = ehz(t)i = ehln y(t)i .

(4.60)

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109

Note that, if we know the moment functions of random process y(t) as functions of time, i.e., if we know functions hyn (t)i (n = 1, 2, . . .), we know also the statistical characteristics of random process z(t) = ln y(t). Indeed,  

n  D n ln y(t) E n2 2 y (t) = e = exp n hln y(t)i + σln y (t) , 2 and, consequently, 1 n  ln y (t) , n

hln y(t)i = lim

n→0

σln2 y (t) = lim

n→∞

2 n  ln y (t) . n2

(4.61)

Further, analyzing statistical topography of random processes and fields, we will use, in line with the integral distribution function, functions Z∞ V(y, t) =

dy P(y , t) and Y(y, t) = 0

0

y

Z∞

y0 dy0 P(y0 , t),

y

closely related to the probability integral Pr(z). For function V(y, t), we have Z∞

1

V(y, t) = p 2πσ 2 (t) 1 =√ 2π  = Pr

y

Z∞ ln y−hz(t)i σ (t)

(
2 ) ln y0 − hz(t)i dy0 exp − y0 2σ 2 (t)

 2   z ln y − hz(t)i dz exp − = Pr − 2 σ (t)

  1 hz(t)i 1 ln e . σ (t) y

(4.62)

For function Y(y, t), we obtain the expression 1

Y(y, t) = p 2πσ 2 (t)

Z∞ y

(

ln y0 − hz(t)i dy exp − 2σ 2 (t)


2 )

0

1 2 = √ ehz(t)i+σ (t)/2 2π

Z∞

 2 z dz exp − 2

ln y−hz(t)i−σ 2 (t) σ (t)

 ln y − hz(t)i − σ 2 (t) =e Pr − σ (t)    1 1 2 hz(t)i+σ (t)/2 hz(t)i+σ 2 (t) =e Pr ln e . σ (t) y hz(t)i+σ 2 (t)/2



(4.63)

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Lectures on Dynamics of Stochastic Systems

Note that for random process e y(t) = y0 y(t), i.e., e y(t) = y0 ez(t) , instead of Eqs. (4.62), (4.63) we have the expressions    y0 hz(t)i 1 e ln e , V (y, t) = Pr σ (t) y    1 y0 hz(t)i+σ 2 (t) 2 e . Y (y, t) = y0 ehz(t)i+σ (t)/2 Pr ln e σ (t) y

4.2.6

(4.64)

Discontinuous Random Processes

Consider now some examples of discontinuous processes. The discontinuous processes are the random functions that change their time-dependent behavior at discrete instants t1 , t2 , . . . given statistically. The description of discontinuous processes requires first of all either the knowledge of the statistics of these instants, or the knowledge of the statistics of number n(0, t) of instants ti falling in time interval (0, t). In the latter case, we have the equality n(0, t) = n(0, t0 ) + n(t0 , t),

0 ≤ t0 ≤ t.

The quantity n(0, t) by itself is a random process, and Fig. 4.2 shows its possible realization. The set of points of discontinuity t1 , t2 , . . . of process z(t) is called the stream of points. In what follows, we will consider Poisson’s stationary stream of points in which the probability of falling n points in interval (t1 , t2 ) is specified by Poisson’s formula h in n(t1 , t2 ) Pn(t1 ,t2 )=n = e−n(t1 ,t2 ) , (4.65) n! with the mean number of points in interval (t1 , t2 ) given by the formula n(t1 , t2 ) = ν|t1 − t2 |, n(0, t) 4 3 2 1 0

Figure 4.2

t1 t2

t3

A possible realization of process n(0, t).

t4

t

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111

where ν is the mean number of points per unit time. It is assumed here that the numbers of points falling in nonoverlapping intervals are statistically independent and the instants at which points were fallen in interval (t1 , t2 ) under the condition that their total number was n are also statistically independent and uniformly distributed over the interval (t1 , t2 ). The length of the interval between adjacent points of discontinuity satisfies the exponential distribution. Poisson stream of points is an example of the Markovian processes. Consider now random processes whose points of discontinuity form Poisson’s streams of points. Currently, three types of such process – Poisson’s process, telegrapher’s process, and generalized telegrapher’s process – are mainly used to model problems in physics. Below, we focus our attention on these processes.

Poisson’s (Impulse) Random Process Poisson’s (impulse) random process z(t) is the process described by the formula

z(t) =

n X

ξi g(t − ti ),

(4.66)

i=1

where random quantities ξi are statistically independent and distributed with probability density p(ξ ); random points tk are uniformly distributed on interval (0, T), so that their number n obeys Poisson’s law with parameter n¯ = νT; and function g(t) is the deterministic function that describes the pulse envelope (g(t) = 0 for t < 0). The characteristic functional of Poisson’s random process z(t) assumes the form  t   t   Z  Z  8[t; v(τ )] = exp ν dt0 W  dτ v(τ )g(t − t0 ) − 1 ,  

(4.67)

t0

0

where Z∞ W(v) =

dξ p(ξ )eiξ v,

−∞

is the characteristic function of random quantity ξ . Consequently, functional 2[t; v(τ )] is given by the formula 2[t; v(τ )] = ln 8[t; v(τ )] =ν

Zt 0

dt0

Z∞

−∞

 



dξ p(ξ ) exp iξ 

Zt t0

  dτ v(τ )g(t − t0 ) − 1 ,  

(4.68)

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and cumulant functions assume the form min{t Z1 ,...,tn }

 Kn (t1 , . . . , tn ) = ν ξ n

dt0 g(t − t0 ) . . . g(tn − t0 ).

(4.69)

0

We consider two types of Poisson’s processes important for applications. n P t > 0, ξi θ (t − ti ). In this case, , i.e., z(t) = t < 0, i=1

 Kn (t1 , . . . , tn ) = ν ξ n min{t1 , . . . , tn }.

1. Let g(t) = θ (t) =



1, 0,

If additionally ξ = 1, then process z(t) ≡ n(0, t), and we have Kn (t1 , . . . , tn ) = ν min{t1 , . . . , tn }, 2[t; v(τ )] = ν

Zt dt

0

 

 exp i



0

Zt t0

  dτ v(τ ) − 1 .  

(4.70)

2. Let now g(t) = δ(t). In this case, process z(t) =

n X

ξi δ(t − ti )

i=1

is usually called the shot noise process. This process is a special case of the delta-correlated processes (see Sect. 5.5, page 130). For such a process, functional 2[t; v(τ )] is 2[t; v(τ )] = ν

Z∞

Zt dτ 0

n o dξ p(ξ ) eiξ v(τ ) − 1

(4.71)

−∞

and cumulant functions assume the form

 Kn (t1 , . . . , tn ) = ν ξ n δ(t1 − t2 )δ(t2 − t3 ) . . . δ(tn−1 − tn ). Note that Poisson’s process z(t) with arbitrary given impulse function g(t) can be always expressed in terms of Poisson’s delta-correlated random process zδ (t) by the formula Zt z(t) =

dτ g(t − τ )zδ (τ ).

(4.72)

0

Telegrapher’s Random Process Consider now statistical characteristics of telegrapher’s random process (Fig. 4.3) defined by the formula z(t) = a(−1)n(0,t)

(z(0) = a, z2 (t) ≡ a2 ),

(4.73)

where n(t1 , t2 ) is the random sequence of integers equal to the number of points of discontinuity in interval (t1 , t2 ). We consider two cases.

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113

z (t ) a t −a

Figure 4.3

A possible realization of telegrapher’s random process.

1. We will assume first that amplitude a is the deterministic quantity. For the two first moment functions of process z(t), we have the expressions ∞ X

hz(t)i = a

(−1)n(0,t) Pn(0,t) = ae−2n(0,t) = ae−2νt,

n(0,t)=0

D E D E hz(t1 )z(t2 )i = a (−1)n(0,t1 )+n(0,t2 ) = a2 (−1)n(t2 ,t1 ) 2

= a2 e−2n(t2 ,t1 ) = a2 e−2ν(t1 −t2 )

(t1 ≥ t2 ).

(4.74)

The higher moment functions for t1 > t2 > . . . > tn satisfy the recurrence relationship Mn (t1 , . . . , tn ) = hz(t1 ) . . . z(tn )i D E = a2 (−1)n(0,t1 )+n(0,t2 )+n(0,t3 )+···+n(0,tn ) D ED E = a2 (−1)n(t2 ,t1 ) (−1)n(0,t3 )+···+n(0,tn ) = hz(t1 )z(t2 )i Mn−2 (t3 , . . . , tn ).

(4.75)

This relationship is very similar to Eq. (4.42) for the Gaussian process with correlation function B(t1 , t2 ). The only difference is that the right-hand side of Eq. (4.75) coincides with only one term of the sum in Eq. (4.42), namely, with the term that corresponds to the above order of times. Consider now the characteristic functional of this process  t +  Z  8a [t; v(τ )] = exp i dτ z(τ )v(τ ) ,   *

0

where index a means that amplitude a is the deterministic quantity. Expanding the characteristic functional in the functional Taylor series and using recurrence formula (4.75), we obtain the integral equation

8a [t; v(τ )] = 1 + ia

Zt

dt1 e−2νt1 v(t1 )

0 2

Zt1

Zt

−a

dt1 0

0

dt2 e−2ν(t1 −t2 ) v(t1 )v(t2 )8a [t2 ; v(τ )].

(4.76)

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Lectures on Dynamics of Stochastic Systems

Differentiating Eq. (4.76) with respect to t, we obtain the integro-differential equation d 8a [t; v(τ )] = iae−2νt v(t) − a2 v(t) dt

Zt

dt1 e−2ν(t−t1 ) v(t1 )8a [t1 ; v(τ )].

(4.77)

0

2. Assume now amplitude a is the random quantity with probability density p(a). To obtain the characteristic functional of process z(t) in this case, we should average Eq. (4.77) with respect to random amplitude a. In the general case, such averaging cannot be performed analytically. Analytical averaging of Eq. (4.77) appears to be possible only if probability density of random amplitude a has the form 1 [δ (a − a0 ) + δ (a + a0 )] , (4.78) 2

 with hai = 0 and a2 = a20 (in fact, this very case is what is called usually telegrapher’s process). As a result, we obtain the integro-differential equation p(a) =

d 8[t; v(τ )] = −a20 v(t) dt

Zt

dt1 e−2ν(t−t1 ) v(t1 )8[t1 ; v(τ )].

(4.79)

0

Now, we dwell on an important limiting theorem concerning telegrapher’s random processes. Consider the random process ξN (t) = z1 (t) + · · · + zN (t), where all zk (t) are statistically independent telegrapher’s processes with zero-valued means and correlation functions hz(t)z(t + τ )i =

σ 2 −α|τ | e . N

In the limit N → ∞, we obtain that process ξ(t) = limN→∞ ξN (t) is the Gaussian random process with the exponential correlation function hξ(t)ξ(t + τ )i = σ 2 e−α|τ | , i.e., the Gaussian Markovian process. Thus, process ξN (t) for finite N is the finite-numberof-states process approximating the Gaussian Markovian process.

Generalized Telegrapher’s Random Process Consider now generalized telegrapher’s process defined by the formula z(t) = an(0,t) .

(4.80)

Here, n(0, t) is the sequence of integers described above and quantities ak are assumed statistically independent with distribution function p(a). Figure 4.4 shows a possible realization of such a process.

Random Quantities, Processes, and Fields

z (t )

115

a7

a2 a0

a6 a1 a5 t

a3 a4

Figure 4.4

a8

A possible realization of generalized telegrapher’s random process.

For process z(t), we have hz(t)i =

∞ X

 ak δk,n(0,t) = hai,

k=0

hz(t1 )z(t2 )i =

∞ X ∞ X

 hak al i δk,n(0,t1 ) δl,n(0,t2 )

k=0 k=0

(∞ ) ∞ D E X X







 = a2 δk,n(0,t1 ) δ0,n(t2 ,t1 ) + 1 − δk,n(0,t2 ) δ0,n(t2 ,t1 ) k=0

k=0

D E D E  = a2 e−ν(t1 −t2 ) + a2 1 − e−ν(t1 −t2 )

(t1 > t2 ),

and so on. In addition, the probability of absence of points of discontinuity in interval (t2 , t1 ) is given by the formula

 Pn(t2 ,t1 )=0 = δ0,n(t2 ,t1 ) = e−ν|t1 −t2 | . Earlier, we mentioned that Poisson’s stream of points and processes based on these streams are the Markovian processes. Below, we consider this important class of random processes in detail.

4.3

Markovian Processes

4.3.1

General Properties

In the foregoing section, we considered the characteristic functional that describes all statistical characteristics of random process z(t). Specification of the argument of the functional in the form v(t) =

n X k=1

vk δ(t − tk )

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Lectures on Dynamics of Stochastic Systems

transforms the characteristic functional into the joint characteristic function of random quantities zk = z(tk ) *

(

8n (v1 , . . . , vn ) = exp i

n X

)+ vk z(tk )

,

k=1

whose Fourier transform is the joint probability density of process z(t) at discrete instants Pn (z1 , t1 ; . . . ; zn , tn ) = hδ(z(t1 ) − z1 ) . . . δ(z(tn ) − zn )i .

(4.81)

Assume that the above instants are ordered according to the line of inequalities t1 ≥ t2 ≥ · · · ≥ tn . Then, by definition of the conditional probability, we have Pn (z1 , t1 ; . . . ; zn , tn ) = pn (z1 , t1 |z2 , t2 ; . . . ; zn , tn )Pn−1 (z2 , t2 ; . . . ; zn , tn ), (4.82) where pn is the conditional probability density of the value of process z(t) at instant t1 under the condition that function z(t) was equal to zk at instants tk for k = 2, . . . , n (z(tk ) = zk , k = 2, . . . , n). If process z(t) is such that the conditional probability density for all t1 > t2 is unambiguously determined by the value z2 of the process at instant t2 and is independent of the previous history, i.e., if pn (z1 , t1 |z2 , t2 ; . . . ; zn , tn ) = p(z1 , t1 |z2 , t2 ),

(4.83)

then this process is called the Markovian process, or the memoryless process. In this case, function p(z, t|z0 , t0 ) = hδ(z(t) − z)|z(t0 ) = z0 i

(t > t0 ),

(4.84)

is called the transition probability density. Setting t = t0 in Eq. (4.84), we obtain the equality p(z, t0 |z0 , t0 ) = δ(z − z0 ). Substituting expression (4.82) in Eq. (4.81), we obtain the recurrence formula for the n-time probability density of process z(t). Iterating this formula, we find the

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117

expression of probability density Pn in terms of the one-time probability density (t1 ≥ t2 ≥ · · · ≥ tn ) Pn (z1 , t1 ; . . . ; zn , tn ) = p(z1 , t1 |z2 , t2 ) . . . p(zn−1 , tn−1 |zn , tn )P(tn , zn ).

(4.85)

Substituting expression (4.82) in Eq. (4.81), we obtain the recurrence formula for the n-time probability density of process z(t). Iterating this formula, we find the relationship of probability density Pn with the one-time probability density (t1 ≥ t2 ≥ · · · ≥ tn ) Pn (z1 , t1 ; . . . ; zn , tn ) = p(z1 , t1 |z2 , t2 ) . . . p(zn−1 , tn−1 |zn , tn )P(tn , zn ).

(4.86)

Nevertheless, statistical analysis of stochastic equations often requires the knowledge of the characteristic functional of random process z(t).

4.3.2

Characteristic Functional of the Markovian Process

For the Markovian process z(t), no closed equation can be derived in the general case for the characteristic functional 8[t; v(τ )] = hϕ[t; v(τ )]i, where  t   Z  ϕ[t; v(τ )] = exp i dτ z(τ )v(τ ) .   0

Instead, we can derive the closed equation for the functional 9[z, t; v(τ )] = hδ(z(t) − z)ϕ[t; v(τ )]i,

(4.87)

describing correlations of process z(t) with its prehistory. The characteristic functional 8[t; v(τ )] can be obtained from functional 9[z, t; v(τ )] by the formula 8[t; v(τ )] =

Z∞

dz9[z, t; v(τ )].

(4.88)

−∞

To derive the equation in functional 9[z, t; v(τ )], we note that the following equality ϕ[t; v(τ )] = 1 + i

Zt

dt1 z(t1 )v(t1 )ϕ[t1 ; v(τ )]

(4.89)

0

holds. Substituting Eq. (4.89) in Eq. (4.87), we obtain the expression 9[z, t; v(τ )] = P(z, t) + i

Zt 0

dt1 v(t1 ) hδ(z(t) − z)z(t1 )ϕ[t1 ; v(τ )]i,

(4.90)

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Lectures on Dynamics of Stochastic Systems

where P(z, t) = hδ(z(t) − z)i is the one-time probability density of random quantity z(t). We rewrite Eq. (4.90) in the form 9[z, t; v(τ )] = P(z, t) Zt +i

dt1 v(t1 )

Z∞

dz1 z1 hδ(z(t) − z)δ(z(t1 ) − z1 )ϕ[t1 ; v(τ )]i.

−∞

0

(4.91) Taking into account the fact that process z(t) is the Markovian process, we can perform averaging in Eq. (4.91) to obtain the closed integral equation

9[z, t; v(τ )] = P(z, t) + i

Zt

dt1 v(t1 )

Z∞

dz1 z1 p(z, t|z1 , t1 )9[z1 , t1 ; v(τ )],

−∞

0

(4.92) where p(z, t; z0 , t0 ) is the transition probability density. Integrating Eq. (4.92) with respect to z, we obtain an additional relationship between the characteristic functional 8[t; v(τ )] and functional 9[z, t; v(τ )]. This relationship has the form 1 d 8[t; v(τ )] = iv(t) dt

Z∞

dz1 z1 9[z1 , t; v(τ )] = 9[t; v(τ )].

(4.93)

−∞

Multiplying Eq. (4.92) by z and integrating the result over z, we obtain the relationship between functionals 9[t; v(τ )] and 9[z, t; v(τ )] 9[t; v(τ )] = hz(t)i + i

Zt

dt1 v(t1 )

0

Z∞

dz1 hz(t)|z1 , t1 i 9[z1 , t1 ; v(τ )].

(4.94)

−∞

Equation (4.92) is generally a complicated integral equation whose explicit form depends on functions P(z, t) and p(z, t; z0 , t0 ), i.e., on parameters of the Markovian process. In some specific cases, the situation is simplified. For example, for telegrapher’s process, we have from Eq. (4.74) hz(t)|z1 , t1 i = z1 e−2ν(t−t1 ) , and we obtain Eq. (4.24), page 95.

hz(t)i = 0,

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119

Problems Problem 4.1 Derive expression for the characteristic functional of Poisson’s random process (4.67). Solution The characteristic functional of Poisson’s random process has the form

 t +  Z  8[t; v(τ )] = exp i dτ v(τ )z(τ ) .   *

0

(4.95)

z

We will average Eq. (4.95) in two stages. At first, we perform averaging over random quantity ξ and positions of random points tk :  t + n  Z  X exp i dτ v(τ ) ξk g(τ − tk )  

** 8[t; v(τ )] =

k=1

0

+

ξi ,tk

n

 n + * ZT Z∞  Zt  1 dt0 dξ exp iξ dτ v(τ )g(τ − t0 )  =    T 0

−∞

0

n

 t n + * ZT Z 1 dt0 W  dτ v(τ )g(τ − t0 ) , =  T t0

0

(4.96)

n

R∞ where W(v) = −∞ dξ p(ξ )eiξ v is the characteristic function of random quantity ξ . Let t < T. Then Eq. (4.96) can be rewritten in the form n +  t  * Z t Z T − t 1 dt0 W  dτ v(τ )g(τ − t0 ) + . 8[t; v(τ )] =  T T t0

0

(4.97)

n

Average now Eq. (4.97) over Poisson’s distribution pn = quantity n:

e−n (n)n of random n!

 t  n + * Z t Z ∞ −n X e T − t n 1 0 0   8[t; v(τ )] = dt W dτ v(τ )g(τ − t ) + (n) n! T T 0

t0

0

 

   t  t Z Z  n 0 0 = exp −n + dt W  dτ v(τ )g(τ − t ) + T − t .   T 0

t0

n

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As a result, we obtain the expression (4.67) for the characteristic functional of Poisson’s random process.

Problem 4.2 Show that random process ξN (t) = z1 (t) + · · · + zN (t), where all zk (t) are statistically independent telegrapher’s processes with zerovalued means and correlation functions hz(t)z(t + τ )i =

σ 2 −α|τ | e N

in the limit N → ∞ is the Gaussian Markovian random process ξ(t) = lim ξN (t) N→∞

with the exponential correlation function hξ(t)ξ(t + τ )i = σ 2 e−α|τ | . Remark Consider the differential equation for the characteristic functional of

random process ξN (t)  t +  Z  8N [t; v(τ )] = exp i dτ ξN (τ )v(τ ) .   *

0

Solution The characteristic functional of process zk (t) satisfies Eq. (4.79)

σ2 d 8[t; v(τ )] = − v(t) dt N

Zt

dt1 e−α(t−t1 ) v(t1 )8[t1 ; v(τ )],

0

from which follows that 8[t; v(τ )] → 1 for N → ∞. For the characteristic functional of random process ξN (t), we have the expression  t + *  Z  8N [t; v(τ )] = exp i dτ ξN (τ )v(τ ) = {8[t; v(τ )]}N .   0

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121

Consequently, it satisfies the equation d ln 8N [t; v(τ )] = −σ 2 v(t) dt

Zt

dt1 e−α(t−t1 ) v(t1 )

0

8[t1 ; v(τ )] . 8[t; v(τ )]

In the limit N → ∞, we obtain the equation d ln 8∞ [t; v(τ )] = −σ 2 v(t) dt

Zt

dt1 e−α(t−t1 ) v(t1 ),

0

which means that process ξ(t) is the Gaussian random process.

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Lecture 5 Correlation Splitting

5.1

General Remarks

For simplicity, we content ourselves here with the one-dimensional random processes (extensions to multidimensional cases are obvious). We need the ability of calculating correlation hz(t)]R[z(τ )]i, where R[z(τ )] is the functional that can depend on process z(t) both explicitly and implicitly. To calculate this average, we consider auxiliary functional R[z(τ ) + η(τ )], where η(t) is arbitrary deterministic function, and calculate the correlation hz(t)R[z(τ ) + η(τ )]i.

(5.1)

The correlation of interest will be obtained by setting η(τ ) = 0 in the final result. We can expand the above auxiliary functional R[z(τ ) + η(τ )] in the functional Taylor series with respect to z(τ ). The result can be represented in the form  ∞  Z δ  dτ z(τ ) R [z(τ ) + η(τ )] = exp R [η(τ )],  δη(τ )  −∞

where we introduced the functional shift operator. With this representation, we can obtain the following expression for correlation (5.1)   δ hz(t)R[z(τ ) + η(τ )]i =  t; hR[z(τ ) + η(τ )]i, (5.2) iδη(τ ) where functional  ∞ +  Z  z(t) exp i dτ z(τ )v(τ )   δ −∞ + = [t; v(τ )] = *  ∞ 2[v(τ )]. iδv(t)  Z  exp i dτ z(τ )v(τ )   *

−∞

Lectures on Dynamics of Stochastic Systems. DOI: 10.1016/B978-0-12-384966-3.00005-2 c 2011 Elsevier Inc. All rights reserved. Copyright

(5.3)

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Lectures on Dynamics of Stochastic Systems

Here 2[v(τ )] = ln 8[v(τ )] and 8[v(τ )] is the characteristic functional of random process z(t). Replacing differentiation with respect to η(τ ) by differentiation with respect to z(τ ) and setting η(τ ) = 0, we obtain the expression   hz(t)R[z(τ )i =  t;

  δ R[z(τ )] . iδz(τ )

(5.4)

If we expand functional 2[v(τ )] in the functional Taylor series (4.32), page 98, and differentiate the result with respect to v(t), we obtain Z∞ ∞ n Z∞ X i dt1 . . . dtn Kn+1 (t, t1 , . . . , tn )v(t1 ) . . . v(tn ) [t; v(τ )] = n! n=0

−∞

−∞

and expression (5.4) will assume the form

 z(t0 )R[z(τ )] =

  Z∞ Z∞ ∞ X 1 δ n R[z(τ )] dt1 . . . dtn Kn+1 (t0 , t1 , . . . , tn ) . n! δz(t1 ) . . . δz(tn ) n=0

−∞

(5.5)

−∞

In physical problems satisfying the condition of dynamical causality in time, statistical characteristics of the solution at instant t depend on the statistical characteristics of process z(τ ) for 0 ≤ τ ≤ t, which are completely described by the characteristic functional + *  Zt   8[t; v(τ )] = exp{2[t; v(τ )]} = exp i dτ z(τ )v(τ ) .   0

In formulas hold also for calculating statistical averages

0this case, the obtained z(t )R[t; z(τ )] for t0 < t, τ ≤ t, i.e., we have the equality  

0  z(t )R[t; z(τ )] =  t0 , t;

  δ R[t; z(τ )] iδz(τ )

(0 < t0 < t),

(5.6)

where δ 2[t; v(τ )] iδv(t0 )] Zt ∞ n Zt X i = dt1 . . . dtn Kn+1 (t0 , t1 , . . . , tn )v(t1 ) . . . v(tn ). n!

[t0 , t; v(τ )] =

n=0

0

0

(5.7)

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125

For t0 = t − 0, formula (5.6) holds as before, i.e.,   hz(t)R[t; z(τ )]i =  t, t;

  δ R[t; z(τ )] . iδz(τ )

(5.8)

However, expansion (5.7) not always gives the correct result in the limit t0 → t − 0 (which means that the limiting process and the procedure of expansion in the functional Taylor series can be non-commutable). In this case,  t + Z  z(t) exp i dτ z(τ )v(τ )   d 0 + = 2[t; v(τ )], [t, t; v(τ )] = *  t Z iv(t)dt   exp i dτ z(τ )v(τ )   *

(5.9)

0

and statistical averages in Eqs. (5.6) and (5.8) can be discontinuous at t0 = t − 0. Consider several examples of random processes.

5.2

Gaussian Process

In the case of the Gaussian random process z(t), all formulas obtained in the previous section become significantly simpler. In this case, the logarithm of characteristic functional 8[v(τ )] is given by Eq. (4.41), page 103, (we assume that the mean value of process z(t) is zero), and functional 2[t, v(τ )] assumes the form Z∞ Z∞

1 2[t, v(τ )] = − 2

dτ1 dτ2 B(τ1 , τ2 )v(τ1 )v(τ2 ).

−∞ −∞

As a consequence, functional [t; v(τ )] (5.3) is the linear functional

[t; v(τ )] = i

Z∞

dτ1 B(t, τ1 )v(τ1 ),

(5.10)

−∞

and Eq. (5.2) assumes the form

hz(t)R[z(τ ) + η(τ )]i =

Z∞ −∞

dτ1 B(t, τ1 )

δ hR[z(τ ) + η(τ )]i. δη(τ1 )

(5.11)

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Replacing differentiation with respect to η(τ ) by differentiation with respect to z(τ ) and setting η(τ ) = 0, we obtain the equality Z∞

hz(t)R[z(τ )]i =

−∞

 δ R[z(τ )] dτ1 B(t, τ1 ) δz(τ1 ) 

(5.12)

commonly known in physics as the Furutsu–Novikov formula [1, 46]. One can easily obtain the multi-dimensional extension of Eq. (5.12); it can be written in the form   Z



 δR[z] 0 0 , (5.13) zi1 ,...,in (r)R[z] = dr zi1 ,...,in (r)zj1 ,...,jn (r ) δzj1 ,...,jn (r0 ) where r stands for all continuous arguments of random vector field z(r) and i1 , . . . , in are the discrete (index) arguments. Repeated index arguments in the right-hand side of Eq. (5.13) assume summation. Formulas (5.12) and (5.13) extend formulas (4.17), (4.24) to the Gaussian random processes. If random process z(τ ) is defined only on time interval [0, t], then functional 2[t, v(τ )] assumes the form 1 2[t, v(τ )] = − 2

Zt Zt 0

dτ1 dτ2 B(τ1 , τ2 )v(τ1 )v(τ2 ),

(5.14)

0

and functionals [t0 , t; v(τ )] and [t, t; v(τ )] are the linear functionals δ [t , t; v(τ )] = 2[t, v(τ )] = i iδv(t0 ) 0

d [t, t; v(τ )] = 2[t, v(τ )] = i iv(t)dt

Zt

dτ B(t0 , τ )v(τ ),

0

Zt

(5.15) dτ B(t, τ )v(τ ).

0

As a consequence, Eqs. (5.6), (5.8) assume the form

0  z(t )R[t, z(τ )] =

Zt

dτ B(t0 , τ )

0



δR[z(τ )] δz(τ )



(t0 6 t)

(5.16)

that coincides with Eq. (5.12) if the condition δR[t; z(τ )] =0 δz(τ ) holds.

for τ < 0,

τ >t

(5.17)

Correlation Splitting

5.3

127

Poisson’s Process

The Poisson process is defined by Eq. (4.66), page 111, and its characteristic functional is given by Eq. (4.67), page 111. In this case, formulas (5.7) and (5.9) for functionals [t0 , t; v(τ )] and [t, t; v(τ )] assume the forms δ 2[t, v(τ )] iδv(t0 )  t  Zt0 Z ˙  dτ1 v(τ1 )g(τ1 − τ ), = −i dτ g(t0 − τ )W

[t0 , t; v(τ )] =

τ

0

d 2[t, v(τ )] iv(t)dt  t  Zt Z ˙  dτ1 v(τ1 )g(τ1 − τ ), = −i dτ g(t − τ )W

[t, t; v(τ )] =

(5.18)

τ

0

Z ∞ dW(v) =i dξ ξ p(ξ )eiξ v . dv −∞ Changing the integration order, we can rewrite equalities (5.18) in the form

˙ where W(v) =

[t0 , t; v(τ )] Z∞ =

dξ ξ p(ξ )

−∞

Zt0 0

   Zt  dτ g(t0 − τ ) exp iξ dτ1 v(τ1 )g(τ1 − τ )  

(t0 6 t). (5.19)

τ

As a result, we obtain that correlations of Poisson’s random process z(t) with functionals of this process are described by the expression

0  z(t )R[t; z(τ )] =ν

Z∞

−∞

dξ ξ p(ξ )

Zt0

 dτ 0 g(t0 − τ 0 ) R[t; z(τ ) + ξ g(τ − τ 0 )]

(t0 6 t). (5.20)

0

As we mentioned earlier, random process n(0, t) describing the number of jumps during temporal interval (0, t) is the special case of Poisson’s process. In this case, p(ξ ) = δ(ξ − 1) and g(t) = θ(t), so that Eq. (5.20) assumes the extra-simple form hn(0, t)R[t; n(0, τ )]i = ν

Zt0

 dτ R[t; n(0, τ ) + θ(t0 − τ )]

(t0 6 t).

(5.21)

0

Equality (5.21) extends formula (4.25) for Poisson’s random quantities to Poisson’s random processes.

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Lectures on Dynamics of Stochastic Systems

5.4

Telegrapher’s Random Process

Now, we dwell on telegrapher’s random process defined by formula (4.73), page 112, −z(t) = a(−1)n(0,t) , where a is the deterministic quantity. As we mentioned earlier, calculation of correlator hz(t)R[t; z(τ )]i for τ ≤ t assumes the knowledge of the characteristic functional of process z(t); unfortunately, the characteristic functional is unavailable in this case. We know only Eqs. (4.77) and (4.79), page 114, that describe the relationship between the functional  t + Z   1 d 8[t; v(τ )] = z(t) exp i dτ z(τ )v(τ ) 9[t; v(τ )] =   iv(t) dt *

0

and the characteristic functional 8[t; v(τ )] in the form  t +  Z  z(t) exp i dτ z(τ )v(τ )  

*

0

= ae−2νt + ia2

 t +  Z1  dt1 e−2ν(t−t1 ) v(t1 ) exp i dτ z(τ )v(τ ) .  

Zt

*

0

0

This relationship is sufficient to split correlator hz(t)R[t; z(τ )]i, i.e., to express it immediately in terms of the average functional hR[t; z(τ )]i. Indeed, the equality  t + Z δ  hz(t)R[t; z(τ ) + η(τ )]i = z(t) exp R[t; η(τ )] dτ z(τ )  δη(τ )  *

0

= aR[t; η(τ )]e−2νt 2

Zt

+a

dt1 e−2ν(t−t1 )

0

δ hR[t; z(τ )θ(t1 − τ ) + η(τ )]i, δη(t1 ) (5.22)

where η(t) is arbitrary deterministic function, holds for any functional R[t; z(τ )] for τ ≤ t. The final result is obtained by the limit process η → 0: hz(t)R[t; z(τ )]i = aR[t; 0]e−2νt 2

Zt

+a

0

dt1 e−2ν(t−t1 )



 δ e R[t, t1 ; z(τ )] , δz(t1 )

(5.23)

Correlation Splitting

129

where functional e R[t, t1 ; z(τ )] is defined by the formula e R[t, t1 ; z(τ )] = R[t; z(τ )θ(t1 − τ + 0)].

(5.24)

In the case of random quantity a distributed according to probability density distribution (4.78), page 114, additional averaging (5.23) over a results in the equality hz(t)R[t; z(τ )]i =

a20

Zt

−2ν(t−t1 )

dt1 e 0



 δ e R[t, t1 ; z(τ )] . δz(t1 )

(5.25)

Formula (5.25) is very similar to the formula for splitting the correlator of the Gaussian process z(t) characterized by the exponential correlation function (i.e., the Gaussian Markovian process) with functional R[t; z(τ )]. The difference consists in the fact that the right-hand side of Eq. (5.25) depends on the functional that is cut by the process z(τ ) rather than on functional R[t; z(τ )] itself. Let us differentiate Eq. (5.23) with respect to time t. Taking into account the fact that there is no need in differentiating with respect to the upper limit of the integral in the right-hand side of Eq. (5.23), we obtain the expression [47]     d d + 2ν hz(t)R[t; z(τ )]i = z(t) R[t; z(τ )] (5.26) dt dt usually called the differentiation formula. One essential point should be noticed. Functional R[t; z(τ )] in differentiation formula (5.26) is an arbitrary functional and can simply coincide with process z(t − 0). In the general case, the realization of telegrapher’s process is the generalized function. The derivative of this process is also the generalized function (the sequence of delta-functions), so that z(t)

d 1d 2 z(t) 6= z (t) ≡ 0 dt 2 dt

in the general case. These generalized functions, as any generalized functions, are defined only in terms of functionals constructed on them. In the case of our interest, such functionals are the average quantities denoted by angle brackets h· · · i, and the above differentiation formula describes the differential constraint between different functionals related to random process z(t) and its one-sided derivatives for t → t − 0, such as dz/dt, d2 z/dt2 . For example, formula (5.26) allows derivation of equalities, such as   D E d z(t) z(τ ) = 2ν z2 , dτ τ =t−0



 D E d2 z(t) 2 z(τ ) = 4ν 2 z2 . dτ τ =t−0

It is clear that these formulas can be obtained immediately by differentiating the corre

 lation function z(t)z(t0 ) with respect to t0 (t0 < t) followed by limit process t0 → t−0.

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Lectures on Dynamics of Stochastic Systems

Now, we dwell on some extensions of the above formulas for the vector Markovian process z(t) = {z1 (t), . . . , zN (t)}. For example, all above Markovian processes having the exponential correlation function D E hz(t)z(t + τ )i = z2 e−α|τ | (5.27) satisfy the equality     ∂ ∂ + αk hz1 (t) . . . zk (t)R[t; z(τ )]i = z1 (t) . . . zk (t) R[t; z(τ )] . ∂t ∂t

(5.28)

Formula (5.28) defines the rule of factoring the operation of differentiating with respect to time out of angle brackets of averaging; in particular, we have    n ∂n ∂ z1 (t) . . . zk (t) n R[t; z(τ )] = + αk hz1 (t) . . . zk (t)R[t; z(τ )]i, (5.29) ∂t ∂t where k = 1, . . . , N.

5.5

Delta-Correlated Random Processes

Random processes z(t) that can be treated as delta-correlated processes are of special interest in physics. The importance of this approximation follows first of all from the fact that it is physically obvious in the context of many physical problems and the corresponding dynamic systems allow obtaining closed equations for probability densities of the solutions to these systems. For the Gaussian delta-correlated (in time) process, correlation function has the form B(t1 , t2 ) = hz(t1 )z(t2 )i = B(t1 )δ(t1 − t2 ),

(hz(t)i = 0).

In this case, functionals 2[t; v(τ )], [t0 , t; v(τ )] and [t, t; v(τ )] introduced by Eqs. (5.14), (5.15) are 1 2[t; v(τ )] = − 2

Zt

dτ B(τ )v2 (τ ),

0

[t , t; v(τ )] = iB(t )v(t0 ), 0

0

[t, t; v(τ )] =

i B(t)v(t), 2

and Eqs. (5.16) assume significantly simpler forms  

0  δ 0 z(t )R[t; v(τ )] = B(t ) R[t; v(τ )] (0 < t0 < t), δz(t0 )   1 δ hz(t)R[t; v(τ )]i = B(t) R[t; v(τ )] . 2 δz(t)

(5.30)

Correlation Splitting

131

Formulas (5.30) show that statistical averages of the Gaussian delta-correlated process considered here are discontinuous at t0 = t. This discontinuity is dictated entirely by the fact that this process is delta-correlated; if the process at hand is not deltacorrelated, no discontinuity occurs (see Eq. (5.16)). Poisson’s delta-correlated random process corresponds to the limit process g(t) → δ(t). In this case, the logarithm of the characteristic functional has the simple form (see Eq. (4.71)); as a consequence, Eqs. (5.19) for functionals [t0 , t; v(τ )] and [t, t; v(τ )] assume the forms [t , t; v(τ )] = i 0

Z∞

dξ ξ p(ξ )eiξ v(t ) 0

(t0 < t),

−∞

Z∞

ν [t, t; v(τ )] = iv(t)

Z∞ Zξ h i iξ v(t) dξ ξ p(ξ ) e −1 =ν dξ p(ξ ) dηeiηv(t) ,

−∞

−∞

0

and we obtain the following expression for the correlation of Poisson random process z(t) with a functional of this process



 z(t )R[t; z(τ )] = ν 0

Z∞

 dξ ξ p(ξ ) R[t; z(τ ) + ξ δ(τ − τ 0 )]

(t0 6 t),

−∞

hz(t)R[t; z(τ )]i = ν

Z∞

−∞

dξ p(ξ )

Zξ

(5.31) dη hR[t; z(τ ) + ηδ(t − τ )]i.

0

These expressions also show that statistical averages are discontinuous at t0 = t. As in the case of the Gaussian process, this discontinuity is dictated entirely by the fact that this process is delta-correlated. In the general case of delta-correlated process z(t), we can expand the logarithm of characteristic functional in the functional Taylor series t

∞ n Z X i dτ Kn (τ )vn (τ ), 2[t; v(τ )] = n! n=1

0

where cumulant functions assume the form Kn (t1 , . . . , tn ) = Kn (t1 )δ(t1 − t2 ) . . . δ(tn−1 − tn ).

(5.32)

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As can be seen from Eq. (5.32), a characteristic feature of these processes consists in the validity of the equality ˙ v(τ )] = 2[t; ˙ v(t)] 2[t;



˙ v(τ )] = 2[t;

 d 2[t; v(τ )] , dt

(5.33)

which is of fundamental significance. This equality shows that, in the case of arbitrary delta-correlated process, quantity 2[t; v(τ )] appears not a functional, but simply a function of time t. In this case, functionals [t0 , t; v(τ )] and [t, t; v(τ )] are

[t0 , t; v(τ )] =

∞ n X i Kn+1 (t0 )vn (t0 ) n!


t0 < t ,

n=0

[t, t; v(τ )] =

∞ X n=0

in Kn+1 (t)vn (t), (n + 1)!

and formulas for correlation splitting assume the forms  n  ∞ n


0  X i 0 δ R[t; z(τ )] z(t )R[t; z(τ )] = Kn+1 (t ) t0 < t , n! δzn (t0 ) n=0  n  ∞ X δ R[t; z(τ )] in hz(t)R[t; z(τ )]i = Kn+1 (t) . (n + 1)! δzn (t)

(5.34)

n=0

These formulas describe the discontinuity of statistical averages at t0 = t in the general case of delta-correlated processes. Note that, for t0 > t, delta-correlated processes satisfy the obvious equality

0 

 z(t )R[t; z(τ )] = z(t0 ) hR[t; z(τ )]i.

(5.35)

Now, we dwell on the concept of random fields delta-correlated in time. We will deal with vector field f (x, t), where x describes the spatial coordinates and t is the temporal coordinate. In this case, the logarithm of characteristic functional can be expanded in the Taylor series with coefficients expressed in terms of cumulant functions of random field f (x, t) (see Secttion 4.2). In the special case of cumulant functions Kni1 ,...,im (x1 , t1 ; . . . , xn , tn ) = Kni1 ,...,im (x1 , . . . , xn ; t1 )δ(t1 − t2 ) . . . δ(tn−1 − tn ), (5.36)

Correlation Splitting

133

we will call field f (x, t) the random field delta-correlated in time t. In this case, functional 2[t; ψ(x0 , τ )] assumes the form t

Z Z ∞ n Z X i dτ . . . dx1 . . . dxn Kni1 ,...,im (x1 , . . . , xn ; τ ) 2[t; ψ(x , τ )] = n! 0

n=1

0

× ψi1 (x1 , τ ) . . . ψim (xn , τ ),

(5.37)

An important feature of this functional is the fact that it satisfies the equality similar to Eq. (5.33): ˙ ψ(x0 , τ )] = 2[t; ˙ ψ(x0 , t)]. 2[t;

5.5.1

(5.38)

Asymptotic Meaning of Delta-Correlated Processes and Fields

The nature knows no delta-correlated processes. All actual processes and fields are characterized by a finite temporal correlation radius, and delta-correlated processes and fields result from asymptotic expansions in terms of their temporal correlation radii. We illustrate the appearance of delta-correlated processes using the stationary Gaussian process with correlation radius τ0 as an example. In this case, the logarithm of the characteristic functional is described by the expression 2[t; v(τ )] = −

Zt

dτ1 v(τ1 )

0

Zτ1 0

  τ1 − τ2 dτ2 B v(τ2 ). τ0

(5.39)

Setting τ1 − τ2 = ξ τ0 , we transform Eq. (5.39) to the form 2[t; v(τ )] = −τ0

Zt

dτ1 v(τ1 )

0

τZ1 /τ0

dξ B (ξ ) v(τ1 − ξ τ0 ).

0

Assume now that τ0 → 0. In this case, the leading term of the asymptotic expansion in parameter τ0 is given by the formula 2[t; v(τ )] = −τ0

Z∞

dξ B (ξ )

0

Zt

dτ1 v2 (τ1 )

0

that can be represented in the form 2[t; v(τ )] = −B

eff

Zt 0

dτ1 v2 (τ1 ),

(5.40)

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Lectures on Dynamics of Stochastic Systems

where Beff =

Z∞

 dτ B

0

τ τ0

 =

1 2

Z∞

 dτ B

−∞

 τ . τ0

(5.41)

Certainly, asymptotic expression (5.40) holds only for the functions v(t) that vary during times about τ0 only slightly, rather than for arbitrary functions v(t). Indeed, if we specify this function as v(t) = vδ(t − t0 ), asymptotic expression (5.40) appears invalid; in this case, we must replace Eq. (5.39) with the expression 1 2[t; v(τ )] = − B(0)v2 2

(t > t0 )

corresponding to the characteristic function of process z(t) at a fixed time t = t0 . Consider now correlation hz(t)R[t; z(τ )]i given, according to the Furutsu–Novikov formula (5.16), page 126, by the expression hz(t)R[t; z(τ )]i =

Zt 0



t − t1 dt1 B τ0



 δ R[t; z(τ )] . δz(t1 )

The change of integration variable t − t1 → ξ τ0 transforms this expression to the form  Zt/τ0 hz(t)R[t; z(τ )]i = τ0 dξ B (ξ ) 0

 δ R[t; z(τ )] , δz(t − ξ τ0 )

(5.42)

which grades for τ0 → 0 into the equality obtained earlier for the Gaussian deltacorrelated process   δ eff hz(t)R[t; z(τ )]i = B R[t; z(τ )] δz(t) provided that the variational derivative in Eq. (5.42) varies only slightly during times of about τ0 . Thus, the approximation of process z(t) by the delta-correlated one is conditioned by small variations of functionals of this process during times approximating the process temporal correlation radius. Consider now telegrapher’s and generalized telegrapher’s processes. In the case of telegrapher’s process, the characteristic functional satisfies Eq. (4.79), page 114. The correlation radius of this process is τ0 = 1/2ν, and, for ν → ∞ (τ0 → 0), this equation grades for sufficiently smooth functions v(t) into the equation a2 d 8[t; v(τ )] = − 0 v2 (t)8[t; v(τ )], dt 2ν

(5.43)

Correlation Splitting

135

which corresponds to the Gaussian delta-correlated process. If we additionally assume that a20 → ∞ and lim a2 /2ν ν→∞ 0

= σ02 ,

then Eq. (5.43) appears independent of parameter ν. Of course, this fact does not mean that telegrapher’s process loses its telegrapher’s properties for ν → ∞. Indeed, for ν → ∞, the one-time probability distribution of process z(t) will as before correspond to telegrapher’s process, i.e., to the process with two possible states. As regards the correlation function and higher-order moment functions, they will possess for ν → ∞ all properties of delta-functions in view of the fact that n o  0, −2ν|τ | lim 2νe = ∞, ν→∞

if τ 6= 0, if τ = 0.

Such functions should be considered the generalized functions; their delta-functional behavior will manifest itself in the integrals of them. As Eq. (5.43) shows, limit process ν → ∞ is equivalent for these quantities to the replacement of process z(t) by the Gaussian delta-correlated process. This situation is completely similar to the approximation of the Gaussian random process with a finite correlation radius τ0 by the delta-correlated process for τ0 → 0.

Problems Problem 5.1 Split the correlator hg(ξ )f (ξ )iξ , where ξ is the random quantity. Solution

hg(ξ + η1 )f (ξ + η2 )iξ        d d d d hg(ξ + η1 )i hf (ξ + η2 )i. = exp 2 + −2 −2 idη1 idη2 idη1 idη2

Problem 5.2

 Split the correlator eωξ f (ξ ) ξ , where parameter ω can assume complex values. Solution

     

ωξ  1 d d hf (ξ + η)iξ . e f (ξ + η) ξ = exp 2 ω+ −2 i dη idη

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Lectures on Dynamics of Stochastic Systems

Problem 5.3



 Split the correlator eωξ ξ and ξ eωξ ξ assuming that ξ is the Gaussian random quantity with zero-valued mean. Solution

 2 2

ωξ  ω σ , e ξ = exp 2



ωξ

ξe



ξ

 2 2 ω σ = ωσ exp . 2 2

Problem 5.4

 Split the correlator eωξ f (ξ ) ξ assuming that ξ is the Gaussian random quantity with zero-valued mean. Solution

 2 2D E

ωξ  ω σ f (z + ωσ 2 ) . e f (ξ ) ξ = exp 2

Problem 5.5 Split the correlator hF[z(τ )]R[z(τ )]i. Solution

hF[z(τ ) + η1 (τ )]R[z(τ ) + η2 (τ )]i          δ δ 1 δ 1 δ 1 + −2 −2 = exp 2 i δη1 (τ ) δη2 (τ ) i δη1 (τ ) i δη2 (τ ) × hF[z(τ ) + η1 (τ )]i hR[z(τ ) + η2 (τ )]i, where η1 (τ ) and η2 (τ ) are arbitrary deterministic functions.

Problem 5.6 Split the correlator hF[z(τ )]R[z(τ )]i assuming that z(τ ) is the Gaussian random process. Solution hF[z(τ ) + η1 (τ )]R[z(τ ) + η2 (τ )]i  ∞ ∞  Z Z δ δ  hF[z(τ ) + η1 (τ )]i hR[z(τ ) + η2 (τ )]i, = exp dτ1 dτ2 B(τ1 , τ2 )  δη1 (τ1 ) δη2 (τ2 )  −∞ −∞

Correlation Splitting

137

or  hF[z(τ )]R[z(τ ) + η(τ )]i = F z(τ ) +

Z∞

−∞

 δ  hR[z(τ ) + η(τ )]i. dτ1 B(τ, τ1 ) δη(τ 1 )

Problem 5.7 D R∞ E Split the correlator ei −∞ dτ z(τ )v(τ ) R[z(τ )] assuming that z(τ ) is the Gaussian random process. Solution

+ *  Z∞ D R∞ E ei −∞ dτ z(τ )v(τ ) R[z(τ )] = 8[v(τ )] R z(τ ) + i dτ1 B(τ, τ1 )v(τ1 ) . −∞

Problem 5.8 E D Rt Split the correlator ei 0 dτ z(τ )v(τ ) R[t; z(τ )] assuming that z(τ ) is the Gaussian random process. Solution

D

+ *  Zt E ei 0 dτ z(τ )v(τ ) R[t; z(τ )] = 8[t; v(τ )] R t; z(τ ) + i dτ1 B(τ, τ1 )v(τ1 ) , Rt

0

where 8[t; v(τ )] is the characteristic functional of the Gaussian random process z(t).

Problem 5.9 Calculate the characteristic functional of process z(t) = ξ 2 (t), where ξ(t) is the Gaussian random process with the parameters hξ(t)i = 0,

hξ(t1 )ξ(t2 )i = B(t1 − t2 ).

Instruction The characteristic functional

8[t; v(τ )] = hϕ[t; ξ(τ )]i,

 t   Z  where ϕ[t; ξ(τ )] = exp i dτ v(τ )ξ 2 (τ )   0

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satisfies the stochastic equation D E d 8[t; v(τ )] = iv(t) ξ 2 (t)ϕ[t; ξ(τ )] . dt One needs to obtain the integral equation for quantity 9(t1 , t) = hξ(t1 )ξ(t)ϕ[t; ξ(τ )]i, 9(t1 , t) = B(t1 − t)8[t; v(τ )] + 2i

Zt

dτ B(t1 − τ )v(τ )9(τ, t),

0

and represent function 9(t1 , t) in the form 9(t1 , t) = S(t1 , t)8[t; v(τ )]. Solution The characteristic functional 8[t; v(τ )] has the form

 t   Z  8[t; v(τ )] = exp i dτ v(τ )S(τ, τ ) ,   0

where S(t, t) =

∞ X

S(n) (t, t),

n=0

S(n) (t, t) = (2i)n

Zt 0

dτ1 . . .

Zt

dτn v(τ1 ) . . . v(τn )B(t − τ1 )B(τ1 − τ2 ) . . . B(τn − t).

0

Consequently, the n-th cumulant of process z(t) = ξ 2 (t) is expressed in terms of quantity S(n−1) (t, t).

Problem 5.10 Calculate the characteristic functional of process z(t) = ξ 2 (t), where ξ(t) is the Gaussian Markovian process with the parameters hξ(t)i = 0,

hξ(t1 )ξ(t2 )i = B(t1 − t2 ) = σ 2 exp{−α|t1 − t2 |}.

Solution The characteristic functional 8[t; v(τ )] has the form

 t   Z  8[t; v(τ )] = exp i dτ v(τ )S(τ, τ ) ,   0

Correlation Splitting

139

where function S(t, t) can be described in terms of the initial-value problem h i d S(t, t) = −2α S(t, t) − σ 2 + 2iv(t)S2 (t, t), S(t, t)|t=0 = σ 2 . dt

Problem 5.11 Show that the expression hz(t1 )z(t2 )R[z(τ )]i = hz(t1 )z(t2 )i hR[z(τ )]i, holds for arbitrary functional R[z(τ )] and τ ≤ t2 ≤ t1 under the condition that z(t) is telegrapher’s process [48]. Expand functional R[z(τ )] in the Taylor series in z(τ ) and use formulas Eq. (4.75).

Instruction

Problem 5.12 Show that the expression [48] hF[z(τ1 )]z(t1 )z(t2 )R[z(τ2 )]i = hF[z(τ1 )]i hz(t1 )z(t2 )i hR[z(τ2 )]i + hF[z(τ1 )]z(t1 )i hz(t2 )R[z(τ2 )]i, holds for any τ1 ≤ t1 ≤ t2 ≤ τ2 and arbitrary functionals F[z(τ1 )] and R[z(τ2 )] under the condition that z(t) is telegrapher’s process with random parameter a distributed according to the probability density p(a) =

1 [δ(a − a0 ) + δ(a + a0 )]. 2

Instruction In terms of the Taylor expansion in z(τ ), functional R[z(τ2 )] can be represented in the form X X R[z(τ2 )] = + , 2k

2k+1

where the first sum consists of terms with the even number of co-factors z(τ ) and the second sum consists of terms with the odd number of such co-factors. The desired result immediately follows from the fact that * + * + X X hR[z(τ2 )]i = , hz(t2 )R[z(τ2 )]i = z(t2 ) . 2k

2k+1

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Problem 5.13 Show that the equality

0  0 z(t )R[t; z(τ )] = e−2ν(t −t) hz(t)R[t; z(τ )]i

(t0 > t)

holds for telegrapher’s process z(t). Instruction Rewrite the correlator z(t0 )R[t; z(τ )] (t0 ≥ t, τ < t) in the form





0   1

z(t )R[t; z(τ )] = 2 z(t0 )z(t)z(t)R[t; z(τ )] . a0

Lecture 6 General Approaches to Analyzing Stochastic Systems

In this Lecture, we will consider basic methods of determining statistical characteristics of solutions to the stochastic equations. Consider a linear (differential, integro-differential, or integral) stochastic equation. Averaging of such an equation over an ensemble of realizations of fluctuating parameters will not result generally in a closed equation for the corresponding average value. To obtain a closed equation, we must deal with an additional extended space whose dimension appears to be infinite in most cases. This approach makes it possible to derive the linear equation for average quantity of interest, but this equation will contain variational derivatives. Consider some special types of dynamic systems.

6.1

Ordinary Differential Equations

Assume dynamics of vector function x(t) is described by the ordinary differential equation d x(t) = v(x, t) + f (x, t), dt

x(t0 ) = x0 .

(6.1)

Here, function v(x, t) is the deterministic function and f (x, t) is the random function. The solution to Eq. (6.1) is a functional of f ( y, τ ) + v( y, τ ) with τ ∈ (t0 , t), i.e., x(t) = x[t; f ( y, τ ) + v( y, τ )]. From this fact follows the equality δ δ ∂F(x(t)) δxi (t) F (x(t)) = F (x(t)) = δfj (y, τ ) δvj ( y, τ ) ∂xi δfj ( y, τ ) valid for arbitrary function F(x). In addition, we have δ δ xi (t) = xi (t) = δij δ (x(t)−y). δfj ( y,t − 0) δvj (y,t − 0) Lectures on Dynamics of Stochastic Systems. DOI: 10.1016/B978-0-12-384966-3.00006-1 c 2011 Elsevier Inc. All rights reserved. Copyright

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The corresponding Liouville equation for the indicator function ϕ(x, t) = δ(x(t) − x) follows from Eq. (6.1) and has the form ∂ ∂ ϕ(x, t) = − {[v(x, t) + f (x, t)] ϕ(x, t)} , ϕ(x, t0 ) = δ(x − x0 ), ∂t ∂x

(6.2)

from which follows the equality δ ∂ δ ϕ(x, t) = ϕ(x, t) = − {δ(x − y)ϕ(x, t)}. δf ( y, t − 0) δv( y,t − 0) ∂x

(6.3)

Using this equality, we can rewrite Eq. (6.2) in the form, which may look at first sight more complicated 

 Z ∂ ∂ δ + v(x, t) ϕ(x, t) = dyf ( y, t) ϕ(x, t). ∂t ∂x δv( y, t)

(6.4)

Consider now the one-time probability density for solution x(t) of Eq. (6.1) P(x, t) = hϕ(x, t)i = hδ(x(t) − x)i. Here, x(t) is the solution of Eq. (6.1) corresponding to a particular realization of random field f (x, t), and angle brackets h· · · i denote averaging over an ensemble of realizations of field f (x, t). Averaging Eq. (6.4) over an ensemble of realizations of field f (x, t), we obtain the expression 

 Z ∂ ∂ δ h f ( y, t)ϕ(x, t)i. + v(x, t) P(x, t) = dy ∂t ∂x δv( y, t)

(6.5)

Quantity h f ( y, t)ϕ(x, t)i in the right-hand side of Eq. (6.5) is the correlator of random field f (y, t) with function ϕ(x, t), which is a functional of random field f ( y, τ ) and is given either by Eq. (6.2), or by Eq. (6.4). The characteristic functional  t + * Z  Z  8[t, t0 ; u( y, τ )] = exp i dτ dyf ( y, τ )u( y, τ )   t0

= exp {2[t, t0 ; u( y, τ )]} exhaustively describes all statistical characteristics of random field f (y, τ ) for τ ∈ (t0 , t).

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143

We split correlator h f ( y, t)ϕ(x, t)i using the technique of functionals. Introducing functional shift operator with respect to field v( y, τ ), we represent functional ϕ[x, t; f (y, τ ) + v( y, τ )] in the operator form ϕ[x, t; f ( y, τ ) + v( y, τ )]  t Z Z = exp  dτ dyf ( y, τ ) t0

 δ  ϕ[x, t; v( y, τ )]. δv( y, τ )

With this representation, the term in the right-hand side of Eq. (6.5) assumes the form +  δ fj ( y, t) exp dτ dy0 f ( y0 , τ )  δv( y0 , τ )  Z δ t0 +  t dy P(x, t) * δvj ( y, t)  Z R δ exp dτ dy0 f ( y0 , τ )  δv( y0 , τ )   t Z

*

R

t0

 ˙ t t, t0 ; =2

 δ P(x, t), iδv( y, τ )

where we introduced the functional ˙ t [t, t0 ; u( y, τ )] = 2

d ln 8[t, t0 ; u( y, τ )]. dt

Consequently, we can rewrite Eq. (6.5) in the form 

   ∂ ∂ δ ˙ + v(x, t) P(x, t) = 2t t, t0 ; P(x, t). ∂t ∂x iδv( y, τ )

(6.6)

Equation (6.6) is the closed equation with variational derivatives in the functional space of all possible functions {v(y, τ )}. However, for a fixed function v(x, t), we arrive at the unclosed equation 

     ∂ δ ∂ ˙ t t, t0 ; + v(x, t) P(x, t) = 2 ϕ[x, t; f ( y, τ )] , ∂t ∂x iδf ( y, t)

(6.7)

P(x, t0 ) = δ(x − x0 ). Equation (6.7) is the exact consequence of the initial dynamic equation (6.1). Statistical characteristics of random field f (x, t) appear in this equation only through ˙ t [t, t0 ; u( y, τ )] whose expansion in the functional Taylor series in powers functional 2 of u(x, τ ) depends on all space-time cumulant functions of random field f (x, t).

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As we mentioned earlier, Eq. (6.7) is unclosed in the general case with respect to function P(x, t), because quantity   δ ˙ δ(x(t) − x) 2t t, t0 ; iδf ( y, τ ) appeared in averaging brackets depends on the solution x(t) (which is a functional of random field f ( y, τ )) for all times t0 < τ < t. However, in some cases, the variational derivative in Eq. (6.7) can be expressed in terms of ordinary differential operators. In such conditions, equations like Eq. (6.7) will be the closed equations in the corresponding probability densities. The corresponding examples will be given below. Proceeding in a similar way, one can obtain the equation similar to Eq. (6.7) for the m-time probability density that refers to m different instants t1 < t2 < · · · < tm Pm (x1 , t1 ; . . . ; xm , tm ) = hϕm (x1 , t1 ; . . . ; xm , tm )i ,

(6.8)

where the indicator function is defined by the equality ϕm (x1 , t1 ; . . . ; xm , tm ) = δ(x(t1 ) − x1 ) . . . δ(x(tm ) − xm ). Differentiating Eq. (6.8) with respect to time tm and using then dynamic equation (6.1), one can obtain the equation   ∂ ∂ + v(xm ,tm ) Pm (x1 , t1 ; . . . ; xm , tm ) ∂tm ∂xm     (6.9) δ ˙ tm tm , t0 ; ϕm (x1 , t1 ; . . . ; xm , tm ) . = 2 iδf ( y, τ ) No summation over index m is performed here. The initial value to Eq. (6.9) can be derived from Eq. (6.8). Setting tm = tm−1 in Eq. (6.8), we obtain the equality Pm (x1 , t1 ; . . . ; xm , tm−1 ) = δ(xm − xm−1 )Pm−1 (x1 , t1 ; . . . ; xm−1 , tm−1 ), which just determines the initial value for Eq. (6.9).

6.2

Completely Solvable Stochastic Dynamic Systems

Consider now several dynamic systems that allow sufficiently adequate statistical analysis for arbitrary random parameters.

6.2.1

Ordinary Differential Equations

Multiplicative Action As the first example, we consider the vector stochastic equation with initial value d x(t) = z(t)g(t)F(x), dt

x(0) = x0 ,

(6.10)

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145

where g(t) and Fi (x), i = 1, . . . , N, are the deterministic functions and z(t) is the random process whose statistical characteristics are described by the characteristic functional  t + *  Z  8[t; v(τ )] = exp i dτ z(τ )v(τ ) = e2[t;v(τ )] .   0

Equation (6.10) has a feature that offers a possibility of determining statistical characteristics of its solutions in the general case of arbitrarily distributed process z(t). The point is that introduction of new ‘random’ time Zt T=

dτ z(τ )g(τ )

0

reduces Eq. (6.10) to the equation looking formally deterministic d x(T) = F(x), dT

x(0) = x0 ,

so that the solution to Eq. (6.10) has the following structure  t  Z x(t) = x(T) = x  dτ z(τ )g(τ ) .

(6.11)

0

Varying Eq. (6.11) with respect to z(τ ) and using Eq. (6.10), we obtain the equality d δ x(t) = g(τ ) x(T) = g(τ )F(x(t)). δz(τ ) dT

(6.12)

Thus, the variational derivative of solution x(t) is expressed in terms of the same solution at the same time. This fact makes it possible to immediately write the closed equations for statistical characteristics of problem (6.10). Let us derive the equation for the one-time probability density P(x, t) = hδ(x(t) − x)i . It has the form     δ ∂ ˙ P(x, t) = 2t t; δ (x(t)−x) . ∂t iδz(τ )

(6.13)

Consider now the result of operator δ/δz(τ ) applied to the indicator function ϕ(x, t) = δ(x(t) − x). Using formula (6.12), we obtain the expression δ ∂ δ (x(t)−x) = −g(τ ) {F(x)ϕ(x, t)} . δz(τ ) ∂x

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Consequently, we can rewrite Eq. (6.13) in the form of the closed operator equation   ∂ ∂ ˙ t t; ig(τ ) F(x) P(x, t), P(x, 0) = δ(x − x0 ), P(x, t) = 2 (6.14) ∂t ∂x whose particular form depends on the behavior of process z(t). For the two-time probability density P(x, t; x1 , t1 ) = hδ(x(t) − x)δ(x(t1 ) − x1 )i we obtain similarly the equation (for t > t1 ) ∂ P(x, t; x1 , t1 ) ∂t    ∂ ∂ ˙ = 2t t; ig(τ ) F(x) + θ(t1 −τ ) F(x1 ) P(x, t; x1 , t1 ) ∂x ∂x1

(6.15)

with the initial value P(x, t1 ; x1 , t1 ) = δ(x − x1 )P(x1 , t1 ), where function P(x1 , t1 ) satisfies Eq. (6.14). One can see from Eq. (6.15) that multidimensional probability density cannot be factorized in terms of the transition probability (see Sect. 4.3), so that process x(t) is not the Markovian process. The particular forms of Eqs. (6.14) and (6.15) are governed by the statistics of process z(t). If z(t) is the Gaussian process whose mean value and correlation function are

 hz(t)i = 0, B(t, t0 ) = z(t)z(t0 ) , then functional 2[t; v(τ )] has the form 1 2[t; v(τ )] = − 2

Zt

Zt dt1

0

dt2 B(t1 , t2 )v(t1 )v(t2 )

0

and Eq. (6.14) assumes the following form ∂ P(x, t) = g(t) ∂t

Zt

dτ B(t, τ )g(τ )

0

∂ ∂ Fj (x) Fk (x)P(x, t) ∂xj ∂xk

(6.16)

and can be considered as the extended Fokker–Planck equation. The class of problems formulated in terms of the system of equations d x(t) = z(t)F(x) − λx(t), dt

x(0) = x0 ,

(6.17)

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147

where F(x) are the homogeneous polynomials of power k, can be reduced to problem (6.10). Indeed, introducing new functions x(t) = e x(t)e−λt , we arrive at problem (6.10) with function g(t) = e−λ(k−1)t . In the important special case with k = 2 and functions F(x) such that xF(x) = 0, the system of equations (6.10) describes hydrodynamic systems with linear friction (see, e.g., [24]). In this case, the interaction between the components appears to be random. If λ = 0, energy conservation holds in hydrodynamic systems for any realization of process z(t). For t → ∞, there is the steady-state probability distribution P(x), which is, under the assumption that no additional integrals of motion exist, the uniform distribution over the sphere xi2 = E0 . If additional integrals of motion exist (as it is the case for finite-dimensional approximation of the two-dimensional motion of liquid), the domain of the steady-state probability distribution will coincide with the phase space region allowed by the integrals of motion. Note that, in the special case of the Gaussian process z(t) appeared in the onedimensional linear equation of type Eq. (6.17) d x(t) = −λx(t) + z(t)x(t), dt

x(0) = 1,

which determines the simplest logarithmic-normal random process, we obtain, instead of Eq. (6.16), the extended Fokker–Planck equation 

 Zt ∂ ∂ ∂ ∂ + λ x P(x, t) = dτ B(t, τ ) x xP(x, t), ∂t ∂x ∂x ∂x 0

(6.18)

P(x, 0) = δ(x − 1).

Additive Action Consider now the class of linear equations d x(t) = A(t)x(t) + f (t), dt

x(0) = x0 ,

(6.19)

where A(t) is the deterministic matrix and f (t) is the random vector function whose characteristic functional 8[t; v(τ )] is known. For the probability density of the solution to Eq. (6.19), we have     ∂ ∂ δ ˙ P(x, t) = − δ (x(t)−x) . (Aik (t)xk P(x, t)) + 2t t; ∂t ∂xi iδf (τ )

(6.20)

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Lectures on Dynamics of Stochastic Systems

In the problem under consideration, the variational derivative (for τ < t) the linear equation with the initial value d δ δ xi (t) = Aik (t) xk (t), dt δf (τ ) δf (τ )

δx(t) also satisfies δf (τ )

δ = δil . xi (t) δfl (τ ) t=τ

(6.21)

Equation (6.21) has no randomness and governs Green’s function Gil (t, τ ) of homogeneous system (6.19), which means that δ xi (t) = Gil (t, τ ). δfl (τ ) As a consequence, we have δ ∂ δ(x(t) − x) = − Gkl (t, τ )δ(x(t) − x), δfl (τ ) ∂xk and Eq. (6.20) appears to be converted into the closed equation   ∂ ∂ ∂ ˙ t t; iGkl (t, τ ) P(x, t) = − P(x, t). (Aik (t)xk P(x,t)) + 2 ∂t ∂xi ∂xk

(6.22)

From Eq. (6.22) follows that any moment of quantity x(t) will satisfy a closed linear equation that will include only a finite number of cumulant functions whose order will not exceed the order of the moment of interest. For the two-time probability density P(x, t; x1 , t1 ) = hδ(x(t) − x)δ(x(t1 ) − x1 )i, we quite similarly obtain the equation ∂ ∂ P(x, t; x1 , t1 ) = − (Aik (t)xk P(x,t; x1 ,t1 )) ∂t ∂xi   ∂ ˙ + 2t t; i {Gkl (t, τ ) + Gkl (t1 , τ )} P(x, t; x1 , t1 ) ∂xl

(6.23) (t > t1 )

with the initial value P(x, t1 ; x1 , t1 ) = δ(x − x1 )P(x1 , t1 ), where P(x1 , t1 ) is the one-time probability density satisfying Eq. (6.22). From Eq. (6.23) follows that x(t) is not the Markovian process. The particular form of Eqs. (6.22) and (6.23) depends on the structure of functional 8[t; v(τ )], i.e., on the behavior of random function f (t).

General Approaches to Analyzing Stochastic Systems

149

For the Gaussian vector process f (t) whose mean value and correlation function are

 h f (t)i = 0, Bij (t, t0 ) = fi (t)fj (t0 ) , Eq. (6.22) assumes the form of the extended Fokker–Planck equation ∂ ∂ P(x, t) = − (Aik (t)xk P(x,t)) ∂t ∂xi Zt +

dτ Bjl (t, τ )Gkj (t, t)Gml (t, τ )

0

∂2 P(x, t). ∂xk ∂xm

(6.24)

Consider the dynamics of a particle under random forces in the presence of friction [49] as an example of such a problem. Inertial Particle Under Random Forces The simplest example of particle diffusion under the action of random external force f (t) and linear friction is described by the linear system of equations (1.75), page 36

d v(t) = −λ [v(t) − f (t)] , dt

d r(t) = v(t), dt r(0) = 0,

(6.25)

v(0) = 0.

The stochastic solution to Eqs. (6.25) has the form v(t) = λ

Zt

−λ(t−τ )

dτ e

f (τ ),

Zt r(t) =

0

h i dτ 1 − e−λ(t−τ ) f (τ ).

(6.26)

0

In the case of stationary random process f (t) with the correlation tensor

 fi (t)fj (t0 ) = Bij (t − t0 ) and temporal correlation radius τ0 determined from the relationship Z∞

dτ Bii (τ ) = τ0 Bii (0),

0

Eq. (6.25) allows obtaining analytical expressions for correlators between particle velocity components and coordinates

 vi (t)vj (t) = λ

Zt

h i dτ Bij (τ ) e−λτ − e−λ(2t−τ ) ,

0



 1d

ri (t)rj (t) = ri (t)vj (t) = 2 dt

Zt 0

(6.27) i  h dτ Bij (τ ) 1 − e−λt 1 − e−λ(t−τ ) .

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In the steady-state regime, when λt  1 and t/τ0  1, but parameter λτ0 can be arbitrary, particle velocity is the stationary process with the correlation tensor

 vi (t)vj (t) = λ

Z∞

dτ Bij (τ )e−λτ ,

(6.28)

0



 and correlations ri (t)vj (t) and ri (t)rj (t) are as follows

 ri (t)vj (t) =

Z∞

dτ Bij (τ ),



 ri (t)rj (t) = 2t

Z∞

dτ Bij (τ ).

(6.29)

0

0

If we additionally assume that λτ0  1, the correlation tensor grades into

 vi (t)vj (t) = Bij (0),

(6.30)

which is consistent with (6.25), because v(t) ≡ f (t) in this limit. If the opposite condition λτ0  1 holds, then

 vi (t)vj (t) = λ

Z∞

dτ Bij (τ ).

0

This result corresponds to the delta-correlated approximation of random process f (t). Introduce now the indicator function of the solution to Eq. (6.25) ϕ(r, v, t) = δ (r(t) − r) δ (v(t) − v), which satisfies the Liouville equation   ∂ ∂ ∂ ∂ + v − λ v ϕ(r, v, t) = −λf (t) ϕ(r, v, t), ∂t ∂r ∂v ∂v

(6.31)

ϕ(r, v; 0) = δ (r) δ (v). The mean value of the indicator function ϕ(r, v, t) over an ensemble of realizations of random process f (t) is the joint one-time probability density of particle position and velocity P(r, v, t) = hϕ(r, v; t)i = hδ (r(t) − r) δ (v(t) − v)if . Averaging Eq. (6.31) over an ensemble of realizations of random process f (t), we obtain the unclosed equation   ∂ ∂ ∂ ∂ + v − λ v P(r, v, t) = −λ h f (t)ϕ(r, v, t)i, ∂t ∂r ∂v ∂v (6.32) P(r, v; 0) = δ (r) δ (v).

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This equation contains correlation h f (t)ϕ(r, v, t)i and is equivalent to the equality 

     . δ ∂ ∂ ∂ + v − λ v P(r, v, t) = 2 t; ϕ(r, v, t) , ∂t ∂r ∂v iδf (τ ) P(r, v; 0) = δ (r) δ (v) ,

(6.33)

where functional 2 [t; v(τ )] is related to the characteristic functional of random process f (t)  t +  Z  8 [t; ψ(τ )] = exp i dτ ψ(τ )f (τ ) = e2[t;ψ(τ )]   *

0

by the formula .

2 [t; ψ(τ )] =

d d ln 8 [t; ψ(τ )] = 2 [t; ψ(τ )] . dt dt

Functional 2[t; ψ(τ )] can be expanded in the functional power series t

t

Z ∞ n Z X i (n) dt1 . . . dtn Ki1 ,...,in (t1 , . . . , tn )ψi1 (t1 ) . . . ψin (tn ), 2[t; ψ(τ )] = n! n=1

0

0

where functions (n)

Ki1 ,...,in (t1 , . . . , tn ) =

1 δn 2[t; ψ(τ )] n i δψi1 (t1 ) . . . δψin (tn ) ψ=0

are the n-th order cumulant functions of random process f (t). Consider the variational derivative   ∂ δrk (t) ∂ δvk (t) δ ϕ(r, v, t) = − + ϕ(r, v, t). δfj (t0 ) ∂rk δfj (t0 ) ∂vk δfj (t0 )

(6.34)

In the context of dynamic problem (6.25), the variational derivatives of functions r(t) and v(t) in Eq. (6.34) can be calculated from Eqs. (6.26) and have the forms δvk (t) 0 = λδkj e−λ(t−t ) , 0 δfj (t )

h i δrk (t) −λ(t−t0 ) = δ 1 − e . kj δfj (t0 )

Using Eq. (6.35), we can now rewrite Eq. (6.34) in the form h  i δ −λ(t−t0 ) ∂ −λ(t−t0 ) ∂ ϕ(r, v; t) = − 1 − e + λe ϕ(r, v, t), δf (t0 ) ∂r ∂v

(6.35)

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after which Eq. (6.33) assumes the closed form 

 ∂ ∂ ∂ + v − λ v P(r, v, t) ∂t ∂r ∂v   h i . −λ(t−t0 ) ∂ −λ(t−t0 ) ∂ P(r, v, t), + λe = 2 t; i 1 − e ∂r ∂v

(6.36)

P(r, v; 0) = δ (r) δ (v). Note that from Eq. (6.36) follows that equations for the n-th order moment functions include cumulant functions of orders not higher than n. Assume now that f (t) is the Gaussian stationary process with the zero mean value and correlation tensor

 Bij (t − t0 ) = fi (t)fj (t0 ) . In this case, the characteristic functional of process f (t) is    1 Zt Zt  8 [t; ψ(τ )] = exp − dt1 dt2 Bij (t1 − t2 )ψi (t1 )ψj (t2 ) ,  2  0

0

.

functional 2[t; ψ(τ )] is given by the formula .

2 [t; ψ(τ )] = −ψi (t)

Zt

dt0 Bij (t − t0 )ψj (t0 ),

0

and Eq. (6.36) appears to be an extension of the Fokker–Planck equation 

 Zt ∂ ∂ ∂2 ∂ + v − λ v P(r, v, t) = λ2 dτ Bij (τ )e−λτ P(r, v, t) ∂t ∂r ∂v ∂vi ∂vj 0

+λ

Zt

  dτ Bij (τ ) 1 − e−λτ

0

∂2 P(r, v, t), ∂vi ∂rj

(6.37)

P(r, v; 0) = δ (r) δ (v). Equation (6.37) is the exact equation and remains valid for arbitrary times t. From this equation follows that r(t) and v(t) are the Gaussian functions. For moment

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functions of processes r(t) and v(t), we obtain in the ordinary way the system of equations 

 d

ri (t)rj (t) = 2 ri (t)vj (t) , dt   Zt  



 d + λ ri (t)vj (t) = vi (t)vj (t) + λ dτ Bij (τ ) 1 − e−λτ , dt 0



(6.38)

 Zt

 d 2 + 2λ vi (t)vj (t) = 2λ dτ Bij (τ )e−λτ . dt 0

From system (6.38) follows that steady-state values of all one-time correlators for λt  1 and t/τ0  1 are given by the expressions

 vi (t)vj (t) = λ

Z∞

dτ Bij (τ )e−λτ ,



 ri (t)vj (t) = Dij ,

0

(6.39)

 ri (t)rj (t) = 2tDij , where Z∞ Dij =

dτ Bij (τ )

(6.40)

0

is the spatial diffusion tensor, which agrees with expressions (6.28) and (6.29). Remark Temporal correlation tensor and temporal correlation radius of process v(t). We can additionally calculate the temporal correlation radius of velocity v(t), i.e., the scale of correlator vi (t)vj (t1 ) . Using equalities (6.35), we obtain for t1 < t the equation



 Zt1

 d 0 2 + λ vi (t)vj (t1 ) = λ dt0 Bij (t − t0 )e−λ(t1 −t ) dt 0 2 λ(t−t1 )

=λ e

Zt

dτ Bij (τ )e−λτ

(6.41)

t−t1

with the initial value



 vi (t)vj (t1 ) |t=t1 = vi (t1 )vj (t1 ) .

(6.42)

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In the steady-state regime, i.e., for λt  1 and λt1  1, but at fixed difference (t − t1 ), we obtain the equation with initial value (τ = t − t1 ) 

 Z∞

 d 2 λτ + λ vi (t + τ )vj (t) = λ e dτ1 Bij (τ1 )e−λτ1 , dτ τ



 vi (t + τ )vj (t) τ =0 = vi (t)vj (t) .

(6.43)

One can easily write the solution to Eq. (6.43); however, our interest here concerns only the temporal correlation radius τv of random process v(t). To obtain this quantity, we integrate Eq. (6.43) with respect to parameter τ over the interval (0, ∞). The result is λ

Z∞



 dτ vi (t + τ )vj (t) = vi (t)vj (t) + λ

0

Z∞

  dτ1 Bij (τ1 ) 1 − e−λτ1 ,

0

and we, using Eq. (6.39), arrive at the expression D E τv v2 (t) = Dii = τ0 Bii (0),

(6.44)

i.e., τ0 Bii (0) τv = 2  = v (t)

τ0 Bii (0) λ

R∞

=

dτ Bii (τ )e−λτ

  τ0 , 

1/λ,

for λτ0  1, (6.45) for λτ0  1.

0

Integrating Eq. (6.37) over r, we obtain the closed equation for the probability density of particle velocity 

 Zt ∂ ∂ ∂2 2 − λ v P(v, t) = λ dτ Bij (τ )e−λτ P(r, v, t), ∂t ∂v ∂vi ∂vj 0

P(r, v; 0) = δ (v) . The solution to this equation corresponds to the Gaussian process v(t) with correlation tensor (6.27), which follows from the fact that the second equation of system (6.25) is closed. It can be shown that, if the steady-state probability density exists under the condition λt  1, then this probability density satisfies the equation ∂ − vP(v, t) = λ ∂v

Z∞ 0

dτ Bij (τ )e−λτ

∂2 P(v, t), ∂vi ∂vj

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and the rate of establishing this distribution depends on parameter λ. The equation for the probability density of particle coordinate P(r, t) cannot be derived immediately from Eq. (6.37). Indeed, integrating Eq. (6.37) over v, we obtain the equality ∂ ∂ P(r, t) = − ∂t ∂r For function 

R

Z

P(r, 0) = δ (r).

vP(r, v, t)dv,

(6.46)

vk P(r, v, t)dv, we have the equality

Z Z ∂ ∂ +λ vk P(r, v, t)dv = − vk vP(r, v, t)dv ∂t ∂r Zt   ∂ P(r, t), − λ dτ Bkj (τ ) 1 − e−λτ ∂rj

(6.47)

0

and so on, i.e., this approach results in an infinite system of equations. Random function r(t) satisfies the first equation of system (6.25) and, if we would know the complete statistics of function v(t) (i.e., the multi-time statistics), we could calculate all statistical characteristics of function r(t). Unfortunately, Eq. (6.37) describes only one-time statistical quantities, and only the infinite system of equations similar to Eqs. (6.46), (6.47), and so on appears equivalent to the multi-time statistics of function v(t). Indeed, function r(t) can be represented in the form Zt r(t) =

dt1 v(t1 ),

0

so that the spatial diffusion coefficient in the steady-state regime assumes, in view of Eq. (6.44), the form Z∞ D E 1d D2 E r (t) = dτ hv(t + τ )v(t)i = τv v2 (t) = Dii = τ0 Bii (0), 2 dt

(6.48)

0

from which follows that it depends on the temporal correlation radius τv and the variance of random function v(t). However, in the case of this simplest problem, we know both variances and all correlations of functions v(t) and r(t) (see Eqs. (6.27)) and, consequently, can draw the equation for the probability density of particle coordinate P(r, t). This equation is the diffusion equation ∂ ∂2 P(r, t) = Dij (t) P(r, t), ∂t ∂ri ∂rj

P(r, 0) = δ (r),

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where  1 



 1d

ri (t)rj (t) = ri (t)vj (t) + rj (t)vi (t) 2 dt 2 Zt i  h = dτ Bij (τ ) 1 − e−λt 1 − e−λ(t−τ ) .

Dij (t) =

0

is the diffusion tensor (6.27). Under the condition λt  1, we obtain the equation ∂2 ∂ P(r, t) = Dij P(r, t), ∂t ∂ri ∂rj

P(r, 0) = δ (r)

(6.49)

with the diffusion tensor Z∞ Dij =

dτ Bij (τ ).

(6.50)

0

Note that conversion from Eq. (6.37) to the equation in probability density of particle coordinate (6.49) with the diffusion coefficient (6.50) corresponds to the so-called Kramers problem. Under the assumption that λτ0  1, where τ0 is the temporal correlation radius of process f (t), Eq. (6.37) becomes simpler 

 Zt ∂ ∂ ∂ ∂2 2 + v − λ v P(r, v, t) = λ dτ Bij (τ ) P(r, v, t), ∂t ∂r ∂v ∂vi ∂vj 0

P(r, v; 0) = δ (r) δ (v), and corresponds to the approximation of random function f (t) by the delta-correlated process. If t  τ0 , we can replace the upper limit of the integral with the infinity and proceed to the standard diffusion Fokker–Planck equation 

 ∂ ∂ ∂2 ∂ + v − λ v P(r, v, t) = λ2 Dij P(r, v, t), ∂t ∂r ∂v ∂vi ∂vj

(6.51)

P(r, v; 0) = δ (r) δ (v), with diffusion tensor (6.50). In this approximation, the combined random process {r(t), v(t)} is the Markovian process. Under the condition λt  1, there are the steady-state equation for the probability density of particle velocity −λ

∂2 ∂ vP(v) = λ2 Dij P(v), ∂v ∂vi ∂vj

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and the nonsteady-state equation for the probability density of particle coordinate ∂2 ∂ P(r, t) = Dij P(r, t), P(r, 0) = δ (r). ∂t ∂ri ∂rj Consider now the limit λτ0  1. In this case, we can rewrite Eq. (6.37) in the form 

   ∂2 ∂ ∂ ∂ + v − λ v P(r, v, t) = λBij (0) 1 − e−λt P(r, v, t) ∂t ∂r ∂v ∂vi ∂vj − Bij (0) 1 − e

−λt





∂2 P(r, v, t) + λ ∂vi ∂rj

Zt

dτ Bij (τ )

0

∂2 P(r, v, t), ∂vi ∂rj

P(r, v; 0) = δ (r) δ (v) . Integrating this equation over r, we obtain the equation for the probability density of particle velocity 

   ∂2 ∂ ∂ − λ v P(v, t) = λBij (0) 1 − e−λt P(v, t), ∂t ∂v ∂vi ∂vj P(v; 0) = δ (v),

and, under the condition λt  1, we arrive at the steady-state Gaussian probability density with variance

 vi (t)vj (t) = Bij (0). As regards the probability density of particle position, it satisfies, under the condition λt  1, the equation ∂ ∂2 P(r, t) = Dij P(r, t), P(r, 0) = δ (r) ∂t ∂ri ∂rj

(6.52)

with the same diffusion coefficient as previously. This is a consequence of the fact that Eq. (6.44) is independent of parameter λ. Note that this equation corresponds to the limit process λ → ∞ in Eq. (6.25) d r(t) = v(t), dt

v(t) = f (t),

r(0) = 0.

In the limit λ → ∞ (or λτ0  1), we have the equality v(t) ≈ f (t),

(6.53)

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and all multi-time statistics of random functions v(t) and r(t) will be described in terms of statistical characteristics of process f (t). In particular, the one-time probability

density  of particle velocity v(t) is the Gaussian probability density with variance vi (t)vj (t) = Bij (0), and the spatial diffusion coefficient is Z∞ 1d D2 E r (t) = dτ Bii (τ ) = τ0 Bii (0). D= 2 dt 0

As we have seen earlier, in the case of process f (t) such that it can be correctly described in the delta-correlated approximation (i.e., if λτ0  1), the approximate equality (6.53) appears inappropriate to determine statistical characteristics of process v(t). Nevertheless, Eq. (6.52) with the same diffusion tensor remains as before valid for the one-time statistical characteristics of process r(t), which follows from the fact that Eq. (6.48) is valid for any parameter λ and arbitrary probability density of random process f (t). Above, we considered several types of stochastic ordinary differential equations that allow obtaining closed statistical description in the general form. It is clear that similar situations can appear in dynamic systems formulated in terms of partial differential equations.

6.2.2

Partial Differential Equations

First of all, we note that the first-order partial differential equation 

 ∂ ∂ + z(t)g(t) F(x) ρ(x, t) = 0, ∂t ∂x

(6.54)

is equivalent to the system of ordinary differential equations (6.10), page 144 and, consequently, also allows the complete statistical description for arbitrary given random process z(t). Solution ρ(r, t) to Eq. (6.54) is a functional of random process z(t) of the form ρ(x, t) = ρ[x, t; z(τ )] = ρ(x, T[t; z(τ )]), with functional T[t; z(τ )] =

Zt

dτ z(τ )g(τ ),

0

so that Eq. (6.54) can be rewritten in the form 

 ∂ ∂ + F(x) ρ(x, T) = 0. ∂T ∂x

(6.55)

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159

The variational derivative is expressed in this case as follows δ dρ(x, T) δT[t; z(τ )] δ ρ(x, t) = ρ(x, T[t; z(τ )]) = δz(τ ) δz(τ ) dT δz(τ ) = θ(t − τ )g(τ )

dρ(x, T) ∂ = −θ(t − τ )g(τ ) F(x)ρ(x, t). dT ∂x

Consider now the class of nonlinear partial differential equations whose parameters are independent of spatial variable x,     ∂ ∂ ∂q ∂ ∂q + z(t) q(x, t) = F t, q, , ⊗ ,... , (6.56) ∂t ∂x ∂x ∂x ∂x where z(t) is the vector random process and F is the deterministic function. Solution to this equation is representable in the form     Zt  Zt ∂  Q (x, t) = Q x − dτ z(τ ), t, (6.57) q(x, t) = exp − dτ z(τ )  ∂x  0

0

where we introduced the shift operator. Function Q(x, t) satisfies the equation    Zt ∂  ∂Q(x, t) exp − dτ z(τ )  ∂x  ∂t 0          Zt    Zt  ∂Q  ∂ ∂ = F t, exp − dτ z(τ ) Q , exp − dτ z(τ ) ,   ∂x  ∂x  ∂x 0 0      t Z  ∂ ∂  ∂Q  ⊗ exp − dτ z(τ ) ,...,. (6.58)  ∂x ∂x  ∂x 0

The  be factored out from arguments of function  shift operator can ∂q ∂ ∂q F t, q, , ⊗ , . . . , and Eq. (6.58) assumes the form of the deterministic ∂x ∂x ∂x equation   ∂Q(x, t) ∂Q ∂ ∂Q = F t, Q, , ⊗ exp ,... . (6.59) ∂t ∂x ∂x ∂x Thus, the variational derivative can be expressed in the form ! Rt δQ x − dτ z(τ ), t δq(x, t) ∂ 0 = = −θ(t − τ ) q(x, t), δz(τ ) δz(τ ) ∂x

(6.60)

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i.e., the variational derivatives of the solution to problem (6.56) are expressed in terms of spatial derivatives. Consider the Burgers equation with random drift z(t) as a specific example:   ∂ ∂2 ∂ (6.61) v(x, t) + (v + z(t)) v(x, t) = ν 2 v(x, t). ∂t ∂x ∂x In this case, the solution has the form     Zt  Zt ∂  V(x, t) = V x − dτ z(τ ), t, v(x, t) = exp − dτ z(τ )  ∂x  0

0

where function V(x, t) satisfies the standard Burgers equation   ∂ ∂2 ∂ V(x, t) = ν 2 V(x, t). V(x, t) + V(x, t) ∂t ∂x ∂x In this example, the variational derivative is given by the expression   Zt δ δ ∂ v(x, t) = V x − dτ z(τ ), t = −θ(t − τ ) v(x, t). δzi (τ ) δzi (τ ) ∂xi

(6.62)

(6.63)

0

In the case of such problems, statistical characteristics of the solution can be determined immediately by averaging the corresponding expressions constructed from the solution to the last equation. Unfortunately, there is only limited number of equations that allow sufficiently complete analysis. In the general case, the analysis of dynamic systems appears possible only on the basis of various asymptotic and approximate techniques. In physics, techniques based on approximating actual random processes and fields by the fields delta-correlated in time are often and successfully used.

6.3

Delta-Correlated Fields and Processes

In the case of random field f (x, t) delta-correlated in time, the following equality holds (see Sect. 5.5, page 130) ˙ t [t, t0 ; v( y, τ )] ≡ 2 ˙ t [t, t0 ; v( y, t)], 2 and situation becomes significantly simpler. The fact that field f (x, t) is deltacorrelated means that t

Z Z ∞ n Z X i 2[t, t0 ; v( y, τ )] = dτ dy1 . . . dyn Kni1 ,...,in ( y1 , . . . , yn ; τ ) n! i=1

t0

× vi1 ( y1 , τ ) . . .vin ( yn , τ ),

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which, in turn, means that field f (x, t) is characterized by cumulant functions of the form Kni1 ,...,in ( y1 , t1 ; . . . ; yn , tn ) = Kni1 ,...,in ( y1 , . . . , yn ;t1 )δ(t1 − t2 ) . . . δ(tn−1 − tn ). In view of (6.3), page 142, Eqs. (6.7) and (6.9) appear to be in this case the closed operator equations in functions P(x, t), p(x, t|x0 , t0 ), and Pm (x1 , t1 ; . . . ; xm , tm ). Indeed, Eq. (6.7), page 143 is reduced to the equation     ∂ ∂ ∂ ˙ + v(x, t) P(x,t) = 2t t, t0 ; i δ( y − x) P(x, t), ∂t ∂x ∂x (6.64) P(x, 0) = δ(x − x0 ), whose concrete form is governed by functional 2[t, t0 ; v( y, τ )], i.e., by statistical behavior of random field f (x, t). Correspondingly, Eq. (6.9) for the m-time probability density is reduced to the operator equation (t1 ≤t2 ≤ . . . ≤tm ) 

 ∂ ∂ + v(xm , tm ) Pm (x1 ,t1 ; . . . ; xm , tm ) ∂t ∂xm   ∂ ˙ tm tm , t0 ; i δ( y − xm ) Pm (x1 , t1 ; . . . ; xm , tm ), =2 ∂xm

(6.65)

Pm (x1 , t1 ; . . . ; xm , tm−1 ) = δ(xm − xm−1 )Pm−1 (x1 , t1 ; . . . ; xm−1 , tm−1) , We can seek the solution to Eq. (6.65) in the form Pm (x1 , t1 ; . . . ; xm , tm ) = p(xm , tm |xm−1 , tm−1 )Pm−1 (x1 , t1 ; . . . ; xm−1 , tm−1 ).

(6.66)

Because all differential operations in Eq. (6.65) concern only tm and xm , we can substitute Eq. (6.66) in Eq. (6.65) to obtain the following equation for the transition probability density     ∂ ∂ ∂ ˙ t t, t0 ; i δ( y − x) p(x, t|x0 , t0 ), + v(x, t) p(x, t|x0 ,t0 ) =2 ∂t ∂x ∂x (6.67) p(x, t|x0 , t0 )|t→t0 = δ(x − x0 ). Here, we denoted variables xm and tm as x and t and variables xm−1 and tm−1 as x0 and t0 . Using formula (6.66) (m − 1) times, we obtain the relationship Pm (x1 , t1 ; . . . ; xm , tm ) = p(xm , tm |xm−1 , tm−1 ) . . . p(x2 , t2 |x1 , t1 )P(x1 , t1 ),

(6.68)

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where P(x1 , t1 ) is the one-time probability density governed by Eq. (6.64). Equality (6.68) expresses the many-time probability density in terms of the product of transition probability densities, which means that random process x(t) is the Markovian process. The transition probability density is defined in this case as follows: p(x, t|x0 , t0 ) = hδ(x(t|x0 , t0 ) − x)i . Special models of parameter fluctuations can significantly simplify the obtained equations. For example, in the case of the Gaussian delta-correlated field f (x, t), the correlation tensor has the form (h f (x, t)i = 0) Bij (x, t; x0 , t0 ) = 2δ(t − t0 )Fij (x, x0 ; t). Then, functional 2[t, t0 ; v( y, τ )] assumes the form 2[t, t0 ; v( y, τ )] = −

Zt

Z dτ

Z dy1

dy2 Fij ( y1 , y2 ; τ )vi ( y1 , τ )vj (y2 , τ ),

t0

and Eq. (6.64) reduces to the Fokker–Planck equation   ∂ ∂2 ∂ [vk (x,t) + Ak (x,t)] P(x, t) = [Fkl (x, x,t)P(x,t)] , + ∂t ∂xk ∂xk ∂xl

(6.69)

where ∂ 0 Ak (x, t) = Fkl (x, x ;t) . ∂xl0 x0 =x In view of the special role that the Gaussian delta-correlated field f (x, t) plays in physics, we give an alternative and more detailed discussion of this approximation commonly called the approximation of the Gaussian delta-correlated field in Lecture 8. We illustrate the above general theory by the examples of several equations.

6.3.1

One-Dimensional Nonlinear Differential Equation

Consider the one-dimensional stochastic equation d x(t) = f (x, t) + z(t)g(x, t), dt

x(0) = x0 ,

(6.70)

where f (x, t) and g(x, t) are the deterministic functions and z(t) is the random function of time. For indicator function ϕ(x, t) = δ(x(t) − x), we have the Liouville equation   ∂ ∂ ∂ + f (x, t) ϕ(x, t) = −z(t) {g(x, t)ϕ(x, t)}, ∂t ∂x ∂x

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163

so that the equation for the one-time probability density P(x, t) has the form 

     ∂ ∂ δ ˙ + f (x, t) P(x, t) = 2t t, ϕ(x, t) . ∂t ∂x iδz(τ )

In the case of the delta-correlated random process z(t), the equality ˙ t [t, v(τ )] = 2 ˙ t [t, v(t)] 2 holds. Taking into account the equality ∂ δ ϕ(x, t) = − {g(x, t)ϕ(x, t)} , δz(t − 0) ∂x we obtain the closed operator equation     ∂ ∂ ∂ ˙ t t, i g(x, t) P(x, t). + f (x, t) P(x, t) = 2 ∂t ∂x ∂x

(6.71)

For the Gaussian delta-correlated process, we have 1 2[t, v(τ )] = − 2

Zt

dτ B(τ )v2 (τ ),

(6.72)

0

and Eq. (6.71) assumes the form of the Fokker–Planck equation   ∂ ∂ ∂ ∂ 1 + f (x, t) P(x, t) = B(t) g(x, t) g(x, t)P(x, t). ∂t ∂x 2 ∂x ∂x For Poisson’s delta-correlated process z(t), we have  ∞  Zt Z  2[t, v(τ )] = ν dτ dξ p(ξ )eiξ v(τ ) − 1 ,   0

(6.73)

(6.74)

−∞

and Eq. (6.71) reduces to the form 

 ∞    Z ∂ ∂ ∂ + f (x, t) P(x, t) = ν dξ p(ξ )e−ξ ∂x g(x,t) −1 P(x, t).   ∂t ∂x −∞

If we set g(x, t) = 1, Eq. (6.70) assumes the form d x(t) = f (x, t) + z(t), dt

x(0) = x0 .

(6.75)

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In this case, the operator in the right-hand side of (6.75) is the shift operator, and Eq. (6.75) assumes the form of the Kolmogorov–Feller equation 

 Z∞ ∂ ∂ + f (x, t) P(x, t) = ν dξ p(ξ )P(x − ξ, t) − νP(x, t). ∂t ∂x −∞

Define now g(x, t) = x, so that Eq. (6.70) reduces to the form d x(t) = f (x, t) + z(t)x(t), dt

x(0) = x0 .

In this case, Eq. (6.75) assumes the form  ∞    Z  ∂ ∂ ∂ + f (x, t) P(x, t) = ν dξ p(ξ )e−ξ ∂x x −1 P(x, t).   ∂t ∂x

(6.76)

−∞

To determine the action of the operator in the right-hand side of Eq. (6.76), we expand it in series in ξ  n ∞ o n X ∂ (−ξ )n ∂ x −ξ ∂x x P(x, t) − 1 P(x, t) = e n! ∂x n=1

and consider the action of every term. Representing x in the form x = eϕ , we can transform this formula as follows (the fact that x is the alternating quantity is insignificant here) ∞ X (−ξ )n

∂n ϕ e P(eϕ , t) n! ∂ϕ n n=1 n o ∂ = e−ϕ e−ξ ∂ϕ − 1 eϕ P(eϕ , t) = e−ξ P(eϕ−ξ , t) − P(eϕ , t). e−ϕ

Reverting to variable x, we can represent Eq. (6.76) in the final form of the integrodifferential equation similar to the Kolmogorov–Feller equation 

 Z∞ ∂ ∂ + f (x, t) P(x, t) = ν dξ p(ξ )e−ξ P(xe−ξ , t) − νP(x, t). ∂t ∂x −∞

In Lecture 4, page 89, we mentioned that formula Zt x(t) = 0

dτ g(t − τ )z(τ )

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165

relates Poisson’s process x(t) with arbitrary impulse function g(t) to Poisson’s deltacorrelated random process z(t). Let g(t) = e−λt . In this case, process x(t) satisfies the stochastic differential equation d x(t) = −λx(t) + z(t) dt and, consequently, both transition probability density and one-time probability density of this process satisfy, according to Eq. (6.75), the equations ∂ P(x, t) = b L(x)P(x, t), ∂t

∂ p(x, t|x0 , t0 ) = b L(x)p(x, t|x0 , t0 ), ∂t where operator

 ∞  Z  ∂ ∂ b L(x) = λ x + ν dξ p(ξ )e−ξ ∂x − 1 .   ∂x

(6.77)

−∞

6.3.2

Linear Operator Equation

Consider now the linear operator equation d x(t) = b A(t)x(t) + z(t)b B(t)x(t), dt

x(0) = x0 ,

(6.78)

where b A(t) and b B(t) are the deterministic operators (e.g., differential operators with respect to auxiliary variables or regular matrixes). We will assume that function z(t) is the random delta-correlated function. Averaging system (6.78), we obtain, according to general formulas,     d δ ˙ t t, hx(t)i = b A(t) hx(t)i + 2 x(t) . dt iδz(t)

(6.79)

Then, taking into account the equality δ x(t) = b B(t)x(t) δz(t − 0) that follows immediately from Eq. (6.78), we can rewrite Eq. (6.79) in the form   d ˙ t t, −ib hx(t)i = b A(t) hx(t)i + 2 B hx(t)i. dt

(6.80)

Thus, in the case of linear system (6.78), equations for average values also are the linear equations.

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We can expand the logarithm of the characteristic functional 2[t; v(τ )] of deltacorrelated processes in the functional Fourier series t

2[t; v(τ )] =

∞ n Z X i dτ Kn (τ )vn (τ ), n! n=1

(6.81)

0

where Kn (t) determine the cumulant functions of process z(t). Substituting Eq. (6.81) in Eq. (6.80), we obtain the equation ∞ X n  1 d hx(t)i = b B(t) hx(t)i. A(t) hx(t)i + Kn (t) b dt n!

(6.82)

n=1

If there exists power l such that b Bl (t) = 0, then Eq. (6.82) assumes the form l−1 X n  d 1 hx(t)i = b B(t) hx(t)i. A(t) hx(t)i + Kn (t) b dt n!

(6.83)

n=1

In this case, the equation for average value depends only on a finite number of cumulants of process z(t). This means that there is no necessity in knowledge of probability distribution of function z(t) in the context of the equation for average value; sufficient information includes only certain cumulants of process and knowledge of the fact that process z(t) can be considered as the delta-correlated random process. Statistical description of an oscillator with fluctuating frequency is a good example of such system in physics.

Problems Problem 6.1 Starting from the backward Liouville equation (3.8), page 71 describing the behavior of dynamic system (6.1), page 141 as a function of initial values t0 and x0 , derive the backward equation for probability density P(x, t|x0 , t0 ) = hδ(x(t|x0 , t0 ) − x)i. Solution



∂ ∂ + v(x0 , t0 ) ∂t0 ∂x0



  ˙ P(x, t|x0 , t0 ) = 2t0 t, t0 ;

  δ δ (x(t|x0 , t0 ) − x) , iδf ( y, τ )

where ˙ t0 [t, t0 ; u( y, τ )] = 2

d ln 8[t, t0 ; u( y, τ )] dt0

and 8[t, t0 ; u( y, τ )] is the characteristic functional of field f (x0 , t0 ).

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Problem 6.2 Derive the equation for probability density of the solution to the one-dimensional problem on passive tracer diffusion ∂ ∂ ρ(x, t) + v(t) f (x)ρ(x, t) = 0, ∂t ∂x where v(t) is the Gaussian stationary random process with the parameters  D E

 hv(t)i = 0, v(t)v(t0 ) = B(t − t0 ) Bv (0) = v2 (t) and f (x) is the deterministic function. Instruction Average the Liouville equation (3.19), page 74 that can be rewritten

in the form    ∂ ∂f (x) ∂ ∂ ϕ(x, t; ρ) = −v(t) f (x) − 1+ ρ ϕ(x, t; ρ), ∂t ∂x ∂x ∂ρ ϕ(x, 0; ρ) = δ(ρ0 (x) − ρ) over an ensemble of realizations of random process v(t). Use the Furutsu– Novikov formula (5.16), page 126 for functional ϕ[x, t; ρ; v(τ )] and take into account Eq. (6.12), page 145 that assumes in this problem the form δ ∂ϕ(x, T(t), ρ) ϕ [x, t; ρ; v(τ )] = θ(t − t0 ) 0 δv(t ) ∂T    ∂ ∂f (x) ∂ = −θ(t − t0 ) f (x) − 1+ ρ ϕ(x, t; ρ), ∂x ∂x ∂ρ where T[t, v(τ )] =

Z

iside step function.

t

dτ v(τ ), is the new “random” time, and θ(t) is the Heav-

0

Solution

∂ P(x, t; ρ) = ∂t

Zt

dτ B(τ )



0

  ∂ df (x) ∂ f (x) − 1+ ρ ∂x dx ∂ρ

  ∂ df (x) ∂ × f (x) − ρ P(x, t; ρ), ∂x dx ∂ρ P(0, x; ρ) = δ(ρ0 (x) − ρ),

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which can be rewritten in the form ∂ P(x, t; ρ) = ∂t

Zt

dt B(t ) 0

0

∂ 2 ∂ ∂ df (x) ∂ f (x) − f (x) ρ ∂x ∂x ∂x dx ∂ρ



0

     ) ∂ ∂ ∂ df (x) df (x) 2 ∂ f (x) 1+ ρ + ρ 1+ ρ − P(x, t; ρ), dx ∂x ∂ρ dx ∂ρ ∂ρ P(x, 0; ρ) = δ(ρ0 (x) − ρ). Remark Note that, if function f (x) varies along x with characteristic scale k−1 and

is a periodic function (‘fast variation’), then additional averaging over x results in an equation for ‘slow’ spatial variations ∂ P(x, t; ρ) = ∂t

Zt

( dt B(t ) 0

0

f 2 (x)

0

)   ∂2 df (x) 2 ∂ 2 2 + ρ P(x, t; ρ), dx ∂x2 ∂ρ 2

P(x, 0; ρ) = δ(ρ0 (x) − ρ). If, moreover, the initial condition is independent of x (a homogeneous initial state), the average probability density will be also independent of x and satisfy the equation   Zt ∂ df (x) 2 ∂2 P(t; ρ) = dt0 B(t0 ) 2 ρ 2 P(t; ρ), ∂t dx ∂ρ

P(0; ρ) = δ(ρ0 − ρ).

0

The solution to this equation is the probability density of the lognormal random process. For t  τ0 this equation grades into the equation ∂ ∂2 P(t; ρ) = Dρ 2 ρ 2 P(t; ρ), ∂t ∂ρ

P(0; ρ) = δ(ρ0 − ρ),

(6.84)

where Dρ is the diffusion coefficient in ρ-space, df (x) Dρ = dx 

2 Z∞

dt0 B(t0 ) = σ 2 τ0



df (x) dx

2

,

0

and τ0 is the temporal correlation radius of random process v(t). Equation (6.84) can be rewritten in the form ∂ ∂ ∂ ∂ P(t; ρ) = Dρ ρP(t; ρ) + Dρ ρ ρP(t; ρ), ∂t ∂ρ ∂ρ ∂ρ

P(0; ρ) = δ(ρ0 − ρ). (6.85)

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169

Features of a logarithmically normal random process will be considered in detail later, in Sect. 8.3.3, page 200.

Problem 6.3 Derive the equation for the average of the field satisfying the Burgers equation with random shear ∂2 ∂ ∂ q(x, t) + (q + z(t)) q(x, t) = ν 2 q(x, t), ∂t ∂x ∂x

(6.86)

where z(t) is the stationary Gaussian process with correlation function

 B(t − t0 ) = z(t)z(t0 ) . Instruction Averaging Eq. (6.86), use the Furutsu–Novikov formula

Zt hz(t)q(x, t)i = 0

 δ dτ B(t − τ ) q(x, t) , δz(τ ) 

and the expression hq[z(τ ) + η1 (τ )]q[z(τ ) + η2 (τ )]i  t  Zt Z  2 δ = exp dτ1 dτ2 B(τ1 − τ2 )  δη1 (τ1 )δη2 (τ2 )  0

0

× hq[z(τ ) + η1 (τ )]q[z(τ ) + η2 (τ )]i to split correlations (see Lecture 5 and Problem 5.5, page 136) and rewrite these formulas, in view of Eq. (6.63), in the form Zt hz(t)q(x, t)i = −

dτ B(τ )

0

∂ hq(x, t)i , ∂x

 t   Z D E ∂2  2 hq(x, t + η1 )i hq(x, t + η2 )i |η=0 q (x, t) = exp 2 dτ (t − τ ) B(τ )  ∂η1 ∂η2  0

 n 2  n ∞ n Zt X 2  ∂ hq(x, = dτ (t − τ ) B(τ ) t)i . n! ∂xn n=0

0

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Solution

 t n  n 2 Z ∞ ∂ 1 ∂ X 2n  ∂ hq(x, t)i + hq(x, dτ (t − τ )B(τ ) t)i ∂t 2 ∂x n! ∂xn n=0

 = ν +

Zt

0

 dτ B(τ )

0

∂2 hq(x, t)i . ∂x2

(6.87)

Problem 6.4 Derive Eq. (6.87) for average field starting from the one-dimensional Burger’s equation with random drift (6.86) and Eq. (6.57), where z(t) is the stationary Gaussian process with correlation function

 B(t − t0 ) = z(t)z(t0 ) . Instruction Use equality

hq(x, t)i = exp

= exp

  1 Zt 2  t Z 

0

0

Zt dτ1 0

 ∂2  dτ2 B(τ1 − τ2 ) 2 Q(x, t) ∂x 

 ∂2  dτ (t − τ )B(τ ) 2 Q(x, t). ∂x 

Problem 6.5 Under the assumption that random process z(t) is delta-correlated, derive the equation for probability density of the solution to stochastic equation (1.30), page 19 describing the stochastic parametric resonance. Instruction Rewrite Eq. (1.30) in the form of the system of equations

d d x(t) = y(t), y(t) = −ω02 [1 + z(t)]x(t), dt dt x(0) = x0 , y(0) = y0 .

(6.88)

Solution

    . ∂ ∂ ∂ ∂ P(x, y, t) = −y + ω02 x P(x, y, t) + 2t t; −iω02 x P(x, y, t), ∂t ∂x ∂y ∂y P(x, y, 0) = δ(x − x0 )δ(y − y0 ),

(6.89)

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171

.

d 2 [t; v(τ )], 2 [t; v(τ )] = ln 8 [t; v(τ )], and 8 [t; v(τ )] dt is the characteristic functional of random process z(t).

where 2t [t; v(τ )] =

Problem 6.6 Starting from system of equations (6.88), derive the equation for the average of quantity Ak (t) = xk (t)yN−k (t)

(k = 0, . . . , N)

under the assumption that z(t) is the delta-correlated random process. Solution

d hAk (t)i = k hAk−1 (t)i − ω02 (N − k) hAk+1 (t)i dt N X   1 Kn Bn kl hAl (t)i (k = 0, . . . , N), + n! n=1

where Kn are the cumulants of random process z(t) and matrix Bij = −ω02 (N − i)δi,j−1 . The square of this matrix has the form b B2ij = −ω04 (N − i)(N − j + i)δi,j−2 and so N+1 forth for higher powers, so that b B ≡ 0.

Problem 6.7 Derive the Fokker–Planck equation for system of equations (6.88) in the case of the Gaussian delta-correlated process z(t) with the parameters

 hz(t)i = 0, z(t)z(t0 ) = 2σ 2 τ0 δ(t − t0 ). Solution

  ∂ ∂ ∂ ∂2 P(x, y, t) = −y + ω02 x P(x, y, t) + Dω02 x2 2 P(x, y, t), ∂t ∂x ∂y ∂y (6.90) P(x, y, 0) = δ (x − x0 ) δ (y − y0 ) , where D = σ 2 τ0 ω02 is the coefficient of diffusion in space {x, y/ω0 }.

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Problem 6.8 Obtain the solutions for the first- and second-order moments, which follow from Eq. (6.90) for D/ω0  1 and x(0) = 0, y(0) = ω0 . Solution

hx(t)i = sin ω0 t,

hy(t)i = ω0 cos ω0 t,

which coincides with the solution to system (6.88) for absent fluctuations, and   D E 1 3D Dt −Dt/2 2 cos(2ω0 t) + e −e sin(2ω0 t) , x (t) = 2 4ω0  i ω0 D h Dt −Dt/2 −Dt/2 hx(t)y(t)i = 2e sin(2ω0 t) + e −e cos(2ω0 t) , 4 ω0   D E ω2  D 0 2 Dt −Dt/2 cos(2ω0 t) − y (t) = e +e sin(2ω0 t) 2 4ω0 (6.91) Remark Solution (6.19) includes terms increasing with time, which corresponds to statistical parametric build-up of oscillations in a dynamic system (6.88) because of fluctuations of frequency. Solutions to a statistical problem (6.88) have two characteristic temporal scales t1 ∼ 1/ω0 and t2 ∼ 1/D. The first temporal scale corresponds to the period of oscillations in system (6.88) without fluctuations (fast processes), and the second scale characterizes slow variations caused in statistical characteristics by fluctuations (slow processes). The ratio of these scales is small:

t1 /t2 = D/ω0  1. We can explicitly obtain slow variations of statistical characteristics of processes x(t) and y(t) by excluding fast motions by averaging the corresponding quantities over the period T = 2π/ω0 . Denoting such averaging with the overbar, we have

 1 x2 (t) = eDt , 2

hx(t)y(t)i = 0,



 ω2 y2 (t) = 0 eDt . 2

Problem 6.9 Starting from (6.90), derive

the third-order equation for average potential energy of an oscillator hU(t)i = x2 (t) .

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173

Solution

d d3 hU(t)i + 4ω02 hU(t)i − 4Dω02 hU(t)i = 0, dt dt3

(6.92)

Problem 6.10 Derive the Fokker–Planck equation for the stochastic oscillator with linear friction d d x(t) = y(t), y(t) = −2γ y(t) − ω02 [1 + z(t)]x(t), dt dt x(0) = x0 , y(0) = y0 ,

(6.93)

and the condition of stochastic excitation of second moments. Solution

  ∂ ∂ ∂ ∂ ∂2 P(x, y, t) = 2γ y − y + ω02 x P(x, y, t) + Dω02 x2 2 P(x, y, t), ∂t ∂y ∂x ∂y ∂y P(x, y, 0) = δ(x − x0 )δ(y − y0 ), where D = σ 2 τ0 ω02 is the diffusion coefficient. If we substitute in the system of equations for second moments the solutions proportional to exp{λt}, we obtain the characteristic equation in λ of the form λ3 + 6γ λ2 + 4(ω02 + 2γ 2 )λ + 4ω02 (2γ − D) = 0. Under the condition 2γ < D,

(6.94)

second moments are the functions exponentially increasing with time, which means the statistical parametric excitation of second moments.

Problem 6.11 Obtain steady-state values of second moments formed under the action of random forces on the stochastic parametric oscillator with friction d x(t) = y(t), x(0) = x0 , dt d y(t) = −2γ y(t) − ω02 [1 + z(t)]x(t) + f (t), dt

(6.95) y(0) = y0 .

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Here, f (t) is the Gaussian delta-correlated process statistically independent of process z(t) and having the parameters h f (t)i = 0,



 f (t)f (t0 ) = 2σf2 τf δ(t − t0 ),

where σf2 is the variance and τf is the temporal correlation radius. Solution In the case of stochastic system of equations (6.95), the one-time probability density satisfies the Fokker–Planck equation

  ∂ ∂ ∂ ∂ P(x, y, t) = 2γ y − y + ω02 x P(x, y, t) ∂t ∂y ∂x ∂y + Dω02 x2

∂2 ∂2 2 P(x, y, t) + σ τ P(x, y, t), f f ∂y2 ∂y2

P(x, y, 0) = δ (x − x0 ) δ (y − y0 ). Consequently, the steady-state solution for second moments exists for t → ∞ under the condition (6.94) and has the form hx(t)i = 0, hx(t)y(t)i = 0,

hy(t)i = 0, D E x2 (t) =

σf2 τf ω02 (D − 2γ )

,

D E y2 (t) =

σf2 τf D − 2γ

.

Problem 6.12 Starting from (6.89), derive the Kolmogorov–Feller equation in the case of the Poisson delta-correlated random process z(t). Solution

  ∂ ∂ 2 ∂ P(x, y, t) = −y + ω0 x P(x, y, t) ∂t ∂x ∂y +ν

Z∞

−∞

(6.96) dξ p(ξ )P(t; x, y + ξ ω02 x) − νP(x, y, t).

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175

Remark For sufficiently small ξ , Eq. (6.96) grades into the Fokker–Planck equation (6.90) with the diffusion coefficient

D=

1 D 2E 2 ν ξ ω0 x. 2

Problem 6.13 Derive the equation for the characteristic functional of the solution to linear parabolic equation (1.89), page 39 by averaging Eq. (3.48), page 82. Solution Characteristic functional of the problem solution

 8[x; v, v∗ ] = ϕ[x; v, v∗ ] , where   Z   ϕ[x; v, v∗ ] = exp i dR0 u(x, R0 )v(R0 )+u∗ (x, R0 )v∗ (R0 ) , satisfies the equation     ∂ δ ∗ ∗ ˙ 8[x; v, v ] = 2x x; ϕ[x; v,v ] ∂x iδε(ξ, R0 )  Z  i δ δ 0 0 ∗ 0 + dR v(R )1R0 − v (R )1R0 ∗ 0 2k δv(R0 ) δv (R ) ∗ × 8[x; v, v ], (6.97) where  x + Z Z   d ˙ x x; ψ(ξ, R0 ) = 2 ln exp i dξ dR0 ε(ξ, R0 )ψ(ξ, R0 )   dx *





0

is the derivative of the characteristic functional logarithm of field ε(x, R).

Problem 6.14 Starting from Eq. (6.97), derive the equation for the characteristic functional of the solution to linear parabolic equation (1.89), page 39 assuming that ε(x, R) is the homogeneous random field delta-correlated in x.

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Solution Characteristic functional 8[x; v, v∗ ] of the problem solution satisfies

the equation   k ∂ ∗ 0 ˙ b 8[x; v, v ] = 2x x, M (R ) 8[x; v, v∗ ] ∂x 2  Z  δ i δ 0 0 ∗ 0 dR v(R )1R0 + −v (R )1R0 ∗ 0 2k δv(R0 ) δv (R ) × 8[x; v, v∗ ]

(6.98)

with the Hermitian operator b (R0 ) = v(R0 ) M

δ δ − v∗ (R0 ) ∗ 0 . δv(R0 ) δv (R )

Here, functional  x + * Z  Z   d ˙ x x; ψ(ξ, R0 ) = 2 ln exp i dξ dR0 ε(ξ, R0 )ψ(ξ, R0 )   dx 

0

is the derivative of the logarithm of the characteristic functional of filed ε(x, R). Remark Equation (6.98) yields the equations for the moment functions of field

u(x, R), Mm,n (x; R1 , . . . , Rm ; R01 , . . . , R0n ) 

= u(x, R1 ) . . . u(x, Rm )u∗ (x, R01 ) . . . u∗ (x, R0n ) , (for m = n, these functions are usually called the coherence functions of order 2n)   n m X ∂ i X Mm,n = 1Rp − 1R0q  Mm,n ∂x 2k p=1

q=1



  m n X X 1 ˙ x x,  +2 δ(R0 − R0q ) Mm,n . δ(R0 − Rp )− k p=1

q=1

(6.99)

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177

Problem 6.15 Assuming that ε(x, R) is the homogeneous Gaussian random process deltacorrelated in x with the correlation function Bε (x, R) = A(R)δ(x),

Z∞ A(R) =

dxBε (x, R)

−∞

and starting from the solution to the previous problem, derive the equations for the characteristic functional of wave field u(x, R), which is the solution to linear parabolic equation (1.89), page 39, and moment functions of this field [50]. Solution

Z Z k2 ∂ b (R0 )M b (R)8[x; v, v∗ ] 8[x; v, v∗ ] = − dR0 dRA(R0 − R)M ∂x 8   Z i δ δ 0 0 ∗ 0 0 0 + dR v(R )1R −v (R )1R ∗ 0 8[x; v, v∗ ], 2k δv(R0 ) δv (R )   m n X i X ∂xMm,n = 1Rp − 1R0q  Mm,n 2k p=1

−

q=1

k2 Q(R1 , . . . , Rm ; R01 , . . . , R0m )Mm,n , 8

where Q(R1 , . . . , Rm ; R01 , . . . , R0m ) = =

m X m X i=1 j=1

A(Ri − Rj )−2

m X n X

A(Ri − R0j )+

i=1 j=1

n X n X

A(R0i − R0j ).

i=1 j=1

Problem 6.16 Derive the equation for the one-time characteristic functional of the solution to stochastic nonlinear integro-differential equation (1.102), page 44. Solution Characteristic functional of the velocity field

 8[t; z(k0 )] = 8[t; z] = ϕ[t; z(k0 )] ,

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where  Z  ϕ[t; z(k0 )] = exp i dk0b u(k0 , t)z(k0 ) , satisfies the unclosed variational-derivative equation ∂ 8[t; z] = ∂t

Z

 Z Z 1 δ2 dkzi (k) − dk1 dk2 3i,αβ (k1 , k2 , k) 2 δzα (k1 )δzβ (k2 )

− νk2

     δ δ ˙ t t; 8[t; z] + 2 ϕ[t; z] , δzi (k) iδf (κ, τ )

where  t + * Z  Z  d ˙ t [t; ψ(κ, τ )] = ln exp i dτ dκf (κ, τ )ψ(κ, τ ) 2   dt 0

is the derivative of the characteristic functional logarithm of external forces b f (k, t).

Problem 6.17 Assuming that f (x, t) is the homogeneous stationary random field deltacorrelated in time, derive the equation for the one-time characteristic functional of the solution to stochastic nonlinear integro-differential equation (1.102), page 44. Consider the case of the Gaussian field f (x, t) [1]. Solution

∂ 8[t; z] = ∂t

 Z Z 1 δ2 dkzi (k) − dk1 dk2 3i,αβ (k1 , k2 , k) 2 δzα (k1 )δzβ (k2 )  δ ˙ t [t; z(k)] 8[t; z], − νk2 8[t; z] + 2 (6.100) δzi (k) Z

where  t + * Z  Z  d ˙ t [t; ψ(κ, τ )] = ln exp i dτ dκ b 2 f (κ, τ )ψ(κ, τ )   dt 0

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179

is the derivative of the characteristic functional logarithm of external forces b f (k, t). If we assume now that f (x, t) is the Gaussian random field homogeneous and isotropic in space and stationary in time with the correlation tensor

 Bij (x1 − x2 , t1 − t2 ) = fi (x1 , t1 )fj (x2 , t2 ) , then the field b f (k, t) will also be the Gaussian stationary random field with the correlation tensor

 1 b fi (k, t + τ )b fj (k0 , t) = Fij (k, τ )δ(k + k), 2

where Fij (k, τ ) is the external force spatial spectrum given by the formula Fij (k, τ ) = 2(2π)3

Z

dxBij (x, τ )e−ikx .

In view of the fact that forces are spatially isotropic, we have Fij (k, τ ) = F(k, τ )1ij (k). As long as field b f (k, t) is delta-correlated in time, we have that F(k, τ ) = F(k)δ(τ ), so that functional 2 [t; ψ(κ, τ )] is given by the formula 1 2 [t; ψ(κ, τ )] = − 4

Zt

Z dτ

dκ F(κ)1ij (κ)ψi (κ, τ )ψj (−κ, τ ),

0

and Eq. (6.100) assumes the closed form 1 ∂ 8[t; z] = − ∂t 4 Z −

Z

dkF(k)1ij (k)zi (k)zj (−k)8[t; z]

 Z Z 1 δ2 8[t; z] dkzi (k) dk1 dk2 3i,αβ (k1 , k2 , k) 2 δzα (k1 )δzβ (k2 )

+ νk2

δ δzi (k)

Z

dkzi (k)

 δ 8[t; z]. δzi (k)

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Problem 6.18 Average the linear stochastic integral equation S(r, r0 ) = S0 (r, r0 ) Z Z Z + dr1 dr2 dr3 S0 (r, r1 )3(r1 , r2 , r3 ) [η(r2 ) + f (r2 )] S(r3 , r0 ), Solution

G[r, r0 ; η(r)] = S0 (r, r0 ) Z Z Z + dr1 dr2 dr3 S0 (r, r1 )3(r1 , r2 , r3 )η(r2 )G[r3 , r0 ; η(r)]   Z Z Z δ G[r3 , r0 ; η(r)]. + dr1 dr2 dr3 S0 (r, r1 )3(r1 , r2 , r3 )r2 iδη(r) Here,

 G[r, r0 ; η(r)] = S[r, r0 ; f (r) + η(r)] and functional r [v(r)] =

δ 2 [v(r)] , iδv(r)

 Z  2 [v(r)] = ln exp i drf (r)v(r) .

Problem 6.19 Assuming that random external force f (r, t) is the homogeneous stationary Gaussian field, derive the equation for the characteristic functional of space-time harmonics of turbulent velocity field (1.103), page 45 [51]. Solution Characteristic functional of the velocity field



 Z 

4 0 0 0 8[z] = exp i d K b u K z K and functional *

Gij



+ δui (K) ϕ[z] K, K ; z = δb fj (K 0 ) 0



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181

satisfy the closed system of functional equations Z δ 1 8[z] = − Fij (K) d4 K 1 zα (K 1 ) Gαj [K 1 , −K; z] δzi (K) 2 Z Z 1 δ 2 8[z] αβ − d4 K 1 d4 K 2 3i (K 1 , K 2 , K) , 2 δzα (K 1 ) δzβ (K 2 )   (iω + νk2 )Gij K, K 0 ; z Z Z   δ αβ 4 + d K 1 d4 K 2 3i (K 1 , K 2 , K) Gβj K 2 , K 0 ; z δzα (K 1 )
= δij δ 4 K − K 0 8[z], (iω + νk2 )

 1 where b fi (K 1 )b fj (K 2 ) = δ 4 (K 1 + K 2 ) Fij (K 1 ), and Fij (K) is the space-time 2 spectrum of external forces. Remark Expansion of functionals 8[z] and Gij K, K 0 ; z in the functional Taylor





series in z (K) determines
velocity field cumulants and correlators of Green’s
function analog Sij K, K 0 = δui (K) /δfj K 0 with the velocity field.

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Lecture 7 Stochastic Equations with the Markovian Fluctuations of Parameters

In the preceding Lecture, we dealt with the statistical description of dynamic systems in terms of general methods that assumed knowledge of the characteristic functional of fluctuating parameters. However, this functional is unknown in most cases, and we are forced to resort either to assumptions on the model of parameter fluctuations, or to asymptotic approximations. The methods based on approximating fluctuating parameters with the Markovian random processes and fields with a finite temporal correlation radius are widely used. Such approximations can be found, for example, as solutions to the dynamic equations with delta-correlated fluctuations of parameters. Consider such methods in greater detail by the examples of the Markovian random processes. Consider stochastic equations of the form d x(t) = f (t, x, z(t)) , dt

x(0) = x0 ,

(7.1)

where f (t, x, z(t)) is the deterministic function and z(t) = {z1 (t), . . . , zn (t)} is the Markovian vector process. Our task consists in the determination of statistical characteristics of the solution to Eq. (7.1) from known statistical characteristics of process z(t). In the general case of arbitrary Markovian process z(t), we cannot judge about process x(t). We can only assert that the joint process {x(t), z(t)} is the Markovian process with joint probability density P(x, z, t). If we could solve the equation for P(x, z, t), then we could integrate the solution over z to obtain the probability density of the solution to Eq. (7.1), i.e., function P(x, t). In this case, process x(t) by itself would not be the Markovian process. There are several types of process z(t) that allow us to obtain equations for density P(x, t) without solving the equation for P(x, z, t). Among these processes, we mention first of all the telegrapher’s and generalized telegrapher’s processes, Markovian Lectures on Dynamics of Stochastic Systems. DOI: 10.1016/B978-0-12-384966-3.00007-6 c 2011 Elsevier Inc. All rights reserved. Copyright

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processes with finite number of states, and Gaussian Markovian processes. Below, we discuss the telegrapher’s and Gaussian Markovian processes in more detail as examples of processes widely used in different applications.

7.1

Telegrapher’s Processes

Recall that telegrapher’s random process z(t) (the two-state, or binary process) is defined by the equality z(t) = a(−1)n(0,t) , where random quantity a assumes values a = ± a0 with probabilities 1/2 and n(t1 , t2 ), t1 < t2 is Poisson’s integer-valued process with mean value n(t1 , t2 ) = ν|t1 − t2 |. Telegrapher’s process z(t) is stationary in time and its correlation function

 0 z(t)z(t0 ) = a20 e−2ν|t−t | has the temporal correlation radius τ0 = 1/ (2ν). For splitting the correlation between telegrapher’s process z(t) and arbitrary functional R[t; z(τ )], where τ ≤ t, we obtained the relationship (5.25), (page 129) hz(t)R[t; z(τ )]i =

a20

Zt 0

dt1 e−2ν(t−t1 )



 δ e R[t, t1 ; z(τ )] , δz(t1 )

(7.2)

where functional e R[t, t1 ; z(τ )] is given by the formula e R[t, t1 ; z(τ )] = R[t; z(τ )θ(t1 − τ + 0)] (t1 < t).

(7.3)

Formula (7.2) is appropriate for analyzing stochastic equations linear in process z(t). Functional R[t; z(τ )] is the solution to a system of differential equations of the first order in time with initial values at t = 0. Functional e R[t, t1 ; z(τ )] will also satisfy the same system of equations with product z(t)θ(t1 − t) instead of z(t). Consequently, we obtain that functional e R[t, t1 ; z(τ )] = R[t; 0] for all times t > t1 ; moreover, it satisfies the same system of equations for absent fluctuations (i.e., at z(t) = 0) with the initial value e R[t1 , t1 ; z(τ )] = R[t1 ; z(τ )]. Another formula convenient in the context of stochastic equations linear in random telegrapher’s process z(t) concerns the differentiation of a correlator of this process with arbitrary functional R[t; z(τ )](τ ≤ t) (5.26)   d d hz(t)R[t; z(τ )]i = −2ν hz(t)R[t; z(τ )]i + z(t) R[t; z(τ )] . dt dt

(7.4)

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185

Formula (7.4) determines the rule of factoring the differentiation operation out of averaging brackets   n  d dn + 2ν hz(t)R[t; z(τ )]i . z(t) n R[t; z(τ )] = dt dt

(7.5)

We consider some special examples to show the usability of these formulas. It is evident that both methods give the same result. However, the method based on the differentiation formula appears more practicable. The first example concerns the system of linear operator equations d x(t) = b A(t)x(t) + z(t)b B(t)x(t), dt

x(0) = x0 ,

(7.6)

where b A(t) and b B(t) are certain differential operators (they may include differential operators with respect to auxiliary variables). If operators b A(t) and b B(t) are matrices, then Eqs. (7.6) describe the linear dynamic system. Average Eqs. (7.6) over an ensemble of random functions z(t). The result will be the vector equation d ψ (t), hx(t)i = b A(t) hx(t)i + b B(t)ψ dt

hx(0)i = x0 ,

(7.7)

where we introduced new functions ψ (t) = hz(t)x(t)i . We can use formula Eqs. (7.4) for these functions; as a result, we obtain the equality   d d ψ (t) + z(t) x(t) . ψ (t) = −2νψ (7.8) dt dt Substituting now derivative dx/dt (7.6) in Eq. (7.8), we obtain the equation for the vector function ψ (t)   D E d ψ (t) + b + 2ν ψ (t) = b A(t)ψ B(t) z2 (t)x(t) . (7.9) dt Because z2 (t) ≡ a20 for telegrapher’s process, we obtain finally the closed system of linear equations for vectors hx(t)i and ψ (t) d ψ (t), hx(0)i = x0 , hx(t)i = b A(t) hx(t)i + b B(t)ψ dt   d ψ (t) + a20b + 2ν ψ (t) = b A(t)ψ B(t) hx(t)i , ψ (0) = (0). dt

(7.10)

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If operators b A(t) and b B(t) are the time-independent matrices A and B, we can solve system (7.10) using the Laplace transform. After the Laplace transform, system (7.10) grades into the algebraic system of equations ψ p = x0 , (pE − A) hxip − Bψ ψ p − a20 B hxip = 0, [(p + 2ν) E − A]ψ

(7.11)

where E is the unit matrix. From this system, we obtain solution hxip in the form  hxip = (pE − A) − a20 B

−1 1 B x0 . (p + 2ν)E − A

(7.12)

Assume now operators b A(t) and b B(t) in Eq. (7.6) be the matrices. If only one component is of interest, we can obtain for it the operator equation  b L

 n−1 X dj di d x(t) + bij (t) i z(t) j x(t) = f (t), dt dt dt

(7.13)

i+j=0

where  b L

d dt

 =

n X

ai (t)

i=0

di dti

and n is the order of matrices A and B in Eq. (7.6). The initial conditions for x are included in function f (t) through the corresponding derivatives of delta function. Note that function f (t) can depend additionally on derivatives of random process z(t) at t = 0, i.e., f (t) is also the random function statistically related to process z(t). Averaging Eq. (7.13) over an ensemble of realizations of process z(t) with the use of formula (7.5), we obtain the equation     d d d b hx(t)i + M , + 2ν hz(t)x(t)i = hf (t)i , (7.14) L dt dt dt where M [p, q] =

n−1 X

bij (t)pi qj .

i+j=0

However, Eq. (7.14) is unclosed because of the presence of correlation hz(t)x(t)i. Multiplying Eq. (7.13) by z(t) and averaging the result, we obtain the equation for correlation hz(t)x(t)i     d d d 2 b hx(t)i = hz(t)f (t)i . L + 2ν hz(t)x(t)i + a0 M + 2ν, (7.15) dt dt dt

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187

System of equations (7.14) and Eq. (7.15) is the closed system. If functions ai (t) and bij (t) are independent of time, this system can be solved using Laplace transform. The solution is as follows: hxip =

b L(p + 2ν) h f ip − M[p, p + 2ν] hz f ip . b L(p)b L(p + 2ν) − a2 M[p + 2ν, p]M[p, p + 2ν]

(7.16)

0

7.2

Gaussian Markovian Processes

Here, we consider several examples associated with the Gaussian Markovian processes. Define random process z(t) by the formula z(t) = z1 (t) + · · · + zN (t),

(7.17)

where zi (t) are statistically independent telegrapher’s processes with correlation functions D E

 0 zi (t)zj (t0 ) = δij z2 e−α|t−t | (α = 2ν).

 If we set z2 = σ 2 /N, then this process grades for N → ∞ into the Gaussian Markovian process with correlation function

 D E 0 zi (t)zj (t0 ) = z2 e−α|t−t | . Thus, process z(t) (7.17) approximates the Gaussian Markovian process in terms of the Markovian process with a finite number of states. It is evident that the differentiation formula and the rule of factoring the derivative out of averaging brackets for process z(t) assume the forms     d d + αk hz1 (t) · · · zk (t)R[t; z(τ )]i = z1 (t) · · · zk (t) R[t; z(τ )] , dt dt (7.18)    n dn d z1 (t) · · · zk (t) n R[t; z(τ )] = + αk hz1 (t) · · · zk (t)R[t; z(τ )]i , dt dt where R[t; z(τ )] is an arbitrary functional of process z(t) (τ ≤ t). Consider again Eq. (7.6), which we represent here in the form d x(t) = b A(t)x(t) + [z1 (t) + · · · + zN (t)]b B(t)x(t), dt

x(0) = x0 ,

(7.19)

and introduce vector-function Xk (t) = hz1 (t) . . . zk (t)x(t)i ,

k = 1, . . . , N;

X0 (t) = hx(t)i .

(7.20)

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Using formula (7.18) for differentiating correlations (7.20) and Eq. (7.19), we obtain the recurrence equation for xk (t) (k = 0, 1, . . . , N)

 d Xk (t) = b A(t)Xk (t) + z1 (t) · · · zk (t)[z1 (t) + · · · + zN (t)]b B(t)x(t) dt D E =b A(t)Xk (t) + k z2 b B(t)Xk−1 (t) + (N − k)b B(t)Xk+1 (t),

(7.21)

with the initial condition Xk (0) = x0 δk,0 . Thus, the mean value of the solution to system Eq. (7.19) satisfies the closed system of (N + 1) vector equations. If operators b A(t), b B(t) are time-independent matrices, system (7.21) can be easily solved using the Laplace transform. It is clear that such

 a solution will have the form of a finite segment of continued fraction. If we set z2 = σ 2 /N and proceed to the limit N → ∞, then random process (7.17) will grade, as was mentioned earlier, into the Gaussian Markovian process and solution to system (7.21) will assume the form of the infinite continued fraction.

Problems Problem 7.1 Assuming that z(t) is telegrapher’s random process, find average potential energy of the stochastic oscillator described by the equation   d3 d d 2 d 2 U(t) + 4ω0 U(t) + 2ω0 z(t) U(t) + z(t)U(t) = 0, dt dt dt dt3 d d2 U(t) U(t) x(0) = 0, = 0, = 2y20 . dt dt2 t=0 t=0 Solution

  d3 d 2 d 2 hU(t)i hU(t)i hz(t)U(t)i = 0, + 4ω + 4ω + 2ν 0 0 dt dt dt3  3   d d 2 + 2ν hz(t)U(t)i + 4ω0 + 2ν hz(t)U(t)i dt dt   d + 4ω02 a20 + 2ν hU(t)i = 0. dt

(7.22)

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189

System of equations (7.22) can be solved using Laplace transform, and we obtain the solution in the form hUip = 2y20

L(p + 2ν) , L(p)L(p + 2ν) − a20 M 2 (p)

  L(p) = p p2 + 4ω02 ,

(7.23)

  M(p) = 4ω02 p2 + ν .

Remark In the limiting case of great parameters ν and a20 , but finite ratio a20 /2ν =

σ 2 τ0 , we obtain from the second equation of system (7.22) hz(t)U(t)i = −

ω02 σ 2 τ0 hU(t)i . ν

Consequently, average potential energy hU(t)i satisfies in this limiting case the closed third-order equation d3 d hU(t)i + 4ω02 hU(t)i − 4ω04 σ 2 τ0 hU(t)i = 0, dt dt3 which coincides with Eq. (6.92), page 173, and corresponds to the Gaussian delta-correlated process z(t).

Problem 7.2 Assuming that z(t) is telegrapher’s random process, obtain the steady-state probability distribution of the solution to the stochastic equation d x(t) = f (x) + z(t)g(x), dt

x(0) = x0 .

Solution Under the condition that | f (x)| < a0 |g(x)|, we have

( Z ) C|g(x)| 2ν f (x) P(x) = 2 exp dx 2 , a0 g2 (x) − f 2 (x) a20 a0 g2 (x) − f 2 (x)

(7.24)

where the positive constant C is determined from the normalization condition.

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Remark In the limit ν → ∞ and a20 → ∞ under the condition that a20 τ0 = const

(τ0 = 1/ (2ν)), probability distribution (7.24) grades into the expression ( Z ) C f (x) 2ν P(x) = exp dx 2 |g(x)| g (x) a20

corresponding to the Gaussian delta-correlated process z(t), i.e., the Gaussian

 process with correlation function z(t)z(t0 ) = 2a20 τ0 δ(t − t0 ).

Problem 7.3 Assuming that z(t) is telegrapher’s random process, obtain the steady-state probability distribution of the solution to the stochastic equation d x(t) = −x + z(t), dt

x(0) = x0 .

(7.25)

Solution

P(x) =

1 (1 − x2 )ν−1 B(ν, 1/2)

(|x| < 1),

(7.26)

where B(ν, 1/2) is the beta-function. This distribution has essentially different behaviors for ν > 1, ν = 1 and ν < 1. One can easily see that this system resides mainly near state x = 0 if ν > 1, and near states x = ±1 if ν < 1. In the case ν = 1, we obtain the uniform probability distribution on segment [−1, 1] (Fig. 7.1). P(x) P(x)

−1

P(x)

ν >1

1

x

−1

ν <1

ν =1

1

x

−1

1

x

Figure 7.1 Steady-state probability distribution (7.26) of the solution to the Eq. (7.25) versus parameter ν.

Lecture 8 Approximations of Gaussian Random Field Delta-Correlated in Time

In the foregoing Lectures, we considered in detail the general methods of analyzing stochastic equations. Here, we give an alternative and more detailed consideration of the approximation of the Gaussian random delta-correlated (in time) field in the context of stochastic equations and discuss the physical meaning of this widely used approximation.

8.1

The Fokker–Planck Equation

Let vector function x(t) = {x1 (t), x2 (t), . . . , xn (t)} satisfies the dynamic equation d x(t) = v(x, t) + f (x, t), dt

x(t0 ) = x0 ,

(8.1)

where vi (x, t) (i = 1, . . . , n) are the deterministic functions and fi (x, t) are the random functions of (n + 1) variable that have the following properties: (a) fi (x, t) is the Gaussian random field in the (n + 1)-dimensional space (x, t); (b) h fi (x, t)i = 0.

For definiteness, we assume that t is the temporal variable and x is the spatial variable. Statistical characteristics of field fi (x, t) are completely described by correlation tensor

 Bij (x, t; x0 , t0 ) = fi (x, t)fj (x0 , t0 ) . Because Eq. (8.1) is the first-order equation with the initial value, its solution satisfies the dynamic causality condition δ xi (t) = 0 δfj (x0 , t0 )

for t0 < t0

andt0 > t,

Lectures on Dynamics of Stochastic Systems. DOI: 10.1016/B978-0-12-384966-3.00008-8 c 2011 Elsevier Inc. All rights reserved. Copyright

(8.2)

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which means that solution x(t) depends only on values of function fj (x, t0 ) for times t0 preceding time t, i.e., t0 ≤ t0 ≤ t. In addition, we have the following equality for the variational derivative δ xi (t) = δij δ(x(t) − x0 ). δfj (x0 , t − 0)

(8.3)

Nevertheless, the statistical relationship between x(t) and function values fj (x, t00 ) for consequent times t00 > t can exist, because such function values fj (x, t00 ) are correlated with values fj (x, t0 ) for t0 ≤ t. It is obvious that the correlation between function x(t) and consequent values fj (x, t00 ) is appreciable only for t00 − t ≤ τ0 , where τ0 is the correlation radius of field f (x, t) with respect to variable t. For many actual physical processes, characteristic temporal scale T of function x(t) significantly exceeds correlation radius τ0 (T  τ0 ); in this case, the problem has small parameter τ0 /T that can be used to construct an approximate solution. In the first approximation with respect to this small parameter, one can consider the asymptotic solution for τ0 → 0. In this case values of function x(t0 ) for t0 < t will be independent of values f (x, t00 ) for t00 > t not only functionally, but also statistically. This approximation is equivalent to the replacement of correlation tensor Bij with the effective tensor 0 0 0 0 Beff ij (x, t; x , t ) = 2δ(t − t )Fij (x, x ; t).

(8.4)

Here, quantity Fij (x, x0 , t) is determined from the condition that integrals of 0 0 0 Bij (x, t; x0 , t0 ) and Beff ij (x, t; x , t ) over t coincide 1 Fij (x, x , t) = 2 0

Z∞

dt0 Bij (x, t; x0 , t0 ),

−∞

which just corresponds to the passage to the Gaussian random field delta-correlated in time t. Introduce the indicator function ϕ(x, t) = δ(x (t) − x),

(8.5)

where x(t) is the solution to Eq. (8.1), which satisfies the Liouville equation 

 ∂ ∂ ∂ + v(x, t) ϕ(x, t) = − f (x, t)ϕ(x, t) ∂t ∂x ∂x

(8.6)

and the equality δ δfj

(x0 , t

− 0)

ϕ(x, t) = −

∂  δ(x − x0 )ϕ(x, t) . ∂xj

(8.7)

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193

The equation for the probability density of the solution to Eq. (8.1) P(x, t) = hϕ(x, t)i = hδ(x(t) − x)i can be obtained by averaging Eq. (8.6) over an ensemble of realizations of field f (x, t),   ∂ ∂ ∂ + v(x, t) P(x, t) = − h f (x, t)ϕ(x, t)i. (8.8) ∂t ∂x ∂x We rewrite Eq. (8.8) in the form   ∂ ∂ + v(x, t) P(x, t) ∂t ∂x   Z Zt δϕ(x, t) ∂ 0 0 0 0 dx dt Bij (x, t; x , t ) , =− ∂xi δfj (x0 , t0 )

(8.9)

t0

where we used the Furutsu–Novikov formula (5.13), page 126,   Z Z δR[t; f (y, τ )] h fk (x, t)R[t; f (y, τ )]i = dx0 dt0 Bkl (x, t; x0 , t0 ) δ fl (x0 , t0 )

(8.10)

for the correlator of the Gaussian random field f (x, t) with arbitrary functional R[t; f (y, τ )] of it and the dynamic causality condition (8.2). Equation (8.9) shows that the one-time probability density of solution x(t) at instant t is governed by functional dependence of solution x(t) on field f (x0 , t) for all times in the interval (t0 , t). In the general case, there is no closed equation for the probability density P(x, t). However, if we use approximation (8.4) for the correlation function of field f (x, t), there appear terms related to variational derivatives δϕ[x, t; f (y, τ )]/δfj (x0 , t0 ) at coincident temporal arguments t0 = t − 0,     Z ∂ ∂ ∂ δϕ(x, t) + v(x, t) P(x, t) = − dx0 Fij (x, x0 , t) . ∂t ∂x ∂xi δfj (x0 , t − 0) According to Eq. (8.7), these variational derivatives can be expressed immediately in terms of quantity ϕ[x, t; f (y, τ )]. Thus, we obtain the closed Fokker-Planck equation 

 ∂ ∂ ∂2 [vk (x, t) + Ak (x, t)] P(x, t) = [Fkl (x, x, t)P(x, t)], + ∂t ∂xk ∂xk ∂xl

where ∂ 0 Ak (x, t) = Fkl (x, x , t) . 0 ∂xl x0 =x

(8.11)

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Equation (8.11) should be solved with the initial condition P(x, t0 ) = δ(x − x0 ), or with a more general initial condition P(x, t0 ) = W0 (x) if the initial conditions are also random, but statistically independent of field f (x, t). The Fokker–Planck equation (8.11) is a partial differential equation and its further analysis essentially depends on boundary conditions with respect to x whose form can vary depending on the problem at hand. Consider the quantities appeared in Eq. (8.11). In this equation, the terms containing Ak (x, t) and Fkl (x, x0 , t) are stipulated by fluctuations of field f (x, t). If field f (x, t) is stationary in time, quantities Ak (x) and Fkl (x, x0 ) are independent of time. If field f (x, t) is additionally homogeneous and isotropic in all spatial coordinates, then Fkl (x, x, t) = const, which corresponds to the constant tensor of diffusion coefficients, and Ak (x, t) = 0 (note however that quantities Fkl (x, x0 , t) and Ak (x, t) can depend on x because of the use of a curvilinear coordinate systems).

8.2

Transition Probability Distributions

Turn back to dynamic system (8.1) and consider the m-time probability density Pm (x1 , t1 ; . . . ; xm , tm ) = hδ(x(t1 ) − x1 ) · · · δ(x(tm ) − xm )i

(8.12)

for m different instants t1 < t2 < · · · < tm . Differentiating Eq. (8.12) with respect to time tm and using then dynamic equation (8.1), dynamic causality condition (8.2), definition of function Fkl (x, x0 , t), and the Furutsu–Novikov formula (8.10), one can obtain the equation similar to the Fokker–Planck equation (8.11), ∂ Pm (x1 , t1 ; . . . ; xm , tm ) ∂tm n X ∂ [vk (xm , tm ) + Ak (xm , tm )] Pm (x1 , t1 ; . . . ; xm , tm ) + ∂xmk =

k=1 n n XX k=1 l=1

∂2 [Fkl (xm , xm , tm )Pm (x1 , t1 ; . . . ; xm , tm ))]. ∂xmk ∂xml

(8.13)

No summation over index m is performed here. The initial value to Eq. (8.13) can be determined from Eq. (8.12). Setting tm = tm−1 in (8.12), we obtain Pm (x1 , t1 ; . . . ; xm , tm−1 ) = δ(xm − xm−1 )Pm−1 (x1 , t1 ; . . . ; xm−1 , tm−1 ).

(8.14)

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Equation (8.13) assumes the solution in the form Pm (x1 , t1 ; . . . ; xm , tm ) = p(xm , tm |xm−1 , tm−1 )Pm−1 (x1 , t1 ; . . . ; xm−1 , tm−1 ).

(8.15)

Because all differential operations in Eq. (8.13) concern only tm and xm , we can find the equation for the transitional probability density by substituting Eq. (8.15) in Eq. (8.13) and (8.14):   ∂ ∂ [vk (x, t) + Ak (x, t)] p(x, t|x0 , t0 ) + ∂t ∂xk =

∂2 [Fkl (x, x, t)p(x, t|x0 , t0 )] ∂xk ∂xl

(8.16)

with initial condition p(x, t|x0 , t0 )|t→t0 = δ(x − x0 ), where p(x, t|x0 , t0 ) = hδ(x(t) − x)|x(t0 ) = x0 i. In Eq. (8.16) we denoted variables xm and tm as x and t, and variables xm−1 and tm−1 as x0 and t0 . Using formula (8.15) (m − 1) times, we obtain the relationship Pm (x1 , t1 ; . . . ; xm , tm ) = p(xm , tm |xm−1 , tm−1 ) · · · p(x2 , t2 |x1 , t1 )P(x1 , t1 ),

(8.17)

where P(x1 , t1 ) is the one-time probability density governed by Eq. (8.11). Equality (8.17) expresses the multi-time probability density as the product of transitional probability densities, which means that random process x(t) is the Markovian process. Equation (8.11) is usually called the forward Fokker–Planck equation. The backward Fokker–Planck equation (it describes the transitional probability density as a function of the initial parameters t0 and x0 ) can also be easily derived. Indeed, in Lecture 3 (page 69), we obtained the backward Liouville equation (3.8), page 71, for indicator function   ∂ ∂ ∂ + v(x0 , t0 ) ϕ(x, t|x0 , t0 ) = −f (x0 , t0 ) ϕ(x, t|x0 , t0 ), (8.18) ∂t0 ∂x0 ∂x0 with the initial value ϕ(x, t|x0 , t) = δ(x − x0 ). This equation describes the dynamic system evolution in terms of initial parameters t0 and x0 . From Eq. (8.18) follows the equality similar to Eq. (8.7), δ δfj

(x0 , t

0

+ 0)

ϕ(x, t|x0 , t0 ) = δ(x − x0 )

∂ ϕ(x, t|x0 , t0 ). ∂x0j

(8.19)

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Averaging now the backward Liouville equation (8.18) over an ensemble of realizations of random field f (x, t) with the effective correlation tensor (8.4), using the Furutsu–Novikov formula (8.10), and relationship (8.19) for the variational derivative, we obtain the backward Fokker–Planck equation 

∂ ∂ + [vk (x0 , t0 ) + Ak (x0 , t0 )] ∂t0 ∂x0k = −Fkl (x0 , x0 , t0 )



p(x, t|x0 , t0 )

∂2 p(x, t|x0 , t0 ), ∂x0k ∂x0l

(8.20)

p(x, t|x0 , t) = δ(x − x0 ). The forward and backward Fokker–Planck equations are equivalent. The forward equation is more convenient for analyzing the temporal behavior of statistical characteristics of the solution to Eq. (8.1). The backward equation appears to be more convenient for studying statistical characteristics related to initial values, such as the time during which process x(t) resides in certain spatial region and the time at which the process arrives at region’s boundary. In this case the probability of the fact that random process x(t) rests in spatial region V is given by the integral Z G(t; x0 , t0 ) = dxp(x, t|x0 , t0 ), V

which, according to Eq. (8.20), satisfies the closed equation 

∂ ∂ + [vk (x0 , t0 ) + Ak (x0 , t0 )] ∂t0 ∂x0k



G(t; x0 , t0 )

∂2 G(t; x0 , t0 ), = −Fkl (x0 , x0 , t0 ) ∂x0k ∂x0l  1 (x0 ∈ V), G(t; x0 , t0 ) = 0 (x0 ∈ / V).

(8.21)

For Eq. (8.21), we must formulate additional boundary conditions, which depend on characteristics of both region V and its boundaries.

8.3

The Simplest Markovian Random Processes

There are only few Fokker–Planck equations that allow an exact solution. First of all, among them are the Fokker–Planck equations corresponding to the stochastic equations that are themselves solvable in the analytic form. Such problems often allow determination of not only the one-point and transitional probability densities, but also the characteristic functional and other statistical characteristics important for practice. The simplest special case of Eq. (8.11) is the equation that defines the Wiener random process. In view of the significant role that such processes plays in physics (for

Approximations of Gaussian Random Field Delta-Correlated in Time

197

example, they describe the Brownian motion of particles), we consider the Wiener process in detail.

8.3.1

Wiener Random Process

The Wiener random process is defined as the solution to the stochastic equation d w(t) = z(t), dt

w(0) = 0,

where z(t) is the Gaussian process delta-correlated in time and described by the parameters hz(t)i = 0,



 z(t)z(t0 ) = 2σ 2 τ0 δ(t − t0 ). t

Z The solution to this equation w(t) =

dτ z(τ ) is the continuous Gaussian nonsta-

0

tionary random process with the parameters hw(t)i = 0,

8.3.2



 w(t)w(t0 ) = 2σ 2 τ0 min(t, t0 ).

Wiener Random Process with Shear

Consider a more general process that includes additionally the drift dependent on parameter α w(t; α) = −αt + w(t),

α > 0.

Process w(t; α) is the Markovian process, and its probability density P(w, t; α) = hδ(w(t; α) − w)i satisfies the Fokker–Planck equation 

 ∂ ∂ ∂2 −α P(w, t; α) = D 2 P(w, t; α), ∂t ∂w ∂w

P(w, 0; α) = δ(w),

(8.22)

where D = σ 2 τ0 is the diffusion coefficient. The solution to this equation has the form of the Gaussian distribution   1 (w + αt)2 P(w, t; α) = √ exp − . 4Dt 2 π Dt

(8.23)

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The corresponding integral distribution function defined as the probability of the event that w(t; α) < w is given by the formula F(w, t; α) =

Zw −∞

! t , dwP(w, t; α) = Pr √ +α 2D 2Dt r

w

(8.24)

where function Pr(z) is the probability integral (4.20), page 94. In this case, the typical realization curve of the Wiener random process with shear is the linear function of time in accordance with Eqs. (4.60), page 108 w∗ (t; α) = −αt. In addition to the initial value, supplement Eq. (8.22) with the boundary condition P(w, t; α)|w=h = 0,

(t > 0).

(8.25)

This condition breaks down realizations of process w(t; α) at the instant they reach boundary h. For w < h, the solution to the boundary-value problem (8.22), (8.25) (we denote it as P(w, t; α, h)) describes the probability distribution of those realizations of process w(t; α) that survived instant t, i.e., never reached boundary h during the whole temporal interval. Correspondingly, the norm of the probability density appears not unity, but the probability of the event that t < t∗ , where t∗ is the instant at which process w(t; α) reaches boundary h for the first time Zh

dwP(w, t; α, h) = P(t < t∗ ).

(8.26)

−∞

Introduce the integral distribution function and probability density of random instant at which the process reaches boundary h F(t; α, h) = P(t∗ < t) = 1 − P(t < t∗ ) = 1 −

Zh

dwP(w, t; α, h),

−∞

(8.27)

∂ ∂ P(t; α, h) = F(t; α, h) = − P(w, t; α, h)|w=h . ∂t ∂w If α > 0, process w(t; α) moves on average out of boundary h; as a result, probability P(t < t∗ ) (8.26) tends for t → ∞ to the probability of the event that process w(t; α) never reaches boundary h. In other words, limit Zh lim

t→∞ −∞

dwP(w, t; α, h) = P (wmax (α) < h)

(8.28)

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199

is equal to the probability of the event that the process absolute maximum wmax (α) = max w(t; α) t∈(0,∞)

is less than h. Thus, from Eq. (8.28) follows that the integral distribution function of the absolute maximum wmax (α) is given by the formula

F(h; α) = P (wmax (α) < h) = lim

Zh

t→∞ −∞

dwP(w, t; α, h).

(8.29)

After we solve boundary-value problem (8.22), (8.25) by using, for example, the reflection method, we obtain P(w, t; α, h)      hα (w − 2h + αt)2 (w + αt)2 1 exp − − exp − − . = √ 4Dt D 4Dt 2 π Dt

(8.30)

Substituting this expression in Eq. (8.27), we obtain the probability density of instant t∗ at which process w(t; α) reaches boundary h for the first time   1 (h + αt)2 P(t; α, h) = . exp − √ 4Dt 2Dt π Dt Finally, integrating Eq. (8.30) over w and setting t → ∞, we obtain, in accordance with Eq. (8.29), the integral distribution function of absolute maximum wmax (α) of process w(t; α) in the form [1, 2]   hα F(h; α) = P (wmax (α) < h) = 1 − exp − . D

(8.31)

Consequently, the absolute maximum of the Wiener process has the exponential probability density P(h; α) = hδ (wmax (α) − h)i =

  α hα exp − . D D

The Wiener random process offers the possibility of constructing other processes convenient for modeling different physical phenomena. In the case of positive quantities, the simplest approximation is the logarithmic-normal (lognormal) process. We will consider this process in greater detail.

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8.3.3

Logarithmic-Normal Random Process

We define the lognormal process (logarithmic-normal process) by the formula   Zt   (8.32) y(t; α) = ew(t;α) = exp −αt + dτ z(τ ) ,   0

where z(t) is the Gaussian white noise process with the parameters

 hz(t)i = 0, z(t)z(t0 ) = 2σ 2 τ0 δ(t − t0 ). The lognormal process satisfies the stochastic equation d y(t; α) = {−α + z(t)} y(t; α), dt

y(0; α) = 1.

The one-time probability density of the lognormal process is given by the formula


 P(y, t; α) = hδ (y(t; α) − y)i = δ ew(t;α) − y 1 1 = hδ (w(t; α) − ln y)i = P(w, t; α)|w=ln y , y y where P(w, t; α) is the one-time probability density of the Wiener process with a drift, which is given by Eq. (8.23), so that ) ( (
) ln2 yeαt 1 1 (ln y + αt)2 = √ , P(y, t; α) = √ exp − exp − 4Dt 4Dt 2y πDt 2y πDt (8.33) where D = σ 2 τ0 . Note that the one-time probability density of random processe y(t; α) = 1/y(t; α) is also lognormal and is given by the formula (
) ln2 e ye−αt 1 exp − P(e y, t; α) = √ , 4Dt 2e y πDt

(8.34)

which coincides with Eq. (8.33) with parameter α of opposite sign. Correspondingly, the integral distribution functions are given, in accordance with Eq. (8.24), by the expressions 

1

F(y, t; α) = P (y(t; α) < y) = Pr √ ln ye 2Dt

±αt

where Pr(z) is the probability integral (4.20), page 94.


,

(8.35)

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201

Figure 8.1 shows the curves of the lognormal probability densities (8.33) and (8.34) for α/D = 1 and dimensionless times τ = Dt = 0.1 and 1. Figure 8.2 shows these probability densities at τ = 1 in logarithmic scale. Structurally, these probability distributions are absolutely different. The only common feature of these distributions consists in the existence of long flat tails that appear in distributions at τ = 1; these tails increase the role of high peaks of processes y(t; α) and e y(t; α) in the formation of the one-time statistics. Having only the one-point statistical characteristics of processes y(t; α) ande y(t; α), one can obtain a deeper insight into the behavior of realizations of these processes on the whole interval of times (0, ∞) [2, 5]. In particular,

2.0 τ =1

1.0 τ = 0.1

0.5 0.5

τ = 0.1

0.6 ∼ P(y)

P(y)

1.5

0.8

1.0

1.5

0.4 0.2

τ =1

2.0

1

2

y (a)

∼ y (b)

3

4

Figure 8.1 Logarithmic-normal probability densities (8.33) (a) and (8.34) (b) for α/D = 1 and dimensionless times τ = 0.1 and 1.

2.0 P (y) 1.5

1.0

∼) P (y

−2.0

−1.5

−1.0

−0.5

0.5

0.5

1.0

Figure 8.2 Probability densities of processes y(t; α) (solid line) and e y(t; α) (dashed line) at τ = 1 in common logarithmic scale.

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(1) The lognormal process y(t; α) is the Markovian process and its one-time probability density (8.33) satisfies the Fokker–Planck equation 

 ∂ ∂ ∂ ∂ − α y P(y, t; α) = D y yP(y, t; α), ∂t ∂y ∂y ∂y

P(y, 0; α) = δ(y − 1).

(8.36)

From Eq. (8.36), one can easily derive the equations for the moment functions of processes y(t; α) and e y(t; α); solutions to these equations are given by the formulas (n = 1, 2, . . .)

n  y (t; α) = en(n−α/D)Dt ,

n  e y (t; α) =



 1 = en(n+α/D)Dt , yn (t; α)

(8.37)

from which follows that moments exponentially grow with time. From Eq. (8.36), one can easily obtain the equality (8.38)

hln y(t)i = −αt. Consequently, parameter α appeared in Eq. (8.38) can be rewritten in the form −α =

1 hln y(t)i t

or

α=

1 hlne y(t)i. t

(8.39)

Note that many investigators give great attention to the approach based on the Lyapunov analysis of stability of solutions to deterministic ordinary differential equations d x(t) = A(t)x(t). dt This approach deals with the upper limit of problem solution λx(t) = lim

t→+∞

1 ln |x(t)| t

called the characteristic index of the solution. In the context of this approach applied to stochastic dynamic systems, these investigators often use statistical analysis at the last stage to interpret and simplify the obtained results; in particular, they calculate statistical averages such as

 1 hln |x(t)|i. λx(t) = lim t→+∞ t

(8.40)

Parameter α (8.38) is the Lyapunov exponent of the lognormal random process y(t). (2) From the integral distribution functions, one can calculate the typical realization curves of lognormal processes y(t; α) and e y(t; α) y∗ (t) = ehln y(t)i = e−αt ,

e y ∗ (t) = ehlney(t)i = eαt ,

(8.41)

which, in correspondence with Eqs. (4.60), page 108, are the exponentially decaying curve in the case of process y(t; α) and the exponentially increasing curve in the case of process e y(t; α).

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203

Consequently, the exponential increase of moments of random processes y(t; α) and e y(t; α) are caused by deviations of these processes from the typical realization curves y ∗ (t; α) and e y∗ (t; α) towards both large and small values of y and e y. As it follows from Eq. (8.37) at α/D = 1, the average value of process y(t; D) is independent of time and is equal to unity. Despite this fact, according to Eq. (8.35), the probability of the event that y < 1 for Dt  1 rapidly approaches the unity by the law r P (y(t; D) < 1) = Pr

Dt 2

! =1− √

1 πDt

e−Dt/4 ,

i.e., the curves of process realizations run mainly below the level of the process average hy(t; D)i = 1, which means that large peaks of the process govern the behavior of statistical moments of process y(t; D). Here, we have a clear contradiction between the behavior of statistical characteristics of process y(t; α) and the behavior of process realizations. (3) The behavior of realizations of process y(t; α) on the whole temporal interval can also be evaluated with the use of the p-majorant curves Mp (t, α) whose definition is as follows [2, 5]. We call the majorant curve the curve Mp (t, α) for which inequality y(t; α) < Mp (t, α) is satisfied for all times t with probability p, i.e.,  P y(t; α) < Mp (t, α) for all t ∈ (0, ∞) = p. The above statistics (8.31) of the absolute maximum of the Wiener process with a drift w(t; α) make it possible to outline a wide enough class of the majorant curves. Indeed, let p be the probability of the event that the absolute maximum wmax (β) of the auxiliary process w(t; β) with arbitrary parameter β in the interval 0 < β < α satisfies inequality w(t; β) < h = ln A. It is clear that the whole realization of process y(t; α) will run in this case below the majorant curve Mp (t, α, β) = Ae(β−α)t

(8.42)

with the same probability p. As may be seen from Eq. (8.31), the probability of the event that process y(t; α) never exceeds the majorant curve (8.42) depends on this curve parameters according to the formula p = 1 − A−β/D . This means that we derived the one-parameter class of exponentially decaying majorant curves Mp (t, α, β) =

1 (1 − p)D/β

e(β−α)t .

(8.43)

Notice the remarkable fact that, despite statistical average hy(t; D)i = 1 remains constant and higher-order moments of process y(t; D) are exponentially increasing functions, one can always select an exponentially decreasing majorant curve (8.43) such that realizations of process y(t; D) will run below it with arbitrary predetermined probability p < 1.

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In particular, inequality (τ = Dt) y(t; D) < M1/2 (t, D, D/2) = M(τ ) = 4e−τ/2

(8.44)

is satisfied with probability p = 1/2 for any instant t from interval (0, ∞). Figure 8.3 schematically shows the behaviors of a realization of process y(t; D) and the majorant curve (8.44). This schematic is an additional fact in favor of our conclusion that the exponential growth of moments of process y(t; D) with time is the purely statistical effect caused by averaging over the whole ensemble of realizations. Note that the area below the exponentially decaying majorant curves has a finite value. Consequently, high peaks of process y(t; α), which are the reason of the exponential growth of higher moments, only insignificantly contribute to the area below realizations; this area appears finite for almost all realizations, which means that the peaks of the lognormal process y(t; α) are sufficiently narrow. (4) In this connection, it is of interest to investigate immediately the statistics of random area below realizations of process y(t; α) Sn (t; α) =

Zt

dτ yn (τ ; α).

(8.45)

0

This function satisfies the system of stochastic equations d Sn (t; α) = yn (t; α), Sn (0; α) = 0, dt d y(t; α) = {−α + z(t)} y(t; α), y(0; α) = 1, dt

(8.46)

4

3

2 M (τ ) y (τ )

1

1

2

3

τ

Figure 8.3 Schematic behaviors of a realization of process y(t; D) and majorant curve M(τ ) (8.44).

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205

so that the two-component process {y(t; α), Sn (t; α)} is the Markovian process whose onetime probability density P(Sn , y, t; α) = hδ (Sn (t; α) − Sn ) δ (y(t; α) − y)i and transition probability density satisfy the Fokker–Planck equation 

 ∂ ∂ ∂ ∂ ∂ + yn − α y P(Sn , y, t; α) = D y yP(Sn , y, t; α), ∂t ∂Sn ∂y ∂y ∂y

(8.47)

P(Sn , y, 0; α) = δ(Sn )δ(y − 1). Unfortunately, Eq. (8.47) cannot be solved analytically, which prevents us from studying the statistics of process Sn (t; α) exhaustively. However, for the one-time statistical averages of process Sn (t; α), i.e., averages at a fixed instant, the corresponding statistics can be studied in sufficient detail. With this goal in view, we rewrite Eq. (8.45) in the form

Sn (t; α) =

0

Zt = 0

Zτ

 

Zt

dτ exp −nατ + n 

0

  dτ1 z(τ1 ) 

 

dτ exp −nα(t − τ ) + n 

Zt−τ 0

  dτ1 z(t − τ − τ1 ) , 

from which it follows that quantity Sn (t; α) in the context of the one-time statistics is statistically equivalent to the quantity (see section 4.2.1, page 95)

Sn (t; α) =

Zt

−nα(t−τ )+n

dτ e

t−τ R

dτ1 z(τ +τ1 )

0

.

(8.48)

0

Differentiating now Eq. (8.48) with respect to time, we obtain the statistically equivalent stochastic equation d Sn (t; α) = 1 − n{α − z(t)}Sn (t; α), dt

Sn (0; α) = 0,

whose one-time statistical characteristics are described by the one-time probability density P(Sn , t; α) = hδ (Sn (t; α) − Sn )i that satisfies the Fokker–Planck equation 

 ∂ ∂ ∂ ∂ ∂ + − nα Sn P(Sn , t; α) = n2 D Sn Sn P(Sn , y, t; α). ∂t ∂Sn ∂Sn ∂Sn ∂Sn

As may be seen from Eq. (8.49), random integrals Sn (α) =

Z∞ 0

dτ yn (τ ; α)

(8.49)

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are distributed according to the steady-state probability density P(Sn ; α) = 

n2 D

α/nD

  1 1 exp − 2 , α n DSn 1+α/D Sn 0 D

where 0(z) is the gamma function. In the special case n = 1, quantity S(α) = S1 (α) =

Z∞

dτ y(τ ; α)

0

has the following probability density P(S; α) =

Dα/D 0

1 α D



S1+α/D

 1 exp − . DS

(8.50)

If we set now α = D, then the steady-state probability density and the corresponding integral distribution function will have the form     1 1 1 P(S; D) = exp − , F(S; D) = exp − . (8.51) DS DS DS2 The time-dependent behavior of the probability density of random process e S(t, α) =

Z∞

dτ y(τ ; α)

(8.52)

t

gives an additional information about the behavior of realizations of process y(t; α) with time t. The integral in the right-hand side of Eq. (8.52) can be represented in the form e S(t, α) = y(t; α)

Z∞ 0

  Zτ   dτ exp −ατ + dτ1 z(τ1 + t) .  

(8.53)

0

In Eq. (8.53), random process y(t; α) is statistically independent of the integral factor, because they depend functionally on process z(τ ) for nonoverlapping intervals of times τ ; in addition, the integral factor by itself appears statistically equivalent


to random quantity S(α). Consequently, the one-time probability density P(e S, t; α) = δ e S(t; α) − e S of random process e S(t, α) is described by the expression P(e S, t; α) =

Z∞ Z∞ 0 0

dydSδ(yS − e S)P(y, t; α) =

Z∞

e  S dy P(y, t; α)P ;α , y y

(8.54)

0

where e P(y,  t; α) is the one-time probability density of lognormal process y(t; α) (8.33) and S P ; α is the probability density (8.50) of random area. y

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207

The corresponding integral distribution function


F(e S, t; α) = P e S(t; α) < e S =

ZeS

de SP(e S, t; α)

0

is given by the integral

F(e S, t; α) =

Z∞

dyP(y, t; α)F

e  S ;α , y

0

where F(S; α) is the integral distribution function of random area S(t; α). In the special case α = D, we obtain, according to Eq. (8.33) and (8.51), the expression 1 F(e S, t; D) = √ 2 π Dt

Z∞ 0

) (
ln2 yeDt dy y exp − − y 4Dt De S

from which follows that the probability of the event that inequality e S(t; D) < e S is satisfied monotonously tends to unity with increasing Dt for any predetermined value of De S. This is an additional evidence in favor of the fact that every separate realization of the lognormal process tends to zero with increasing Dt, though moment functions of process y(t; α) show the exponential growth caused by large spikes. (5) Now, we dwell on positive random fields E(r, t) closely related to lognormal processes whose one-point probability density is described by the equation ∂ ∂2 − D0 2 ∂t ∂r

!

 ∂ ∂ ∂ P(r, t; E) = α E + D E E P(r, t; E), ∂E ∂E ∂E P(r, 0; E) = δ (E − E0 (r)), 

(8.55)

where D0 is the coefficient of diffusion in r-space, and coefficients α and D characterize diffusion in E-space. Parameter α can be both positive and negative. In the context of the one-point characteristics, the change of sign of parameter α means transition from random field E(r, t) to random field e E(r, t) = 1/E(r, t). For definiteness, we will term field E(r, t) energy. The solution to this equation has the form ) ∂2 P(r, t; E) = exp D0 t 2 e P(r, t; E), ∂r (

where function e P(r, t; E) satisfies the equation (cf. with Eq. (8.36))   ∂ ∂ ∂ ∂ e P(r, t; E) = α E + D E E e P(r, t; E), ∂t ∂E ∂E ∂E e P(r, 0; E) = δ (E − E0 (r)) .

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As may be seen, dependence of function e P(r, t; E) on parameter r is determined solely by initial value E0 (r), i.e., e P(r, t; E) ≡ e P(t; E|E0 (r)) P(t; E|E0 (r)) is the lognormal probability density of random process and, hence, function e E(t, |E0 (r)) dependent on r parametrically, e P(t; E|E0 (r)) =

(  ) ln2 Eeαt /E0 (r) 1 exp − . √ 4Dt 2E π Dt

Thus, the solution to Eq. (8.55) has the form ( ) (  ) ln2 Eeαt /E0 (r) ∂2 1 exp D0 t 2 exp − P(r, t; E) = . √ 4Dt ∂r 2E πDt

(8.56)

A consequence of equation (8.55) or expression (8.56) are relationships for spatial integrals of moment functions (for all n > 0 and n < 0) Z

 dr En (r, t) = en(nD−α)t

Z

drE0n (r),

which are independent of the coefficient of diffusion in r-space (coefficient D0 ), and, in particular, the expression for average total energy in the whole space, Z

dr hE(r, t)i = eγ t

Z

drE0 (r),

where γ = D−α. If the initial energy distribution is homogeneous, E0 (r) = E0 , probability density (8.56) is independent of r and is given by Eq. (8.33), (  ) ln2 Eeαt /E0 1 exp − . P(t; E) = √ 4Dt 2E πDt

(8.57)

Thus, in this case, the one-point statistical characteristics of energy E(r, t) are statistically equivalent to statistical characteristics of random process E(t) = y(t; α) (8.32). A characteristic feature of distribution (8.57) consists in a long flat tail appeared for Dt  1, which is indicative of an increased role of great peaks of process E(t; α) in the formation of the one-point statistics. For this distribution, all moments of energy are exponential functions and, in particular, hE(t)i = E0 eγ t ,

γ =D−α

at n = 1, and hln (E(t)/E0 )i = −αt,

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209

so that the typical realization curve determining the behavior of energy at arbitrary spatial point in specific realizations of random process E(t) is the exponential quantity E∗ (t) = E0 e−αt increasing or decreasing with time. For α < 0, the typical realization curve exponentially increases with time, which is evidence of general increase of energy at every spatial point. Otherwise, for α > 0, the typical realization curve exponentially decreases at every spatial point, which is indicative of cluster structure of energy field. Figure 8.4 schematically shows random realizations of energy field for different signs of parameter α. The knowledge of the one-point probability density of energy (8.56) yields general knowledge of the spatial structure of energy. In particular, functionals of the energy field such as total average volume (in the three-dimensional case) or area (in the twodimensional case) of the region in which E(r, t) > E and total average energy contained in this region are given by the equalities Z∞ hV(t, E)i =

Z de E

E Z∞

hE(t, E)i =

Z

e Ede E

dr P(r, t; e E),

dr P(r, t; e E).

E

E (r, 0)

1

r

α >0

r

E ∗ (t )

E (r, t1)

t

t1

r

r

α <0 E (r, t1)

E (r, 0)

1

E ∗ (t ) t1

Figure 8.4

t

Schematic behavior of random realization of energy for α > 0 and α < 0.

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The values of these functionals are independent of the coefficient of diffusion in r-space (coefficient D0 ), and, using probability density (8.56), we obtain the expressions    1 E0 (r) −αt dr Pr √ ln e , E 2Dt

Z hV(t, E)i =

(8.58) hE(t, E)i = eγ t

 E0 (r) (2D−α)t 1 ln , e dr E0 (r) Pr √ E 2Dt 



Z

where probability integral Pr(z) is defined by Eq. (4.20), page 94. Asymptotic expressions of function Pr(z) for z → ∞ and z → −∞ (4.23), page 95, offer a possibility of studying temporal evolution of these functionals. Namely, average volume asymptotically decays (for α > 0) with time (t → ∞) according to the law hV(t, E)i ≈

1 α

r

D 2 e−α t/4D πEα/D t

Z

q dr

α/D

E0

(r).

On the contrary, for α < 0, average volume occupies the whole space for t → ∞. Asymptotic behavior of total energy for t → ∞ has the form (in the most interesting case of γ > 0, α < D) hE(t, E)i ≈ eγ t

Z

 1 drE0 (r) 1 − (2D − α)

s

D πt



E E0 (r)

(2D−α)/D

 2

e−(2D−α)

,

t/4 

which means that 100% of total average energy is contained in clusters for α > 0. In the case of homogeneous initial conditions, the corresponding expressions without integration over r present specific values of volume occupied by large peaks and their total energy per unit volume: hV(t, E)i = Pr



hE(t, E)i = E0

eγ t

√1 2Dt

ln



E0 (r) −αt E e



, (8.59)

Pr



√1 2Dt

ln



E0 (r) (2D−α)t E e



,

where γ = D − α. If we take section at level E > E0 , then the initial values of these quantities will be equal to zero at the initial instant, hV(0, E)i = 0 and hE(0, E)i = 0. Then, spatial disturbances of energy field appear with time and, for t → ∞, they are given by the following asymptotic expressions:

hV(t, E)i ≈

       

1 α

s

D πt



E0 E

α/D

e−α

2

t/4D

(α > 0),

s       1 D E |α|/D −α 2 t/4D   e (α < 0), 1 − |α| πt E0

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211

and, for D > α,  1 hE(t, E))i ≈ E0 eγ t 1 − (2D − α)

s

D πt



E E0

(2D−α)/D

 2

e−(2D−α)

.

t/4D 

Thus, for α > 0, specific total volume tends to zero and specific total energy increases with time as average energy does, which is evidence of clustering the field of energy in this case. For α < 0, no clustering occurs and specific volume occupies the whole space in which specific energy increases with time as average energy does.

8.4

Applicability Range of the Fokker–Planck Equation

To estimate the applicability range of the Fokker–Planck equation, we must include into consideration the finite-valued correlation radius τ0 of field f (x, t) with respect to time. Thus, smallness of parameter τ0 /T is the necessary but generally not sufficient condition in order that one can describe the statistical characteristics of the solution to Eq. (8.1), using the approximation of the delta-correlated random field of which a consequence is the Fokker–Planck equation. Every particular problem requires more detailed investigation. Below, we give a more physical method called the diffusion approximation. This method also leads to the Markovian property of the solution to Eq. (8.1); however, it considers to some extent the finite value of the temporal correlation radius. Here, we emphasize that the approximation of the delta-correlated random field does not reduce to the formal replacement of random field f (x, t) in Eq. (8.1) with the random field with correlation function (8.4). This approximation corresponds to the construction of an asymptotic expansion in temporal correlation radius τ0 of filed f (x, t) for τ0 → 0. It is in such a limited process that exact average quantities like

 f (x, t)R[t; f (x0 , τ )] grade into the expressions obtained by the formal replacement of the correlation tensor of field f (x, t) with the effective tensor (8.4).

8.4.1

Langevin Equation

We illustrate the above speculation with the example of the Langevin equation that allows an exhaustive statistical analysis. This equation has the form d x(t) = −λx(t) + f (t), dt

x(t0 ) = 0

(8.60)

and assumes that the sufficiently fine smooth function f (t) is the stationary Gaussian process with zero-valued mean and correlation function

 f (t)f (t0 ) = Bf (t − t0 ).

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For any individual realization of random force f (t), the solution to Eq. (8.60) has the form Zt x(t) =

dτ f (τ )e−λ(t−τ ) .

t0

Consequently, this solution x(t) is also the Gaussian process with the parameters

hx(t)i = 0,

 x(t)x(t0 ) =

Zt0

Zt dτ1 t0

dτ2 Bf (τ1 − τ2 )e−λ(t+t −τ1 −τ2 ) . 0

t0

In addition, we have, for example, Zt−t0 h f (t)x(t)i = dτ Bf (τ )e−λτ . 0

Note that the one-point probability density P(x, t) = hδ(x(t) − x)i for Eq. (8.60) satisfies the equation 

 Zt−t0 ∂2 ∂ ∂ dτ Bf (τ )e−λτ 2 P(x, t), − λ x P(x, t) = ∂t ∂x ∂x

P(x, t0 ) = δ(x),

0

which rigorously follows from Eq. (6.24), page 149. As a consequence, we obtain

 d 2  x (t) = −2λ x2 (t) + 2 dt

Zt−t0 dτ Bf (τ )e−λτ . 0

For t0 → −∞, process x(t) grades into the stationary Gaussian process with the following one-time statistical parameters (hx(t)i = 0) σx2

 1 = x2 (t) = λ

Z∞

dτ Bf (τ )e

−λτ

,



f (t)x(t) = 

Z∞

0

dτ Bf (τ )e−λτ .

0

In particular, for exponential correlation function Bf (t), Bf (t) = σf2 e−|τ |/τ0 , we obtain the expressions hx(t)i = 0,

D E x2 (t) =

σf2 τ0 λ(1 + λτ0 )

,

h f (t)x(t)i =

σf2 τ0 1 + λτ0

,

(8.61)

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213

which grade into the asymptotic expressions

2  σf2 τ0 , x (t) = λ

h f (t)x(t)i = σf2 τ0

(8.62)

for τ0 → 0. Multiply now Eq. (8.60) by x(t). Assuming that function x(t) is sufficiently fine function, we obtain the equality x(t)

d 1d 2 x(t) = x (t) = −λx2 (t) + f (t)x(t). dt 2 dt

Averaging this equation over an ensemble of realizations of function f (t), we obtain the equation



 1 d 2  x (t) = −λ x2 (t) + f (t)x(t) , 2 dt

(8.63)

whose steady-state solution (it corresponds to the limit process t0 → −∞ and τ0 → 0)

2  1

 x (t) = f (t)x(t) λ coincides with Eq. (8.61) and (8.62). Taking into account the fact that δx(t)/δf (t − 0) = 1, we obtain the same result for correlation h f (t)x(t)i by using the formula h f (t)x(t)i =

Zt −∞

δ x(t) dτ Bf (t − τ ) δf (τ ) 

 (8.64)

with the effective correlation function 2 Beff f (t) = 2σf τ0 δ(t).

Earlier, we mentioned that statistical characteristics of solutions to dynamic problems in the approximation of the delta-correlated random process (field) coincide with the statistical characteristics of the Markovian processes. However, one should clearly understand that this is the case only for statistical averages and equations for these averages. In particular, realizations of process x(t) satisfying the Langevin equation (8.60) drastically differ from realizations of the corresponding Markovian process. The latter satisfies Eq. (8.60) in which function f (t) in the right-hand side is the ideal white noise with the correlation function Bf (t) = 2σf2 τ0 δ(t); moreover, this equation must be treated in the sense of generalized functions, because the Markovian processes are not differentiable in the ordinary sense. At the same time, process x(t) – whose

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Lectures on Dynamics of Stochastic Systems

statistical characteristics coincide with the characteristics of the Markovian process – behaves as a sufficiently fine function and is differentiable in the ordinary sense. For example, x(t)

1d 2 d x(t) = x (t), dt 2 dt

and we have for t0 → −∞ in particular   d x(t) x(t) = 0. dt

(8.65)

On the other hand, in the case of the ideal Markovian process x(t) satisfying (in the sense of generalized functions) the Langevin equation (8.60) with the white noise in the right-hand side, Eq. (8.65) makes no sense at all, and the meaning of the relationship   D E d x(t) x(t) = −λ x2 (t) + h f (t)x(t)i dt

(8.66)

depends on the definition of averages. Indeed, if we will treat Eq. (8.66) as the limit of the equality   d x(t + 1) x(t) = −λ hx(t)x(t + 1)i + h f (t)x(t + 1)i dt

(8.67)

for 1 → 0, the result will be essentially different depending on whether we use limit processes 1 → +0, or 1 → −0. For limit process 1 → +0, we have lim h f (t)x(t + 1)i = 2σf2 τ0 ,

1→+0

and, taking into account Eq. (8.64), we can rewrite Eq. (8.67) in the form   d x(t + 0) x(t) = σf2 τ0 . dt

(8.68)

On the contrary, for limit process 1 → −0, we have h f (t)x(t − 0)i = 0 because of the dynamic causality condition, and Eq. (8.67) assumes the form   d x(t − 0) x(t) = −σf2 τ0 . dt

(8.69)

Approximations of Gaussian Random Field Delta-Correlated in Time

215

Comparing Eq. (8.65) with Eq. (8.68) and (8.69), we see that, for the ideal Markovian process described by the solution to the Langevin equation with the white noise in the right-hand side and commonly called the Ohrnstein–Ulenbeck process, we have     d 1d D 2 E d x (t) . x(t + 0) x(t) 6= x(t − 0) x(t) 6= dt dt 2 dt Note that equalities (8.68) and (8.69) can also be obtained from the correlation function hx(t)x(t + τ )i =

σf2 τ0 λ

e−λ|τ |

of process x(t). To conclude with a discussion of the approximation of the delta-correlated random process (field), we emphasize that, in all further examples, we will treat the statement ‘dynamic system (equation) with the delta-correlated parameter fluctuations’ as the asymptotic limit in which these parameters have temporal correlation radii small in comparison with all characteristic temporal scales of the problem under consideration.

8.5

Causal Integral Equations

In problems discussed earlier, we succeeded in deriving the closed statistical description in the approximation of the delta-correlated random field due to the fact that each of these problems corresponded to a system of the first-order (in temporal coordinate) differential equations with given initial conditions at t = 0. Such systems possess the dynamic causality property, which means that the solution at instant t depends only on system parameter fluctuations for preceding times and is independent of fluctuations for consequent times. However, problems described in terms of integral equations that generally cannot be reduced to a system of differential equations can also possess the causality property. In short, we illustrate this fact by the simplest example of the one-dimensional causal equation (t > t0 ) G(t; t ) = g(t; t ) + 3 0

0

Zt

dτ g(t; τ )z(τ )G(τ ; t0 ),

(8.70)

t0

where function g(t; t0 ) is Green’s function for the problem with absent parameter fluctuations, i.e., for z(t) = 0, and we assume that z(t) is the Gaussian delta-correlated random function with the parameters hz(t)i = 0,



 z(t)z(t0 ) = 2Dδ(t − t0 )

(D = σz2 τ0 ).

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Averaging then Eq. (8.70) over an ensemble of realizations of random function z(t), we obtain the equation

 G(t; t0 ) = g(t; t0 ) + 3

Zt

 dτ g(t; τ ) z(τ )G(τ ; t0 ) .

(8.71)

t0

Taking into account equality δ G(t; t0 ) = g(t; t)3G(t; t0 ) δz(t)

(8.72)

following from Eq. (8.70), we can rewrite the correlator in the right-hand side of Eq. (8.71) in the form

 z(τ )G(τ ; t0 ) = D





 δ G(τ ; t0 ) = 3Dg(τ ; τ ) G(τ ; t0 ) . δz(τ )

As a consequence, Eq. (8.71) grades into the closed integral equation for average Green’s function

 G(t; t0 ) = g(t; t0 ) + 32 D

Zt

 dτ g(t; τ )g(τ ; τ ) G(τ ; t0 ) ,

(8.73)

t0

which has the form of the Dyson equation (in the terminology of the quantum field theory)

 G(t; t0 ) = g(t; t0 ) + 3

Zt

dτ g(t; τ )

t0

Zτ

 dτ 0 Q(τ ; τ 0 ) G(τ 0 ; t0 ) ,

t0

or

 G(t; t0 ) = g(t; t0 ) + 3

Zt

(8.74)

dτ hG(t; τ )i

t0

Zτ

dτ 0 Q(τ ; τ 0 )g(τ 0 ; t0 ),

t0

with the mass function Q(τ ; τ 0 ) = 32 Dg(τ ; τ )δ(τ − τ 0 ). Derive now the equation for the correlation function

 0(t, t0 ; t1 , t10 ) = G(t; t0 )G∗ (t1 ; t10 )

(t > t0 ,

t1 > t10 ),

Approximations of Gaussian Random Field Delta-Correlated in Time

217

where G∗ (t; t0 ) is complex conjugated Green’s function. With this goal in view, we multiply Eq. (8.70) by G∗ (t1 ; t10 ) and average the result over an ensemble of realizations of random function z(t). The result is the equation that can be symbolically represented as



 0 = g G∗ + 3g zGG∗ . (8.75) Taking into account the Dyson equation (8.74) hGi = {1 + hGi Q}g, we apply operator {1 + hGi Q} to Eq. (8.75). As a result, we obtain the symbolic-form equation

 

 0 = hGi G∗ + hGi 3 zGG∗ − Q0 , which can be represented in common variables as



 0(t, t0 ; t1 , t10 ) = G(t; t0 ) G∗ (t1 ; t10 )   Zt δG∗ (t1 ; t10 ) δG(τ ; t0 ) ∗ G (t1 ; t10 ) + 2G(τ ; t0 ) + 3D dτ hG(t; τ )i δz(τ ) δz(τ ) 0

− 32 D

Zt

dτ hG(t; τ )i g(τ ; τ )0(τ, t0 ; t1 , t10 ).

(8.76)

0

Deriving Eq. (8.76), we used additionally Eq. (5.30), page 130, for splitting correlations between the Gaussian delta-correlated process z(t) and functionals of this process    δ   R[t; z(τ )] (t0 = t, τ < t), D   δz(t)

0  z(t )R[t; z(τ )] >=    δ    2D R[t; z(τ )] (t0 < t, τ < t). δz(t0 ) Taking into account Eq. (8.72) and equality δ G∗ (t1 ; t10 ) = 3G∗ (t1 ; τ )G∗ (τ ; t10 ), δz(τ ) following from Eq. (8.70) we can rewrite Eq. (8.76) as



 0(t, t0 ; t1 , t10 ) = G(t; t0 ) G∗ (t1 ; t10 ) 2

Zt

+ 2|3| D 0

 dτ hG(t; τ )i G∗ (t1 ; τ )G(τ ; t0 )G∗ (τ ; t10 ) .

(8.77)

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Now, we take into account the fact that function G∗ (t1 ; τ ) functionally depends on random process z(e τ ) for e τ ≥ τ while functions G(τ ; t0 ) and G∗ (τ ; t10 ) depend on it for e τ ≤ τ . Consequently, these functions are statistically independent in the case of the delta-correlated process z(e τ ), and we can rewrite Eq. (8.77) in the form of the closed equation (t1 ≥ t) 



0(t, t0 ; t1 , t10 ) = G(t; t0 ) G∗ (t1 ; t10 ) 2

Zt

+ 2|3| D

 dτ hG(t; τ )i G∗ (t1 ; τ ) 0(τ ; t0 ; τ ; t10 ).

(8.78)

0

8.6

Diffusion Approximation

Applicability of the approximation of the delta-correlated random field f (x, t) (i.e., applicability of the Fokker–Planck equation) is restricted by the smallness of the temporal correlation radius τ0 of random field f (x, t) with respect to all temporal scales of the problem under consideration. The effect of the finite-valued temporal correlation radius of random field f (x, t) can be considered within the framework of the diffusion approximation. The diffusion approximation appears to be more obvious and physical than the formal mathematical derivation of the approximation of the delta-correlated random field. This approximation also holds for sufficiently weak parameter fluctuations of the stochastic dynamic system and allows us to describe new physical effects caused by the finite-valued temporal correlation radius of random parameters, rather than only obtaining the applicability range of the delta-correlated approximation. The diffusion approximation assumes that the effect of random actions is insignificant during temporal scales about τ0 , i.e., the system behaves during these time intervals as the free system. Again, assume vector function x(t) satisfies the dynamic equation (8.1), page 191, d x(t) = v(x, t) + f (x, t), dt

x(t0 ) = x0 ,

(8.79)

where v(x, t) is the deterministic vector function and f (x, t) is the random statistically homogeneous and stationary Gaussian vector field with the statistical characteristics h f (x, t)i = 0,

 Bij (x, t; x0 , t0 ) = Bij (x − x0 , t − t0 ) = fi (x, t)fj (x0 , t0 ) .

Introduce the indicator function ϕ(x, t) = δ(x(t) − x),

(8.80)

where x(t) is the solution to Eq. (8.79) satisfying the Liouville equation (8.6)   ∂ ∂ ∂ + v(x, t) ϕ(x, t) = − f (x, t)ϕ(x, t). (8.81) ∂t ∂x ∂x

Approximations of Gaussian Random Field Delta-Correlated in Time

219

As earlier, we obtain the equation for the probability density of the solution to Eq. (8.79) P(x, t) = hϕ(x, t)i = hδ(x(t) − x)i by averaging Eq. (8.81) over an ensemble of realizations of field f (x, t) 

 ∂ ∂ ∂ + v(x, t) P(x, t) = − hf (x, t)ϕ(x, t)i, ∂t ∂x ∂x

(8.82)

P(x, t0 ) = δ(x − x0 ). Using the Furutsu–Novikov formula (8.10), page 193, h fk (x, t)R[t; f (y, τ )]i =

Z

0

dx

Z

 δ dt Bkl (x, t; x , t ) R[t; f (y, τ )] δ fl (x0 , t0 ) 0

0

0



valid for the correlation between the Gaussian random field f (x, t) and arbitrary functional R[t; f (y, τ )] of this field, we can rewrite Eq. (8.82) in the form 

   Z Zt ∂ ∂ ∂ δ 0 + v(x, t) P(x, t) = − dx dt0 Bij (x, t; x0 , t0 ) ϕ(x, t) . ∂t ∂x ∂xi δ fj (x0 , t0 ) t0

(8.83) In the diffusion approximation, Eq. (8.83) is the exact equation, and the variational derivative and indicator function satisfy, within temporal intervals of about temporal correlation radius τ0 of random field f (x, t), the system of dynamic equations   ∂ δϕ(x, t) ∂ δϕ(x, t) =− v(x, t) , ∂t δfi (x0 , t0 ) ∂x δfi (x0 , t0 ) δϕ(x, t) ∂  δ(x − x0 )ϕ(x, t0 ) , =− 0 0 δfi (x , t ) t=t0 ∂xi ∂ ∂ ϕ(x, t) = − {v(x, t)ϕ(x, t)}, ∂t ∂x

(8.84)

ϕ(x, t)|t=t0 = ϕ(x, t0 ).

The solution to problem (8.83), (8.84) holds for all times t. In this case, the solution x(t) to problem (8.79) cannot be considered as the Markovian vector random process because its multi-time probability density cannot be factorized in terms of the transition probability density. However, in asymptotic limit t  τ0 , the diffusionapproximation solution to the initial dynamic system (8.79) will be the Markovian random process, and the corresponding conditions of applicability are formulated as smallness of all statistical effects within temporal intervals of about temporal correlation radius τ0 .

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Problems Problem 8.1 Assuming that ε(x, R), is the homogeneous isotropic delta-correlated Gaussian field with zero-valued mean and correlation function

 Bε (x − x0 , R − R0 ) = ε(x, R)ε(x0 , R0 ) = A(R − R0 )δ(x − x0 ), derive the Fokker–Planck equation for the diffusion of rays satisfying system of equations (1.92), page 42, d R(x) = p(x), dx

d 1 p(x) = ∇ R ε(x, R). dx 2

(8.85)

Solution Function P(x; R, p) = hδ(R(x) − R)δ(p(x) − p)i satisfies the Fokker–

Planck equation 

 ∂ ∂ +p P(x; R, p) = D1R P(x; R, p) ∂x ∂R

1 with the diffusion coefficient D = − 1R A(R)|R=0 = π 2 8 Here,

∞

Z 0

dκκ 3 8ε (0, κ).

Z∞ Z 1 8ε (q, κ) = dx dRBε (x, R)e−i(qx+κR) , (2π)3 −∞ Z A(R) = 2π dκ8ε (0, κ)eiκR , is the three-dimensional spectral density of random field ε(x, R). Solution corresponding to the initial condition P(0; R, p) = δ(R)δ(p) is the Gaussian probability density with the parameters

 2

 Rj (x)Rk (x) = Dδjk x3 , Rj (x)pk (x) = Dδjk x2 , 3

 pj (x)pk (x) = 2Dδjk x.

Problem 8.2 Assuming that ε(x, R) is the homogeneous isotropic delta-correlated Gaussian field with zero-valued mean and correlation function

 Bε (x − x0 , R − R0 ) = ε(x, R)ε(x0 , R0 ) = A(R − R0 )δ(x − x0 ),

Approximations of Gaussian Random Field Delta-Correlated in Time

221

derive, starting from Eqs. (8.85), the longitudinal correlation function of ray displacement. Solution

 

 1 R(x)R(x0 ) = 2D(x0 )2 x − x0 . 3

Problem 8.3 Consider the diffusion of two rays described by the system of equations d Rν (x) = pν (x), dx

d 1 p (x) = ∇ Rν ε(x, Rν ), dx ν 2

where index ν = 1, 2 denotes the number of the corresponding ray. Solution The joint probability density

 P(x; R1 , p1 , R2 , p2 ) = δ(R1 (x) − R1 )δ(p1 (x) − p1 )δ(R2 (x) − R2 )δ(p2 (x) − p2 ) satisfies the Fokker–Planck equation 

 ∂ ∂ ∂ + p1 + p2 P(x; R1 , p1 , R2 , p) ∂x ∂R1 ∂R2   ∂ ∂ , P(x; R1 , p1 , R2 , p), =b L ∂p1 ∂p2

where operator ∂ ∂ b L , ∂p1 ∂p2 



"

   ∂ 2 ∂ 2 + κ ∂p1 ∂p2    ∂ ∂ κ . + 2 cos [κ(R1 − R2 )] κ ∂p1 ∂p1

π = 4

Z

dκ8ε (0, κ)

κ

Problem 8.4 Consider the relative diffusion of two rays. Solution Probability density of relative diffusion of two rays

 P(x; R, p) = δ(R1 (x) − R2 (x) − R)δ(p1 (x) − p2 (x) − p) ,

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satisfies the Fokker–Planck equation 

 ∂ ∂ ∂2 +p P(x; R, p) = Dαβ (R) P(x; R, p), ∂x ∂R ∂pα ∂pβ

(8.86)

where Dαβ (R) = 2π

Z

dκ [1 − cos (κR)] κα κβ 8ε (0, κ).

Remark If l0 is the correlation radius of random field ε(x, R), than under the

condition R  l0 we have Dαβ (R) = 2Dδαβ , i.e., relative diffusion of rays is characterized by the diffusion coefficient exceeding the diffusion coefficient of a single ray by a factor of 2, which corresponds to statistically independent rays. In this case, the joint probability density of relative diffusion is Gaussian. In the general case, Eq. (8.86) can hardly be solved in the analytic form. The only clear point is that the solution corresponding to variable diffusion coefficient is not the Gaussian distribution. Asymptotic case R  l0 can be analyzed in more detail. In this case, we can expand function {1 − cos (κR)} in the Taylor series to obtain the diffusion matrix in the form   Dαβ (R) = π B R2 δaβ + 2Rα Rβ ,

(8.87)

where π B= 4

Z∞

dκκ 5 8ε (0, κ).

0

From Eq. (8.86) with coefficients (8.87) follow three equations for moment functions D E d D 2 E p (x) = 8πB R2 (x) , dx D E d hR(x)p(x)i = p2 (x) . dx

d D 2 E R (x) = 2 hR(x)p(x)i, dx

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223

These equations can be easily solved. From the solution follows that, for x from the interval in which αx  1 (α = (16πB)1/3 ), but R20 eαx  l02 (such an interval always exists for sufficiently the rays under conside

 small distances R0 between  ration), quantities R2 (x) , hR(x)p(x)i, and p2 (x) are exponentially increasing functions of x.

Problem 8.5 Derive an equation for the probability density of magnetic field with homogeneous initial conditions with the use of the simplest model of velocity field (1.3), (1.4), page 4, u(r, t) = v(t)f (kx), where v(t) is the stationary Gaussian vector random process with the correlation tensor

 vi (t)vj (t0 ) = 2σ 2 δij τ0 δ(t − t0 ), where σ 2 is the variance of every velocity component, τ0 is the temporal correlation radius of these components, and f (kx) is a deterministic function. Solution In the dimensionless variables

t → k2 σ 2 τ0 t,

x → kx,

 vi (t)vj (t0 ) → 2δij δ(t − t0 ),

the indicator function of the transverse portion of magnetic field is described by Eq. (3.27), page 76. Averaging this equation over an ensemble of realizations of vector function v(t), we obtain the equation   ∂ ∂ ∂ ∂ 0 2 P(x, t; H) = f (x) − f (x)f (x) H P(x, t; H) ∂t ∂x ∂x ∂H     0 2 ∂ ∂ ∂ 0 H −f (x)f (x) + f (x) H P(x, t; H) + 1+ ∂H ∂x ∂H  2 ∂ ∂ + f 0 (x) Ex0 P(x, t; H). ∂Hi ∂Hi Remark If function f (x) varies along x with characteristic scale k−1 and is a

periodic function (‘fast variation’), then additional averaging over x results in an

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equation for ‘slow’ spatial variations   ∂ ∂ ∂ 2 f (x) P(x, t; H) = P(x, t; H) ∂t ∂x ∂x   ∂ ∂ ∂2 2 0 H HP(x, t; H) + [f 0 (x)]2 Ex0 P(x, t; H). + [f (x)] 1 + ∂H ∂H ∂H2 Taking into account the fact that function P(x, t; H) is independent of x in view of the initial conditions, we obtain the equation   ∂ ∂ ∂ ∂ ∂2 P(t; H), P(t; H) = D H+ H H P(t; H) + DEx0 ∂t ∂H ∂H ∂H ∂H2 where D = [f 0 (x)]2 . Then, introducing the normalized time (t → Dk2 σ 2 τ0 t), we obtain the equation in the final form   ∂ ∂ ∂ ∂ ∂2 P(t; H) = H+ H H P(t; H) + Ex0 P(t; H). (8.88) ∂t ∂H ∂H ∂H ∂H2

Problem 8.6 Starting from Eq. (8.88), derive an equation for the probability density of generated energy of the transverse portion of magnetic field with the use of the simplest model of velocity field (1.3), (1.4), page 4. Solution Multiplying Eq. (8.88) by δ(H2 − E) and integrating over H, we obtain

the equation   ∂ ∂ ∂ ∂ ∂ ∂ P(t; E) = 2 E + 4 E E P(t; E) + 4Ex0 E P(t; E). ∂t ∂E ∂E ∂E ∂E ∂E (8.89) Introduce dimensionless energy by normalizing it by Ex0 . The result will be the equation   ∂ ∂ ∂ ∂ ∂ ∂ P(t; E) = 2 E + 4 E E P(t; E) + 4 E P(t; E), (8.90) ∂t ∂E ∂E ∂E ∂E ∂E with the initial condition P(0; E) = δ(E − β), where β = E⊥0 /Ex0 .

Approximations of Gaussian Random Field Delta-Correlated in Time

225

Remark It should be remembered that problem (8.90) is the boundary-value prob-

lem, and boundary conditions are the requirements of vanishing the probability density at E = 0 and E = ∞ for arbitrary time t > 0. In addition, Eq. (8.90) generally has a steady-state solution independent of parameter β and characterized by infinite moments, P(∞; E) =

1 2 (E + 1)3/2

.

(8.91)

It should be kept in mind that this distribution is a generalized function and is valid on interval (0, ∞). As for point E = 0, the probability density must vanish at this point. Note that, at β = 1 and Ex0 = 0, probability density of energy normalized by E⊥0 satisfies Eq. (8.90) without the last term and, consequently, the corresponding probability density is lognormal,   1 1 2  2t  P(t; E) = √ exp − ln Ee . (8.92) 16t 4 πtE This structure of probability density will evidently be valid only for very small time intervals during which the generating term is still insignificant. The solid lines in Fig. 8.5 show the solutions to Eq. (8.90) obtained numerically for parameter β = 1 at (a) t = 0.05 and (b) t = 0.25. The dashed lines in Fig. 8.5 show the lognormal distribution (8.92). These curves show that the term responsible for energy generation in Eq. (8.90) prevails even for small times. Figure 8.6 shows the solution to 8.90) (the solid lines) for β = 0 (i.e., the probability density of the generated magnetic energy) at t = 10 and the asymptotic curve (8.91) (the dashed line). These curves indicate that only the tail of the solution to Eq. (8.90) verges towards the asymptotic solution (8.91) (see Fig. 8.6a), and this process is very slow (see Fig. 8.6b). The curves intersect because of normalization of the probability densities under consideration. It seems that probability density P(t; E) will approach the tail of the steady-state probability density with further increasing time. 1.5 P(t ; E )

P(t ; E )

1

0.5

0

0

0.5 E (a)

Figure 8.5

1

1 0.5 0

0

0.5 E

1

(b)

Probability distributions at β = 1 at times (a) t = 0.05 and (b) t = 0.25.

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1

0.03

P(t ; E )

P(t ; E )

0.02 0.5

0.01

0

0

4 E (a)

8

0

6

10

15

20

E (b)

Figure 8.6 Probability distributions at β = 0 at time t = 10 for energy in intervals (a) (0, 8) and (b) (6, 20).

Problem 8.7 Find the relationship between fields q(r, t) and p(r, t) whose moment functions are given by the expressions

n 

 q (r, t) = exp {n(n + 0, 5)t}, pn (r, t) = exp {n(n − 0, 5)t}. Which field possesses the property of clustering? Solution q(r, t) = 1/p(r, t) and clustering occurs for field p(r, t).

Problem 8.8 Show that integral equation (8.70) is equivalent to the variational derivative equation δ G(t, t0 ) = G(t, τ )3G(τ, t0 ) δz(τ )

(t0 ≤ τ ≤ t)

with the initial condition G(t, t0 ) z(τ )=0 = g(t, t0 ).

Problem 8.9 Construct the diffusion approximation for the problem on the dynamics of a particle with linear friction under random forces, which is described by the

Approximations of Gaussian Random Field Delta-Correlated in Time

227

stochastic system (1.12), page 8, d d r(t) = v(t), v(t) = −λv(t) + f (r, t), dt dt r(0) = r0 , v(0) = v0 . Solution The one-time probability density of particle position and velocity

P(r, v, t) = hδ(r(t) − r)δ(v(t) − v)i satisfies the Fokker–Planck equation 

   ∂ ∂ ∂ ∂ ∂ ∂ (1) (2) + v − λ v P(r, v, t) = Dij (v) P(r, v, t). + Dij (v) ∂t ∂r ∂v ∂vi ∂vj ∂rj (8.93)

with the diffusion coefficients (1) Dij (v)

Z∞ =

−λτ

dτ e 0

(2)

Dij (v) =

1 λ

Z∞

 Bij

  1  λτ e − 1 v, τ , λ (8.94)

 dτ 1 − e−λτ Bij 

0



  1  λτ e − 1 v, τ . λ

Problem 8.10 In the diffusion approximation, find the steady-state probability distribution of the velocity of a particle described by dynamic system (1.12), page 8, in the onedimensional case assuming that the force correlation function has the form Bf (x, t) =



σf2 exp

 |x| |t| − − . l0 τ0

Solution From Eq. (8.93) follows that steady-state probability density of par-

ticle’s velocity (i.e., probability density in limit t → ∞) satisfies the equation −λvP(v) = D(v)

∂ P(v), ∂v

(8.95)

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where, in accordance with Eq. (8.94), Z∞ D(v) =

dτ e−λτ B

0



  1  λτ e − 1 v, τ . λ

For a sufficiently small friction (λτ0  1), the solution to Eq. (8.95) has the form (

 ) λv2 2 |v|τ0 P(v) = C exp − 2 1+ . 3 l0 2σf τ0

(8.96)

For small particle velocity |v|τ0  l0 , probability distribution (8.96) grades into the Gaussian distribution corresponding to the approximation of the deltacorrelated (in time) random field f (x, t). However, in the opposite limiting case |v|τ0  l0 , probability distribution (8.96) decreases significantly faster than in the case of the approximation of the delta-correlated (in time) random field f (x, t), namely, ) λv2 |v| , P(v) = C exp − 2 3σf l0 (

(8.97)

which corresponds to the diffusion coefficient decreasing according to the law D(v) ∼ 1/|v| for great particle velocities. Physically, it means that the effect of random force f (x, t) on faster particles is significantly smaller than on slower ones.

Lecture 9 Methods for Solving and Analyzing the Fokker–Planck Equation

The Fokker–Planck equations for the one-point probability density (8.11), page 193, and for the transitional probability density (8.16), page 195, are the partial differential equations of parabolic type, so that we can use methods from the theory of mathematical physics equations to solve them. In this context, the basic methods are those such as the method of separation of variables, Fourier transformation with respect to spatial coordinates, and other integral transformations.

9.1

Integral Transformations

Integral transformations are very practicable for solving the Fokker–Planck equation. Indeed, earlier we mentioned the convenience of the Fourier transformation in Eq. (8.11), page 193, if the tensor of diffusion coefficients Fkl (x, x; t) is independent of x. Different integral transformations related to eigenfunctions of the diffusion operator b L=

∂2 Fkl (x, x; t) ∂xk ∂xl

can be used in other situations. For example, in the case of the Legendre operator b L=

∂ ∂ 2 (x − 1) , ∂x ∂x

it is quite natural to use the integral transformation related to the Legendre functions. This transformation is called the Meler–Fock transform and is defined by the formula Z∞ F(µ) =

dx f (x)P−1/2+iµ (x)

(µ > 0),

1 Lectures on Dynamics of Stochastic Systems. DOI: 10.1016/B978-0-12-384966-3.00009-X c 2011 Elsevier Inc. All rights reserved. Copyright

(9.1)

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where P−1/2+iµ (x) is the complex index Legendre function of the first kind, which satisfies the equation   d 1 d 2 2 (x − 1) P−1/2+iµ (x) = − µ + P−1/2+iµ (x). dx dx 4

(9.2)

The inversion of the transform (9.1) has the form f (x) =

Z∞

dµ µ tanh(πµ)F(µ)P−1/2+iµ (x)

(1 ≤ x ≤ ∞),

(9.3)

0

where F(µ) is given by formula (9.1). Another integral transformation called the Kontorovich–Lebedev transform, is related to the diffusion operator b L=

∂ 2∂ x ∂x ∂x

and has the form F(τ ) =

Z∞

dx f (x)Kiτ (x)

(τ > 0),

(9.4)

0

where Kiτ (x) is the imaginary index McDonald’s function of the first kind, which satisfies the equations   2 d 2 d 2 2 x 2 + x − x + τ Kiτ (x) = 0, dx dx (9.5) 

   d 2d d x −x Kiτ (x) = x2 − τ 2 Kiτ (x). dx dx dx

The corresponding inversion has the form 2 f (x) = 2 π x

9.2

Z∞

dτ sinh(πτ )F(τ )Kiτ (x).

(9.6)

0

Steady-State Solutions of the Fokker–Planck Equation

In previous sections, we discussed the general methods for solving the Fokker–Planck equation for both transition and one-point probability densities. However, the problem of the one-point probability density can have peculiarities related to the possible

Methods for Solving and Analyzing the Fokker–Planck Equation

231

existence of the steady-state solution; in a number of cases, such a solution can be obtained immediately. The steady-state solution, if it exists, is independent of initial values and is the solution of the Fokker–Planck equation in the limit t → ∞. There are two classes of problems for which the steady-state solution of the Fokker–Planck equation can be easily found. These classes deal with one-dimensional differential equations and with the Hamiltonian systems of equations. We will consider them in greater detail.

9.2.1

One-Dimensional Nonlinear Differential Equation

The one-dimensional nonlinear systems are described by the stochastic equation d x(t) = f (x) + z(t)g(x), dt

x(0) = x0 ,

(9.7)

where z(t) is, as earlier, the Gaussian delta-correlated process with the parameters hz(t)i = 0,



 z(t)z(t0 ) = 2Dδ(t − t0 )

(D = σz2 τ0 ).

The corresponding Fokker–Planck equation has the form 

 ∂ ∂ ∂ ∂ + f (x) P(x, t) = D g(x) g(x)P(x, t). ∂t ∂x ∂x ∂x

(9.8)

The steady-state probability distribution P(x), if it exists, satisfies the equation f (x)P(x) = Dg(x)

d g(x)P(x) dx

(9.9)

(we assume that P(x) is distributed over the whole space, i.e., for −∞ < x < ∞) whose solution is as follows P(x) =

 Z  C 1 f (x) exp dx 2 , |g(x)| D g (x)

where constant C is determined from the normalization condition Z∞ dxP(x) = 1. −∞

In the special case of the Langevin equation (8.60), page 211, f (x) = −λx, g(x) = 1,

(9.10)

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Eq. (9.10) grades into the Gaussian probability distribution r   λ λ exp − x2 . P(x) = 2π D 2D

9.2.2

(9.11)

Hamiltonian Systems

Another type of dynamic system that allows us to obtain the steady-state probability distribution for N particles is described by the Hamiltonian system of equations with linear friction and external random forces d ∂ ri (t) = H({ri }, {pi }), dt ∂pi ∂ d p (t) = − H({ri }, {pi }) − λpi + f i (t), dt i ∂ri

(9.12)

where i = 1, 2, . . . , N; H({ri }, { pi }) =

N X p2 (t) i

2

i=1

+ U(r1 , . . . , rN )

is the Hamiltonian function; {ri }, { pi } stand for the totalities of all quantities r(t) and p(t), i.e., {ri } = {r1 , . . . , rN },

{ pi } = { p1 , . . . , pN };

λ is a constant coefficient (friction), and random forces f i (t) are the Gaussian deltacorrelated random vector functions with the correlation tensor D E β fiα (t) fj (t0 ) = 2Dδij δαβ δ(t − t0 ), D = σf2 τ0 . (9.13) Here, α and β are the vector indices. The system of equations (9.12) describes the Brownian motion of a system of N interacting particles. The corresponding indicator function ϕ({ri }, {pi }, t) satisfies the equation N

N

N

k=1

k=1

k=1

X X ∂ X ∂ ∂ ϕ({ri }, {pi }, t) + {H, ϕ}(k) − λ { pk ϕ} = { f (t)ϕ}, ∂t ∂pk ∂pk k which is a generalization of Eq. (3.58), page 85, and, consequently, the Fokker–Planck equation for the joint probability density of the solution to system (9.12) has the form N

X ∂ P({ri }, {pi }, t) + {H, P({ri }, {pi }, t)}(k) ∂t k=1

−λ

N X k=1

N X ∂  ∂2 pk P({ri }, {pi }, t) = D P({ri }, {pi }, t), ∂pk ∂p2k k=1

(9.14)

Methods for Solving and Analyzing the Fokker–Planck Equation

233

where {ϕ, ψ}(k) =

∂ψ ∂ϕ ∂ϕ ∂ψ − ∂pk ∂rk ∂pk ∂rk

is the Poisson bracket for the kth particle. One can easily check that the steady-state solution to Eq. (9.14) is the canonical Gibbs distribution   λ P({ri }, { pi }) = C exp − H({ri }, { pi }) . (9.15) D The specificity of this distribution consists in the Gaussian behavior with respect to momenta and statistical independence of particle coordinates and momenta. Integrating Eq. (9.15) over all r, we can obtain the Maxwell distribution that describes velocity fluctuations of the Brownian particles. The case U(r1 , . . . , rN ) = 0 corresponds to the Brownian motion of a system of free particles. If we integrate probability distribution (9.15) over momenta (velocities), we obtain the Boltzmann distribution of particle coordinates   λ P({ri }) = C exp − U({ri }) . (9.16) D In the case of sufficiently strong friction, the equilibrium distribution (9.15) is formed in two stages. First, the Gaussian momentum distribution (the Maxwell distribution) is formed relatively quickly and then, the spatial distribution (the Boltzmann distribution) is formed at a much slower rate. The latter stage is described by the Fokker–Planck equation N

∂ 1X ∂ P({ri }, t) = ∂t λ ∂rk k=1



 N ∂U({ri }) D X ∂2 P({ri }, t) + 2 P({ri }, t), ∂rk λ ∂r2k k=1

(9.17)

which is usually called the Einstein–Smolukhovsky equation. Derivation of Eq. (9.17) from the Fokker–Planck equation (9.14) is called Kramers problem (see, e.g., the corresponding discussion in Sect. 6.2.1, page 144), where dynamics of particles under a random force is considered as an example). Note that Eq. (9.17) statistically corresponds to the stochastic equation 1 ∂ 1 d ri (t) = − U({ri }) + f i (t), dt λ ∂ri λ

(9.18)

which, nevertheless, cannot be considered as the limit of Eq. (9.12) for λ → ∞. In the one-dimensional case, Eqs. (9.12) are simplified and assume the form of the system of two equations d x(t) = y(t), dt

d ∂ y(t) = − U(x) − λy(t) + f (t). dt ∂x

(9.19)

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Lectures on Dynamics of Stochastic Systems

The corresponding steady-state probability distribution has the form   λ P(x, y) = C exp − H(x, y) , D

9.2.3

H(x, y) =

y2 + U(x). 2

(9.20)

Systems of Hydrodynamic Type

In Sect. 1.1.3, page 10, we considered the general dynamics of simplest hydrodynamic-type systems (HTS). Now, we consider these systems in terms of the statistical description. Hydrodynamic-type systems with linear friction are described by the dynamic equations d vi (t) = Fi (v) − λ(i) vi (t) dt

(i = 1, . . . , N),

(9.21)

where λ(i) is the friction coefficient of the i-th component of the N-dimensional vector v 1 and Fi (v) is the function quadratic in v and having the following properties: (a) vi Fi (v) = 0, energy conservation holds at λ(i) = 0: (b)

v2 (t) d E(t) = 0, E(t) = i ; dt 2

∂ Fi (v) = 0, ∂vi conditions of the Liouville theorem are satisfied at λ(i) = 0, and this equality is the equation of incompressibility in the phase space.

Equilibrium Thermal Fluctuations in the Hydrodynamic-Type Systems Here, we dwell on a class of phenomena closely related to the Brownian motion; namely, we dwell on equilibrium thermal fluctuations in solids. Microscopic equations describe the behavior of physical systems only in terms of spatial scales large in comparison with the molecule free path in the medium and temporal scales great in comparison with the time intervals between molecule collisions. This means that macroscopic equations adequately describe the behavior of systems only on average. However, in view of molecule thermal motion, macroscopic variables are in general terms stochastic variables, and a complete macroscopic theory must describe not only system behavior on average, but also the fluctuations about the average. 1

In the general case, the dissipative term in Eq. (9.21) has the form λik vk . However, we can always choose the coordinate system in which two positively defined quadratic forms – energy E = v2i /2 and dissipation λik vi vk – have the diagonal representation. The form of Eq. (9.21) assumes the use of namely such a coordinate system.

Methods for Solving and Analyzing the Fokker–Planck Equation

235

Such a description can be performed in terms of macroscopic variables by supplementing the corresponding macroscopic equations with the ‘external forces’ specified as the Gaussian random fields delta-correlated in time (this approach is closely related to the fluctuation – dissipation theorem, or the Callen–Welton theorem). See [53, 54] for the corresponding correlation theories of equilibrium thermal fluctuations in electrodynamics, hydrodynamics, and viscoelastic media. In HTS, equilibrium thermal fluctuations are described by Eq. (9.21) supplemented with external forces fi (t) d vi (t) = Fi (v) − λ(i) vi (t) + fi (t) dt

(i = 1, . . . , N).

(9.22)

External forces are assumed to be the Gaussian random functions delta-correlated in time with the correlation tensor of the form

 2 fi (t)fj (t0 ) = 2δij σ(i) δ(t − t0 )

(h f (t)i = 0).

(9.23)

Note that one can consider Eqs. (9.22) as the system describing the Brownian motion in HTS. Such a treatment assumes that coefficients λ(i) are certain effective friction coefficients. For example, at N = 3, system (9.22) describes (in the velocity space) the rotary Brownian motion of a solid in a medium, and quantities λ(i) vi (t) play the role of the corresponding resistance forces. The probability density of solution v(t) to Eqs. (9.22), i.e., function P(v, t) = hδ(v(t) − v)i, satisfies the Fokker–Planck equation 2 ∂ ∂ ∂ 2 ∂ {Fi (v)P(v, t)} + λ(i) {vi P(v, t)} + σ(i) P(v, t) = − P(v, t). ∂t ∂vi ∂vi ∂v2i

(9.24)

The steady-state, initial data-independent solution to Eq. (9.24) must behave like the Maxwell distribution that corresponds to the uniform distribution of energy over the degrees of freedom ) v2i P(v) = C exp − , 2kT (

(9.25)

where k is the Boltzmann constant and T is the equilibrium temperature in the system. 2 Substituting Eq. (9.25) in Eq. (9.24), we obtain the expression for σ(i) 2 σ(i) = λ(i) kT

(9.26)

called the Einstein formula. Here, we used that, in view of conditions (a) and (b), ∂ {Fi (v)P(v)} = 0. ∂vi

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Thus, Eqs. (9.23) and (9.26) completely determine statistics of external forces, and the nonlinear term in Eq. (9.22) plays no role for the one-time fluctuations of v(t). Namely this feature constitutes the subject matter of the fluctuation – dissipation theorem (the Callen–Welton theorem) in the context of HTS. Note that Eqs. (9.22) with the correlation relationships (9.23) describe not only equilibrium thermal fluctuations in HTS, but also the interaction of small-scale motions (such as microturbulence, for example) with the motions of larger scales. If such an HTS allows the description in terms of phenomenological equations (9.22) 2 = σ 2 = const, the steady-state probability distribution of with λ(i) = λ = const, σ(i) v will have the form similar to distribution (9.25):  λ 2 P(v) = C exp − 2 vi . 2σ 

(9.27)

Noises in Hydrodynamic-Type Systems Under the Regular Force We illustrate statistical description of HTS by the example of the simplest system with three degrees of freedom (S3 ), which is described by system of equations (1.23), page 14: d v0 (t) = v22 (t) − v21 (t) − v0 (t) + R + f0 (t), dt d v1 (t) = v0 (t)v1 (t) − v1 (t) + f1 (t), dt d v2 (t) = −v0 (t)v2 (t) − v2 (t) + f2 (t). dt

(9.28)

We assume that fi (t) are the Gaussian random forces delta-correlated in time with the correlation tensor

 fi (t)fj (t0 ) = 2δij σ 2 δ(τ ).

In the absence of external forces, this system is equivalent to the Euler equations of the dynamics of a gyroscope with isotropic friction, which is excited by a constant moment of force relative to the unstable axis. The steady-state solution to this system depends on parameter R (an analog to the Reynolds number), and the critical value of this parameter is Rcr = 1. For R < 1, it has the stable steady-state solution v1 s-s = vs-s = 0,

v0 s-s = R.

(9.29)

For R > 1, this solution becomes unstable with respect to small disturbances of parameters, and new steady-state regime is formed: v0 s-s = 1,

v2 s-s = 0,

√ v1 s-s = ± R − 1.

(9.30)

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237

Here, we have an element of randomness because quantity v1 s-s can be either positive or negative, depending on the amplitude of small disturbances. As was mentioned earlier, the steady-state probability distribution at R = 0 has the form ) ( v2i (9.31) P(v) = C exp − 2 . 2σ Assume now R 6= 0 and R < 1. In this case, we can easily obtain that fluctuations of components relative to their steady-state values (e vi = vi − vi s-s ) satisfy the system of equations d e v0 (t) = e v22 (t) −e v21 (t) −e v0 (t) + f0 (t), dt d e v1 (t) = e v0 (t)e v1 (t) − (1 − R)e v1 (t) + f1 (t), dt

(9.32)

d e v2 (t) = −e v0 (t)e v2 (t) − (1 + R)e v2 (t) + f2 (t). dt Statistical characteristics of the solution to system (9.32) can be determined using the perturbation theory with respect to small parameter σ 2 . The second moments of componentse vi will then be described by the linearized system of equations (9.32); the second moment being determined, mean values can be obtained by averaging system (9.32) directly. The corresponding steady-state variances of fluctuations e vi have the form D E e v20 (t) = σ 2 ,

D E e v21 (t) =

σ2 , 1−R

D E e v22 (t) =

σ2 . 1+R

(9.33)

Note that expressions (9.33) hold for R  1. With increasing R, both intensity and temporal correlation radius of component e v1 increase, while the intensity of component e v2 decreases. In this process, maximum fluctuations of quantity e v1 occur when the dynamic system gets over the critical regime. Consider now the case of R > 1. The√steady-state probability distribution of component v1 has two maxima near v1 = ± R − 1 (they correspond to the stable steadystate positions) and a minimum v1 = 0 that corresponds to the unstable position. This probability distribution is formed by ensemble averaging over realizations of random forces fi (t). In a single realization, the system arrives with a probability of 1/2 at one of the stable positions corresponding to distribution maxima. In this case, averaging over time or ensemble averaging over the force realizations that bring the system to this state will form the probability distribution near the maximum, and we can determine statistical characteristics of the solution using the perturbation theory in parameter σ 2 . Assume the system arrives at the state corresponding to the stable steady-state posi√ tion v2 s-s = 0, v1 s-s = R − 1. Then, fluctuations relative to this state will satisfy the

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system of equations √ d e v0 (t) = e v22 (t) −e v21 (t) − 2 R − 1e v1 (t) −e v0 (t) + f0 (t), dt √ d e v1 (t) = e v0 (t)e v1 (t) + R − 1e v0 (t) + f1 (t), dt

(9.34)

d e v2 (t) = −e v0 (t)e v2 (t) − 2e v2 (t) + f2 (t). dt For R  1, the second moments of fluctuations e vi are obtained from the linearized system (9.34) and mean values are then determined by direct averaging of Eqs. (9.34). Using this procedure, we obtain that, for R  1, the steady-state variances and correlation coefficients of triplet components have the form σ2 he v0 (t)e v1 (t)i = − √ , R D E 3 D E σ2 e v21 (t) = σ 2 , e v22 (t) = . 2 2

he v0 (t)e v2 (t)i = he v1 (t)e v2 (t)i = 0, D E e v20 (t) = 3σ 2 ,

(9.35)

As we mentioned earlier, these statistical characteristics correspond either to averaging over time, or to ensemble averaging over realizations of forces fi (t) that bring the system to the steady-state position. It should be noted that if the system has arrived at one of the most probable states under the action of certain realization of forces fi (t), it will be transferred into another most probable state after the lapse of certain time T (the greater T, the greater R and the less σ 2 ) in view of availability of sufficiently great values of fi (t). This process simulated for a realization of random forces is shown in Fig. 1.5 on page 14. Consider now fluctuations of components vi (t) at critical regime (i.e., at R = 1). Equations for the fluctuations relative to the state v1 CT = v2 CT = 0, v0 CT = R can be obtained by setting R = 1 either in Eq. (9.32), or in Eq. (9.34). In this case, fluctuations of componente v2 (t) can be described in terms of the linearized equation (σ 2  1), and the corresponding steady-state variance is given by the formula D E σ2 e v22 (t) = . 2 Components e v0 (t) and e v1 (t) are described by the nonlinear system of equations d e v0 (t) = −e v21 (t) −e v0 (t) + f0 (t), dt d e v1 (t) = e v0 (t)e v1 (t) + f1 (t), dt and no linearization is allowed here.

(9.36)

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Averaging system (9.36), we obtain the following relationships for steady-state values of fluctuations D E he v0 (t)e v1 (t)i = 0, he v0 (t)i = − e v21 (t) . As we noted earlier, intensity of fluctuationse v1 increases in the critical regime and, consequently, quantity he v0 (t)i increases. At the same time, the variance of fluctuation e v0 (t) remains nearly intact. Indeed, as follows from Eqs. (9.36), the variance of steadystate fluctuation e v0 (t) is given by the formula D E e v20 (t) = 2σ 2 . It becomes clear that fluctuations e v0 (t) and e v1 (t) can be estimated from the simplified system d e v0 (t) = −e v21 (t) −e v0 (t) + f0 (t), dt d e v1 (t) = he v0 (t)ie v1 (t) + f1 (t) dt obtained by the replacement of quantity e v0 (t) in the second equation of system (9.36) by he v0 (t)i. Using this system, we obtain that steady-state fluctuations e v0 (t) and e v1 (t) satisfy the expressions D E e v21 (t) = − he v0 (t)i = σ. In a similar way, we obtain the estimate of the temporal correlation radius of quantity e v1 (t): τ ∼ 1/σ. Earlier, we considered the behavior of a triplet under the assumption that random forces act on all variables. Let us see what will be changed if only one unstable component v0 (t) is affected by the random force (we assume that the force acting on component v0 (t) has regular (R) and random components). In this case, the dynamic system assumes the form (R > 1) d v0 (t) = −v21 (t) − v0 (t) + R + f (t), dt d v1 (t) = v0 (t)v1 (t) − v1 (t). dt

(9.37)

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In system (9.37), we omitted the terms related to component v2 (t), because one can easily see that it will not be excited in the problem at hand. Represent component v0 (t) as v0 (t) = 1 + e v0 (t). The system of equations (9.37) assumes then the form d e v0 (t) = −v21 (t) −e v0 (t) + (R − 1) + f (t), dt (9.38) d v1 (t) = e v0 (t)v1 (t), dt from which follows that temporal evolution of component v1 (t) depends on its initial value. If v1 (0) > 0, then v1 (t) > 0 too. In this case, we can represent v1 (t) in the form v1 (t) = eϕ(t) and rewrite the system of equations (9.38) in the Hamiltonian form (9.19) d ∂U(ϕ) e v0 (t) = − −e v0 (t) + f (t), dt ∂ϕ

d ϕ(t) = e v0 (t), dt

(9.39)

where U(ϕ) =

1 2ϕ e − (R − 1) ϕ. 2

Here, variable ϕ(t) plays the role of particle’s coordinate and variable v0 (t) plays the role of particle’s velocity. The solid line in Fig. 9.1 shows the behavior of function U(ϕ). At point ϕ0 = √ ln R − 1, this function has a minimum U(ϕ0 ) =

1 (R − 1) [1 − ln(R − 1)] 2

√ corresponding to the stable equilibrium state v1 = R − 1. Thus, the steady-state probability distribution of ϕ(t) and e v0 (t) is similar to the Gibbs distribution (9.20)   1 P(e v0 , ϕ) = C exp − H(e v0 , ϕ) , D

H(e v0 , ϕ) =

e v20 + U(ϕ). 2

(9.40)

From Eq. (9.40) follows that, for R > 1, the steady-state probability distribution is composed of two independent steady-state distributions, of which the distribution of component v0 (t) of system (9.37) is the Gaussian distribution   (v0 − 1)2 P(v0 ) = √ exp − , 2D 2π D 1

(9.41)

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241

U (ϕ)

ϕ0 U (ϕ 0)

ϕ

Figure 9.1 Potential function U(ϕ). The dashed lines show curve U(ϕ) = straight line U(ϕ) = −(R − 1)ϕ.

1 2

exp{2ϕ} and

and the distribution of quantity ϕ(t) is the non-Gaussian distribution. If we turn back to variable v1 (t), we obtain the corresponding steady-state probability distribution in the form ( ) R−1 v21 D −1 exp − P(v1 ) = const v1 . (9.42) 2D As may be seen from Eq. (9.42), no steady-state probability distribution exists for component v1 (t) in critical regime(R = 1), which contrasts with the above case when random forces acted on all components of the triplet. Earlier, we mentioned that two stages are characteristic of the formation of distribution (9.40). First, the Maxwellian distribution with respect to v0 (9.41) is rapidly formed and the distribution with respect to ϕ is then formed at a much slower rate. The second stage is described by the Einstein–Smolukhovsky equation (9.17) that, in terms of variable v1 , assumes the form   i o ∂ ∂ n h 2 ∂ ∂ [v1 P(v1 , t)] . P(v1 , t) = v1 v1 − (R − 1) P(v1 , t) + σ 2 v1 ∂t ∂v1 ∂v1 ∂v1 (9.43) This equation is the Fokker–Planck equation for the stochastic dynamic equation h i d v1 (t) = −v1 (t) v21 (t) − (R − 1) + v1 (t)f (t). dt At critical regime R = 1, this equation assumes the form d v1 (t) = −v31 (t) + v1 (t)f (t), dt from which follows that, despite strong nonlinear friction, its solution is stochastically unstable due to the specific form of the random term. Stochastic stabilization occurs,

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as we saw earlier, due to inclusion of random forces into the equations for other components.

9.3

Boundary-Value Problems for the Fokker–Planck Equation (Hopping Phenomenon)

The Fokker–Planck equations are the partial differential equations and they generally require boundary conditions whose particular form depends on the problem under consideration. One can proceed from both forward and backward Fokker–Planck equations, which are equivalent. Consider one typical example. Consider the nonlinear oscillator with friction described by the equation d d2 x(t) + λ x(t) + ω02 x(t) + βx3 (t) = f (t) dt dt2

(β, λ > 0)

(9.44)

and assume that random force f (t) is the delta-correlated random function with the parameters hf (t)i = 0,

 f (t)f (t0 ) = 2Dδ(t − t0 )

(D = σf2 τ0 ).

At λ = 0 and f (t) = 0, this equation is called the Duffing equation. We can rewrite Eq. (9.44) in the standard form of the Hamiltonian system in funcd tions x(t) and v(t) = x(t), dt d ∂ x(t) = H(x, v), dt ∂v

d ∂ v(t) = − H(x, v) − λv + f (t), dt ∂x

where H(x, v) =

v2 + U(x), 2

U(x) =

ω02 x2 x4 +β 2 4

is the Hamiltonian. According to Eq. (9.20), the steady-state solution to the corresponding Fokker– Planck equation has the form   λ P(x, v) = C exp − H(x, v) . D

(9.45)

It is clear that this distribution is the product of two independent distributions, of which one – the steady-state probability distribution of quantity v(t) – is the Gaussian distribution and the other – the steady-state probability distribution of quantity x(t) – is

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243

the non-Gaussian distribution. Integrating Eq. (9.45) over v, we obtain the steady-state probability distribution of x(t) !) ( x4 λ ω02 x2 . +β P(x) = C exp − D 2 4 This distribution is maximum at the stable equilibrium point x = 0. Consider now the equation d2 d x(t) + λ x(t) − ω02 x(t) + βx3 (t) = f (t) dt dt2

(β, λ > 0).

(9.46)

In this case again, the steady-state probability distribution has the form (9.45), where now H(x, v) =

v2 + U(x), 2

U(x) = −

ω02 x2 x4 +β . 2 4

The steady-state probability distribution of x(t) assumes now the form ( !) ω02 x2 λ x4 P(x) = C exp − − +β D 2 4

(9.47)

q and has maxima at points x = ± ω02 /β and a minimum at point x = 0; the maxima correspond to the stable equilibrium points of problem (9.46) for f (t) = 0 and the minimum, to the instable equilibrium point. Figure 9.2 shows the behavior of the probability distribution (9.47). As we mentioned earlier, the formation of distribution (9.47) is described by the Einstein–Smolukhovsky equation (9.17), which has in this case the form 1 ∂ ∂ P(x, t) = ∂t λ ∂x



 1 ∂2 ∂U(x) P(x, t) + 2 2 P(x, t). ∂x λ ∂x

(9.48)

This equation is statistically equivalent to the dynamic equation 1 ∂U(x) 1 d x(t) = − + f (t). dt λ ∂x λ

(9.49)

P (x )

−√ ω 2 /β 0

Figure 9.2

Probability distribution (9.47).

√ ω 20 /β

x

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Probability distribution (9.47) corresponds to averaging over an ensemble of realizations of random process f (t). If we deal with a single realization, the system arrives at one of states corresponding to the distribution maxima with a probability of 1/2. In this case, averaging over time will form the probability distribution around the maximum position. However, after a lapse of certain time T (the longer, the smaller D), the system will be transferred in the vicinity of the other maximum due to the fact that function f (t) can assume sufficiently large values. For this reason, temporal averaging will form probability distribution (9.48) only if averaging time t  T. s Introducing dimensionless coordinate x → rewrite Eq. (9.48) in the form ∂ ∂ P(x, t) = ∂t ∂x



ω02 λ x and time t → 2 t, we can β ω0

 ∂2 ∂U(x) P(x, t) + µ 2 P(x, t), ∂x ∂x

(9.50)

where µ=

βD , λω04

U(x) = −

x4 x2 + . 2 4

In this case, the equivalent stochastic equation (9.49) assumes the form of Eq. (1.15), page 10 d ∂U(x) x(t) = − + f (t). dt ∂x

(9.51)

Estimate the time required for the system to switch from a most probable state x = −1 to the other x = 1. Assume the system described by stochastic equation (9.51) was at a point from the interval (a, b) at instant t0 . The corresponding probability for the system to leave this interval G(t; x0 , t0 ) = 1 −

Zb

dxp(x, t|x0 , t0 )

a

satisfies Eq. (8.21) following from the backward Fokker–Planck equation (8.20), page 196, i.e., the equation ∂U(x0 ) ∂ ∂2 ∂ G(t; x0 , t0 ) = G(t; x0 , t0 ) − µ 2 G(t; x0 , t0 ) ∂t0 ∂x0 ∂x0 ∂x0 with the boundary conditions G(t; x0 , t) = 0,

G(t; a, t0 ) = G(t; b, t0 ) = 1.

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245

Taking into account the fact that G(t; x0 , t0 ) = G(t − t0 ; x0 ) in our problem, we can denote (t − t0 ) = τ and rewrite the boundary-value problem in the form ∂U(x0 ) ∂ ∂2 ∂ G(τ ; x0 ) = G(τ ; x0 ) − µ 2 G(τ ; x0 ), ∂τ ∂x0 ∂x0 ∂x0   G(0; x0 ) = 0, G(τ ; a, ) = G(τ ; b) = 1 lim G(τ ; x0 ) = 0 .

(9.52)

τ →∞

From Eq. (9.52), one can easily see that average time required for the system to leave interval (a, b) T(x0 ) =

Z∞

∂G(τ ; x0 ) ∂τ

dτ τ

0

satisfies the boundary-value problem µ

d2 T(x0 ) dU(x0 ) dT(x0 ) = −1, − dx0 dx0 dx02

T(a) = T(b) = 0.

(9.53)

Equation (9.53) can be easily solved, and we obtain that the average time required for the system under random force to switch its state from x0 = −1 to x0 = 1 (this time is usually called the Kramers time) is given by the expression 1 T= µ

Zξ

Z1 dξ −1

−∞

   Z1 C(µ) 1 1 [U(ξ ) − U(η)] = dξ exp U(ξ ) , dη exp µ µ µ 

0

(9.54) Z∞ where C(µ) = −∞

√ T ≈ 2π exp



 1 dξ exp U(ξ ) . For µ  1, we obtain µ 

 1 , 4µ

i.e., the average switching time increases exponentially with decreasing the intensity of fluctuations of the force.

9.4

Method of Fast Oscillation Averaging

If fluctuations of dynamic system parameters are sufficiently small, we can analyze the Fokker–Planck equation using different asymptotic and approximate techniques.

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Here, we consider in greater detail the method that is most often used in the statistical analysis. Assume the stochastic system is described by the dynamic equations d e) + z(t)B(x, φ e), x(t) = A(x, φ dt d e) + z(t)D(x, φ e), φ(t) = C(x, φ dt

(9.55)

where e(t) = ω0 t + φ(t), φ e), B(x, φ e), C(x, φ e), and D(x, φ e) are the periodic functions of variable functions A(x, φ e φ , and z(t) is the Gaussian delta-correlated process with the parameters hz(t)i = 0,



 z(t)z(t0 ) = 2Dδ(t − t0 ),

D = σ 2 τ0 .

Variables x(t) and φ(t) can mean the vector module and phase, respectively. The Fokker–Planck equation corresponding to system of equations (9.55) has the form ∂ ∂ ∂ e)P(x, φ, t) − e)P(x, φ, t) P(x, φ, t) = − A(x, φ C(x, φ ∂t ∂x ∂φ  2 ∂ ∂ e e +D B(x, φ ) + D(x, φ ) P(x, φ, t). ∂x ∂φ

(9.56)

Commonly, Eq. (9.56) is very complicated for immediately analyzing the joint probability density. We rewrite this equation in the form ∂ ∂ ∂ e)P(x, φ, t) − e)P(x, φ, t) P(x, φ, t) = − A(x, φ C(x, φ ∂t ∂x ∂φ   e) ∂B(x, φ e) ∂ ∂B2 (x, φ e −D + D(x, φ ) P(x, φ, t) e ∂x 2∂x ∂φ   e) e) ∂ ∂D(x, φ ∂D2 (x, φ e −D B(x, φ ) + ) P(x, φ, t) e ∂φ ∂x 2∂ φ  2  ∂ 2 ∂2 ∂2 2 e) + 2 e)D(x, φ e) + e) P(x, φ, t). +D B (x, φ B(x, φ D (x, φ ∂x∂φ ∂x2 ∂φ 2 (9.57) e) and C(x, φ e) are sufficiently small and flucNow, we assume that functions A(x, φ tuation intensity of process z(t) is also small. In this case, statistical characteristics of system of equations (9.55) only slightly vary during times ∼ 1/ω0 . To study these small variations (accumulated effects), we can average Eq. (9.57) over the period of

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247

all oscillating functions. Assuming that function P(x, φ, t) remains intact under averaging, we obtain the equation ∂ ∂ ∂ e) P(x, φ, t) − e) P(x, φ, t) P(x, φ, t) = − A(x, φ C(x, φ ∂t ∂x ∂φ ! ) ( e) ∂B(x, φ e) e) ∂ ∂D(x, φ ∂ ∂B2 (x, φ e e + D(x, φ ) + B(x, φ ) P(x, φ, t) −D e ∂x 2∂x ∂φ ∂x ∂φ   2 ∂2 ∂2 2 ∂ 2 e) + 2 e)D(x, φ e) + e) P(x, φ, t), (9.58) B (x, φ B(x, φ D (x, φ +D ∂x∂φ ∂x2 ∂φ 2 where the overbar denotes quantities averaged over the oscillation period. Integrating Eq. (9.58) over φ, we obtain the Fokker–Planck equation for function P(x, t) ∂ ∂ ∂ e) P(x, t) + D P(x, t) = − A(x, φ ∂t ∂x ∂x +D

! e) ∂B(x, φ e) ∂B2 (x, φ e) P(x, t) + D(x, φ e 2∂x ∂φ

∂ 2 ∂ e) P(x, t). B (x, φ ∂x ∂x

(9.59)

Note that quantity x(t) appears to be the one-dimensional Markovian random process in this approximation. e) ∂D(x, φ e) = 0, and C(x, φ e)D(x, φ e) = e) = const, B(x, φ If we assume that B(x, φ ∂x e) = const in Eq. (9.58), then processes x(t) and φ(t) become statistically indeD2 (x, φ pendent, and process φ(t) becomes the Markovian Gaussian process whose variance is the linear increasing function of time t. This means that probability distribution of e) = 0). quantity φ(t) on segment [0, 2π] becomes uniform for large t (at C(x, φ

Problems Problem 9.1 Solve the Fokker–Planck equation ∂ ∂ ∂ p(x, t|x0 , t0 ) = D (x2 − 1) p(x, t|x0 , t0 ) ∂t ∂x ∂x p(x, t0 |x0 , t0 ) = δ(x − x0 ). Instruction Use the integral Meler–Fock transform.

(x ≥ 1),

(9.60)

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Solution

p(x, t|x0 , t0 ) =

Z∞

    1 dµ µ tanh(πµ) exp −D µ2 + (t − t0 ) 4

0

P−1/2+iµ (x)P−1/2+iµ (x0 ). If x0 = 1 at the initial instant t0 = 0, then Z∞ P(x, t) =

    1 dµ µ tanh(πµ) exp −D µ2 + (t − t0 ) P−1/2+iµ (x). 4

0

Problem 9.2 In the context of singular stochastic problem (1.29), page 18, for λ = 1 d x(t) = −x2 (t) + f (t), dt

x(0) = x0 ,

(9.61)

where f (t) is the Gaussian delta-correlated process with the parameters hf (t)i = 0,

 f (t)f (t0 ) = 2Dδ(t − t0 )

(D = σf2 τ0 ),

estimate average time required for the system to switch from state x0 to state (−∞) and average time between two singularities. Solution

hT(x0 )i =

Zx0

Z∞ dη exp

dξ −∞

ξ

√ 121/6 hT(∞)i = π 0 3

   1 3 ξ − η3 , 3

  1 ≈ 4.976. 6

Problem 9.3 Find the steady-state probability distribution for stochastic problem (9.61). Instruction Solve the Fokker–Planck equation under the condition that function x(t) is discontinuous and defined for all times t in such a way that its value of

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249

−∞ at instant t → t0 − 0 is immediately followed by its value of ∞ at instant t → t0 + 0. This behavior corresponds to the boundary condition of the Fokker– Planck equation formulated as continuity of probability flux density. Solution

Zx P(x) = J

   1 3 3 ξ −x , dξ exp 3

(9.62)

−∞

where J=

1 hT(∞)i

is the steady-state probability flux density. Remark From Eq. (9.62) follows the asymptotic formula

P(x) ≈

1 hT(∞)i x2

(9.63)

for great x.

Problem 9.4 Show that asymptotic formula (9.63) is formed by discontinuities of function x(t). Instruction Near discontinuous points tk , represent the solution x(t) in the form

x(t) =

1 . t − tk

Problem 9.5 Derive the Fokker–Planck equation for slowly varying amplitude and phase of the solution to the problem d x(t) = y(t), dt

d y(t) = −2γ y(t) − ω02 [1 + z(t)]x(t), dt

where z(t) is the Gaussian process with the parameters

 hz(t)i = 0, z(t)z(t0 ) = 2σ 2 τ0 δ(t − t0 ).

(9.64)

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Instruction Replace functions x(t) and y(t) with new variables (oscillation amplitude and phase) according to the equalities

x(t) = A(t) sin (ω0 t + φ(t)) ,

y(t) = ω0 A(t) cos (ω0 t + φ(t))

and represent amplitude A(t) in the form A(t) = eu(t) . Solution

∂ D ∂ ∂ P(u, φ, t) = γ P(u, φ, t) − P(u, φ, t) ∂t ∂u 4 ∂u 3D ∂ 2 D ∂2 P(t; u, φ) + P(u, φ, t), + 2 8 ∂u 8 ∂φ 2

(9.65)

where D = σ 2 τ0 ω02 . From Eq. (9.65) follows that statistical characteristics of oscillation amplitude and phase (averaged over the oscillation period) are statistically independent and have the Gaussian probability densities (

) (u − hu(t)i)2 P(u, t) = p exp − , 2σu2 (t) 2πσu2 (t) ( ) 1 (φ − φ0 )2 P(φ, t) = q exp − , 2σφ2 (t) 2πσ 2 (t) 1

(9.66)

φ

where hu(t)i = u0 − γ t + hφ(t)i = φ0 ,

D t, 4

σφ2 (t) =

σu2 (t) =

D t, 4

3D t. 4

Problem 9.6

 Using probability distributions (9.66), calculate quantities hx(t)i and x2 (t) . Derive the condition of stochastic excitation of the second moment. Solution Average value

hx(t)i = e−γ t sin(ω0 t) coincides with the problem solution in the absence of fluctuations.

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251

Quantity n o D E 1 x2 (t) = e(D−2γ )t 1 − e−3Dt/2 cos(2ω0 t) , 2

(9.67)

coincides in the absence of absorption with Eq. (6.91) to terms about D/ω0  1.

Problem 9.7 Find the condition of stochastic parametric excitation of function hAn (t)i. Solution We have

 

n  D nu(t) E 1 A (t) = e = An0 exp −nγ t + n(n + 2)Dt , 8 and, consequently, the stochastic dynamic system is parametrically excited beginning from moment function of order n if the condition 8γ < (n + 2)D is satisfied. Remark The typical realization curve of random amplitude has the form

    D t A (t) = A0 exp − γ − 4 ∗

and exponentially decays in time if absorption is sufficiently small, namely, if 1<4

γ 1 < 1 + n, D 2

which always holds for sufficiently great n, whereas all moment functions of random amplitude A(t) of order n and higher exponentially grow in time. This means that statistics of random amplitude A(t) are formed mainly by large spikes over the exponentially decaying typical realization curve. This fact follows from the lognormal property of random amplitude A(t).

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Lecture 10 Some Other Approximate Approaches to the Problems of Statistical Hydrodynamics

In previous Lectures, we analyzed statistics of solutions to the nonlinear equations of hydrodynamics using the rigorous approach based on deriving and investigating the exact variational differential equations for characteristic functionals of nonlinear random fields. However, this approach encounters severe difficulties caused by the lack of development of the theory of variational differential equations. For this reason, many researchers prefer to proceed from more habitual partial differential equations for different moment functions of fields of interest. The nonlinearity of the input dynamic equations governing random fields results in the appearance of higher moment functions of fields of interest in the equations governing any moment function. As a result, even determination of average field or correlation functions requires, in the strict sense, solving an infinite system of linked equations. Thus, the main problem of this approach consists in cutting the mentioned system of equations on the basis of one or another physical hypothesis. The most known example of such a hypothesis is the Millionshchikov hypothesis according to which the higher moment functions of even orders are expressed in terms of the lower ones by the laws of the Gaussian statistics. The disadvantage of such approaches consists in the fact that the validity of the hypothesis usually cannot be proved; moreover, cutting the system of equations may often yield physically contradictory results, namely, energy spectra of turbulence can appear negative for certain wave numbers. Nevertheless, these approximate approaches provide a deeper insight into physical mechanisms of forming the statistics of strongly nonlinear random fields and make it possible to derive quantitative expressions for fields’ correlation functions and spectra. It seems that the Millionshchikov hypothesis provides correct spectra of the developed turbulence in viscous interval [35]. We emphasize additionally that the mentioned approximate equations reveal many nontrivial effects inherent in nonlinear random fields and have no analogs in the behavior of deterministic fields and waves. Here, we illustrate these methods of analysis by the example of an interesting physical effect, namely, that average flows of Lectures on Dynamics of Stochastic Systems. DOI: 10.1016/B978-0-12-384966-3.00010-6 c 2011 Elsevier Inc. All rights reserved. Copyright

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noncompressible liquids superimposed on the background of developed turbulent pulsations acquire quasi-elastic wave properties. This effect was first mentioned by Moffatt [57] who studied the reaction of turbulence on variations of the transverse gradient of average velocity. Moffatt noted that turbulent medium behaves in some sense like an elastic medium; namely, variations of average velocity profile of a planeparallel flow satisfy the wave equation.

10.1

Quasi-Elastic Properties of Isotropic and Stationary Noncompressible Turbulent Media

Let uT (r, t) be the random velocity field of developed turbulence

 stationary in time and homogeneous and isotropic in space (we assume that uT (r, t) = 0). In actuality, regular average flows are imposed on turbulent pulsations of liquid. We denote U(r, t) the velocity field of these regular flows. Because the turbulent pulsations and average flows interact nonlinearly, we can represent the total velocity field in the form u(r, t) = U(r, t) + uT (r, t) + u0 (r, t),

(10.1)

Here, uT (r, t) is the unperturbed turbulent field in the absence of average flows and u0 (r, t) is the turbulent field

 disturbance caused by the interaction with the regular flow (we assume that u0 (r, t) = 0). Investigation of nonlinear interactions between regular flows and turbulent pulsations (and the analysis of turbulent pulsations uT (r, t) by themselves, though) is very difficult in the general case. However, in the case of weak average flows, i.e., under the condition that U2 (r, t)  2T(t),  i2  1 h T u (r, t) is the average density of turbulent energy pulsation, we where T(t) = 2 can discuss the effect of turbulence on the average flow evolution in sufficient detail by considering the linear approximation in small disturbances of fields U(r, t) and u0 (r, t) under the assumption that the spectrum of turbulent pulsations uT (r, t) is known. The total velocity field (10.1) and its turbulent component uT (r, t) satisfy the Navier–Stokes equation (1.95), page 43. Hence, substituting Eq. (10.1) in Eq. (1.95) and linearizing the equations in regular (average) field U(r, t) and fluctuation component u0 (r, t), we arrive at the approximate system of equations ∂ ∂ ∂ Ui (r, t) + Tik (r, t) = − P(r, t), ∂t ∂rk ∂ri ∂ ∂ 0 [ui (r, t)uk (r, t) − hui (r, t)uk (r, t)i] ui (r, t) + ∂t ∂rk ∂uT (r, t) ∂Ui (r, t) ∂ + uTl (r, t) + Ul (r, t) i = − p0 (r, t), ∂rl ∂rl ∂ri

(10.2)

(10.3)

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255

where Tik (r, t) = hui (r, t)uk (r, t)i is the Reynolds stress tensor and P(r, t) and p0 (r, t) are the average and perturbed turbulent pressure components, respectively. Here and below, we neglect the effect of viscosity on dynamics and statistics of regular component U(r, t) and perturbed field u0 (r, t). In addition, we will bear in mind that fields U(r, t) and u0 (r, t) satisfy the incompressibility condition (1.95), page 43, div U(r, t) = 0,

div u0 (r, t) = 0.

(10.4)

Being combined with Eq. (1.95), page 43, for turbulent pulsations uT (r, t), Eq. (10.3) yields the following equation for the Reynolds stress tensor Tik (r, t) ∂ ∂ hui (r, t)uk (r, t)ul (r, t)i Tik (r, t) + ∂t ∂rl D E ∂U (r, t) D E ∂U (r, t) i k + uTk (r, t)uTl (r, t) + uTi (r, t)uTl (r, t) ∂rl ∂rl

 ∂ uTi (r, t)uTk (r, t) + Ul (r, t) ∂rl   ∂ ∂ 0 − uTk (r, t) p0 (r, t) + uTi (r, t) p (r, t) . ∂ri ∂rk

(10.5)

Here, ui (r, t)uk (r, t)ul (r, t) = uTi (r, t)uTk (r, t)u0l (r, t) + · · · . Pressure pulsations p0 in this equations can be expressed in terms of perturbed velocities U(r, t) and u0 (r, t), p0 (r, t) −1

= −1

(r, r ) 0





∂2  um (r0 , t)un (r0 , t) − um (r0 , t)un (r0 , t) 0 ∂r 0 ∂rm n ) ∂uTl (r0 , t) ∂Um (r0 , t) +2 , 0 ∂rm ∂rl0

where 1−1 (r, r0 ) is the integral operator inverse to the Laplace operator. Equations (10.2), (10.5), and the expression for pressure p0 (r, t) form the system of equations in average field U(r, t) and stress tensor Tik (r, t). These equations are not closed because they depend on higher-order velocity correlators like the triple correlator uTi (r, t)uTk (r, t)u0l (r, t) . Assuming that such correlators only slightly affect the dynamics of perturbations and taking into account conditions (10.4) and statistical homogeneity of field uT (r, t), we reduce the system of equations (10.2), (10.5) to the form (τi (r, t) = ∂r∂ k Tik (r, t)) ∂ ∂ Ui (r, t) + τi (r, t) = − P(r, t), ∂t ∂ri

(10.6)

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D E ∂ 2 U (r, t) ∂ i τi (r, t) + uTk (r, t)uTl (r, t) ∂t ∂rl ∂rk "* !+ T (r0 , t) 0 , t) ∂u ∂ ∂U (r ∂ m l uTk (r, t) 1−1 (r, r0 ) =2 0 ∂rk ∂ri ∂rm ∂rl0 * !+# T (r0 , t) 0 , t) ∂u ∂ ∂U (r m l + uTi (r, t) . 1−1 (r, r0 ) 0 ∂rk ∂rm ∂rl0

(10.7)

This system of equations is closed in U(r, t). The coefficients in the left-hand side of Eq. (10.7) and the integral operator in the right-hand side can be expressed in terms of the correlation tensor of vortex field of unperturbed turbulence uT (r, t) D E Z 0 T T 0 (10.8) ui (r, t)uj (r , t) = dq8ij (q)eiq(r−r ) . In view of the fact that field uT (r, t) is the solenoidal field, we have 8ij (q) = 1ij (q)F(q), where 1ij (q) = δij −

qi qj . q2

We will assume the energy spectrum of turbulent pulsations F(q) known. From Eq. (10.8) follows in particular that the energy of turbulent pulsations is expressed through the energy spectrum by the relationship T=

 Z∞ i2  1 h T u (r, t) = 4π dq q2 F(q), 2 0

and coefficients in the left-hand side of Eq. (10.7) are given by the formula D E 2 uTk (r, t)uTl (r, t) = Tδkl. 3

(10.9)

Finally, using Eq. (10.8), the right-hand side of Eq. (10.7) can be represented in the form Z − dqeiqr Um (q, t)gim (q), (10.10) where U(q, t) is the spatial Fourier transform of average velocity field U(r, t) and tensor gim (q) is given by the formula Z qm pl pk [(qi + pi ) 1kl (q) + (qk + pk ) 1il (q)] F(q). (10.11) gim (p) = 2 dq (q + p)2

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257

From the obvious fact that tensor gim (p) is invariant relative rotations in space, it follows that it must have the form gim (p) = A(p)δim + B(p)

pi pm . p2

(10.12)

Moreover, the term proportional to B(p) disappears in Eq. (10.12) in view of Eq. (10.4) (it expresses the property of incompressibility of average field U(r, t)) and the identity pU( p, t) = 0 following from Eq. (10.4), so that the right-hand side (10.10) of Eq. (10.7) assumes the form Z − dqeiqr Ui (q, t)A(q). Substituting this expression in the right-hand side of Eq. (10.7) and taking into account Eq. (10.9), we rewrite Eq. (10.6), (10.7) in the form ∂ ∂ Ui (r, t) + τi (r, t) = − P(r, t), ∂t ∂ri

(10.13)

2 ∂ τi (r, t) + T1Ui (r, t) = −A(−i∇)Ui (r, t). ∂t 3

The kernel of the integral operator appeared in the second equation can be obtained from comparison of Eq. (10.11) with (10.12). The result is as follows #" # " Z F(q) (qp)2 (qp)2 2 2
p − 2 A( p) = dq 2 q − (qp) − 2 2 . (10.14) q p q + p2 We can perform integration in Eq. (10.14) over angular coordinate to obtain the expression 2

Z∞

A(p) = 2π p

(

q2 − p2 q4 − p4 2q2 dq F(q) − + 8qp5 3p2 4p4

0


3

) q + p . ln q − p

(10.15)

We note additionally that Eq. (10.13) with allowance for Eq. (10.4) yields the identity 1P(r, t) = 0, so that P(r, t) = P0 = const. Then, eliminating quantity τi (r, t) from system of equations (10.11) and (10.4), we arrive at a single equation in the vector field of average velocity of flow U(r, t),   2 ∂2 U(r, t) − T1 + A(−i∇) U(r, t) = 0. 3 ∂t2 The corresponding dispersion equation ω2 (p) =

2 2 Tp − A(p) 3

(10.16)

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can be rewritten in the form ω (p) = 2π p 2

2

Z∞ dq F(q)f

  q , p

0

where
3

x2 − 1 2 − x2 x4 − 1 f (x) = + − 3 8 16x

1 + x . ln 1 − x

Function f (x) has the following asymptotics ( f (x) =

2/3,

x  1,

4/15

x  1.

Moreover, the inequalities 2 4 ≥ f (x) ≥ 3 15 hold for arbitrary x. Thus, the time-dependent development of disturbances in average flow is governed by the hyperbolic equation (10.16). As a result, the turbulent medium possesses certain quasi-elastic properties; namely, disturbances diffuse in the turbulent medium as transverse waves showing dispersion property. The phase and group velocities of these waves vary in the limits r

10.2

2 T≥c≥ 3

r

4 T. 15

Sound Radiation by Vortex Motions

In the previous section, we considered the physical effect immediately related to statistical averaging of the nonlinear system of hydrodynamical equations in the case of noncompressible flow. Here, we consider an effect related to weakly compressible media; namely, we consider the problem on sound radiation by a weakly compressible medium. This problem corresponds to the inclusion of random fields in the linearized equations of hydrodynamics. Note that, within the framework of linear equations, parametric action of turbulent medium yields the equations of acoustics with random refractive index. Turbulent motion of a flow in certain finite spatial region excites acoustic waves outside this region. In the case of a weakly compressible flow, this sound field is as if it

Some Other Approximate Approaches to the Problems of Statistical Hydrodynamics

259

were generated by the static distribution of acoustic quadrupoles whose instantaneous power per unit volume is given by the relationship Tij (r, t) = ρ0 vTi (r, t)vTj (r, t), where ρ0 is the average density and vTi (r, t) are the components of the velocity of flow in turbulent region V inside which the turbulence is assumed homogeneous and isotropic in space and stationary in time (we use the coordinate system in which the whole of the flow is quiescent). Turbulent motions cause the fluctuating waves of density ρ(r, t) that satisfy the wave equation 

 ∂2 ∂2 2 Tij (r, t), −c 1 ρ(r, t) = 0 ∂ri ∂rj ∂t2

(10.17)

where c0 is the sound velocity in the homogeneous portion of the medium, i.e., outside the region of turbulent motions. The solution to this equation has the form of the retarded solution ρ(r, t) =

1 ∂2 2 4πc0 ∂ri ∂rj

Z

  |r − y| dyTij y, t− . c0

V

For distances significantly exceeding linear sizes of turbulent region V, this solution can be reduced to the asymptotic expression   Z 1 ri rj |r − y| ¨ ij y, t − ρ(r, t) = dy T , (10.18) c0 4πc40 x3 V

where T¨ ij ( y, t) =

∂2 Tij ( y, t) . ∂t2

Correspondingly, average energy flux density of acoustic waves excited by turbulent motions

 c30 ρ 2 (r, t) q(r, t) = ρ0 is given by the expression q(r, t) =

ri rj rk rl r6      Z Z |r − y| |r − z| × dy dz T¨ ij y, t− T¨ kl z, t − . c0 c0 1

16π 2 c50

V

V

(10.19)

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Further calculations require the knowledge of the space–time correlators of the velocity field. The current theory of strong turbulence is incapable of yielding the corresponding relationships. For this reason, investigators limit themselves to various plausible hypotheses that allow complete calculations to be performed (see, e.g., [58–60]). In particular, it was shown that, with allowance for incompressibility of flow in volume V and the use of the Kolmogorov–Obukhov hypothesis and a number of simplifying hypotheses for splitting space-time correlators, from Eq. (10.19) follows that both average energy flux density and acoustic power are proportional to ∼ M 5 , where q

 v2 (r, t) M= c0 is the Mach number (significantly smaller than unity). We note that this result can be explained purely hydrodynamically, by analyzing vortex interactions in weakly compressible medium [61]. The simplest soundradiating vortex systems are the pair of vortex lines (radiating cylindrical waves) and the pair of vortex rings (radiating spherical waves).

10.2.1

Sound Radiation by Vortex Lines

Consider two parallel vortex lines separated by distance 2h and characterized by equal intensities κ=

1 πξ σ, 2

where ξ is the vorticity (the size of the vortex uniformly distributed over the area of infinitely small section σ ), so that the circulation about each of vortex line is 0 = 2πκ. We will call these vortex lines simply vortices. In a noncompressible flow, these vortices revolve with angular velocity ω=

κ 2h2

around the center of the line connecting these vortices (see Fig. 10.1a and, e.g., [62]). Select the coordinate system with the origin at a fixed point and z-axis along the vortex line. In this coordinate system, velocity potential ϕ0 (r, t) (v0 (r, t) = −Re∇ϕ0 (r, t)) and squared velocity v20 (r, t) assume the forms h i ϕ0 (r, t) = ik ln r2 e2iθ −h2 e2iωt , v20 (r, t)

4κ 2 r2 = 4 . r + h4 −2r2 h2 cos 2(ωt−θ)

Here reiθ is the radius-vector of the observation point.

(10.20)

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261

a 1(t ) h (t ) 2h (t )

h (t )

(a)

Figure 10.1

a 2(t )

(b)

Schematics of dynamics of (a) vortex lines and (b) vortex rings.

According to the Bernoulli equation, local velocity pulsations described by Eq. (10.20) must produce the corresponding pressure pulsations; in the case of weakly κ is compressible medium, namely, under the condition that M  1, where M = 2hc 0 the Mach number and c0 is the sound velocity, these pressure pulsations will propagate at large distances as sound waves. Encircle the origin with a circle of radius R such that h  R  λ, where λ is the sound wavelength. This is possible because λ π = 1 h M

for M  1.

In region r < R, dynamics of flow approximately coincides with the dynamics of noncompressible flow. In other words, the dynamics of flow is described by Eqs. (10.20) in this region. In region r > R, equations of motion coincide with the standard equations of acoustics (see, e.g., [63])  2  ∂ ∂ 2 −c 1 ϕ(r, t) = 0, p0 (r, t) = ρ0 ϕ(r, t). (10.21) ∂t ∂t2 0 Here, ϕ(r, t) is the potential of velocity in the acoustic wave, p0 (r, t) is the pressure in the wave, and ρ0 is the density of the medium. Taking into account the fact that velocity v0 (r, t) depends on t and θ only in combination 2 (ωt − θ), we will seek the solution to Eq. (10.21) in the form ϕ(r, t) = f (r)e2i(ωt−θ ) ,

(10.22)

Substituting Eq. (10.22) in Eq. (10.21) and solving the resulting equation in function f (r) with allowance for the radiation condition, we obtain ϕ(r, t) = AH22 (2ωr/c0 ) e2i(ωt−θ ) , where H22 (z) is the Hankel function of the second kind and A is a constant.

(10.23)

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For r  c0 /2ω we have the standard divergent cylindrical wave with wavelength λ = πc0 /ω r c exp{2i(ωt − ωr/c0 − θ − 3π/8)}. ϕ(r, t) = A πωr In the opposite case r  λ, we obtain ϕ(r, t) = i

A  c 2 2i(ωt−θ ) e . π ωr

Potential ϕ(r, t) in region h  r  λ must coincide with the oscillating portion of potential ϕ0 (r, t) (see, e.g., [63]), i.e., with (1)

ϕ0 (r, t) = −iκ

h2 2i(ωt−θ ) e . r2

This condition yields the following expression for constant A A = −πκM 2 = −

πκ 3 . 4h2 c20

Consequently, potential ϕ(r, t) in the wave zone assumes the form r ϕ(r, t) = −κM

3/2

πh exp{2i(ωt − ωr/c0 − θ − 3π/8)}. r

(10.24)

The pressure in sound waves can be determined from potential (10.24) using Eq. (10.21). The result is as follows r p (r, t) = −2κωρ0 M 0

3/2

πh exp{2i(ωt − ωr/c0 − θ − 3π/8)}. r

The sound intensity (energy radiated per unit time) can be obtained by integrating along the circle of radius R  λ I=

c0 ρ0

I

D  E κ3 dl p0 (r, t) 2 = 2π 2 ρ0 M 4 2 . h

(10.25)

The radiated energy must coincide with the interaction energy of vortices located in region r < R. Total energy in region r < R is E=

ρ0 2

Z

dSv20 (r, t).

(10.26)

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263

Substituting Eq. (10.20) in (10.26) and discarding infinite terms corresponding to the energy of motion of vortices themselves (we assume vortices the point vortices), we obtain the interaction energy in the form E1 = 4πκ 2 ρ0 ln(R/h).

(10.27)

The interaction energy can vary only at the expense of varying the distance between vortices (h = h(t)), because the circulation remains intact due to the fact that we consider nonviscous medium. Differentiating Eq. (10.27) with respect to time, we obtain energy variation rate I(t) = −4πρ0

κ2 d h(t), h(t) dt

(10.28)

which is just transferred into the energy of acoustic waves. Using Eq. (10.25) and (10.28), we obtain that the distance between vortices satisfies the equation d πκM 4 h(t) = . dt 2h(t)

(10.29)

Integrating Eq. (10.29) with allowance for the fact that M = M(t) =

κ , 2h(t)c0

we obtain h i1/6 h(t) = h0 1 + 6πM 40 ω0 t . Thus, the intensity of radiated sound is proportional to M 4 , I ∼ M 4 . It is obvious that, in the case of statistically distributed system of vortex line pairs, this estimate will remain valid for certain portion of the plane.

10.2.2

Sound Radiation by Vortex Rings

In a noncompressible flow, a vortex ring of intensity κ causes the flow to move with a velocity κ v(r, t) = a 2

Z2π dφ

s×r , r3

0

which follows from the the Biot–Savart law (see, e.g. [62]). Here, s is the unit vector tangent to the vortex ring (it is directed along the vortex vector), a is the radius of the

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ring, and r is the vector specifying observation point position relative to points lying on the ring. In the cylindrical coordinate system with origin at the center of ring and z-axis directed along the ring axis, we have κ vR = a 2

Z2π

cos φ dφ 3 , r

vθ = 0,

0

κ vz = a 2

Z2π dφ

a−Rcos φ , r3

(10.30)

0

where  1/2 r = R2 +z2 +a2 −2Racos φ . Here (R, θ, z) are the coordinates of the radius-vector of the observation point. We now have two vortex rings of equal intensities and equal radii a0 at distance 2h0 . In this case, the front ring will increase in size, while the rear ring will decrease and pursue the front ring (Fig. 10.1b). At a certain instant, it will penetrate through the front ring, and the rings switch places. This phenomenon is called the game of vortex rings. In a weakly compressible flow, these motions of rings produce local regions of compression and rarefaction; these regions propagate in the medium and, for large distances assume the form of spherical acoustic waves. To determine the structure of radiated sound, we must know relative motions of rings in a noncompressible flow. Assume rings have radii a1 (t) and a2 (t) and are separated by distance 2h(t) at certain instant t. The rates of variations of ring radii are equal to radial velocities that rings induce at each other, and the rate of variation of the distance between the rings is equal to the difference of the z-components of velocities induced by the rings. Consequently, we have d a1 (t) = 2κa2 (t)h(t) dt

Zπ dφ 0

d a2 (t) = −2κa1 (t)h(t) dt

cos φ , |a1 (t) − a2 (t)|3

Zπ dφ 0

cos φ , |a1 (t) − a2 (t)|3

i Zπ κh 2 d 1 2 h(t) = − a1 (t)−a2 (t) , dφ dt 2 |a1 (t) − a2 (t)|3 0

where h i1/2 |a1 (t) − a2 (t)| = a21 (t)+a22 (t) + 4h2 (t) − 2a1 (t)a2 (t) cos φ . Equations (10.31) should be solved with the initial conditions at t = 0 a1 (0) = a2 (0) = a0 ,

h(0) = h0 .

(10.31)

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265

The first pair of equations immediately yields the relationship between a1 (t) and a2 (t); namely, a21 (t) + a22 (t) = 2a20 .

(10.32)

This relationship shows that the moment of inertia of rings relative to the z-axis is conserved. The integrals in the right-hand side of Eq. (10.31) can be expressed in terms of elliptic functions. If rings are far from each other (γ = h0 /a0  1), they interact only slightly, and we can assume that, in the first approximation, they move independently with the velocities determined by the ring areas. In the opposite limiting case γ  1 (namely this case will be considered in what follows), the rings interact actively. The integrals in the right-hand side of Eq.(10.31) are mainly contributed by the neighborhood of point φ = 0. For this reason, we can replace cos φ in the numerators of the first pair of equations by unity. Thus we obtain the second integral of motion 4h2 (t) + [a1 (t)−a2 (t)]2 = 4h20 .

(10.33)

Integral (10.33) means that the distance between points on different rings at the same polar angle is the conserved quantity. In view of the existence of integrals (10.32) and (10.33), we can reduce system (10.31) to a single equation in variable θ(t) determined by the equalities π  π  √ √ − γ sin θ(t) , a2 (t) = 2a0 sin − γ sin θ(t) , a1 (t) = 2a0 cos 4 4 h(t) = h0 cos θ(t). (10.34) With this definition, conservation laws (10.32) and (10.33) are satisfied automatically (in the first order with respect to γ ). Substituting Eq. (10.34) in Eq. (10.31), expanding the result in the series, and calculating the integral, we obtain θ(t) =

κ t. 2h20

Consequently, the ring radii and the distance between rings are given by the expression a1 (t) = a0 (1 + γ sin(ωt)) , a2 (t) = a0 (1 − γ sin(ωt)) , h(t) = h0 cos(ωt),

(10.35)

where ω=

κ , 2h20

as in the above case of vortex lines.

(10.36)

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We note that the infinitely thin rings move with an infinite velocity. However, actual vortex rings move with a finite velocity significantly smaller than the sound velocity, but the dynamics of relative motion of rings only slightly differs from that we just obtained. The fact that angular velocity (10.36) coincides with the corresponding angular velocity in the case of vortex lines shows that the points lying on rings at equal polar angles revolve relative the center point of the line connecting them at the rotational speed coinciding with the rotational speed of vortex lines located at the same distance and having the same intensity as the rings. To study the structure of sound radiated by the system of rings, we need to know the velocity field for large distances from the system. We associate the coordinate system with the point located at the center of the segment connecting the centers of the rings. The velocity of flow outside the rings is given by the formula κ v(r, t) = a1 (t) 2

Z2π dφ 0

s1 × r1 r31

κ + a2 (t) 2

Z2π dφ 0

s2 × r2 r32

.

(10.37)

Here, s1 and s2 are the unit vectors tangent to the vortex rings and r1 and r2 are the vectors specifying observation point position relative to points lying on the rings. For large distances from the rings, the oscillating parts of the velocity have the form (in the cylindrical coordinates)
3κ R 4z2 − R2  2 2  (1) (1) h a1 −a2 , vθ (t) = 0, vR (t) = 4 (R2 + z2 )3/2 (10.38)
 2 2  3κ z 2z − 3R v(1) h a21 −a22 . z (t) = 4 (R2 +z2 )3/2 Introducing potential by the formula v(1) = −∇ϕ (1) , we obtain ϕ (1) (t) = −

  κ R2 − 2z2 2 2 h a − a . 1 2 4 (R2 + z2 )5/2

(10.39)

Substituting Eq. (10.35) in Eq. (10.39) and changing to spherical coordinates, we can rewrite Eq. (10.39) in the complex form κ 1 − 3 cos2 θ 2iωt ϕ (1) (t) = i a30 γ 2 e . 2 r3

(10.40)

Now, we will proceed similar to the case of two vortex lines. We encircle the origin by a sphere of radius L such that a0  L  λ, where λ is the wavelength of radiated sound waves. We will use the fact that the motion of flow inside the sphere approximately coincides with the motion of noncompressible flow. Outside the sphere, the equations of motion will have the form of Eq. (10.21) with the only difference that now 1 is the spatial Laplace operator.

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267

Represent potential ϕ in the form ϕ(r, θ) = f (r, θ)e2iωt . Substituting this expression in Eq. (10.21) and taking into account that we must obtain divergent spherical waves for r  λ , we obtain f (r, θ) in the form f (r, θ) =

  ∞ X 2ωr An (2) Pn (cos θ) , √ Hn+1/2 c0 r

n=0 (2)

where Hn+1/2 (z) is the Hankel function of the second kind and Pn (z) is the Legendre polynomial. Comparing potential ϕ(r, θ) for r  λ with ϕ (1) (r, θ), we obtain that n = 2 and √ 2 π κγ 2 a30 ω5/2 A2 = . 5/2 3 c0 Consequently, potential and pressure in the wave zone have the forms κ 2h0 c0

2κ ϕ(r, θ) = 3



4iκ 3



p0 (r, θ) =

2

    ωr 3π a0  2 1 − 3 cos θ exp 2i ωt− − , r c0 4

2

    a0 ω  ωr 3π ρ0 1 − 3 cos2 θ exp 2i ωt− − , r c0 4

κ 2h0 c0

so that the angular distribution of energy radiated per unit time is 8π κ 3 a20 I(θ) = 9 h30



κ 2h0 c0

5

 2 ρ0 1 − 3 cos2 θ .

Vortex rings radiate energy as a quadrupole; the main portion of energy is radiated in a cone about z-axis with corner angle 106◦ in both positive and negative z-directions. Integrating over θ, we obtain that total energy radiated per unit time is given by the expression I=

64π κ 3 a20 45 h30



κ 2h0 c0

5

ρ0 .

Thus, intensity of sound radiated by a pair of vortex rings is proportional to M 5 . In the case of statistically distributed system of pairs of vortex rings, this proportionality will remain valid for certain portion of space, which agrees with estimates obtained in [58–60].

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Part III Examples of Coherent Phenomena in Stochastic Dynamic Systems

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Lecture 11 Passive Tracer Clustering and Diffusion in Random Hydrodynamic and Magnetohydrodynamic Flows

11.1

General Remarks

Diffusion of such passive fields as scalar density (particle concentration) field and magnetic field is an important problem of the theory of turbulence in magnetohydrodynamics. The basic stochastic equations for the scalar density field ρ(r, t) and the magnetic field H(r, t) are the continuity equation 

 ∂ ∂ + U(r, t) ρ(r, t) = µ1ρ(r, t), ∂t ∂r

ρ(r, 0) = ρ0 (r),

(11.1)

and the induction equation (1.59), page 30, subject to the dissipative effects 

   ∂ ∂ ∂ + U(r, t) H(r, t) = H(r, t) · U(r, t) + µH 1H(r, t), ∂t ∂r ∂r

(11.2)

where µρ is the dynamic molecular diffusion coefficient, and µH = c2 /4πσ is the dynamic diffusion coefficient associated with the conductivity of the flow σ . In these equations, U(r, t) = u0 (r, t) + u(r, t); u0 (r, t) is the deterministic component of the velocity field (mean flow), and u(r, t) is the random component. In the general case, random field u(r, t) can be composed of both solenoidal (for which div u(r, t) = 0) and potential (for which div u(r, t) 6= 0) components. Field H(r, t) is here nondivergent, i.e., div H(r, t) = 0. Dynamic systems (11.1) and (11.2) are conservative and both the total scalar mass Z M = M(t) =

Z drρ(r, t) =

drρ0 (r) = const

Lectures on Dynamics of Stochastic Systems. DOI: 10.1016/B978-0-12-384966-3.00011-8 c 2011 Elsevier Inc. All rights reserved. Copyright

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and the magnetic flux Z Z dr H(r, t) = dr H0 (r) = const remain constant during the evolution. For homogeneous initial conditions, the following equalities are a corollary of the conservatism of dynamic systems (11.1) and (11.2): hρ(r, t)i = ρ0 ,

hH(r, t)i = H0 ,

where h· · · i denotes averaging over an ensemble of realizations of random field {u(r, t)}. If initial conditions are homogeneous, a statistical average of a quantity R depending, for example, on magnetic field dr hf (H(r, t))i changes over in the general case to hf (H(r, t))i, Z dr hf (H(r, t))i ⇔ hf (H(r, t))i, which means that quantity hf (H(r, t))i is a specific quantity per unit volume and is, hence, an integral quantity. For example, quantity H2 (r, t) is the average specific energy per unit volume in the case of homogeneous initial conditions, but it is the average density of energy in the case of inhomogeneous initial conditions. Integral quantities characterize dynamic systems at large, they allow separating the processes of field generation, which offers a possibility of discarding dynamic details related to advection of these quantities by random velocity field [64]. Our interest consists in the temporal evolution of different one-point statistical characteristics of density field ρ(r, t), its gradient p(r, t) = ∇ρ(r, t), magnetic field H(r, t), and its different spatial derivatives from given initial distributions ρ0 (r) and H0 (r) (and, in particular, homogeneous ones, ρ0 (r) = ρ0 and H0 (r) = H0 ). Such characteristics are probability densities of these fields by themselves, isotropization (decrease of anisotropy) of vector correlations, helicity of these vector fields, and their dissipation. The effect of dynamic diffusion can be neglected during the initial stages of the development of diffusion. In this case, Eqs. (11.1) and (11.2) become simpler and assume the form   ∂ ∂U(r, t) ∂ + U(r, t) ρ(r, t) + ρ(r, t) = 0, (11.3) ∂t ∂r ∂r     ∂ ∂U(r, t) ∂ ∂ + U(r, t) H(r, t) + H(r, t) = H(r, t) · U(r, t). (11.4) ∂t ∂r ∂r ∂r In these conditions, such nonstationary and stochastic phenomena occur as mixing, generation and rapid increase (in time) of smaller-scale disturbances (e.g., stochastic (turbulent) dynamo), and, in certain situations, clustering in the phase or physical space. Clustering of a field (such as tracer density, magnetic field energy, etc.) is the

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273

appearance of compact regions, where the content of this field is increased, which are surrounded by regions with decreased content of this field. For example, in the case of homogeneous initial conditions, the density field remains constant in time if liquid is noncompressible (nondivergent velocity field) and, consequently, the gradient of density is identically equal to zero at any spatial point. On the contrary, in compressible flows (divergent velocity field), density field is always clustered, which naturally results in great gradients of the density field. We have seen a similar pattern in the example of the simplest problem whose solution is given in Fig. 1.12, page 27. The characteristic time of the formation of cluster structure of the density field is determined by the relationship Dp t ∼ 1, where quantity Dp given by Eq. (4.59), page 108, depends on the potential component of spectral density of the velocity field. As for the magnetic field, clustering of its energy can occur under certain conditions, and we have seen an example of such clustering in Fig. 1.14, page 33, in the case of the simplest potential velocity field. However, general generation of magnetic field occurs even in noncompressible flows of liquid; this generation causes intense variability of the field structure, and different moment functions of both magnetic field by itself and its spatial derivatives increase exponentially in time. As a result, field dissipation related to higher-order derivatives is rapidly increased at a certain time instant, and the effects of dynamic diffusion become governing. The above equations correspond to the Eulerian description of the evolution of the density field and magnetic field. Equations (11.3), (11.4) are the first-order partial differential equations and can be solved by the method of characteristics. Introducing characteristic curves r(t) satisfying the equations of particle motion d r(t) = U(r, t), dt

r(0) = r0 ,

(11.5)

we can change over from Eq. (11.3) to the ordinary differential equation ∂U(r, t) d ρ(t) = − ρ(t), ρ(0) = ρ0 (r0 ). dt ∂r

(11.6)

Solutions to Eqs. (11.5) and (11.6) have an obvious geometric interpretation. They describe the concentration behavior around a fixed tracer particle moving along trajectory r = r(t). As may be seen from Eq. (11.6), the concentration in divergent flows varies: it increases in regions where medium is dense and decreases in regions where medium is rarefied. Solutions to system (11.5), (11.6) depend on characteristic parameter r0 (the initial coordinate of the particle) r(t) = r(t|r0 ),

ρ(t) = ρ(t|r0 ),

(11.7)

which we will separate by the bar. Components of vector r0 are called the Lagrangian coordinates of a particle; they unambiguously specify the position of arbitrary particle. Equations (11.5), (11.6) correspond in this case to the Lagrangian description of

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concentration evolution. The first of equalities (11.7) specify the relationship between the Eulerian and Lagrangian descriptions. Solving it in r0 , we obtain the relationship that expresses the Lagrangian coordinates in terms of the Eulerian ones r0 = r0 (r, t).

(11.8)

Then, using Eq. (11.8), to eliminate r0 in the last equality in (11.7), we turn back to the concentration in the Eulerian description ρ(r, t) = ρ(t|r0 (r, t)) =

Z

dr0 ρ(t|r0 )j(t|r0 )δ (r(t|r0 ) − r),

(11.9)

where we introduced a new function called divergence

∂ri (t|r0 )

,

j(t|r0 ) = det ||jik (t|r0 )|| = det ∂r0k which is the quantitative measure of the degree of compression (extension) of physically infinitely small liquid particles. One can easily obtain that it satisfies the equation d ∂U(r, t) j(t|r0 ) = j(t|r0 ), dt ∂r

j(0|r0 ) = 1.

(11.10)

Comparing Eq. (11.6) with Eq. (11.10), we see that ρ(t|r0 ) =

ρ0 (r0 ) . j(t|r0 )

(11.11)

Thus, we can rewrite Eq. (11.9) as the equality ρ(r, t) =

Z

dr0 ρ0 (r0 )δ (r(t|r0 ) − r)

(11.12)

specifying the relationship between the Lagrangian and Eulerian characteristics. Delta function in the right-hand side of Eq. (11.12) is the indicator function for the position of the Lagrangian particle; as a consequence, after averaging Eq. (11.12) over an ensemble of realizations of random velocity field, we arrive at the well-known relationship between the average concentration in the Eulerian description and the one-time probability density P(r, t|r0 ) = hδ (r(t|r0 ) − r)i of the Lagrangian particle Z hρ(r, t)i =

dr0 ρ0 (r0 )P (r, t|r0 ).

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275

The relationship between the spatial correlation function of the concentration field in the Eulerian description hR(r1 , r2 , t)i = hρ(r1 , t)ρ(r2 , t)i, where R(r1 , r2 , t) = ρ(r1 , t)ρ(r2 , t), and the joint probability density function of positions of two particles P(r1 , r2 , t|r01 , r02 ) = hδ (r1 (t|r01 ) − r1 ) δ (r2 (t|r02 ) − r2 )i can be obtained similarly hR(r1 , r2 , t)i =

Z

Z

dr02 ρ0 (r01 )ρ0 (r02 )P(r1 , r2 , t|r01 , r02 ).

dr01

For the nondivergent velocity field (div U(r, t) = 0), both particle divergence and particle concentration are invariant, i.e., j(t|r0 ) = 1,

ρ(t|r0 ) = ρ0 (r0 ).

We assume in the general case that the random component of the velocity field is the divergent (div u(r, t) 6= 0) statistically homogeneous Gaussian random field, which is spherically symmetric (but generally not possessing reflection symmetry) in space and stationary in time with zero-valued average (hu(r, t)i = 0) and correlation and spectral tensors given by the formulas Z

 Bij (r − r1 , τ ) = ui (r, t)uj (r1 , t1 ) = dk Eij (k, τ )eik(r−r1 ) , Z 1 dr Bij (r, τ )e−ikr , Eij (k, τ ) = (2π)d where d is the dimension of space. In view of the suggested symmetry condition, correlation tensor Bij (r, τ ) has vector structure (4.49), page 105, (r − r1 → r) Bij (r, τ ) = Biso ij (r, τ ) + C(r, τ )εijk rk , where Biso ij (r, τ ) = A(r, τ )ri rj + B(r, τ )δij is the isotropic portion of the correlation tensor and εijk is the pseudotensor antisymmetric with respect to an arbitrary pair of indices. The isotropic portion of the correlation tensor corresponds to the spatial spectral tensor of the form p

Eij (k, τ ) = Eijs (k, τ ) + Eij (k, τ ),

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where the spectral components of the tensor of velocity field have the following structure   ki kj ki kj p Eij (k, τ ) = Es (k, τ ) δij − 2 , Eij (k, τ ) = Ep (k, τ ) 2 . k k Here, Es (k, t) and Ep (k, t) are the solenoidal and potential components of the spectral density of the velocity field, respectively. Calculating statistical properties of the concentration field and its gradient, we will approximate the velocity field u(r, t) by the random delta-correlated (in time) process. In the framework of this approximation, correlation tensor Bij (r, t) is approximated by the expression Bij (r, t) = 2Bij (r)δ(t),

(11.13)

where 1 Bij (r) = 2

Z∞

dτ Bij (r, τ ) =

−∞

Z∞

dτ Bij (r, τ ).

0

For the Gaussian field u(r, t) correlation with functionals of it can be split by the Furutsu–Novikov formula (8.10), page 193; in the case of delta-correlated field u(r, t) this formula becomes simpler and assumes the form huk (r, t)R[t; u(r, τ )]i =

Z

 δR[t; u(r, τ )] dr Bkl (r − r ) , δul (r0 , t − 0) 0

0



(11.14)

where 0 ≤ τ ≤ t.

11.2 11.2.1

Particle Diffusion in Random Velocity Field One-Point Statistical Characteristics

Thus, in the Lagrangian representation, the behavior of passive tracer is described in terms of ordinary differential Equations (11.5), (11.6), (11.10). We can easily pass on from these equations to the linear Liouville equation in the corresponding phase space. With this goal in view, introduce the indicator function ϕLag (r, ρ, j, t|r0 ) = δ(r(t|r0 ) − r)δ(ρ(t|r0 ) − ρ)δ(j(t|r0 ) − j),

(11.15)

where we explicitly emphasized the fact that the solution to the initial dynamic equations depends on the Lagrangian coordinates r0 . Differentiating Eq. (11.15) with respect to time and using Eqs. (11.5), (11.6), and (11.10), we arrive at the Liouville

Passive Tracer Clustering and Diffusion

277

equation equivalent to the initial problem 

   ∂ ∂U(r, t) ∂ ∂ ∂ + U(r, t) ϕLag (r, ρ, j, t|r0 ) = ρ − j ϕLag (r, ρ, j, t|r0 ), ∂t ∂r ∂r ∂ρ ∂j (11.16) ϕLag (r, ρ, j, t|r0 ) = δ(r0 − r)δ(ρ0 (r0 ) − ρ)δ(j − 1).

The one-time probability density of the solutions to dynamic problems (11.5), (11.6), and (11.10) coincides with the indicator function averaged over an ensemble of realizations

 P(r, ρ, j, t|r0 ) = ϕLag (t; r, ρ, j|r0 ) . Averaging Eq. (11.16) over an ensemble of realizations of random field u(r, t), using the Furutsu–Novikov formula (11.14), and taking into account the equality δ ϕLag (r, ρ, j, t|r0 ) δuβ (r0 , t − 0)    ∂ ∂ ∂δ(r − r0 ) ∂ 0 = − ρ − j ϕLag (t; r, ρ, j|r0 ), δ(r − r ) + ∂rβ ∂rβ ∂ρ ∂j and Eqs. (4.57), page 107, we arrive at the Fokker–Planck equation for the one-time Lagrangian probability density P(r, ρ, j, t|r0 ) of particle coordinate r(t|r0 ), of particle density ρ(t|r0 ) and of the particle divergence j(t|r0 ): 

   ∂ ∂ 2 ∂ ∂2 ∂2 2 p − D0 1 P(r, ρ, j, t|r0 ) = D ρ −2 ρj + 2 j P(r, ρ, j, t|r0 ), ∂t ∂ρ ∂ρ ∂ρ∂j ∂j P(r, ρ, j, 0|r0 ) = δ(r − r0 )δ(ρ − ρ0 (r0 ))δ(j − 1). (11.17)

The solution to Eq. (11.17) has the form P(r, ρ, j, t|r0 ) = P(r, t|r0 )P(j, t|r0 )P(ρ, t|r0 ).

(11.18)

Function P(r, t|r0 ) is the probability distribution of particle coordinates. This distribution satisfies the equation following from Eq. (11.17), ∂ ∂2 P(r, t|r0 ) = D0 2 P(r, t|r0 ), ∂t ∂r

P(r, 0|r0 ) = δ(r − r0 ),

and, consequently, it is the Gaussian distribution ) (   ∂2 1 (r − r0 )2 P(r, t|r0 ) = exp D0 t 2 δ(r − r0 ) = exp − , 4D0 t ∂r (4π D0 dt)d/2 (11.19)

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where d is the dimension of space. Function P(j, t|r0 ) is the probability distribution of the divergence field in particle’s neighborhood and satisfies the Fokker–Planck equation following from Eq. (11.17), ∂2 ∂ P(j, t|r0 ) = Dp 2 j2 P(j, t|r0 ), ∂t ∂j

P(j, 0|r0 ) = δ(j − 1),

which we rewrite in the form   ∂ ∂ ∂ jP(j, t|r0 ) = Dp j j + 1 jP(j, t|r0 ), ∂t ∂j ∂j

P(j, 0|r0 ) = δ(j − 1).

(11.20)

(11.21)

It is convenient to change the variables in Eq. (11.21) according to the formulas j = eη ,

η = ln j,

(11.22)

so that ∂ ∂ =j . ∂η ∂j After the variable change (11.22), Eq. (11.21) in function jP(j, t|r0 )|j=eη = F(η, t) is reduced to the equation   ∂ ∂ p ∂ F(η, t) = D + 1 F(η, t) ∂t ∂η ∂η

(11.23)

with the initial condition F(η, 0) = jP(j, 0|r0 )|j=eη = eη δ(eη − 1) = eη

δ(η) = δ(η). eη

(11.24)

Consequently, the solution to problem (11.23), (11.24) is the Gaussian probability distribution ( ) 1 (η + τ )2 F(η, t) = √ exp − . (11.25) 4τ 2 πτ In Eq. (11.25) and below, we use the dimensionless time τ = Dp t. Considering now function F(η, t)|η=ln j = jP(j, t|r0 ), we obtain the solution to Eq. (11.20) in the form ( )  2 2  ∂ 1 j + τ (ln ) P(j, t|r0 ) = exp Dp t 2 j2 δ(j − 1) = √ exp − 4τ ∂j 2j πτ ( ) 1 ln2 (jeτ ) . (11.26) = √ exp − 4τ 2j πτ

Passive Tracer Clustering and Diffusion

279

It should be emphasized that the obtained solution (11.18) means that coordinates r(t|r0 ) and divergence j(t|r0 ) are statistically independent in the neighborhood of a particle characterized by the Lagrangian coordinate r0 . Moreover, the logarithmically normal distribution (11.26) means that quantity η(t|r0 ) = ln j(t|r0 ) is distributed according to the Gaussian law with the parameters hη(t|r0 )i = −τ,

ση2 (t) = 2τ.

(11.27)

In particular, the following expressions for the moments of the random divergency field follow from distribution (11.26) (and immediately from Eq. (11.20), though)

n  j (t|r0 ) = en(n−1)τ ,

n = ±1, ±2, . . . .

(11.28)

We note that average divergence is constant, hj(t|r0 )i = 1, and its higher moments increase with time exponentially. Note additionally that, according to Eq. (11.11), page 274, and Eq. (11.28), we have the following expression for the Lagrangian moments of density:

n  ρ (t|r0 ) = ρ0n (r0 )en(n+1)τ , according to which both average density and higher moments of density increase exponentially in the Lagrangian representation. Consequently the random process ρ(t|r0 ) is the logarithmically normal process and its probability density has the form
) ln2 ρe−τ /ρ0 (r0 ) . exp − P(ρ, t|r0 ) = 4τ 2ρ πτ 1 √

(

(11.29)

This probability density can be obtained also as the solution to the Fokker–Planck equation following from Eq. (11.17) ∂ ∂ 2 ∂ P(ρ, τ |r0 ) = ρ P(ρ, τ |r0 ), ∂τ ∂ρ ∂ρ

P(ρ, 0|r0 ) = δ(ρ − ρ0 (r0 )).

(11.30)

Rewrite Eq. (11.30) in the form ∂ ∂ ρP(ρ, τ |r0 ) = ρ ∂τ ∂ρ



ρ

 ∂ − 1 ρP(ρ, τ |r0 ), ∂ρ

P(ρ, 0|r0 ) = δ(ρ − ρ0 (r0 )). Then, the variable change (11.22) reduces Eq. (11.31) in function ρP(ρ, τ |r0 )|ρ=eη = F(η, τ |r0 )

(11.31)

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to the equation ∂ ∂ F(η, τ |r0 ) = ∂τ ∂η



 ∂ − 1 F(η, τ |r0 ) ∂η

(11.32)

with the initial condition F(η, 0|r0 ) = ρP(ρ, 0|r0 )|ρ=eη = eη δ(eη − ρ0 (r0 )) = δ(η − ln ρ0 (r0 )). (11.33) Consequently, the solution of problem (11.32), (11.33) is the Gaussian probability distribution ( ) 1 (η − ln ρ0 (r0 ) − τ )2 exp − . (11.34) F(η, τ |r0 ) = √ 4τ 2 πτ Considering now function F(η, τ |r0 )|η=ln ρ = ρP(ρ, τ |r0 ), we obtain the solution to Eq. (11.30) in form (11.29). In this case, the logarithmically normal distribution (11.34) means that quantity η(t|r0 ) = ln ρ(t|r0 ) is distributed according to the Gaussian law with the parameters hη(t|r0 )i = ln ρ0 (r0 ) + τ,

ση2 (t) = 2τ.

The above paradoxical behavior of statistical characteristics of the divergence and concentration (simultaneous growth of all moment functions in time) is a consequence of the lognormal probability distribution. Indeed, the typical realization curve of random divergence is the exponentially decaying curve j∗ (τ ) = e−τ , in accordance with Eqs. (4.60), page 108. Moreover, realizations of the lognormal process satisfy certain majorant estimates. For example, with probability p = 1/2, we have j(t|r0 ) < 4e−τ/2 throughout arbitrary temporal interval t ∈ (t1 , t2 ). Similarly, the typical realization curve of concentration and its minorant estimate have the following form ρ ∗ (t) = ρ0 eτ ,

ρ(t|r0 ) >

ρ0 τ/2 e . 4

We emphasize that the above Lagrangian statistical properties of a particle in flows containing the potential random component are qualitatively different from the statistical properties of a particle in nondivergent flows where j(t|r0 ) ≡ 1 and particle concentration remains invariant in the vicinity of a fixed particle ρ(t|r0 ) = ρ0 (r0 ) = const. The above statistical estimates mean that the statistics of random processes j(t|r0 ) and ρ(t|r0 ) is formed by the realization spikes relative typical realization curves. At the same time, probability distributions of particle coordinates in essence coincide for both divergent and nondivergent velocity fields.

Passive Tracer Clustering and Diffusion

11.2.2

281

Two-Point Statistical Characteristics

Consider now the joint dynamics of two particles in the absence of mean flow. In this case, the indicator function ϕ(r1 , r2 , t) = δ (r1 (t) − r1 ) δ (r2 (t) − r2 ) satisfies the Liouville equation   ∂ ∂ ∂ ϕ(r1 , r2 , t) = − u1 (r, t) + u2 (r, t) ϕ(r1 , r2 , t). ∂t ∂r1 ∂r2 If we average the indicator function over an ensemble of realizations of field u(r, t), use the Furutsu–Novikov formula (11.14), page 276, and the equality   ∂ ∂ δ 0 0 ϕ(t; r1 , r2 ) = − δ(r1 − r ) + δ(r2 − r ) ϕ(t; r1 , r2 ), δuj (r0 , t − 0) ∂r1j ∂r2j then we obtain that the joint probability density of positions of two particles for isotropic velocity field P(r1 , r2 , t) = hϕ(t; r1 , r2 )i satisfies the Fokker–Planck equation   ∂ ∂2 ∂2 P(r1 , r2 , t) = + Bij (0)P(t; r1 , r2 ) ∂t ∂r1i ∂r1j ∂r2i ∂r2j +2

∂2 Bij (r1 − r2 )P(r1 , r2 , t). ∂r1i ∂r2j

(11.35)

Multiplying now Eq. (11.35) by function δ(r1 − r2 − l) and integrating over r1 and r2 , we obtain that the probability density of relative diffusion of two particles P(l, t) = hδ(r1 (t) − r2 (t) − l)i satisfies the Fokker–Planck equation ∂ ∂2 P(l, t) = Dαβ (l)P(l, t), ∂t ∂lα ∂lβ

P(l, 0) = δ(l − l0 ),

(11.36)

where   Dαβ (l) = 2 Bαβ (0) − Bαβ (l) is the structure matrix of vector field u(r, t), and l0 is the initial distance between the particles.

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In the general case, Eq. (11.36) hardly can be solved analytically. However, if the initial distance between particles l0 is sufficiently small, namely, if l0  lcor , where lcor is the spatial correlation radius of the velocity field u(r, t), we can expand functions Dαβ (l) in the Taylor series to obtain in the first approximation ∂ 2 Bαβ (l) Dαβ (l) = − ∂li ∂lj

li lj . l=0

The use of representation (4.57), page 107, simplifies the diffusion tensor reducing it to the form Dαβ (l) =

h i

1 Ds (d + 1) + Dp δαβ l2 − 2 Ds − Dp lα lβ , d(d + 2)

(11.37)

where d is the dimension of space. Substituting now Eq. (11.37) in Eq. (11.36), multiplying both sides of the resulting equation by ln , and integrating over l, we obtain the closed equation
  s
d n  1 ln l (t) = D (d + 1) + Dp n (d + n − 2) − 2 Ds − Dp n(n − 1) , dt d(d + 2) whose solution shows the exponential growth of all moment functions (n = 1, 2, . . .) in time. In this case, the probability distribution of random process l(t)/l0 will be logarithmic-normal. Note that, multiplying Eq. (11.36) by δ(l(t) − l) and integrating the result over l, we can easily obtain that the probability density of the modulus of vector l(t) Z P(l, t) = hδ(|l(t)| − l)i =

dlδ(|l(t)| − l)P(l, t),

satisfies the equation ∂ ∂ Dii (l) ∂ N(l) ∂2 P(l, t) = − P(l, t) + P(t, l) + 2 N(l)P(l, t), ∂t ∂l l ∂l l ∂l where N(l) = lj li Dij (l)/l2 . Using this equation, we can easily derive the equation in function hln l(t)i,   d Dii (l) N(l) hln l(t)i = −2 2 , dt l2 l whose solution for the tensor Dij (l) of form (11.37) is as follows    l(t) 1 ln = {Ds (d − 1)d − Dp (4 − d)}t. l0 d(d + 2)

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283

As a consequence, in accordance with Eqs. (4.60), page 108, the typical realization curve of the distance between two particles will be the exponential function of time   s 1 p D d (d − 1) − D (4 − d) t , l (t) = l0 exp d(d + 2) ∗



(11.38)

and it is the Lyapunov exponent of the lognormal random process l(t). It appears that this expression in the two-dimensional case (d = 2) l∗ (t) = l0 exp




1 s D − Dp t 4



significantly depends of the sign of the difference (Ds − Dp ). In particular, for the nondivergent velocity field (Dp = 0), we have the exponentially increasing typical realization curve, which means that particle scatter is exponentially fast for small distances between them. This result is valid for times   1 s lcor D t  ln , 4 l0 for which expansion (11.37) holds. In another limiting case of the potential velocity field (Ds = 0), the typical realization curve is the exponentially decreasing curve, which means that particles tend to join. In view of the fact that liquid particles themselves are compressed during this process, we arrive at the conclusion that particles must form clusters, i.e., compact particle concentration zones located merely in rarefied regions, which agrees with the evolution of the realization (see Fig. 1.1b, page 5) obtained by simulating the behavior of the initially homogeneous particle distribution in random potential velocity field (though, for drastically other statistical model of the velocity field). This means that the phenomenon of clustering by itself is independent of the model of a velocity field, although statistical parameters characterizing this phenomenon surely depend on this model. Thus, particle clustering requires that inequality Ds < Dp

(11.39)

be satisfied. In the three-dimensional case (d = 3), Eq. (11.38) grades into l∗ (t) = l0 exp




1 6Ds − Dp t , 15

and typical realization curve will exponentially decay in time under the condition Dp > 6Ds , which is stronger than that in the two-dimensional case.

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In the one-dimensional case p

l∗ (t) = l0 e−D t , and typical realization curve always decays in time.

11.3

Probabilistic Description of Density Field in Random Velocity Field

To describe the local behavior of tracer realizations in random velocity field, we need the probability distribution of tracer concentration, which can be obtained only in the absence of dynamic diffusion. To describe the concentration field in the Eulerian description, we introduce the indicator function ϕ(r, t; ρ) = δ(ρ(r, t) − ρ),

(11.40)

which is similar to function (11.15) and is localized on surface ρ(r, t) = ρ = const in the three-dimensional case or on a contour in the two-dimensional case. The Liouville equation for this function coincides in form with Eq. (3.18), page 74,   ∂ ∂ ∂U(r, t) ∂ [ρϕ(r, t; ρ)], + U(r, t) ϕ(r, t; ρ) = (11.41) ∂t ∂r ∂r ∂ρ so that the on-point probability density of the solution to dynamic equation (11.13) coincides in this case with the ensemble averaged indicator function of random velocity field which we assume, as earlier, to be the homogeneous and isotropic Gaussian random field delta-correlated in time, P(r, t; ρ) = hδ(ρ(r, t) − ρ)i. Averaging Eq. (11.41) with absent average flow over an ensemble of realizations of field u(r, t) using the Furutsu–Novikov formula (11.14), page 276, and the equality for variational derivative   ∂δ(r − r0 ) δ 0 ∂ ϕ(r, t; ρ) = − δ(r − r ) + ϕ(r, t; ρ) δuj (r0 , t − 0) ∂rj ∂rj following from Eq. (11.41) with the use of Eqs. (4.56) and (4.57), page 107, we obtain the equation for probability density of the concentration field 

 ∂ ∂2 − D0 1 P(r, t; ρ|ρ0 (r)) = Dρ 2 ρ 2 P(r, t; ρ|ρ0 (r)), ∂t ∂ρ P(r, 0; ρ|ρ0 (r)) = δ(ρ − ρ0 (r)),

(11.42)

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285

where the diffusion coefficients D0 (in r-space) and Dρ = Dp (in ρ-space) are given by Eqs. (4.54) and (4.59), pages 107 and 108; the vertical bar separates the dependence of the probability density of the Eulerian field ρ(r, t) on the initial distribution ρ0 (r). This equation corresponds to Eq. (8.55), page 207, with parameter α = Dρ . Note that Eq. (11.42) can also be derived by using the following relationship between the one-point probability density of concentration field in the Eulerian representation and the one-time probability density in the Lagrangian description P(r, t; ρ) =

Z∞

Z dr0

jdjP(r, ρ, j, t|r0 ).

(11.43)

0

The solution to Eq. (11.42) has the form   ∂2 P(r, t; ρ|ρ0 (r)), P(r, t; ρ|ρ0 (r)) = exp D0 t 2 e ∂r where function e P(r, t; ρ|ρ0 (r)) satisfies the equation ∂ ∂2 e P(r, t; ρ|ρ0 (r)) = Dρ 2 ρ 2e P(r, t; ρ|ρ0 (r)), ∂t ∂ρ

(11.44)

e P(r, 0; ρ|ρ0 (r)) = δ(ρ − ρ0 (r)) that depends on r parametrically and coincides with Eq. (11.20), page 278, for the Lagrangian probability density of particle divergence (the only difference consists in the initial condition). Consequently, the solution to this equation is as follows (
) ln2 ρeDρ t /ρ0 (r) 1 e p exp − , (11.45) P(r, t; ρ|ρ0 (r)) = 4Dρ t 2ρ πDρ t and probability density of the Eulerian density field assumes the form (
)   ln2 ρeDρ t /ρ0 (r) ∂2 1 p exp D0 t 2 exp − . (11.46) P(r, t; ρ|ρ0 (r)) = 4Dρ t ∂r 2ρ π Dρ t In the case of uniform initial tracer concentration ρ0 (r) = ρ0 = const the probability distribution is independent of r. Consequently, the Eulerian concentration field is in this case the lognormal field; the corresponding probability density and integral distribution function are as follows ( ) 1 ln2 (ρeτ /ρ0 ) P(t; ρ|ρ0 ) = exp − , √ 4τ 2ρ πτ (11.47)   ln (ρeτ /ρ0 ) F(t; ρ|ρ0 ) = Pr , √ 2 τ

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where Pr(z) is the probability integral (4.20), page 94, and we use the dimensionless time τ = Dp t. In the context of the one-point characteristics of density field ρ(r, t), the problem is statistically equivalent in this case to the analysis of a random process; hence, all moment functions beginning from the second one appear to be exponentially increasing functions of time hρ(r, t)i = ρ0 ,



 ρ n (r, t) = ρ0n en(n−1)τ ,

(11.48)

and in accordance with Eqs. (4.60), page 108, the typical realization curve of the concentration field exponentially decays with time at any fixed spatial point ρ ∗ (t) = ρ0 e−τ , which is evidence of cluster behavior of medium density fluctuations in arbitrary divergent flows. The Eulerian concentration statistics at any fixed point is formed due to concentration fluctuations about this curve. Even the above discussion of the one-point probability density of tracer concentration in the Eulerian representation revealed several regularities concerning the temporal behavior of concentration field realizations at fixed spatial points. Now we show that this distribution additionally allows us to reveal certain features in the space–time structure of concentration field realizations. For simplicity, we content ourselves with the two-dimensional case. As was mentioned earlier, the analysis of level lines defined by the equality ρ(r, t) = ρ = const can give important data on the spatial behavior of realizations, in particular, on different functionals of the concentration field, such as total area S(t, ρ) of regions where ρ(r, t) > ρ and total tracer mass M(t, ρ) within these regions. Average values of these functionals can be expressed in terms of the one-point probability density:

hS(t, ρ)i =

Z∞

de ρ

Z

dr P(r, t; ρ e),

ρ

hM(t, ρ)i =

Z∞

ρ ede ρ

Z

dr P(r, t; ρ e).

(11.49)

ρ

Substituting the solution to Eq. (11.46) in these expressions and performing some rearrangement, we easily obtain explicit expressions for these quantities   ρ0 (r)e−τ 1 hS(t, ρ)i = dr Pr √ ln , ρ 2τ    Z 1 ρ0 (r)eτ hM(t, ρ)i = dr ρ0 (r) Pr √ ln , ρ 2τ Z



(11.50)

where probability integral Pr(z) is defined as Eq. (4.20), page 94. Taking into account the asymptotic representation of function Pr(z) (4.23), page 95, we obtain that, for

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287

τ  1, the average area of regions where concentration exceeds level ρ decreases in time according to the law Z p 1 hS(t, ρ)i ≈ √ e−τ/4 dr ρ0 (r), (11.51) πτρ while the average tracer mass within these regions r Z p ρ −τ/4 hM(t, ρ)i ≈ M0 − e dr ρ0 (r) πτ

(11.52)

monotonously tends to the total mass Z M0 = drρ0 (r). This is an additional evidence in favor of the above conclusion that tracer particles tend to join in clusters, i.e., in compact regions of enhanced concentration surrounded with rarefied regions. We illustrate the dynamics of cluster formation with the example of the initially uniform distribution of buoyant tracer over the plane, in which case ρ0 (r) = ρ0 = const. In this case, the average specific area (per unit surface area) of regions within which ρ(r, t) > ρ is s(t, ρ|ρ0 ) =

Z∞ ρ

ln ρ0 e−τ /ρ de ρ P(t; ρ e|ρ0 ) = Pr √ 2τ


! ,

(11.53)

where P(t; ρ) is the solution to Eq. (11.44) independent of r (i.e., function (11.47