Lectures on Dynamics ofStochastic Systems

Edited by

Valery I. Klyatskin

Translated from Russian by

A. Vinogradov

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Contents

Preface ixIntroduction xi

Part I Dynamical Description of Stochastic Systems 1

1 Examples, Basic Problems, Peculiar Features of Solutions 3

1.1 Ordinary Differential Equations: Initial-Value Problems 31.1.1 Particles Under the Random Velocity Field 31.1.2 Particles Under Random Forces 81.1.3 The Hopping Phenomenon 101.1.4 Systems with Blow-Up Singularities 181.1.5 Oscillator with Randomly Varying Frequency

(Stochastic Parametric Resonance) 191.2 Boundary-Value Problems for Linear Ordinary Differential

Equations (Plane Waves in Layered Media) 201.3 Partial Differential Equations 24

1.3.1 Linear First-Order Partial Differential Equations 241.3.2 Quasilinear and Nonlinear First-Order Partial

Differential Equations 331.3.3 Parabolic Equation of Quasioptics (Waves in Randomly

Inhomogeneous Media) 391.3.4 Navier–Stokes Equation: Random Forces in Hydrodynamic

Theory of Turbulence 42Problem 50

2 Solution Dependence on Problem Type, Medium Parameters,and Initial Data 53

2.1 Functional Representation of Problem Solution 532.1.1 Variational (Functional) Derivatives 532.1.2 Principle of Dynamic Causality 59

2.2 Solution Dependence on Problem’s Parameters 602.2.1 Solution Dependence on Initial Data 602.2.2 Imbedding Method for Boundary-Value Problems 62

Problems 65

3 Indicator Function and Liouville Equation 69

3.1 Ordinary Differential Equations 693.2 First-Order Partial Differential Equations 72

3.2.1 Linear Equations 72

vi Contents

3.2.2 Quasilinear Equations 773.2.3 General-Form Nonlinear Equations 79

3.3 Higher-Order Partial Differential Equations 803.3.1 Parabolic Equation of Quasi-Optics 803.3.2 Random Forces in Hydrodynamic Theory of Turbulence 83

Problems 85

Part II Statistical Description of Stochastic Systems 87

4 Random Quantities, Processes, and Fields 89

4.1 Random Quantities and their Characteristics 894.2 Random Processes, Fields, and their Characteristics 95

4.2.1 General Remarks 954.2.2 Statistical Topography of Random Processes

and Fields 994.2.3 Gaussian Random Process 1024.2.4 Gaussian Vector Random Field 1054.2.5 Logarithmically Normal Random Process 1084.2.6 Discontinuous Random Processes 110

4.3 Markovian Processes 1154.3.1 General Properties 1154.3.2 Characteristic Functional of the Markovian Process 117

Problems 119

5 Correlation Splitting 123

5.1 General Remarks 1235.2 Gaussian Process 1255.3 Poisson’s Process 1275.4 Telegrapher’s Random Process 1285.5 Delta-Correlated Random Processes 130

5.5.1 Asymptotic Meaning of Delta-Correlated Processesand Fields 133

Problems 135

6 General Approaches to Analyzing Stochastic Systems 141

6.1 Ordinary Differential Equations 1416.2 Completely Solvable Stochastic Dynamic Systems 144

6.2.1 Ordinary Differential Equations 1446.2.2 Partial Differential Equations 158

6.3 Delta-Correlated Fields and Processes 1606.3.1 One-Dimensional Nonlinear Differential Equation 1626.3.2 Linear Operator Equation 165

Problems 166

7 Stochastic Equations with the Markovian Fluctuations ofParameters 183

7.1 Telegrapher’s Processes 184

Contents vii

7.2 Gaussian Markovian Processes 187Problems 188

8 Approximations of Gaussian Random Field Delta-Correlatedin Time 191

8.1 The Fokker–Planck Equation 1918.2 Transition Probability Distributions 1948.3 The Simplest Markovian Random Processes 196

8.3.1 Wiener Random Process 1978.3.2 Wiener Random Process with Shear 1978.3.3 Logarithmic-Normal Random Process 200

8.4 Applicability Range of the Fokker–Planck Equation 2118.4.1 Langevin Equation 211

8.5 Causal Integral Equations 2158.6 Diffusion Approximation 218Problems 220

9 Methods for Solving and Analyzing the Fokker–Planck Equation 229

9.1 Integral Transformations 2299.2 Steady-State Solutions of the Fokker–Planck Equation 230

9.2.1 One-Dimensional Nonlinear Differential Equation 2319.2.2 Hamiltonian Systems 2329.2.3 Systems of Hydrodynamic Type 234

9.3 Boundary-Value Problems for the Fokker–Planck Equation(Hopping Phenomenon) 242

9.4 Method of Fast Oscillation Averaging 245Problems 247

10 Some Other Approximate Approaches to the Problems of StatisticalHydrodynamics 253

10.1 Quasi-Elastic Properties of Isotropic and StationaryNoncompressible Turbulent Media 254

10.2 Sound Radiation by Vortex Motions 25810.2.1 Sound Radiation by Vortex Lines 26010.2.2 Sound Radiation by Vortex Rings 263

Part III Examples of Coherent Phenomena in StochasticDynamic Systems 269

11 Passive Tracer Clustering and Diffusion in Random Hydrodynamicand Magnetohydrodynamic Flows 271

11.1 General Remarks 27111.2 Particle Diffusion in Random Velocity Field 276

11.2.1 One-Point Statistical Characteristics 27611.2.2 Two-Point Statistical Characteristics 281

11.3 Probabilistic Description of Density Field in Random Velocity Field 284

viii Contents

11.4 Probabilistic Description of Magnetic Field and Magnetic Energyin Random Velocity Field 291

11.5 Integral One-Point Statistical Characteristics of PassiveVector Fields 29811.5.1 Spatial Correlation Function of Density Field 29911.5.2 Spatial Correlation Tensor of Density Field Gradient

and Dissipation of Density Field Variance 30211.5.3 Spatial Correlation Function of Magnetic Field 31011.5.4 On the Magnetic Field Helicity 31311.5.5 On the Magnetic Field Dissipation 315

Problems 319

12 Wave Localization in Randomly Layered Media 325

12.1 General Remarks 32512.1.1 Wave Incidence on an Inhomogeneous Layer 32512.1.2 Source Inside an Inhomogeneous Layer 327

12.2 Statistics of Scattered Field at Layer Boundaries 33012.2.1 Reflection and Transmission Coefficients 33012.2.2 Source Inside the Layer of a Medium 33712.2.3 Statistical Energy Localization 338

12.3 Statistical Theory of Radiative Transfer 33912.3.1 Normal Wave Incidence on the Layer of Random Media 34012.3.2 Plane Wave Source Located in Random Medium 347

12.4 Numerical Simulation 350Problems 352

13 Caustic Structure of Wavefield in Random Media 355

13.1 Input Stochastic Equations and Their Implications 35513.2 Wavefield Amplitude–Phase Fluctuations. Rytov’s Smooth

Perturbation Method 36113.2.1 Random Phase Screen (1x� x) 36513.2.2 Continuous Medium (1x = x) 366

13.3 Method of Path Integral 36713.3.1 Asymptotic Analysis of Plane Wave Intensity Fluctuations 371

13.4 Elements of Statistical Topography of Random Intensity Field 38113.4.1 Weak Intensity Fluctuations 38313.4.2 Strong Intensity Fluctuations 386

Problems 388

References 393

Preface

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

R. Kipling

This book is a revised and more comprehensive re-edition of my book Dynamics ofStochastic Systems, Elsevier, Amsterdam, 2005.

Writing this book, I sourced from the series of lectures that I gave to scientificassociates at the Institute of Calculus Mathematics, Russian Academy of Sciences. Inthe book, I use the functional approach to uniformly formulate general methods ofstatistical description and analysis of dynamic systems. These are described in termsof different types of equations with fluctuating parameters, such as ordinary differen-tial equations, partial differential equations, boundary-value problems, and integralequations. Asymptotic methods of analyzing stochastic dynamic systems – the delta-correlated random process (field) approximation and the diffusion approximation – arealso considered. General ideas are illustrated using examples of coherent phenomenain stochastic dynamic systems, such as clustering of particles and passive tracer in ran-dom velocity field, dynamic localization of plane waves in randomly layered mediaand caustic structure of wavefield in random media.

Working at this edition, I tried to take into account opinions and wishes of readersabout both the style of the text and the choice of specific problems. Various mistakesand misprints have been corrected.

The book is aimed at scientists dealing with stochastic dynamic systems in differ-ent areas, such as acoustics, hydrodynamics, magnetohydrodynamics, radiophysics,theoretical and mathematical physics, and applied mathematics; it may also be usefulfor 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 parametersaffecting system behavior. More detailed formulations of these problems and relevantstatistical analysis can be found in other parts of the book.

The second part is devoted to the general theory of statistical analysis of dynamicsystems 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 delta-correlated random process (field) approximation and diffusion approximation.

The third part deals with the analysis of specific physical problems associated withcoherent phenomena. These are clustering and diffusion of particles and passive tracer(density and magnetic fields) in a random velocity field, dynamic localization of planewaves propagating in layered random media and caustic structure of wavefield in ran-dom multidimensional media. These phenomena are described by ordinary differentialequations and partial differential equations.

The material is supplemented with sections concerning dynamical and statisticaldescriptions of simplest hydrodynamic-type systems, the relationship between con-ventional methods (based on the Lyapunov stability analysis of stochastic dynamicsystems), methods of statistical topography and statistical description of magnetic fieldgeneration 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. KlyatskinMoscow, Russia

Introduction

Different areas of physics pose statistical problems in ever greater numbers. Apartfrom issues traditionally obtained in statistical physics, many applications call forincluding fluctuation effects. While fluctuations may stem from different sources (suchas thermal noise, instability, and turbulence), methods used to treat them are very sim-ilar. In many cases, the statistical nature of fluctuations may be deemed known (eitherfrom physical considerations or from problem formulation) and the physical processesmay be modeled by differential, integro-differential or integral equations.

We will consider a statistical theory of dynamic and wave systems with fluctuatingparameters. These systems can be described by ordinary differential equations, partialdifferential equations, integro-differential equations and integral equations. A popularway to solve such systems is by obtaining a closed system of equations for statisticalcharacteristics, to study their solutions as comprehensively as possible.

We note that wave problems are often boundary-value problems. When this is thecase, one may resort to the imbedding method to reformulate the equations at hand toinitial-value problems, thus considerably simplifying the statistical analysis.

The purpose of this book is to demonstrate how different physical problems descri-bed by stochastic equations can be solved on the basis of a general approach.

In stochastic problems with fluctuating parameters, the variables are functions. Itwould be natural therefore to resort to functional methods for their analysis. We willuse a functional method devised by Novikov [1] for Gaussian fluctuations of parame-ters in a turbulence theory and developed by the author of this book [2] for the generalcase of dynamic systems and fluctuating parameters of an arbitrary nature.

However, only a few dynamic systems lend themselves to analysis yielding solu-tions in a general form. It proved to be more efficient to use an asymptotic methodwhere the statistical characteristics of dynamic problem solutions are expanded inpowers of a small parameter. This is essentially a ratio of the random impact’s cor-relation time to the time of observation or to another characteristic time scale of theproblem (in some cases, these may be spatial rather than temporal scales). This methodis essentially a generalization of the theory of Brownian motion. It is termed the delta-correlated random process (field) approximation.

For dynamic systems described by ordinary differential stochastic equations withGaussian fluctuations of parameters, this method leads to a Markovian problem solv-ing model, and the respective equation for transition probability density has the formof the Fokker–Planck equation. In this book, we will consider in depth the methodsof analysis available for this equation and its boundary conditions. We will analyzesolutions and validity conditions by way of integral transformations. In more com-plicated 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 derivativesof the solution’s characteristic functional. For dynamic problems with non-Gaussianfluctuations of parameters, this method also yields Markovian type solutions. Underthe circumstances, the probability density of respective dynamic stochastic equationssatisfies a closed operator equation.

In physical investigations, Fokker–Planck and similar equations are usually set upfrom rule-of-thumb considerations, and dynamic equations are invoked only to cal-culate the coefficients of these equations. This approach is inconsistent, generallyspeaking. Indeed, the statistical problem is completely defined by dynamic equationsand assumptions on the statistics of random impacts. For example, the Fokker–Planckequation must be a logical sequence of the dynamic equations and some assumptionson the character of random impacts. It is clear that not all problems lend themselvesfor reduction to a Fokker–Planck equation. The functional approach allows one toderive a Fokker–Planck equation from the problem’s dynamic equation along with itsapplicability conditions. In terms of formal mathematics, our approach corresponds tothat of R.L. Stratonovich (see, e.g., [3]).

For a certain class of Markovian random process (telegrapher’s processes, Gaussianprocess and the like), the developed functional approach also yields closed equationsfor the solution probability density with allowance for a finite correlation time of ran-dom interactions.

For processes with Gaussian fluctuations of parameters, one may construct abetter physical approximation than the delta-correlated random process (field)approximation – the diffusion approximation that allows for finiteness of correlationtime radius. In this approximation, the solution is Markovian and its applicability con-dition has transparent physical meaning, namely, the statistical effects should be smallwithin the correlation time of fluctuating parameters. This book treats these issues indepth from a general standpoint and for some specific physical applications.

Recently, the interest of both theoreticians and experimenters has been attracted tothe relation of the behavior of average statistical characteristics of a problem solutionwith the behavior of the solution in certain happenings (realizations). This is espe-cially important for geophysical problems related to the atmosphere and ocean where,generally speaking, a respective averaging ensemble is absent and experimenters, as arule, deal with individual observations.

Seeking solutions to dynamic problems for these specific realizations of mediumparameters is almost hopeless due to the extreme mathematical complexity of theseproblems. At the same time, researchers are interested in the main characteristics ofthese phenomena without much need to know specific details. Therefore, the idea ofusing a well-developed approach to random processes and fields based on ensem-ble averages rather than separate observations proved to be very fruitful. By way ofexample, almost all physical problems of the atmosphere and ocean to some extent aretreated by statistical analysis.

Randomness in medium parameters gives rise to a stochastic behavior of physicalfields. 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

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

saddles. Figure 0.1 shows examples of realizations of two random fields characterizedby different statistical structures.

Common methods of statistical averaging (computing mean-type averages –〈ρ (R, t)〉, space-time correlation function –

⟨ρ (R, t) ρ

(R′, t′

)⟩etc., where 〈· · · 〉

implies averaging over an ensemble of random parameter samples) smooth the quali-tative features of specific samples. Frequently, these statistical characteristics havenothing in common with the behavior of specific samples, and at first glance mayeven seem to be at variance with them. For example, the statistical averaging overall observations makes the field of average concentration of a passive tracer in arandom velocity field ever more smooth, whereas each realization sample tends tobe more irregular in space due to the mixture of areas with substantially differentconcentrations.

Thus, these types of statistical average usually characterize ‘global’ space–timedimensions of the area with stochastic processes, but give no details about the processbehavior inside the area. For this case, details heavily depend on the velocity fieldpattern, specifically, on whether it is divergent or solenoidal. Thus, the first case willshow with total probability that clusters will be formed, i.e. compact areas of enhancedconcentration of tracer surrounded by vast areas of low-concentration tracer. In the

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xiv Introduction

circumstances, all statistical moments of the distance between the particles will growwith time exponentially; that is, on average, a statistical recession of particles will takeplace [4].

In a similar way, in the case of waves propagating in random media, an exponentialspread 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 thedynamic localization of plane waves in layered randomly inhomogeneous media. Inthis phenomenon, the wave field intensity exponentially decays inward to the mediumwith the probability equal to unity when the wave is incident on the half-space of sucha medium, while all statistical moments increase exponentially with distance from theboundary of the medium [5].

These physical processes and phenomena occurring with the probability equal tounity will be referred to as coherent processes and phenomena [6]. This type of sta-tistical coherence may be viewed as some organization of the complex dynamic sys-tem, and retrieval of its statistically stable characteristics is similar to the concept ofcoherence as self-organization of multicomponent systems that evolve from the ran-dom interactions of their elements [7]. In the general case, it is rather difficult to saywhether 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 fluctu-ating parameters, this may be handled by analytical means. In other cases, one mayverify this by performing numerical modeling experiments or analyzing experimentalfindings.

The complete statistic (say, the whole body of all n-point space-time moment func-tions), would undoubtedly contain all the information about the investigated dynamicsystem. In practice, however, one may succeed only in studying the simplest statisticalcharacteristics associated mainly with simultaneous and one-point probability distri-butions. It would be reasonable to ask how with these statistics on hand one wouldlook into the quantitative and qualitative behavior of some system happenings?

This question is answered by methods of statistical topography [8]. These meth-ods were highlighted by Ziman [9], who seems to have coined this term. Statisticaltopography yields a different philosophy of statistical analysis of dynamic stochasticsystems, which may prove useful for experimenters planning a statistical processingof experimental data. These issues are treated in depth in this book.

More details about the material of this book and more exhaustive references can befound in the textbook quoted in reference [2] and recent reviews [6, 10–16].