Klyatskin V.I. Lectures on Dynamics of Stochastic Systems
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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
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BOSTON
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HEIDELBERG
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LONDON
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NEW YORK OXFORD
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PARIS
SAN DIEGO
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SAN FRANCISCO
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SINGAPORE
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SYDNEY
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TOKYO
Elsevier
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First edition 2011
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Contents
Preface ix
Introduction 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 3
1.1.1 Particles Under the Random Velocity Field 3
1.1.2 Particles Under Random Forces 8
1.1.3 The Hopping Phenomenon 10
1.1.4 Systems with Blow-Up Singularities 18
1.1.5 Oscillator with Randomly Varying Frequency
(Stochastic Parametric Resonance) 19
1.2 Boundary-Value Problems for Linear Ordinary Differential
Equations (Plane Waves in Layered Media) 20
1.3 Partial Differential Equations 24
1.3.1 Linear First-Order Partial Differential Equations 24
1.3.2 Quasilinear and Nonlinear First-Order Partial
Differential Equations 33
1.3.3 Parabolic Equation of Quasioptics (Waves in Randomly
Inhomogeneous Media) 39
1.3.4 Navier–Stokes Equation: Random Forces in Hydrodynamic
Theory of Turbulence 42
Problem 50
2 Solution Dependence on Problem Type, Medium Parameters,
and Initial Data 53
2.1 Functional Representation of Problem Solution 53
2.1.1 Variational (Functional) Derivatives 53
2.1.2 Principle of Dynamic Causality 59
2.2 Solution Dependence on Problem’s Parameters 60
2.2.1 Solution Dependence on Initial Data 60
2.2.2 Imbedding Method for Boundary-Value Problems 62
Problems 65
3 Indicator Function and Liouville Equation 69
3.1 Ordinary Differential Equations 69
3.2 First-Order Partial Differential Equations 72
3.2.1 Linear Equations 72
vi Contents
3.2.2 Quasilinear Equations 77
3.2.3 General-Form Nonlinear Equations 79
3.3 Higher-Order Partial Differential Equations 80
3.3.1 Parabolic Equation of Quasi-Optics 80
3.3.2 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 89
4.2 Random Processes, Fields, and their Characteristics 95
4.2.1 General Remarks 95
4.2.2 Statistical Topography of Random Processes
and Fields 99
4.2.3 Gaussian Random Process 102
4.2.4 Gaussian Vector Random Field 105
4.2.5 Logarithmically Normal Random Process 108
4.2.6 Discontinuous Random Processes 110
4.3 Markovian Processes 115
4.3.1 General Properties 115
4.3.2 Characteristic Functional of the Markovian Process 117
Problems 119
5 Correlation Splitting 123
5.1 General Remarks 123
5.2 Gaussian Process 125
5.3 Poisson’s Process 127
5.4 Telegrapher’s Random Process 128
5.5 Delta-Correlated Random Processes 130
5.5.1 Asymptotic Meaning of Delta-Correlated Processes
and Fields 133
Problems 135
6 General Approaches to Analyzing Stochastic Systems 141
6.1 Ordinary Differential Equations 141
6.2 Completely Solvable Stochastic Dynamic Systems 144
6.2.1 Ordinary Differential Equations 144
6.2.2 Partial Differential Equations 158
6.3 Delta-Correlated Fields and Processes 160
6.3.1 One-Dimensional Nonlinear Differential Equation 162
6.3.2 Linear Operator Equation 165
Problems 166
7 Stochastic Equations with the Markovian Fluctuations of
Parameters 183
7.1 Telegrapher’s Processes 184
Contents vii
7.2 Gaussian Markovian Processes 187
Problems 188
8 Approximations of Gaussian Random Field Delta-Correlated
in Time 191
8.1 The Fokker–Planck Equation 191
8.2 Transition Probability Distributions 194
8.3 The Simplest Markovian Random Processes 196
8.3.1 Wiener Random Process 197
8.3.2 Wiener Random Process with Shear 197
8.3.3 Logarithmic-Normal Random Process 200
8.4 Applicability Range of the Fokker–Planck Equation 211
8.4.1 Langevin Equation 211
8.5 Causal Integral Equations 215
8.6 Diffusion Approximation 218
Problems 220
9 Methods for Solving and Analyzing the Fokker–Planck Equation 229
9.1 Integral Transformations 229
9.2 Steady-State Solutions of the Fokker–Planck Equation 230
9.2.1 One-Dimensional Nonlinear Differential Equation 231
9.2.2 Hamiltonian Systems 232
9.2.3 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 245
Problems 247
10 Some Other Approximate Approaches to the Problems of Statistical
Hydrodynamics 253
10.1 Quasi-Elastic Properties of Isotropic and Stationary
Noncompressible Turbulent Media 254
10.2 Sound Radiation by Vortex Motions 258
10.2.1 Sound Radiation by Vortex Lines 260
10.2.2 Sound Radiation by Vortex Rings 263
Part III Examples of Coherent Phenomena in Stochastic
Dynamic Systems 269
11 Passive Tracer Clustering and Diffusion in Random Hydrodynamic
and Magnetohydrodynamic Flows 271
11.1 General Remarks 271
11.2 Particle Diffusion in Random Velocity Field 276
11.2.1 One-Point Statistical Characteristics 276
11.2.2 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 Energy
in Random Velocity Field 291
11.5 Integral One-Point Statistical Characteristics of Passive
Vector Fields 298
11.5.1 Spatial Correlation Function of Density Field 299
11.5.2 Spatial Correlation Tensor of Density Field Gradient
and Dissipation of Density Field Variance 302
11.5.3 Spatial Correlation Function of Magnetic Field 310
11.5.4 On the Magnetic Field Helicity 313
11.5.5 On the Magnetic Field Dissipation 315
Problems 319
12 Wave Localization in Randomly Layered Media 325
12.1 General Remarks 325
12.1.1 Wave Incidence on an Inhomogeneous Layer 325
12.1.2 Source Inside an Inhomogeneous Layer 327
12.2 Statistics of Scattered Field at Layer Boundaries 330
12.2.1 Reflection and Transmission Coefficients 330
12.2.2 Source Inside the Layer of a Medium 337
12.2.3 Statistical Energy Localization 338
12.3 Statistical Theory of Radiative Transfer 339
12.3.1 Normal Wave Incidence on the Layer of Random Media 340
12.3.2 Plane Wave Source Located in Random Medium 347
12.4 Numerical Simulation 350
Problems 352
13 Caustic Structure of Wavefield in Random Media 355
13.1 Input Stochastic Equations and Their Implications 355
13.2 Wavefield Amplitude–Phase Fluctuations. Rytov’s Smooth
Perturbation Method 361
13.2.1 Random Phase Screen (1x x) 365
13.2.2 Continuous Medium (1x = x) 366
13.3 Method of Path Integral 367
13.3.1 Asymptotic Analysis of Plane Wave Intensity Fluctuations 371
13.4 Elements of Statistical Topography of Random Intensity Field 381
13.4.1 Weak Intensity Fluctuations 383
13.4.2 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 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 differen-
tial equations, partial differential equations, boundary-value problems, and integral
equations. Asymptotic methods of analyzing stochastic dynamic systems – the delta-
correlated 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 ran-
dom 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 differ-
ent 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