Lectures on dynamics of stochastic systems

Front cover......Page 1
Lectures on Dynamics of Stochastic Systems......Page 4
Copyright page......Page 5
Table of contents......Page 6
Preface......Page 10
Introduction......Page 12
Part I: Dynamical Description of Stochastic Systems......Page 16
1.1. Ordinary Differential Equations: Initial-Value Problems......Page 18
1.2. Boundary-Value Problems for Linear Ordinary Differential Equations (Plane Waves in Layered Media)......Page 35
1.3. Partial Differential Equations......Page 39
Problem......Page 65
2.1. Functional Representation of Problem Solution......Page 68
2.2. Solution Dependence on Problem\'s Parameters......Page 75
Problems......Page 80
3.1. Ordinary Differential Equations......Page 84
3.2. First-Order Partial Differential Equations......Page 87
3.3. Higher-Order Partial Differential Equations......Page 95
Problems......Page 100
Part II: Statistical Description of Stochastic Systems......Page 102
4.1. Random Quantities and their Characteristics......Page 104
4.2. Random Processes, Fields, and their Characteristics......Page 110
4.3. Markovian Processes......Page 130
Problems......Page 134
5.1. General Remarks......Page 138
5.2. Gaussian Process......Page 140
5.3. Poisson\'s Process......Page 142
5.4. Telegrapher\'s Random Process......Page 143
5.5. Delta-Correlated Random Processes......Page 145
Problems......Page 150
6.1. Ordinary Differential Equations......Page 156
6.2. Completely Solvable Stochastic Dynamic Systems......Page 159
6.3. Delta-Correlated Fields and Processes......Page 175
Problems......Page 181
Lecture 7. Stochastic Equations with the Markovian Fluctuations of Parameters......Page 198
7.1. Telegrapher\'s Processes......Page 199
7.2. Gaussian Markovian Processes......Page 202
Problems......Page 203
8.1. The Fokker–Planck Equation......Page 206
8.2. Transition Probability Distributions......Page 209
8.3. The Simplest Markovian Random Processes......Page 211
8.4. Applicability Range of the Fokker–Planck Equation......Page 226
8.5. Causal Integral Equations......Page 230
8.6. Diffusion Approximation......Page 233
Problems......Page 235
9.1. Integral Transformations......Page 244
9.2. Steady-State Solutions of the Fokker–Planck Equation......Page 245
9.3. Boundary-Value Problems for the Fokker–Planck Equation (Hopping Phenomenon)......Page 257
9.4. Method of Fast Oscillation Averaging......Page 260
Problems......Page 262
Lecture 10. Some Other Approximate Approaches to the Problems of Statistical Hydrodynamics......Page 268
10.1. Quasi-Elastic Properties of Isotropic and Stationary Noncompressible Turbulent Media......Page 269
10.2. Sound Radiation by Vortex Motions......Page 273
Part III: Examples of Coherent Phenomena in\nStochastic Dynamic Systems......Page 284
11.1. General Remarks......Page 286
11.2. Particle Diffusion in Random Velocity Field......Page 291
11.3. Probabilistic Description of Density Field in Random Velocity Field......Page 299
11.4. Probabilistic Description of Magnetic Field and Magnetic Energy in Random Velocity Field......Page 306
11.5. Integral One-Point Statistical Characteristics of Passive Vector Fields......Page 313
Problems......Page 334
12.1. General Remarks......Page 340
12.2. Statistics of Scattered Field at Layer Boundaries......Page 345
12.3. Statistical Theory of Radiative Transfer......Page 354
12.4. Numerical Simulation......Page 365
Problems......Page 367
13.1. Input Stochastic Equations and Their Implications......Page 370
13.2. Wavefield Amplitude–Phase Fluctuations. Rytov\'s Smooth Perturbation Method......Page 376
13.3. Method of Path Integral......Page 382
13.4 Elements of Statistical Topography of Random Intensity Field......Page 396
Problems......Page 403
References......Page 408