Dynamics of Stochastic Systems
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Table of contents
- Cover
- DYNAMICS OF STOCHASTIC SYSTEMSiii
- Copyright Pageiv
- Contents3
- Preface1
- Introduction6
- Part I: Dynamical description of stochastic systems9
- Chapter 1. Examples, basic problems, peculiar features of solutions10
- 1.1 Ordinary differential equations: initial value problems10
- 1.2 Boundary-value problems for linear ordinary differential equations (plane waves in layered media17
- 1.3 Partial differential equations20
- Chapter 2. Solution dependence on problem type, medium parameters, and initial data30
- 2.1 Functional representation of problem solution30
- 2.2 Solution dependence on problem's parameters35
- Problems39
- Chapter 3. Indicator function and Liouville equation42
- 3.1 Ordinary differential equations42
- 3.2 First-order partial differential equations43
- 3.3 Higher-order partial differential equations46
- Problems48
- Part II: Statistical description of stochastic systems49
- Chapter 4. Random quantities, processes and fields50
- 4.1 Random quantities and their characteristics50
- 4.2 Random processes, fields, and their characteristics54
- 4.3 Markovian processes66
- Problems68
- Chapter 5. Correlation splitting70
- 5.1 General remarks70
- 5.2 Gaussian process72
- 5.3 Poisson process73
- 5.4 Telegrapher's random process74
- 5.5 Delta-correlated random processes76
- Problems80
- Chapter 6. General approaches to analyzing stochastic dynamic systems84
- 6.1 Ordinary differential equations84
- 6.2 Completely solvable stochastic dynamic systems87
- 6.3 Delta-correlated fields and processes98
- Problems103
- Chapter 7. Stochastic equations with the Markovian fluctuations of parameters111
- 7.1 Telegrapher's processes112
- 7.2 Gaussian Markovian processes114
- Problems115
- Chapter 8. Gaussian delta-correlated random field (ordinary differential equations)118
- 8.1 The Fokker-Planck equation118
- 8.2 Transition probability distributions120
- 8.3 Applicability range of the Fokker-Planck equation122
- Problems127
- Chapter 9. Methods for solving and analyzing the Fokker-Planck equation134
- 9.1 Wiener random process134
- 9.2 Logarithmic-normal random process137
- 9.3 Integral transformations140
- 9.4 Steady-state solutions of the Fokker-Planck equation141
- 9.5 Boundary-value problems for the Fokker-Planck equation (transfer phenomena)144
- 9.6 Method of fast oscillation averaging147
- Problems148
- Chapter 10. Gaussian delta-correlated random field (causal integral equations)153
- Problems155
- Part III: Examples of coherent phenomena in stochastic dynamic systems156
- Chapter 11. Passive tracer clustering and diffusion in random hydrodynamic flows157
- 11.1 Lagrangian description (particle diffusion)160
- 11.2 Diffusion of passive tracer concentration in random velocity field164
- 11.3 Effect of molecular diffusion171
- Problems173
- Chapter 12. Wave localization in randomly layered media176
- 12.1 Statistics of scattered field at layer boundaries180
- 12.2 Statistical theory of radiative transfer187
- 12.3 Numerical simulation195
- Problems197
- Bibliography200
- Index204
Fluctuating parameters appear in a variety of physical systems and phenomena. They typically come either as random forces/sources, or advecting velocities, or media (material) parameters, like refraction index, conductivity, diffusivity, etc. The well known example of Brownian particle suspended in fluid and subjected to random molecular bombardment laid the foundation for modern stochastic calculus and statistical physics. Other important examples include turbulent transport and diffusion of particle-tracers (pollutants), or continuous densities (''oil slicks''), wave propagation and scattering in randomly inhomogeneous media, for instance light or sound propagating in the turbulent atmosphere.
Such models naturally render to statistical description, where the input parameters and solutions are expressed by random processes and fields.
The fundamental problem of stochastic dynamics is to identify the essential characteristics of system (its state and evolution), and relate those to the input parameters of the system and initial data.
This raises a host of challenging mathematical issues. One could rarely solve such systems exactly (or approximately) in a closed analytic form, and their solutions depend in a complicated implicit manner on the initial-boundary data, forcing and system's (media) parameters . In mathematical terms such solution becomes a complicated "nonlinear functional" of random fields and processes.
Part I gives mathematical formulation for the basic physical models of transport, diffusion, propagation and develops some analytic tools.
Part II sets up and applies the techniques of variational calculus and stochastic analysis, like Fokker-Plank equation to those models, to produce exact or approximate solutions, or in worst case numeric procedures. The exposition is motivated and demonstrated with numerous examples.
Part III takes up issues for the coherent phenomena in stochastic dynamical systems, described by ordinary and partial differential equations, like wave propagation in randomly layered media (localization), turbulent advection of passive tracers (clustering).
Each chapter is appended with problems the reader to solve by himself (herself), which will be a good training for independent investigations.
· This book is translation from Russian and is completed with new principal results of recent research.
· The book develops mathematical tools of stochastic analysis, and applies them to a wide range of physical models of particles, fluids, and waves.
· Accessible to a broad audience with general background in mathematical physics, but no special expertise in stochastic analysis, wave propagation or turbulence
Such models naturally render to statistical description, where the input parameters and solutions are expressed by random processes and fields.
The fundamental problem of stochastic dynamics is to identify the essential characteristics of system (its state and evolution), and relate those to the input parameters of the system and initial data.
This raises a host of challenging mathematical issues. One could rarely solve such systems exactly (or approximately) in a closed analytic form, and their solutions depend in a complicated implicit manner on the initial-boundary data, forcing and system's (media) parameters . In mathematical terms such solution becomes a complicated "nonlinear functional" of random fields and processes.
Part I gives mathematical formulation for the basic physical models of transport, diffusion, propagation and develops some analytic tools.
Part II sets up and applies the techniques of variational calculus and stochastic analysis, like Fokker-Plank equation to those models, to produce exact or approximate solutions, or in worst case numeric procedures. The exposition is motivated and demonstrated with numerous examples.
Part III takes up issues for the coherent phenomena in stochastic dynamical systems, described by ordinary and partial differential equations, like wave propagation in randomly layered media (localization), turbulent advection of passive tracers (clustering).
Each chapter is appended with problems the reader to solve by himself (herself), which will be a good training for independent investigations.
· This book is translation from Russian and is completed with new principal results of recent research.
· The book develops mathematical tools of stochastic analysis, and applies them to a wide range of physical models of particles, fluids, and waves.
· Accessible to a broad audience with general background in mathematical physics, but no special expertise in stochastic analysis, wave propagation or turbulence