# Dynamics of Stochastic Systems

**By**

- Valery Klyatskin

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.

**Audience**

Researchers in physics (fluid dynamics, optics, acoustics, radiophysics), geosciences (ocean, atmosphere physics), applied mathematics (stochastic equations), applications (coherent phenomena). Senior and postgraduate students in different areas of physics, engineering and applied mathematics.

Paperback, 212 Pages

Published: March 2005

Imprint: Elsevier

ISBN: 978-0-444-51796-8

## Contents

- Contents Preface Introduction I Dynamical description of stochastic systems 1 Examples, basic problems, peculiar features of solutions 1.1 Ordinary differential equations: initial value problems1.1.1 Particles under the random velocity field 1.1.2 Systems with blow-up singularities 1.1.3 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 Passive tracer in random velocity field 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 NavierâStokes equation: random forces in hydrodynamic theory of turbulence 2 Solution dependence on problem type, medium parameters, and initial data 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 problemâs parameters 2.2.1 Solution dependence on initial data 2.2.2 Imbedding method for boundary-value problems Problems3 Indicator function and Liouville equation 42 3.1 Ordinary differential equations 3.2 First-order partial differential equations 3.2.1 Linear equations 3.2.2 Quasilinear equations3.2.3 General-form nonlinear equations 3.3 Higher-order partial differential equations 3.3.1 Parabolic equation of quasioptics 3.3.2 Random forces in hydrodynamic theory of turbulence Problems II Statistical description of stochastic systems4 Random quantities, processes and fields 4.1 Random quantities and their characteristics4.2 Random processes, fields, and their characteristics 4.2.1 General remarks 4.2.2 Statistical topography of random processes and fields4.2.3 Gaussian random process 4.2.4 Discontinuous random processes 4.3 Markovian processes 4.3.1 General properties4.3.2 Characteristic functional of the Markovian processProblems 5 Correlation splitting 5.1 General remarks 5.2 Gaussian process5.3 Poisson process 5.4 Telegrapherâs random process5.5 Delta-correlated random processes5.5.1 Asymptotic meaning of delta-correlated processes and fieldsProblems 6 General approaches to analyzing stochastic dynamic systems 6.1 Ordinary differential equations 6.2 Completely solvable stochastic dynamic systems 6.2.1 Ordinary differential equations 6.2.2 Partial differential equations 6.3 Delta-correlated fields and processes6.3.1 One-dimensional nonlinear differential equation6.3.2 Linear operator equation Problems 7 Stochastic equations with the Markovian fluctuations of parameters 7.1 Telegrapherâs processes 7.2 Gaussian Markovian processesProblems 8 Gaussian delta-correlated random field (ordinary differential equations)8.1 The Fokker-Planck equation 8.2 Transition probability distributions 8.3 Applicability range of the FokkerâPlanck equation 8.3.1 Langevin equation 8.3.2 Diffusion approximation Problems 9 Methods for solving and analyzing the Fokker-Planck equation 9.1 Wiener random process 9.2 Logarithmic-normal random process9.3 Integral transformations 9.4 Steady-state solutions of the FokkerâPlanck equation 9.4.1 One-dimensional nonlinear differential equation 9.4.2 Hamiltonian systems9.5 Boundary-value problems for the Fokker-Planck equation (transfer phenomena) 9.6 Method of fast oscillation averaging Problems 10 Gaussian delta-correlated random field (causal integral equations) ProblemsIII Examples of coherent phenomena in stochastic dynamic systems11 Passive tracer clustering and diffusion in random hydrodynamic flows11.1 Lagrangian description (particle diffusion) 11.1.1 One-point statistical characteristics 11.1.2 Two-point statistical characteristics 11.2 Diffusion of passive tracer concentration in random velocity field 11.3 Effect of molecular diffusion Problems 12 Wave localization in randomly layered media 12.1 Statistics of scattered field at layer boundaries 12.1.1 Reflection and transmission coefficients 12.1.2 Source inside the layer of a medium 12.1.3 Statistical energy localization 12.2 Statistical theory of radiative transfer 12.2.1 Normal wave incidence on the layer of random media 12.2.2 Plane wave source located in random medium 12.3 Numerical simulation Problems Bibliography Index