# Stochastic Equations through the Eye of the Physicist

## Basic Concepts, Exact Results and Asymptotic Approximations

**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 and III 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 IV 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), wave propagation in disordered 2D and 3D media.For the sake of reader I provide several appendixes (Part V) that give many technical mathematical details needed in the book.

**Audience**

Researches 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

Hardbound, 556 Pages

Published: May 2005

Imprint: Elsevier

ISBN: 978-0-444-51797-5

## 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 Particle under the random velocity field1.1.2 Particles under the random velocity field1.1.3 Particles under random forces1.1.4 Systems with the blow-up singularities1.1.5 Oscillator with randomly varying frequency (stochastic parametric resonance)1.2 Linear ordinary differential equations: boundary-value problems1.2.1 Plane waves in layered media: a wave incident on a medium layer1.2.2 Plane waves in layered media: the source inside the medium 1.2.3 Plane waves in layered media: the two-layer model1.3 First-order partial differential equations1.3.1 Linear first-order partial differential equations: passive tracer in random velocity field1.3.2 Quasilinear equations1.3.3 Boundary-value problems for nonlinear ordinary differential equations1.3.4 Nonlinear first-order partial differential equations1.4 Partial differential equations of higher orders1.4.1 Stationary problems for Maxwellâs equations1.4.2 The Helmholtz equation (boundary-value problem) and the parabolic equation of quasioptics (waves in randomly inhomogeneous media)1.4.3 The NavierâStokes equation: random forces in hydrodynamic theory of turbulence1.4.4 Equations of geophysical hydrodynamics1.5 Solution dependence on medium parameters and initial value1.5.1 Principle of dynamic causality1.5.2 Solution dependence on initial value2 Indicator function and Liouville equation 2.1 Ordinary differential equations2.2 First-order partial differential equations 2.2.1 Linear equations 2.2.2 Quasilinear equations2.2.3 General-form nonlinear equations2.3 Higher-order partial differential equations2.3.1 Parabolic equation of quasioptics2.3.2 Random forces in hydrodynamic theory of turbulenceII Stochastic equations 3 Random quantities, processes and fields 3.1 Random quantities and their characteristics3.2 Random processes, fields, and their characteristics3.2.1 General remarks3.2.2 Statistical topography of random processes and fields3.2.3 Gaussian random process3.2.4 Discontinuous random processes3.3 Markovian processes3.3.1 General properties3.3.2 Characteristic functional of the Markovian process4 Correlation splitting4.1 General remarks 4.2 Gaussian process4.3 Poisson process4.4 Telegrapherâs random process4.5 Generalized telegrapherâs random process4.6 General-form Markovian processes4.7 Delta-correlated random processes4.7.1 Asymptotic meaning of delta-correlated processes and fields5 General approaches to analyzing stochastic dynamic systems5.1 Ordinary differential equations5.2 Partial differential equations5.2.1 Passive tracer transfer in random field of velocities5.2.2 Parabolic equation of quasi-optics5.2.3 Random forces in the theory of hydrodynamic turbulence5.3 Stochastic integral equations (methods of quantum field theory in the dynamics of stochastic systems)5.3.1 Linear integral equations5.3.2 Nonlinear integral equations5.4 Completely solvable stochastic dynamic systems5.4.1 Ordinary differential equations5.4.2 Partial differential equations5.5 Delta-correlated fields and processes5.5.1 One-dimensional nonlinear differential equation5.5.2 Linear operator equation 5.5.3 Partial differential equations6 Stochastic equations with the Markovian fluctuations of parameters 6.1 Telegrapherâs processes6.1.1 System of linear operator equations6.1.2 One-dimension nonlinear differential equation6.1.3 Particle in the one-dimension potential field6.1.4 Ordinary differential equation of the n-th order6.1.5 Statistical interpretation of telegrapherâs equation6.2 Generalized telegrapherâs process6.2.1 Stochastic linear equation6.2.2 One-dimensional nonlinear differential equation6.2.3 Ordinal differential equation of the n-th order6.3 Gaussian Markovian processes6.3.1 Stochastic linear equation6.3.2 Ordinal differential equation of the n-th order6.3.3 The square of the Gaussian Markovian process6.4 Markovian processes with finite-dimensional phase space6.4.1 Two-state process6.5 Causal stochastic integral equations6.5.1 Telegrapherâs random process6.5.2 Generalized telegrapherâs random process6.5.3 Gaussian Markovian processIII Asymptotic and approximate methods for analyzing stochastic equations7 Gaussian random field delta-correlated in time (ordinary differential equations) 7.1 The FokkerâPlanck equation7.2 Transitional probability distributions7.3 Applicability range of the FokkerâPlanck equation7.3.1 Langevin equation8 Methods for solving and analyzing the Fokker-Planck equation 8.1 System of linear equations8.1.1 Wiener random process8.1.2 Logarithmic-normal random process8.2 Integral transformations8.3 Steady-state solutions of the FokkerâPlanck equation8.3.1 One-dimensional nonlinear differential equation8.3.2 Hamiltonian systems8.3.3 Systems of hydrodynamic type8.4 Boundary-value problems for the Fokker-Planck equation (transfer phenomena) 8.4.1 Transfer phenomena in regular systems8.4.2 Transfer phenomena in singular systems8.5 Asymptotic and approximate methods of solving the Fokker-Plank equation 8.5.1 Asymptotic expansion8.5.2 Method of cumulant expansions8.5.3 Method of fast oscillation averaging9 Gaussian delta-correlated random field (causal integral equations)9.1 Causal integral equation9.2 Statistical averaging10 Diffusion approximationIV Coherent phenomena in stochastic dynamic systems 11 Passive tracer clustering and diffusion in random hydrodynamic flows11.1 General remarks 11.2 Statistical description11.2.1 Lagrangian description (particle diffusion) 11.2.2 Eulerian description 11.3 Additional factors 11.3.1 Plane-parallel mean shear11.3.2 Effect of molecular diffusion 11.3.3 Consideration of finite temporal correlation radius 12 Wave localization in randomly layered media12.1 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 12.1.4 Diffusion approximation 12.2 Statistical description of a wave field in random medium 12.2.1 Normal wave incidence on the layer of random media12.2.2 Plane wave source located in random medium 12.2.3 Numerical simulation 12.3 Eigenvalue and eigenfunction statistics 12.3.1 General remarks 12.3.2 Statistical averaging 12.4 Multidimensional wave problems in layered random media12.4.1 Nonstationary problems 12.4.2 Point source in randomly layered medium 12.5 Two-layer model of the medium12.5.1 Formulation of boundary-value problems 12.5.2 Statistical description 13 Wave propagation in random inhomogeneous medium 13.1 Method of stochastic equation 13.1.1 Stochastic equations and their implication 13.1.2 Delta-correlated approximation of the media parameters 13.1.3 Conditions for the applicability of the delta-correlation approximation of the medium parameters fluctuations, and diffusion approximation for the wave field13.1.4 Wavefield amplitudeâphase fluctuations (the Smooth Perturbation Method) 13.2 Geometrical optics approximation in randomly inhomogeneous media13.2.1 Ray diffusion in random media (Lagrangian description) 13.2.2 Caustics formation in randomly inhomogeneous media 13.2.3 Wavefield amplitudeâphase fluctuations (Eulerian description) 13.3 Method of path integral13.3.1 Statistical description of wavefield13.3.2 Asymptotical analysis of intensity fluctuations of plane wave 13.3.3 Caustical structure of wavefield in random media14 Some problems of statistical hydrodynamics 14.1 Quasicompressible properties of isotropic and stationar noncompressible turbulent media14.2 Radiation of sound by vortex systems14.2.1 Sound radiation by vortex lines14.2.2 Sound radiation by vortex ringsV AppendixA Variation (functional) derivatives B Fundamental solutions of wave problems in empty and layered mediaB.1 The case of empty spaceB.2 The case of layered spaceC Imbedding method in boundary-value wave problems C.1 Boundary-value problems for ordinary differential equations C.2 Stationary boundary-value wave problems C.2.1 One-dimensional stationary boundary-value wave problemsC.2.2 Waves in periodically inhomogeneous media C.2.3 Boundary-value stationary nonlinear wave problem of self-actionC.2.4 Stationary multidimensional boundary-value problem C.3 One-dimensional nonstationary boundary-value wave problem C.3.1 Nonsteady mediumC.3.2 Steady medium C.3.3 One-dimensional nonlinear wave problemBibliography Index