Dynamics of Stochastic Systems

Dynamics of Stochastic Systems

1st Edition - March 17, 2005

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  • Author: Valery Klyatskin
  • eBook ISBN: 9780080504858

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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.

Key Features

· 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


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.

Table of Contents

    • Preface
    • Introduction
    • Part I: Dynamical description of stochastic systems
      • Chapter 1: Examples, basic problems, peculiar features of solutions
        • 1.1 Ordinary differential equations: initial value problems
        • 1.2 Boundary-value problems for linear ordinary differential equations (plane waves in layered media)
        • 1.3 Partial differential equations
      • Chapter 2: Solution dependence on problem type, medium parameters, and initial data
        • 2.1 Functional representation of problem solution
        • 2.2 Solution dependence on problem’s parameters
    • Part II: Statistical description of stochastic systems
      • Chapter 3: Indicator function and Liouville equation
        • 3.1 Ordinary differential equations
        • 3.2 First-order partial differential equations
        • 3.3 Higher-order partial differential equations
      • Chapter 4: Random quantities, processes and fields
        • 4.1 Random quantities and their characteristics
        • 4.2 Random processes, fields, and their characteristics
        • 4.3 Markovian processes
        • Problems
      • Chapter 5: Correlation splitting
        • 5.1 General remarks
        • 5.2 Gaussian process
        • 5.3 Poisson process
        • 5.4 Telegrapher’s random process
        • 5.5 Delta-correlated random processes
        • Problems
      • Chapter 6: General approaches to analyzing stochastic dynamic systems
        • 6.1 Ordinary differential equations
        • 6.2 Completely solvable stochastic dynamic systems
        • 6.3 Delta-correlated fields and processes
        • Problems
      • Chapter 7: Stochastic equations with the Markovian fluctuations of parameters
        • 7.1 Telegrapher’s processes
        • 7.2 Gaussian Markovian processes
        • Problems
      • Chapter 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 Fokker–Planck equation
        • Problems
      • Chapter 9: Methods for solving and analyzing the Fokker-Planck equation
        • 9.1 Wiener random process
        • 9.2 Logarithmic-normal random process
        • 9.3 Integral transformations
        • 9.4 Steady-state solutions of the Fokker-Planck equation
        • 9.5 Boundary-value problems for the Fokker-Planck equation (transfer phenomena)
        • 9.6 Method of fast oscillation averaging
        • Problems
      • Chapter 10: Gaussian delta-correlated random field (causal integral equations)
        • Problems
    • Part III: Examples of coherent phenomena in stochastic dynamic systems
      • Chapter 11: Passive tracer clustering and diffusion in random hydrodynamic flows
        • 11.1 Lagrangian description (particle diffusion)
        • 11.2 Diffusion of passive tracer concentration in random velocity field
        • 11.3 Effect of molecular diffusion
        • Problems
      • Chapter 12: Wave localization in randomly layered media
        • 12.1 Statistics of scattered field at layer boundaries
        • 12.2 Statistical theory of radiative transfer
        • 12.3 Numerical simulation
        • Problems
    • Bibliography
    • Index

Product details

  • No. of pages: 212
  • Language: English
  • Copyright: © Elsevier Science 2005
  • Published: March 17, 2005
  • Imprint: Elsevier Science
  • eBook ISBN: 9780080504858

About the Author

Valery Klyatskin

Born in 1940 in Moscow, USSR, Valery I. Klyatskin received his secondary education at school in Tbilisi, Georgia, finishing in 1957. Seven years later he graduated from Moscow Institute of Physics and Technology (FIZTEX), whereupon he took up postgraduate studies at the Institute of Atmospheric Physics USSR Academy of Sciences, Moscow gaining the degree of Candidate of Physical and Mathematical Sciences (Ph.D) in 1968. He then continued at the Institute as a researcher, until 1978, when he was appointed as Head of the Wave Process Department at the Pacific Oceanological Institute of the USSR Academy of Sciences, based in Vladivostok. In 1992 Valery I. Klyatskin returned to Institute of Atmospheric Physics Russian Academy of Sciences, Moscow when he was appointed to his present position as Chief Scientist. At the same time he is Chief Scientific Consultant of Pacific Oceanological Institute Russian Academy of Sciences, Vladivostok. In 1977 he obtained a doctorate in Physical and Mathematical Sciences and in 1988 became Research Professor of Theoretical and Mathematical Physics, Russian Academy of Science.

Affiliations and Expertise

Russian Academy of Science, Russia

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