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
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I Dynamical description of stochastic systems
1 Examples, basic problems, peculiar features of solutions
2 Indicator function and Liouville equation
3 Random quantities, processes and fields
4 Correlation splitting
5 General approaches to analyzing stochastic dynamic systems
6 Stochastic equations with the Markovian fluctuations of parameters
7 Gaussian random field delta-correlated in time (ordinary differential equations)
8 Methods for solving and analyzing the Fokker-Planck equation
9 Gaussian delta-correlated random field (causal integral equations)
10 Diffusion approximation
11 Passive tracer clustering and diffusion in random hydrodynamic flows
12 Wave localization in randomly layered media
13 Wave propagation in random inhomogeneous medium
14 Some problems of statistical hydrodynamics
A Variation (functional) derivatives
B Fundamental solutions of wave problems in empty and layered media
C Imbedding method in boundary-value wave problems 380
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.