By
Valery Klyatskin, 1988 Research Professor of Theoretical and Mathematical Physics, Russian Academy of Science;
1977 D. Sc. in Physical and Mathematical
Sciences, Acoustical Institute, Russian Academy of Science;
1968 Ph.D. in Physical and Mathematical Sciences, Institute of Atmospheric
Physics Russian Academy of Science;
1964 M.Sc. in Theoretical Physics, Moscow Institute of Physics and Technology (FIZTEX)., Russian
Academy of Science, Russia
Description
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.
