# Mathematics of Random Media

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*Werner E. Kohler; Benjamin S. White*

In recent years, there has been remarkable growth in the mathematics
of random media. The field has deep scientific and technological roots, as
well as purely mathematical ones in the theory of stochastic processes. This
collection of papers by leading researchers provides an overview of this
rapidly developing field.

The papers were presented at the 1989 AMS-SIAM Summer Seminar in Applied
Mathematics, held at Virginia Polytechnic Institute and State University in
Blacksburg, Virginia. In addition to new results on stochastic differential
equations and Markov processes, fields whose elegant mathematical techniques
are of continuing value in application areas, the conference was organized
around four themes:

Systems of interacting particles are normally viewed in connection
with the fundamental problems of statistical mechanics, but have also been used
to model diverse phenomena such as computer architectures and the spread of
biological populations. Powerful mathematical techniques have been developed
for their analysis, and a number of important systems are now well
understood.

Random perturbations of dynamical systems have also been used
extensively as models in physics, chemistry, biology, and engineering. Among
the recent unifying mathematical developments is the theory of large
deviations, which enables the accurate calculation of the probabilities of rare
events. For these problems, approaches based on effective but formal
perturbation techniques parallel rigorous mathematical approaches from
probability theory and partial differential equations. The book includes
representative papers from forefront research of both types.

Effective medium theory, otherwise known as the mathematical theory
of homogenization, consists of techniques for predicting the macroscopic
properties of materials from an understanding of their microstructures. For
example, this theory is fundamental in the science of composites, where it is
used for theoretical determination of electrical and mechanical properties.
Furthermore, the inverse problem is potentially of great technological
importance in the design of composite materials which have been optimized for
some specific use.

Mathematical theories of the propagation of waves in random media
have been used to understand phenomena as diverse as the twinkling of stars,
the corruption of data in geophysical exploration, and the quantum mechanics of
disordered solids. Especially effective methods now exist for waves in
randomly stratified, one-dimensional media. A unifying theme is the
mathematical phenomenon of localization, which occurs when a wave propagating
into a random medium is attenuated exponentially with propagation distance,
with the attenuation caused solely by the mechanism of random multiple
scattering.

Because of the wide applicability of this field of research, this book would
appeal to mathematicians, scientists, and engineers in a wide variety of areas,
including probabilistic methods, the theory of disordered materials, systems of
interacting particles, the design of materials, and dynamical systems driven by
noise. In addition, graduate students and others will find this book useful as
an overview of current research in random media.