Mathematics of Random Media

by Kohler, Werner E.; White, Benjamin S.

• ISBN13: 9780821811337
• ISBN10: 0821811339
• Format: Paperback
• Copyright: 1991-06-01
• Publisher: Amer Mathematical Society

Summary

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 propogating 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.

Table of Contents

The contact process, 1974-1989

Is the contact process dead?

Limiting behavior of a one-dimensional system with long range interactions

Topics in percolation

The lace expansion and the upper critical dimension for percolation

Mandelbrot percolation in two and three dimensions

Intersection probabilities for random walks

A summary of results about cyclic cellular automata

Hydrodynamical behavior of asymmetric attractive particle systems, one example: one-dimensional nearest neighbors asymmetric simple exclusion process

Statistical mechanics applied to random-media problems

A control formulation for constrained Markov processes

Diffusion approximation for a class of transport processes with physical reflection boundary conditions

Random perturbations of dynamical systems: a new approach

Partial differential systems with a small parameter and diffusion process with a discrete component

Asymptotics for the solution of the exterior Dirichlet problem for second-order elliptic operators with small first-order perturbations

Asymptotic behavior of invariant density of diffusion process with small fluctuation

Lyapunov exponents for white and real noise driven two-dimensional systems

Asymptotic methods for Markov jump processes

Uniform solution of Kramers' problem by a direct approach

On random differential delay equations with a small parameter

Homogenization error estimates for random elliptic operators

The effective conductivity of composites

On a general concept in optimal material layout

Microstructure and effective properties of random media

Classical transport in quasiperiodic media

Tuning up a quasicrystal

Localization estimates for off-diagonal disorder

Order statistics and renormalization

Wave and particle propagation through one-dimensional and quasi-one-dimensional random media

Water waves in shallow channels of randomly varying depth

Approximations by delta-correlated random processes and diffusive approximation in stochastic problems

Some aspects of wave propagation theory in layer random media

Reflection of waves by randomly layered media

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