Quantitative Methods for Modeling Properties of Random Media

New York University, New York NY

Investigators

Abstract

Many physical systems are best understood as the collective behavior of many smaller irregularities. An example is a cloud of gas comprised of a huge number of individual particles colliding with each other. On a larger (macroscopic) scale, this very seemingly complicated system may have a much simpler, but still very precise, description (the ideal gas law, for instance) due to the effects of the incalculable number of smaller scale interactions "averaging out." Other examples include the modeling of water moving through a porous rock, or the physical properties of composite materials. The derivation of large-scale, "averaged" laws from complicated small-scale irregularities and interactions lies in the realm of statistical physics. The goal of this project is to develop mathematical tools for a rigorous understanding of such problems under the assumption that the small-scale irregularities are occurring randomly. We would like to answer such questions as: How can the large-scale physical law, including the proportionality constants, be accurately predicted from the behavior of the small-scale interactions? What is the error in the large-scale law, in other words, what length scale is large enough that we observe the averaged behavior rather than seeing the complicated random fluctuations? Such questions are not just important to statistical physics and probability, but they also have wider applications, such as to the design of optimal composite materials or the construction of geothermal power plants. Partial differential equations are used to model many important physical systems, and the focus of the project is to understand the large-scale behavior of solutions to such equations under the assumption that the coefficients exhibit small-scale random oscillations. Obtaining an "averaged" partial differential equation that describes the large-scale behavior of the original, more complicated equation is called "homogenization" in the mathematical literature. If randomness is incorporated in the model, it is usually referred to as "stochastic homogenization." In recent years, researchers have developed a rather complete quantitative theory of stochastic homogenization for the simplest situation: uniformly elliptic equations. These equations model electrostatic properties of composite materials, for instance. The present project aims to go further by obtaining a quantitative theory of homogenization for "degenerate" equations as well as equations in porous media. Particular goals include rigorously justifying the Darcy-Brinkman law for fluid flow in a porous rock with quantitative error bounds; deriving the effective viscosity of a dilute suspension of particles in a fluid; obtaining bounds for the diffusivity of a particle moving randomly on a percolation cluster as well as more precise estimates for the long-time behavior (intermediate asymptotics) of a diffusion in a random medium; the analysis of boundary layers in periodic and stochastic homogenization; and obtaining homogenization results for shape optimization problems. For each of these problems, the goal is to develop a quantitative mathematical theory that provides explicit error bounds and robust analytic techniques.