Stochastic homogenization in amorphous media and applications to exclusion processes

TL;DR

This paper proves homogenization for random walks on amorphous media and applies it to derive hydrodynamic limits of exclusion processes, with implications for electron conduction in disordered solids.

Contribution

It extends two-scale convergence techniques to establish homogenization and hydrodynamic limits for models in amorphous media, including Mott variable range hopping.

Findings

Homogenization of Markov generators on amorphous media.

Hydrodynamic limit for exclusion processes with random walks.

Application to electron conduction in disordered solids.

Abstract

We consider random walks on marked simple point processes with symmetric jump rates and unbounded jump range. We prove homogenization properties of the associated Markov generators. As an application, we derive the hydrodynamic limit of the simple exclusion process given by multiple random walks as above, with hard-core interaction, on a marked Poisson point process. The above results cover Mott variable range hopping, which is a fundamental mechanism of phonon-induced electron conduction in amorphous solids as doped semiconductors. Our techniques, based on an extension of two-scale convergence, can be adapted to other models, as e.g. the random conductance model.