Independent Particles in a Dynamical Random Environment

TL;DR

This paper analyzes the behavior of independent particles in a dynamic random environment, characterizing invariant measures, studying correlations, and proving convergence to equilibrium in low dimensions, with insights into current fluctuations.

Contribution

It introduces a comprehensive characterization of invariant measures for particles in a dynamical environment and analyzes their convergence and fluctuation properties.

Findings

Invariant measures are mixtures of inhomogeneous Poisson product measures.

Convergence to equilibrium is proven in one and two dimensions.

Current fluctuations match classical independent particle results.

Abstract

We study the motion of independent particles in a dynamical random environment on the integer lattice. The environment has a product distribution. For the multidimensional case, we characterize the class of spatially ergodic invariant measures. These invariant distributions are mixtures of inhomogeneous Poisson product measures that depend on the past of the environment. We also investigate the correlations in this measure. For dimensions one and two, we prove convergence to equilibrium from spatially ergodic initial distributions. In the one-dimensional situation we study fluctuations of the net current seen by an observer traveling at a deterministic speed. When this current is centered by its quenched mean its limit distributions are the same as for classical independent particles.