Improved field theoretical approach to noninteracting Brownian particles in a quenched random potential

TL;DR

This paper develops a dynamical field theory for noninteracting Brownian particles in a quenched Gaussian random potential, showing that the system remains ergodic with density correlations decaying to zero over time.

Contribution

It introduces a new diagrammatic perturbation theory based on density fluctuations around the local density, differing from previous approaches that used uniform averages.

Findings

Connected density correlation functions decay to zero, indicating ergodicity.

Exact calculation of the zero-frequency response function.

Comparison reveals differences in diagrammatic structures with previous methods.

Abstract

We construct a dynamical field theory for noninteracting Brownian particles in the presence of a quenched Gaussian random potential. The main variable for the field theory is the density fluctuation which measures the difference between the local density and its average value. The average density is spatially inhomogeneous for given realization of the random potential. It becomes uniform only after averaged over the disorder configurations. We develop the diagrammatic perturbation theory for the density correlation function and calculate the zero-frequency component of the response function exactly by summing all the diagrams contributing to it. From this exact result and the fluctuation dissipation relation, which holds in an equilibrium dynamics, we find that the connected density correlation function always decays to zero in the long-time limit for all values of disorder strength implying that the system always remains ergodic. This nonperturbative calculation relies on the simple diagrammatic structure of the present field theoretical scheme. We compare in detail our diagrammatic perturbation theory with the one used in a recent paper [B.\ Kim, M.\ Fuchs and V.\ Krakoviack, J.\ Stat.\ Mech.\ (2020) 023301], which uses the density fluctuation around the uniform average, and discuss the difference in the diagrammatic structures of the two formulations.