A Tight-Binding Approach to Overdamped Brownian Motion on a Multidimensional Tilted Periodic Potential

TL;DR

This paper develops a tight-binding theoretical framework for analyzing overdamped Brownian motion in multidimensional tilted periodic potentials, simplifying the continuous dynamics into a discrete hopping model.

Contribution

It introduces a novel tight-binding approach to model overdamped Brownian motion, deriving explicit relations for hopping rates and transport properties in tilted periodic potentials.

Findings

Single-band approximation is valid for weak tilting and long times.

Hopping rates depend functionally on tilt and potential well depth.

Derived expressions for drift and diffusion in terms of band eigenvalues.

Abstract

We present a theoretical treatment of overdamped Brownian motion on a multidimensional tilted periodic potential that is analogous to the tight-binding model of quantum mechanics. In our approach we expand the continuous Smoluchowski equation in the localized Wannier states of the periodic potential to derive a discrete master equation. This master equation can be interpreted in terms of hopping within and between Bloch bands and for weak tilting and long times we show that a single-band description is valid. In the limit of deep potential wells, we derive a simple functional dependence of the hopping rates and the lowest band eigenvalues on the tilt. We also provide general expressions for the drift and diffusion in terms of the lowest band eigenvalues.