On mathematical foundation of the Brownian motor theory

TL;DR

This paper provides a rigorous mathematical foundation for Brownian motor theory, deriving homogenization results and formulas for effective parameters using spectral methods, with applications to motion in periodic tubes with dead zones.

Contribution

It offers the first detailed mathematical justification of key aspects of Brownian motor theory, including homogenization theorems and spectral formulas for effective parameters.

Findings

Homogenization theorems for Brownian motion in periodic tubes with drift.

Explicit formulas for effective drift and diffusivity using spectral methods.

Application of formulas to motion in tubes with dead zones.

Abstract

The paper contains mathematical justification of basic facts concerning the Brownian motor theory. The homogenization theorems are proved for the Brownian motion in periodic tubes with a constant drift. The study is based on an application of the Bloch decomposition. The effective drift and effective diffusivity are expressed in terms of the principal eigenvalue of the Bloch spectral problem on the cell of periodicity as well as in terms of the harmonic coordinate and the density of the invariant measure. We apply the formulas for the effective parameters to study the motion in periodic tubes with nearly separated dead zones.