Numerical Methods for Computing Effective Transport Properties of Flashing Brownian Motors

TL;DR

This paper introduces a spectral numerical algorithm for calculating the effective transport properties of flashing Brownian motors, demonstrating its efficiency and equivalence to existing methods through numerical experiments.

Contribution

The paper develops a spectral decomposition-based numerical method for effective transport properties, extending homogenization theory to flashing Brownian motors with multiplicative noise.

Findings

Spectral method effectively computes drift and diffusivity.

Method is efficient for strong multiplicative noise.

WPE method is shown to be equivalent to homogenization-based approach.

Abstract

We develop a numerical algorithm for computing the effective drift and diffusivity of the steady-state behavior of an overdamped particle driven by a periodic potential whose amplitude is modulated in time by multiplicative noise and forced by additive Gaussian noise (the mathematical structure of a flashing Brownian motor). The numerical algorithm is based on a spectral decomposition of the solution to the Fokker-Planck equation with periodic boundary conditions and the cell problem which result from homogenization theory. We also show that the numerical method of Wang, Peskin, Elston (WPE, 2003) for computing said quantities is equivalent to that resulting from homogenization theory. We show how to adapt the WPE numerical method to this problem by means of discretizing the multiplicative noise via a finite-volume method into a discrete-state Markov jump process which preserves many important properties of the original continuous-state process, such as its invariant distribution and detailed balance. Our numerical experiments show the effectiveness of both methods, and that the spectral method can have some efficiency advantages when treating multiplicative random noise, particularly with strong volatility.