Diffusion Coefficient and Mobility of a Brownian Particle in a Tilted Periodic Potential

TL;DR

This paper investigates the relationship between diffusion coefficient and mobility of a Brownian particle in a tilted periodic potential, revealing conditions under which a key inequality holds or is violated.

Contribution

It introduces an alternative formula for the diffusion coefficient and compares it with mobility, highlighting the impact of potential symmetry on their relationship.

Findings

Inequality D ≥ μ k_B T holds for symmetric potentials.

Inequality is violated for asymmetric potentials at small nonzero F.

Analytical and numerical methods confirm the conditions for the inequality.

Abstract

The Brownian motion of a particle in a one-dimensional periodic potential subjected to a uniform external force F is studied. Using the formula for the diffusion coefficient D obtained by other authors and an alternative one derived from the Fokker-Planck equation in the present work, D is compared with the differential mobility \mu = dv/dF where v is the average velocity of the particle. Analytical and numerical calculations indicate that inequality D \ge \mu k_{B}T, with k_{B} the Boltzmann constant and T the temperature, holds if the periodic potential is symmetric, while it is violated for asymmetric potentials when F is small but nonzero.