Diffusion coefficient and mobility of a Brownian particle in a tilted periodic potential

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 Smoluchowski equation in the present work, D is compared with the differential mobility μ = dυ/dF where υ is the average velocity of the particle. Analytical and numerical calculations indicate that inequality D ≥ μk_BT, 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.