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 it 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 mu = dv/dF where v is the average velocity of the particle, Analytical and numerical calculations indicate that inequality D >= 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 tor asymmetric potentials when F is small but nonzero.