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 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. 
Kazuo Sasaki and Satoshi Amari