Biased diffusion in periodic potentials: Three types of force dependence of effective diffusivity and generalized Lifson-Jackson formula

Diffusive transport of particles in a biased periodic potential is characterized by the effective drift velocity and diffusivity, which are functions of the biasing force. We derive a simple exact expression for the effective diffusivity and use it to show that the force dependence of this quantity may be a nonmonotonic function with a maximum [as shown in the work of Reimann et al. Phys. Rev. Lett. 87, 010602 (2001) for periodic sinusoidal potential] or with a minimum, or a monotonic function. The shape of the dependence is determined by the shape of the periodic potential.