Diffusion coefficient and drift velocity in periodic media

We present a novel method of computing the asymptotic drift velocity V and the diffusion coefficient D of a particle diffusing in an arbitrary periodic medium. We focus on a particular case of 1D systems with the nearest-neighbor transition rates and explain in detail how our method works in this case. We show that the problem of finding V and D can be reduced to a simple graph-counting problem. Using this method and a computer-algebra system, we were able to prove Fisher and Kolomeisky's [M.E. Fisher and A.B. Kolomeisky, Physica A 274 (1999) 241] conjecture about the maximal force exerted by a molecular motor (in the linear-response regime) for lattice periods L⩽16.