Analytical approach to the time-dependent probability density function in tilted periodic potentials

In this work we introduce a scheme for the calculation of an approximate closed expression for the time-dependent probability density function for overdamped particles in tilted periodic potentials. Our derivation is based on an ansatz for the solution of the corresponding Fokker-Planck equation and on a self-consistent cumulant calculation. The high accuracy of our expression for the time-dependent probability density function is exhibited by comparisons with Langevin dynamics simulations and exact analytic results for the drift and diffusion coefficients. Good agreement is found both, for large and intermediate times.