Thermally activated escape rate for a Brownian particle in a tilted periodic potential for all values of the dissipation

The translational Brownian motion of a particle in a tilted washboard potential is considered. The dynamic structure factor and longest relaxation time are evaluated from the solution of the governing Langevin equation by using the matrix continued fraction method. The longest relaxation time is compared with the Kramers theory of the escape rate of a Brownian particle from a potential well as extended to the Kramers turnover region by Mel'nikov [Physics Reports 209, 1 (1991)]. It is shown that in the low temperature limit, the universal Mel'nikov expression for the escape rate provides a good estimate of the longest relaxation time for all values of dissipation including the very low damping (VLD), very high damping (VHD), and turnover regimes. For low barriers (where the Mel'nikov method is not applicable) and zero tilt, analytic equations for the relaxation times in the VLD and VHD limits are derived.