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

Abstract

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.

Dispersionless Transport in a Washboard Potential

“…The pendulum provides the mathematical background underlying a number of applications, including mobility in superionic conductors, dynamics of chargedensity waves, ring laser gyroscopes, and phase-locking phenomena in radio engineering. Perhaps most directly relevant for experimental testing, it also models a resistively and capacitively shunted single Josephson junction.This latter correspondence has been invoked in some of the most recent work on transport in tilted periodic potentials [7,8]. An excellent table indicating the precise translation between the parameters of a number of physical systems including the quintessential Josephson junction test bed and our model equation can be found in [7].…”

“…This latter correspondence has been invoked in some of the most recent work on transport in tilted periodic potentials [7,8]. An excellent table indicating the precise translation between the parameters of a number of physical systems including the quintessential Josephson junction test bed and our model equation can be found in [7].…”

“…An excellent table indicating the precise translation between the parameters of a number of physical systems including the quintessential Josephson junction test bed and our model equation can be found in [7].…”

Thermal fluctuations of magnetic nanoparticles: Fifty years after Brown

“…(13)) is concerned, many examples of calculations based on either the analytical and numerical solutions of the Klein-Kramers equation or on numerical simulations of the Brownian dynamics exist. They include the comparison of the turnover formula with the numerical results for the escape out of a single potential well 64,65 and that from both double- 66,67 and multi-well [68][69][70][71] potentials (see Fig. 3).…”

Transport of a heated granular gas in a washboard potential

“…A detailed theory capable of describing the behavior of the IBP for arbitrary values of the damping has been developed by Risken 20,21 who derived an analytical expression for the mobility, µ, as a function of noise level, tilting force and substrate characteristics. A closed formula for the diffusion coefficient exists only for the overdamped situation 22 .…”