The time scale of classical diffusion with arbitrary energy barriers

Abstract

An exact upper bound is established for the longest time scale of classical diffusion in a continuous space x in the presence of an arbitrary external potential energy E( x) and diffusion “constant” D( x). The result characterises, for any initial conditions, the solutions of the partial differential equation ∂W/ ∂t = ∂/ ∂x[− V( x) W+ D( x) ∂W/ ∂x], where V( x) = −[ D( x)/ k B T] ∂E( x)/ ∂x at temperature T. The time scale can be obtained from E( x) and D( x) without determination of the solution for the probability density W( x, t). Examples with various types of barriers in E( x) are considered, together with the classical limit of the Metropolis model of Monte Carlo experiments. The classical limit of the thermal activation law of Arrhenius is established in the low-temperature limit.