Diffusion in a bistable potential at intermediate and high friction

We study the motion of a Brownian particle in a bistable potential for intermediate and high-friction . Following ideas of Titulaer we perform a high-friction expansion of the distribution functionP(v,x,t) in velocity and space. We show (for arbitrary potential) that the expansion coefficients obey simple recursion relations, which allow them to be calculated easily. When terms of order ^–5 are neglected the resulting differential equations can be transformed into Hermitian Schrödinger-type equations. Using the WKB technique we solve these equations analytically for the case of the bistable potential and discuss the various time regimes involved in the system, in particular we show that the final approach to equilibrium is governed by the Kramers rate. Our results become exact in the limit of low temperatures.