Brownian motion in the bistable potential at intermediate and high friction: Relaxation from the instability point

We consider a Brownian particle in a double-well potential at intermediate and high friction and study the evolution of an initial distribution which is located at the top of the barrier. The evolution equation is obtained from the Fokker-Planck-Klein-Kramers equation by an inverse friction expansion and is transformed to a time-dependent Schrödinger equation which we solve approximately using the WKB method. The result becomes exact for temperatures low compared with the barrier height. We calculate explicitly the characteristic time in which the distribution changes from a single peak to a double peak structure and show that the decay process is mainly described by a scaling-function which reduces to Suzuki's scaling solution in the limit of high friction.