An exactly solvable model for Brownian motion : II. Derivation of the Fokker-Planck equation and the master equation

Abstract

As in a previous paper1) an elastically bound particle, linearly coupled with a bath of small oscillators, is considered. At the initial time the bath is chosen in thermal equilibrium with temperature T. In the classical case the distribution function for the momentum and displacement of the particle is calculated exactly. It turns out that under the same assumptions needed for the derivation of the Langevin equation1), this distribution function satisfies a Fokker-Planck equation for times large compared to a transient time τt.

In the quantummechanical case the probability to find the particle at time t in an eigenstate of its unperturbed Hamiltonian is also calculated exactly. If in addition to the above mentioned assumptions the phases of the initial state of the particle are chosen randomly, this probability satisfies a master equation for times large compared to τt = /kT, the quantumstatistical transient time.

The main point in this treatment is that perturbation theory is not applied and that therefore averaging over the bath of small oscillators and over the phases of the state of the particle is only needed at the initial time.