An exactly solvable model for Brownian motion : III. Motion of a heavy mass in a linear chain

Abstract

The theory on Brownian motion, developed in previous papers1) 2) is applied to a linear chain with harmonic coupling between nearest neighbours. All masses are equal except for one which is heavy compared to the others. This heavy particle behaves as a Brownian particle, which is not subject to an external field. The Langevin equation and the Fokker-Planck equation are derived for the classical and quantummechanical case, again without applying perturbation theory. It turns out that in addition to the transient times there exists a time τl, beyond which these equations loose their validity. This limit time is due to the fact that the frequency spectrum has a finite maximum.