An exactly solvable model for Brownian motion : I. Derivation of the Langevin equation

Abstract

The motion of an elastically bound particle, linearly coupled with a bath of N harmonic oscillators is calculated exactly. The interaction is assumed to be rather weak and to vary slowly as function of the frequencies of the oscillators. The bath is chosen at the initial time in thermal equilibrium. It turns out that in the limit N → ∞ the motion of the particle is governed by the Langevin equation of the damped harmonic oscillator. The solution of this Langevin equation has the same behaviour as in the conventional theory of Brownian motion after a transient time τt. In the quantummechanical case an additional transient time appears, τq = h/kT.

The theory is applied to an elastically bound electron in an electromagnetic field. Two cases are considered. First the square of the vector potential is discarded and the electron has an extended charge distribution. In the other case a point electron is considered and its mass is renormalized. The familiar expression for the radiation damping constant is found.