Derivation and solution of a low-friction Fokker-Planck equation for a bound Brownian particle

Abstract

The low-friction region of an anharmonically bound Brownian particle is examined using systematic elimination procedures. We obtain an asymptotic expression for the spectrum of the Fokker-Planck operator. Asymptotic means both small anharmonicities and small friction constants γ compared to the oscillatory frequency ω. We conclude that Kramers' low-friction equation is generally valid only for 0<γ≲0.01 ω and has to be modified for γ≳0.01 ω by including phase-dependent terms. From these the nonlinear part of the force field in connection with a finite temperature is shown to shorten the correlation time of the equilibrium velocity autocorrelation function and to renormalize the frequency of the corresponding spectral density.