On the Fokker-Planck equation for the linear chain

Abstract

The Fokker-Planck equation with retarded friction, derived approximately from the Liouville equation, is considered for the Rubin model of a heavy particle, mass M, in a harmonic lattice of light particles, mass m. We find that the solution is inconsistent with Rubin's exact results, except when both the exact and present results are truncated at lowest order in the mass ratio m/M. This casts some doubt on the validity of retarded Fokker-Planck equation for more general situations. We also consider the so-called initial value term in the statistical mechanical derivation, and find it slowly decaying in time, contrary to what is usually assumed.