Classical limit of master equation for harmonic oscillator coupled to oscillator bath with separable initial conditions

TL;DR

This paper investigates the classical limit of the master equation for a harmonic oscillator coupled to a bath, clarifying the differences in classical equations arising from separable versus correlated initial conditions.

Contribution

It demonstrates that the classical limit for separable initial conditions differs from Adelman's equation and derives the correct Fokker-Planck equation matching the Wigner function's classical limit.

Findings

Classical Langevin equation differs for separable initial conditions.

Derived Fokker-Planck equation matches the classical limit of the quantum Wigner function.

Clarified the link between initial conditions and classical equations in oscillator-bath systems.

Abstract

The equation for the Wigner function describing the reduced dynamics of a single harmonic oscillator, coupled to an oscillator bath, was obtained by Karrlein and Grabert [Phys. Rev. E, vol. 55, 153 (1997)]. It was shown that for some special correlated initial conditions the equation reduces, in the classical limit, to the corresponding classical Fokker-Planck equation obtained by Adelman [J. Chem Phys., vol. 64, 124 (1976)]. However for separable initial conditions the Adelman equations were not recovered. We resolve this problem by showing that, for separable initial conditions, the classical Langevin equation obtained from the oscillator bath model is somewhat different from the one considered by Adelman. We obtain the corresponding Fokker-Planck equation and show that it exactly matches the classical limit of the equation for the Wigner function obtained from the master equation for separable initial conditions. We also discuss why the special correlated initial conditions correspond to Adelman's solution.