Resolving a puzzle concerning fluctuation theorems for forced harmonic oscillators innon-Markovian heat baths

Abstract

A harmonic oscillator that evolves under the action of both a systematictime-dependent force and a random time-correlated force can do workw. This work is a random quantity, and Mai and Dhar have recently shown,using the generalized Langevin equation (GLE) for the oscillator’s positionx, that it satisfies a fluctuation theorem. In principle, the same result could have beenderived from the Fokker–Planck equation (FPE) for the probability density function,P(x,w,t), for the oscillatorbeing at x attime t, havingdone work w. Although the FPE equivalent to the above GLE is easily constructed and solved,one finds, unexpectedly, that its predictions for the mean and variance ofw do not agree with the fluctuation theorem. We show that to resolve this contradiction,it is necessary to construct an FPE that includes the velocity of the oscillator,v, as an additionalvariable. The FPE for P(x,v,w,t) does indeed yield expressions for the mean and variance ofw that agree with the fluctuation theorem.