Quantum Fokker-Planck-Kramers equation and entropy production

We use a canonical quantization procedure to set up a quantum Fokker-Planck-Kramers equation that accounts for quantum dissipation in a thermal environment. The dissipation term is chosen to ensure that the thermodynamic equilibrium is described by the Gibbs state. An expression for the quantum entropy production that properly describes quantum systems in a nonequilibrium stationary state is also provided. The time-dependent solution is given for a quantum harmonic oscillator in contact with a heat bath. We also obtain the stationary solution for a system of two coupled harmonic oscillators in contact with reservoirs at distinct temperatures, from which we obtain the entropy production and the quantum thermal conductance.

Figure 1

Free energy and rate of entropy production as functions of time for a quantum oscillator in contact with a heat reservoir. The dimensionless quantities in the plot are as follows: $\mathrm{\Delta }{F}^{*}=(F-F_0)/\hslash \omega ,{\mathrm{\Pi }}^{*}=\mathrm{\Pi }/k_B\omega$, and $t^{*}=\omega t$. The values of the parameters are $k_{B}T/\hslash \omega =1$ and $\gamma /\omega =1$.