Size effects in long-term quasistatic heat transport

We consider finite-size effects on heat transfer between thermal reservoirs mediated by a quantum system, where the number of modes in each reservoir is finite. Our approach is based on the generalized quantum Langevin equation and the thermal reservoirs are described as ensembles of oscillators within the Drude-Ullersma model. A general expression for the heat current between the thermal reservoirs in the long-time quasistatic regime, when an observation time is of the order of Δ(-1) and Δ is the mode spacing constant of a thermal reservoir, is obtained. The resulting equations that govern the long-time relaxation for the mode temperatures and the average temperatures of the reservoirs are derived and approximate analytical solutions are found. The obtained time dependencies of the temperatures and the resulting heat current reveal peculiarities at t=2πm/Δ with non-negative integers m and the heat current vanishes nonmonotonically when t→∞. The validity of Fourier's law for a chain of finite-size macroscopic subsystems is considered. As is shown, for characteristic times of the order of Δ(-1) the temperatures of subsystems' modes deviate from each other and the validity of Fourier's law cannot be established. In a case when deviations of initial temperatures of the subsystems from their average value are small, t→∞ asymptotic values for the mode temperatures do not depend on a mode's number and are the same as if Fourier's law were valid for all times.