Nonequilibrium statistical thermodynamics of a small quantum system coupled to quantum reservoirs

Abstract

A statistical mechanical formulation where nonequilibrium averages of selected quantum observables are given exactly as averages over local density operators is used to describe the dynamics and thermodynamics of a small quantum system coupled to reservoirs. The local density operators are functionals of constraint variables that play the role of thermodynamic forces that satisfy a set of equations of motion. A focus of the work is on the evolution of the average internal energy of the quantum subsystem that may be subject to external autonomous driving that does work on the system, and on entropy production in the system. The nonequilibrium statistical thermodynamics the quantum subsystem coupled to reservoirs, as well as that for the isolated subsystem and reservoirs, are discussed. When the subsystem is coupled to large quantum reservoirs, the effects of the quantum subsystem on the reservoirs can be captured through boundary conditions on the reservoir equations, and these boundary conditions are derived. The formulation is illustrated by considering the heat transfer through a system comprising a quantum system coupled to two heat reservoirs with different temperatures.

The paper develops a statistical-mechanical description of energy exchange and entropy production for a small quantum system coupled to spatially extended, nonequilibrium quantum reservoirs, optionally subject to a prescribed time-dependent drive. A maximum-entropy local density operator is constrained to reproduce the exact expectation values of selected observables: the subsystem Hamiltonian, the driving operator, and the reservoir energy density including system–reservoir coupling. Time-dependent projection operators yield formally exact, nonlinear evolution equations for these averages and their conjugate constraint fields, with memory kernels expressed through projected flux–flux correlations. The energy balance gives $\Delta U=W+Q$, where work is the integrated expectation of the Hamiltonian’s explicit time derivative and $Q=-\Delta e_0$ includes the reservoir and coupling-energy change. Under Markovian and spatial-localization approximations, the equations reduce to transport relations driven by inverse-temperature differences and gradients. The resulting coarse-grained entropy-production expression is nonnegative, with separate driving, interface, and reservoir contributions. Decomposing the reservoir into bulk and interfacial regions, and imposing a vanishing interfacial balance term, gives boundary conditions coupling reservoir heat equations to subsystem constraint variables. Illustrations cover stepwise driving of an isolated subsystem and steady heat transport between two reservoirs. The exact kernels are acknowledged to be difficult to evaluate, and the sign of entropy production is not established for the full memory-dependent equations. The treatment restricts subsystem resolution to global observables and excludes reservoir particle transport and fluid flow. The heat-transfer illustration assumes simplified transport coefficients rather than evaluating a microscopic junction model.