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Foundations of Physics

Quantum thermodynamics of nonequilibrium. Onsager reciprocity and dispersion-dissipation relations

Gian Paolo Beretta

Published Apr 1, 1987• 39 citations
Publisher
DOI
10.1007/bf00733374

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We propose a model of nonequilibrium quantum transport of particles and energy in a system connected to mesoscopic Fermi reservoirs (mesoreservoir). The mesoreservoirs are in turn thermalized to prescribed temperatures and chemical potentials by a simple dissipative mechanism described by the Lindblad equation. As an example, we study transport in monoatomic and diatomic chains of noninteracting spinless fermions. We show numerically the breakdown of the Onsager reciprocity relation due to the dissipative terms of the model.
Without assuming detailed balance, the linear response theory of systems, in which the notation of a quantum dynamical semigroup can be defined, is discussed. This theory is used to prove a fluctuation-dissipation theorem for nonstationary nonequilibrium states. We propose an extended version of the Liouville equation in which we incorporate the stochastic behavior of the subsystem of the macroscopic motions. For this we introduce a non-Hermitian Hamiltonian and set a Liouville equation for a density matrix which is not Hermitian. We show that a response to external perturbations reduces to a sum of two contributions, one for which an improved fluctuation-dissipation theorem is proved and the other due to stochastic excitations that does not satisfy a reciprocity relation of Onsager type.
We study a quantum XX chain coupled to two heat reservoirs that act on multiple sites and are kept at different temperatures and chemical potentials. The baths are described by Lindblad dissipators, which are constructed by direct coupling to the fermionic normal modes of the chain. Using a perturbative method, we are able to find analytical formulas for all steady-state properties of the system. We compute both the particle or magnetization current and the energy current, both of which are found to have the structure of Landauer's formula. We also obtain exact formulas for the Onsager coefficients. All properties are found to differ substantially from those of a single-site bath. In particular, we find a strong dependence on the intensity of the bath couplings. In the weak-coupling regime, we show that the Onsager reciprocal relations are satisfied.