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Fortschritte der Physik/Progress of Physics

On the Nonequilibrium Statistical Operator Method

Roberto Luzzi, Aurea Rosas Vasconcellos

Published Jan 1, 1990• 107 citations
Publisher

Abstract

We describe the large applicability of the Nonequilibrium Statistical Operator Method (NSOM) for the study of dissipative dynamic systems far from equilibrium. It is shown that the NSOM can be encompassed by a unifying variational principle, which produces a large family of NSO that contains existing examples as particular cases. Further, we review the application of the NSOM for the construction of a nonlinear quantum theory of large scope, and for the generation of a response function theory, for far-from-equilibrium Hamiltonian systems. An accompanying non-equilibrium thermodynamic Green's function theory is briefly described. Also it is shown that the NSOM provides mechano-statistical foundations for phenomenological irreversible thermodynamics, and for the important question of stability of far-from-equilibrium steady states and the emergence of self-organized dissipative structures in condensed matter.

DOI
10.1002/prop.2190381104

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We describe the large applicability of the Nonequilibrium Statistical Operator Method (NSOM) for the study of dissipative dynamic systems far from equilibrium. It is shown that the NSOM can be encompassed by a unifying variational principle, which produces a large family of NSO that contains existing examples as particular cases. Further, we review the application of the NSOM for the construction of a nonlinear quantum theory of large scope, and for the generation of a response function theory, for far-from-equilibrium Hamiltonian systems. An accompanying non-equilibrium thermodynamic Green's function theory is briefly described. Also it is shown that the NSOM provides mechano-statistical foundations for phenomenological irreversible thermodynamics, and for the important question of stability of far-from-equilibrium steady states and the emergence of self-organized dissipative structures in condensed matter.
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A response function theory for many-body systems arbitrarily away from equilibrium is presented. It is based on the n onequilibrium statistical operator method fully described in a previous article (~ev. Brasil .Fis.15, 106 (1985)l. We present a formal theory for evaluation of transition probabilties and the average values of dynamical quan- tities in far - from - equilibrium many-body systems under the action of externa1 perturbations. We also derive a n onequilibrium thermodynamic Green's functions algorithm appropriate for the calculation of response functions and scattering cross sections in terms ofa generalized fluc- tuation - dissipation theorem for far - from - equilibrium systems.
Questions about the route from a nonequilibrium initial state to the final global equilibrium have played an important role since the early days of phenomenological thermodynamics and statistical mechanics. Nowadays, their implications reach from central technical devices of the contemporary human society, like heat engines, refrigerators and computers to recent physics at almost all length scales, from Bose-Einstein-condensation and superconductors to black holes. This work addresses the foundation of macroscopic laws concerning the decay to equilibrium, e.g. the celebrated Fourier's Law, on microscopic Schrödingerian quantum dynamics. Here, a proper treatment requires the usage of modern methods in theoretical physics such as the Theory of Open Quantum Systems, the Kubo Formula in Liouville Space and the novel Hilbert Space Average Method. It turns out that both the relaxation to equilibrium as well as the transport of heat is mainly determined by quantum effects comparable to the role of entanglement in considerations of the global equilibrium within Quantum Thermodynamics. Finally, the foundation of phenomenological thermodynamics on a microscopic theory will hopefully improve our understanding of those most impressive and far-reaching theories and their background and will possibly open the way to overcoming their nanoscopic limits.
A numerical approach is presented that allows to compute nonequilibrium steady-state properties of strongly correlated quantum many-body systems. The method is imbedded in the Keldysh Green's function formalism and is based upon the idea of the variational cluster approach as far as the treatment of strong correlations is concerned. It appears that the variational aspect is crucial as it allows for a suitable optimization of a ``reference'' system to the nonequilibrium target state. The approach is neither perturbative in the many-body interaction nor in the field, that drives the system out of equilibrium, and it allows to study strong perturbations and nonlinear responses of systems in which also the correlated region is spatially extended. We apply the presented approach to nonlinear transport across a strongly correlated quantum wire described by the fermionic Hubbard model. We illustrate how the method bridges to cluster dynamical mean-field theory upon coupling two baths containing an increasing number of uncorrelated sites.
The method of the nonequilibrium statistical operator developed by D. N. Zubarev is employed to analyze and derive generalized transport and kinetic equations. The degrees of freedom in solids can often be represented as a few interacting subsystems (electrons, spins, phonons, nuclear spins, etc.). Perturbation of one subsystem may produce a nonequilibrium state which is then relaxed to an equilibrium state due to the interaction between particles or with a thermal bath. The generalized kinetic equations were derived for a system weakly coupled to a thermal bath to elucidate the nature of transport and relaxation processes. It was shown that the "collision term" had the same functional form as for the generalized kinetic equations for the system with small interactions among particles. The applicability of the general formalism to physically relevant situations is investigated. It is shown that some known generalized kinetic equations (e.g. kinetic equation for magnons, Peierls equation for phonons) naturally emerges within the NSO formalism. The relaxation of a small dynamic subsystem in contact with a thermal bath is considered on the basis of the derived equations. The Schrödinger-type equation for the average amplitude describing the energy shift and damping of a particle in a thermal bath and the coupled kinetic equation describing the dynamic and statistical aspects of the motion are derived and analyzed. The equations derived can help in the understanding of the origin of irreversible behavior in quantum phenomena.
This article traces the development of fluctuation theory and its deep connection to irreversibility, from equilibrium to near-equilibrium, and finally to far-from-equilibrium systems. Classical fluctuation theorems, which capture the statistical behaviour of thermodynamic systems far from equilibrium, are now well established. Their quantum counterparts, however, remain an active area of research. In this review, we highlight recent advances by linking quantum fluctuation theorems with linear response theory, offering new insights into the nature of quantum fluctuations and irreversibility in the near-equilibrium regime. Particular emphasis is placed on dissipated work in quantum systems as a pathway to observing non-classical effects in quantum thermodynamics. Understanding quantum fluctuations is not only essential for clarifying the foundations of irreversibility but also crucial for the development of novel quantum technologies, including quantum computers, sensors, and metrological devices.