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Reviews in Mathematical Physics

QUANTUM MACROSTATISTICAL THEORY OF NONEQUILIBRIUM STEADY STATES

Geoffrey L. Sewell

Published Oct 1, 2005• 9 citations
Publisher

Abstract

We provide a general macrostatistical formulation of nonequilibrium steady states of reservoir driven quantum systems. This formulation is centered on the large scale properties of the locally conserved hydrodynamical observables, and our basic physical assumptions comprise (a) a chaoticity hypothesis for the nonconserved currents carried by these observables, (b) an extension of Onsager's regression hypothesis to fluctuations about nonequilibrium states, and (c) a certain mesoscopic local equilibrium hypothesis. On this basis, we obtain a picture wherein the fluctuations of the hydrodynamical variables about a nonequilibrium steady state execute a Gaussian Markov process of a generalized Onsager–Machlup type, which is completely determined by the position dependent transport coefficients and the equilibrium entropy function of the system. This picture reveals that the transport coefficients satisfy a generalized form of the Onsager reciprocity relations in the nonequilibrium situation and that the spatial correlations of the hydrodynamical observables are generically of long range. This last result constitutes a model-independent quantum mechanical generalization of that obtained for special classical stochastic systems and marks a striking difference between the steady nonequilibrium and equilibrium states, since it is only at critical points that the latter carry long range correlations.

DOI
10.1142/s0129055x05002492

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We provide a general macrostatistical formulation of nonequilibrium steady states of reservoir driven quantum systems. This formulation is centered on the large scale properties of the locally conserved hydrodynamical observables, and our basic physical assumptions comprise (a) a chaoticity hypothesis for the nonconserved currents carried by these observables, (b) an extension of Onsager's regression hypothesis to fluctuations about nonequilibrium states, and (c) a certain mesoscopic local equilibrium hypothesis. On this basis, we obtain a picture wherein the fluctuations of the hydrodynamical variables about a nonequilibrium steady state execute a Gaussian Markov process of a generalized Onsager–Machlup type, which is completely determined by the position dependent transport coefficients and the equilibrium entropy function of the system. This picture reveals that the transport coefficients satisfy a generalized form of the Onsager reciprocity relations in the nonequilibrium situation and that the spatial correlations of the hydrodynamical observables are generically of long range. This last result constitutes a model-independent quantum mechanical generalization of that obtained for special classical stochastic systems and marks a striking difference between the steady nonequilibrium and equilibrium states, since it is only at critical points that the latter carry long range correlations.
We present a general, model-independent, quantum statistical treatment of the connection between the quantum and hydrodynamical pictures of reservoir driven macroscopic systems. This treatment is centred on the large scale properties of locally conserved hydrodynamical observables and is designed to form a bridge between quantum microdynamics and classical macroscopic continuum mechanics, rather than a derivation of the latter from the former. The key assumptions on which the treatment is based are hypotheses of chaoticity and local equilibrium for the hydrodynamical fluctuations around nonequilibrium steady states, together with an extension of Onsager's regrssion hypothesis to those states. On this basis, we establish canonical generalisations of both the Onsager reciprocity relations and the Onsager-Machlup fluctuation theory to nonequilibrium steady states, and we show that the spatial correlations of the hydrodynamical fluctuations are generically of long range in these states.
A quantum-mechanical framework is set up to describe the full counting statistics of particles flowing between reservoirs in an open system under time-dependent driving. A symmetry relation is obtained, which is the consequence of microreversibility for the probability of the nonequilibrium work and the transfer of particles and energy between the reservoirs. In some appropriate long-time limit, the symmetry relation leads to a steady-state quantum fluctuation theorem for the currents between the reservoirs. On this basis, relationships are deduced which extend the Onsager–Casimir reciprocity relations to the nonlinear response coefficients.
We construct a C*-algebraic formulation of the dynamics of a pair of mutually interacting quantum-mechanical systems S̃ and Ŝ, the former being finite and the latter infinite. Our basic assumptions are that: (i) Ŝ, when isolated, satisfies the Dubin-Sewell dynamical axioms; (ii) the coupling between S̃ and Ŝ is energetically bounded and spatially localized; and (iii) the initial states of S̃ and Ŝ are mutually uncorrelated, with S̃ in an arbitrary normal state and Ŝ in a Gibbs state. Our formulation leads to a rigorous theory of (a) the dynamics of a finite open system, i.e., of a finite system (S̃) coupled to an ``infinite reservoir'' (Ŝ), and of (b) the dynamics of an infinite system (Ŝ), driven from equilibrium by a ``signal generator'' (S̃). As regards (a), we show that the state of S̃ always remains normal, and we derive a generalized master equation (in an appropriate Banach space) governing its temporal evolution. As regards (b), we show that the state of Ŝ always corresponds to a (time-dependent) density matrix in the representation space of the algebra of observables for Ŝ, induced by the initial Gibbs state. By formulating the linear part (appropriately defined) of the response of Ŝ to S̃, we generalize the fluctuation-dissipation theorem to infinite systems. Further, we show that the total effect of S̃ on Ŝ reduces to that of a ``classical'' time-dependent external force in cases where the initial state of S̃ possesses certain coherence properties similar to those of the Glauber type.
At large scales of space and time, the nonequilibrium dynamics of local observables in extensive many-body systems is well described by hydrodynamics. At the Euler scale, one assumes that each mesoscopic region independently reaches a state of maximal entropy under the constraints given by the available conservation laws. Away from phase transitions, maximal entropy states show exponential correlation decay, and independence of fluid cells might be assumed to subsist during the course of time evolution. We show that this picture is incorrect: under ballistic scaling, regions separated by macroscopic distances "develop long-range correlations as time passes." These correlations take a universal form that only depends on the Euler hydrodynamics of the model. They are rooted in the large-scale motion of interacting fluid modes and are the dominant long-range correlations developing in time from long-wavelength, entropy-maximized states. They require "the presence of interaction" and "at least two different fluid modes" and are of a fundamentally different nature from well-known long-range correlations occurring from diffusive spreading or from quasiparticle excitations produced in far-from-equilibrium quenches. We provide a universal theoretical framework to exactly evaluate them, an adaptation of the macroscopic fluctuation theory to the Euler scale. We verify our exact predictions in the hard-rod gas, by comparing with numerical simulations and finding excellent agreement.
The nonequilibrium dynamics of generic quantum systems denies a fully understanding up to now, even if the thermalization in the long-time asymptotic state has been explained by the eigenstate thermalization hypothesis which proposes a universal form of the observable matrix elements in the eigenbasis of the Hamiltonian. In this paper, we study the form of the quantum state, i.e., of the density matrix elements. We propose that the density matrix has also a universal form in chaotic systems, which is used to understand the nonequilibrium dynamics in the whole time scale, from the transient regime to the long-time steady limit, and then extends the applicability of eigenstate thermalization hypothesis to true nonequilibrium phenomena such as nonequilibrium steady states. Our assumption is numerically tested in various models, and its intimate relation to the eigenstate thermalization hypothesis is discussed.
The dynamics of a quantum system coupled to a classical environment and subject to constraints that drive it out of equilibrium are described. The evolution of the system is governed by the quantum-classical Liouville equation. Rather than evaluating the evolution of the mixed quantum-classical density operator, we derive exact equations of motion for the nonequilibrium average values of a set of operators or variables, along with correlation function expressions for the dissipative coefficients that enter these equations. These equations are obtained by requiring that the exact nonequilibrium averages are equal to local nonequilibrium averages that depend on auxiliary fields whose values satisfy evolution equations obtained using projection operator methods. The results are illustrated by deriving reaction-diffusion equations coupled to fluid hydrodynamic equations for a solution of quantum particles that can exist in two metastable states. Nonequilibrium steady states are discussed along with the reaction rate and diffusion correlation functions that characterize such states.