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Journal of Physics A Mathematical and Theoretical

Successes and failures in the construction of NESS-states

Heide Narnhofer

Published Apr 6, 2010• 0 citations
Publisher

Abstract

We construct non-equilibrium states by coupling a large but finite system with quasifree evolution to two temperature baths of free fermions or bosons with different temperature. We are interested in the resulting behaviour of the temperature baths as well as in the consequences for the finite system, especially what happens in the limit when the finite system tends to infinity. As a special example we consider the Kronig–Penney model and the tight binding model. Here a heat current remains that changes the baths but does not allow a limiting behaviour when the finite system tends to infinity. For the random versions of these models, especially the Anderson model, the heat current disappears exponentially with the size of the system. We show that a limit state is attained that is independent of the time direction and does not show any kind of symmetry breaking.

DOI
10.1088/1751-8113/43/17/175001

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We construct non-equilibrium states by coupling a large but finite system with quasifree evolution to two temperature baths of free fermions or bosons with different temperature. We are interested in the resulting behaviour of the temperature baths as well as in the consequences for the finite system, especially what happens in the limit when the finite system tends to infinity. As a special example we consider the Kronig–Penney model and the tight binding model. Here a heat current remains that changes the baths but does not allow a limiting behaviour when the finite system tends to infinity. For the random versions of these models, especially the Anderson model, the heat current disappears exponentially with the size of the system. We show that a limit state is attained that is independent of the time direction and does not show any kind of symmetry breaking.
Non-equilibrium steady states (NESS) describe particularly simple and stationary non-equilibrium situations. A possibility to obtain such states is to consider the asymptotic evolution of two infinite heat baths brought into thermal contact. In this work we generalise corresponding results of Doyon~et.~al. (J.\ Phys.\ A 18 (2015) no.9) for free Klein-Gordon fields in several directions. Our analysis is carried out directly at the level of correlation functions and in the algebraic approach to QFT. We discuss non-trivial chemical potentials, condensates, inhomogeneous linear models and homogeneous interacting ones. We shall not consider a sharp contact at initial time, but a smooth transition region. As a consequence, the states we construct will be of Hadamard type, and thus sufficiently regular for the perturbative treatment of interacting models. Our analysis shows that perturbatively constructed interacting NESS display thermodynamic properties similar to the ones of the NESS in linear models. In particular, perturbation theory appears to be insufficient to describe full thermalisation in non-linear QFT. Notwithstanding, we find that the NESS for linear and interacting models is stable under small perturbations, which is one of the characteristic features of equilibrium states.
Abstract Within the rigorous axiomatic framework for the description of quantum mechanical systems with a large number of degrees of freedom, we construct the so-called nonequilibrium steady state for the quasifree fermionic system corresponding to the isotropic XY chain in which a finite sample, subject to a local gauge breaking anisotropy perturbation, is coupled to two thermal reservoirs at different temperatures. Using time dependent and stationary scattering theory, we rigorously prove, from first principles, that the nonequilibrium system under consideration is thermodynamically nontrivial, i. e., that its entropy production rate is strictly positive.
This work concerns itself with the exact study of the dynamical properties of two model systems. After a brief summary of theory and concepts previous work is discussed, and this provides the motivation for the formulation of the first model. This quantum mechanical lattice model and some of its equilibrium properties are described in Chapter II. The dynamical problem to be studied is formulated in Chapter III; this is essentially the study of the time evolution (generated by the Hamiltonian) of a finite system at temperature T1 coupled to an infinite copy of itself at a temperature T2 and acting as a heat bath for the system. The problem is solved for the special case, when the coupling as scaled by the parameter γ, takes the value γ = 1. The general case for arbitrary γ values is treated in Chapter IV. It is shown that the system approaches the equilibrium state of the heat bath in a non-exponential manner, provided the spectrum of the Hamiltonian is continuous and does not have a discrete part. This result is in complete accord with the findings of other work summarised in Chapter I. The mixing properties of the model and behaviour of the relaxation rate in the weak coupling limit are studied in Chapter V. The model is shown to fail to behave as a calorimeter and in view of this result the relevance of the concept of mixing to irreversible behaviour is discussed. The main conclusions and results for the model are summarised at the end of Chapter V. The second model discussed, was first introduced by R.J. Glauber to study the dynamics of the Ising chain. The main feature here is that the time evolution is defined through a Master equation, and the associated stochastic operator. It is shown in Chapter VI that exploiting fully the free fermion character of the stochastic operator for the Glauber model, it is possible to provide a simple method to the study of the dynamics of the Ising Chain.
In this thesis we develop a systematic formalism that allows the calculation of transport properties for small quantum systems coupled to multiple baths with different temperatures and/or chemical potentials. Our approach is based on a generalization of the projector operator technique, previously used to study the evolution towards equilibrium of systems weakly coupled to a single bath. Applying this technique to a system weakly coupled to multiple baths, we find a dynamical equation for the reduced density matrix of the system (the reduced density matrix is obtained by tracing out from the total density matrix the baths’ degrees of freedom). Integration of this dynamical equation gives the time evolution of the biased system. After a time interval long enough compared to the characteristic time-scales of the problem, the system arrives to a non-equilibrium steady-state. The density matrix in this non-equilibrium steady state can also be calculated directly, avoiding the time-integration of the dynamical equation. In the second part of the thesis we study non-equilibrium steady states and heat transport through short spin chains coupled to two thermal baths with different temperature. Two new definitions for the local temperature in nonequilibrium are proposed, based on local spin and energy at different sites. The definition based on the local energy seems to be more reliable. We find that for small biases, the heat and spin currents increase proportionally to the bias, however non-linear behavior is observed for large biases. The temperature profile of the chain depends strongly on the parameters of the system.
We construct a finite bath with variable temperature for quantum thermodynamic simulations in which heat flows between a system S and the bath environment E in time evolution of an initial SE pure state. The bath consists of harmonic oscillators that are not necessarily identical. Baths of various numbers of oscillators are considered; a bath with five oscillators is used in the simulations. The bath has a temperaturelike level distribution. This leads to definition of a system-environment microcanonical temperature T_{SE}(t) which varies with time. The quantum state evolves toward an equilibrium state which is thermal-like, but there is significant deviation from the ordinary energy-temperature relation that holds for an infinite quantum bath, e.g., an infinite system of identical oscillators. There are also deviations from the Einstein quantum heat capacity. The temperature of the finite bath is systematically greater for a given energy than the infinite bath temperature, and asymptotically approaches the latter as the number of oscillators increases. It is suggested that realizations of these finite-size effects may be attained in computational and experimental dynamics of small molecules.
Systems whose boundaries are in contact with heat baths at different temperatures allow for the realization of nonequilibrium steady states (NESS) characterized by distribution functions that are still time independent, yet different from the equilibrium distributions. Such states have interesting properties even in classical statistical mechanics, but recently their quantum analogs have received attention, for both fundamental and applied reasons. This review provides an account of recent progress in this field. It introduces various models to describe the rich NESS phenomenology and covers analytic and numerical methods for analyzing them.