This paper deals with some thermodynamical aspects of a class of models in non equilibrium quantum statistical mechanics (QSM) which are commonly used to describe the behaviour of infinite quantum systems perturbed by a time‐dependent periodic force. We are interested to the thermodynamical properties of the set of non‐equilibrium steady states (NESS) of the perturbed system. We extend the construction of the C‐Liouvillean originally introduce in [1] to our time‐dependent situation to define the C‐Liouvillean of Floquet KF. The NESS are thus described by zero‐resonance eigenvectors of KF*. We also give a time dependent version of the rate of entropy production [2] for the perturbed system based on the relative entropy as defined by Araki [3] in terms of the relative modular operator.
This paper deals with some thermodynamical aspects of a class of models in non equilibrium quantum statistical mechanics (QSM) which are commonly used to describe the behaviour of infinite quantum systems perturbed by a time‐dependent periodic force. We are interested to the thermodynamical properties of the set of non‐equilibrium steady states (NESS) of the perturbed system. We extend the construction of the C‐Liouvillean originally introduce in [1] to our time‐dependent situation to define the C‐Liouvillean of Floquet KF. The NESS are thus described by zero‐resonance eigenvectors of KF*. We also give a time dependent version of the rate of entropy production [2] for the perturbed system based on the relative entropy as defined by Araki [3] in terms of the relative modular operator.
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.
In this thesis we study some non-equilibrium aspects of an interacting, massive scalar field theory treated perturbatively. This is done analysing some properties of the interacting KMS state constructed by Fredenhagen and Lindner [FL14] in the framework of perturbative Algebraic Quantum Field Theory. In the first part we treat the stability of KMS states, namely we check whether the free state evolved with the interacting dynamics converges to the interacting state. In the meantime we also analyse the return to equilibrium, that is the analogous property with the role of the free and interacting quantities exchanged. We prove that those two properties hold if the perturbation potential is of spatial compact support and that they fail otherwise, even if an adiabatic mean is considered. While the stability leads to non-curable divergencies, the analysis of return to equilibrium gives something finite, which is interpreted as a non-equilibrium steady state. The novelty of this non-equilibrium state drove us to try to characterise it in more details. To do so, in the second part we introduce relative entropy and entropy production for perturbative quantum field theory, justifying those definitions by proving their main properties. Furthermore, we showed that they are well-defined in the adiabatic limit if we consider densities. These two definitions allowed to prove that the non-equilibrium steady state is thermodynamically trivial, namely it has zero entropy production. The present thesis is based on [DFP18a, DFP18b].
Nonequilibrium steady-state currents, unlike their equilibrium counterparts, continuously dissipate energy into their physical surroundings leading to entropy production and time-reversal symmetry breaking. This Letter discusses these issues in the context of quantum impurity models. We use simple thermodynamic arguments to define the rate of entropy production sigma and show that sigma has a simple information-theoretic interpretation in terms of nonequilibrium distribution functions. This allows us to show that the entropy production is strictly positive for any nonequilibrium steady state. We conclude by applying these ideas to the resonance level model and the Kondo model.
Non-equilibrium steady states (NESS) of Boson systems with different phases are investigated with the aid of the C^*-algebraic method. The system consists of sample and several free or pure-hopping bosonic reservoirs coupled with each other. Initially, the sample and reservoirs are uncoupled and in equilibrium with temperatures, local densities, and phases. At t=0, the reservoirs couple with each other through the sample. NESS are constructed as time limit of the composition of initial state and time evolution of coupled system. In NESS, Josephson currents and the mean entropy production rates are calculated. The mean entropy production rate does not depend on phase differences. As consequence, the currents provided to phase differences produces no entropy.
We consider the temporal evolution of the entropy S of an isolated system from an initial non-equilibrium state 2· of entropy S = S i to the equilibrium state of maximum entropy S = S max > S i. The application of usual density matrix theory of quantum mechanics leads us to the surprising result P = dS/dt = 0. The entropy does not change in time. This result is valid in case of the equilibrium state. However it is wrong for a non-equilibrium state, because the entropy of the isolated system could not spontaneously increase from S i to S max by entropy production P > 0. This contradicts the second law of thermodynamics. We avoid this discrepancy by changing from quantum mechanics to quantum thermodynamics. Now the entropy production P = dS/dt and the time t are the relevant observables. In order to treat the temporal development of S and P, we at first prepare an initial state i of entropy production P i at time t < 0. Subsequently at ϑ t ≥ 0 the initial state i interacts with a continuum of final states f, which transfers the entropy production from i to f. The temporal evolution of the entropy production P(t) of our system is treated by means of the time-dependent perturbation theory. For that reason we split the operator of entropy production P = P 0 + W into two parts. The operator P 0 describes the unperturbed basic vectors of entropy production | i > and | f >, while the perturbation operator W determines the interaction between | i > and | f >. We expand the time-dependent state function < t | ψ > of the isolated system into the complete set of stationary eigenvectors | i > and | f > and calculate the time-dependent expansion coefficients c i(t) and c(f, t) by means of the dynamic equation of quantum thermodynamics − i k (d/dt) < t | ψ > = < t | P | ψ >. Here k means the Boltzmann constant, which is characteristic of quantum thermodynamics and represents the atomic entropy unit. The expansion coefficients are given by a system of coupled differential equations, from which we deduce an integral-differential equation for the coefficient c i(t) of the initial state i. Unfortunately it has been impossible to obtain an analytic solution of the complicated equation. However approximations are available. Accordingly, in the vicinity of the equilibrium state, the probability of still finding the initial state i of entropy production P i at time t is given by ω i(t) = | c i(t) |2 = exp (−Lt). The rate constant L ≥ σ depends on the density ρ(P f) of final states f of entropy production P f near by P i, and on the squared modulus of the perturbation matrix elements W if = < i | W | f >, which determines the interaction between | i > and | f >. The entropy production P(t) = P i ω i(t) = P i exp (−ϑLt) of the isolated system decreases exponentially and irreversibly with proceeding time t. Its rate-determining step is the transfer of entropy production from i to f. The integral over the entropy production of the final states f vanishes. Therefore the continuum states f do not contribute to the total entropy production P(t). The temporal evolution of the entropy S(t) can easily be calculated by integration of P(t). In conclusion, the entropy production of a non-equilibrium state is finite P(t) > 0 and only vanishes in the case of the equilibrium state P(∞) = 0. Thus, quantum thermodynamics agrees with the second law of thermodynamics.
We presents a unified and concise exposition of key topics in the mathematical theory of open quantum systems, developed within the framework of operator algebras. The manuscript consolidates and extends a series of invited articles originally prepared for the Modern Encyclopedia of Mathematical Physics, combining foundational material with modern perspectives on non-equilibrium quantum statistical mechanics. After introducing the C*- and W*-algebraic formulation of quantum mechanics, the paper reviews quantum dynamical systems, KMS states, and Tomita-Takesaki modular theory, as well as CCR and CAR algebras for bosonic and fermionic systems. Particular emphasis is placed on infinite systems, non-equilibrium steady states, entropy production, and linear response theory. The later sections develop a systematic treatment of small systems coupled to reservoirs, open lattice quantum spin systems, culminating in a detailed discussion of competing notions of quantum entropy production. The presentation highlights structural insights, conceptual clarity, and connections between equilibrium and non-equilibrium phenomena, providing a self-contained reference for researchers and graduate students in mathematical physics.