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].
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].
In the first part, we analyse the properties of an interacting, massive scalar field in an equilibrium state over Minkowski spacetime. We compare the known real- and imaginary-time formalisms of Thermal Field Theory with the recent construction by Fredenhagen and Lindner of a KMS state for perturbative interacting theories in the context of perturbative Algebraic Quantum Field Theory, in the adiabatic limit. In particular, we show that the construction of Fredenhagen and Lindner reduces to the real-time formalism only if the cocycle which intertwines between the free and interacting dynamics can be neglected. Furthermore, the Fredenhagen and Lindner construction reduces to the ordinary imaginary-time formalism if one considers the expectation value of translation invariant observables. We thus conclude that a complete description of thermal equilibrium for interacting scalar fields is generally obtained only by means of the state constructed by Fredenhagen and Lindner, which combines both formalisms of Thermal Field Theory. We also discuss the properties of the expansion of the Fredenhagen and Lindner construction in terms of Feynman diagrams in the adiabatic limit. We finally provide examples showing that the real- and the imaginary-time formalisms fail to describe thermal equilibrium already at first or second order in perturbation theory. The results presented in this part are summarized in (BDP19). In the second part, we discuss the so-called secular effects, characterized by the appearance of polynomial divergences in the large time limit of truncated perturbative expansions of expectation values in Quantum Field Theory. We show that, although such effect is an artifact of perturbation theory, and thus may not be obtained via exactly solving the dynamical equation if possible, they do not represent the breakdown of perturbation theory itself. Instead, we show that the polynomial divergences follow from a bad choice of state, and we present examples of states which produce expectation values whose perturbative expansion does not present secular effects. In particular, we point that it is possible to obtain non time-divergent perturbative expressions from thermal equilibrium states for the interacting theory. This last part is based on a research project which, by the time this thesis was written, had not been concluded yet.
In this thesis interacting relativistic and non relativistic Quantum Field Theories (QFT) are studied, investigating their equilibrium configurations and their long time evolution in the framework of perturbative Algebraic Quantum Field Theory, studying also convergence in a specific model. In the first part, concerning the relativistic case, we study, in formal perturbation theory, interacting KMS states generalising results by Fredenhagen and Lindner to fermionic field theories. As an application their stability is investigated together with the characterisation of the Debye screening effect in plasmas via the explicit solution of the Semiclassical Maxwell equations. Connected to these results, we also initiate a systematic study of secular growths appearing in the truncated perturbative series and affecting the validity of perturbation theory in its long time predicting capability. We restrict to theories on Minkowski spacetime and provide, for interactions depending on time via a switching process, a sufficient condition preventing secular growths in the truncated perturbative series. In the second part, we focus on a model of non relativistic quantum field theory. We construct equilibrium states at positive temperature in the presence of a condensation phase for a gas of non relativistic Bose particles on an infinite space interacting through a localised two body interaction. The convergence is proven, both for the interacting KMS state and for the entropy of the interacting KMS state relative to the free KMS state. The strategy of our proof combines ideas of Algebraic QFT, Constructive QFT and methods of statistical mechanics in the operator algebraic setting. Limits where the localisation of the two-body interaction is removed are eventually discussed, in combination with limits of the other physical parameters of the theory such as temperature, strength of the interaction and chemical potential. Finally, in the appendix, we present results on the Araki-Uhlmann relative entropy in fermionic quantum field theories, as well as an entropy-area relation, obtained via semiclassical methods, for the cosmological horizon in the static patch of de Sitter spacetime sourced by a coherent excitation of a free quantum scalar field. This completes the collection of original results obtained during the PhD.
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.
Relatively small many-body quantum systems are often used as hardware for quantum devices. Most of these devices will operate below the thermal limit where thermodynamics must be treated differently to account for quantum behaviours. It is therefore imperative that the thermodynamic properties of these systems are well understood, especially as they can limit the technologies but also help the fabrication and running of efficient quantum devices. \n \nIn this thesis we study quantum work and entropy production in closed many-body quantum systems out-of-equilibrium. We find that, for the systems studied, the largest average quantum work can be extracted in adiabatic weakly correlated regimes. These regimes are also seen to minimise the entropy produced, making them efficient regimes in which to operate devices based on these systems. \n \nAdiabatic evolutions are important for many quantum devices, and so it is important that they can be accurately characterised. The validity of current methods has been questioned recently, so in this thesis we propose the use of metrics as a good quantitative measure to characterise adiabaticity. We found that the density distance (a more accessible quantity than the wavefunction and its distance measures) alone can determine adiabaticity in a range of quantum systems, even at finite temperature. \n \nHowever, when calculating properties of many-body systems, there are many challenges often resulting in the need to approximate. In this thesis we propose a new style of approximation for quantum thermodynamic properties, taking inspiration from density functional theory (DFT). This new style uses the exact initial state of the system but approximates the dynamics and is seen to be computationally cheap but largely accurate. We test this with non-interacting and DFT approximated dynamics, finding that, surprisingly, the non-interacting dynamics give the most accurate results in most regimes, with the cheapest cost.
Within the framework of relativistic quantum field theory, a novel method is established which allows to distinguish non-equilibrium states admitting locally a thermodynamic interpretation. The basic idea is to compare these states with global equilibrium states (KMS states) by means of local thermal observables. With the help of such observables, the states can be ordered into classes of increasing local thermal stability. Moreover, it is possible to identify states exhibiting certain specific thermal properties of interest, such as a definite local temperature or entropy density. The method is illustrated in a simple model describing the spatio-temporal evolution of a “big heat bang”. 1
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.