We consider a finite quantum system coupled to quasifree thermal reservoirs at differenttemperatures. Under the assumptions of small coupling and exponential decay of the reservoir correlation function, the large deviation generating function is shown to be analytic on a compact set. Our methodis different from the spectral deformation technique which was introduced recently in the study of spinboson-like models. As a corollary, we derive the Gallavotti-Cohen fluctuation relation for the entropy production.
We consider a finite quantum system coupled to quasifree thermal reservoirs at differenttemperatures. Under the assumptions of small coupling and exponential decay of the reservoir correlation function, the large deviation generating function is shown to be analytic on a compact set. Our methodis different from the spectral deformation technique which was introduced recently in the study of spinboson-like models. As a corollary, we derive the Gallavotti-Cohen fluctuation relation for the entropy production.
We study the heat current flowing between two baths consisting of harmonic oscillators interacting with a qubit through a spin-boson coupling. An explicit expression for the generating function of the total heat flowing between the right and left baths is derived by evaluating the corresponding Feynman-Vernon path integral by performing the noninteracting blip approximation (NIBA). We recover the known expression, obtained by using the polaron transform. This generating function satisfies the Gallavotti-Cohen fluctuation theorem, both before and after performing the NIBA. We also verify that the heat conductance is proportional to the variance of the heat current, retrieving the well-known fluctuation dissipation relation. Finally, we present numerical results for the heat current.
The celebrated Evans--Searles, respectively Gallavotti--Cohen, fluctuation theorem concerns certain universal statistical features of the entropy production rate of a classical system in a transient, respectively steady, state. In this paper, we review and compare several possible extensions of these fluctuation theorems to quantum systems. We present, in an introductory way, the approach proposed in a recent series of works [Lett. Math. Phys. 114, 32 (2024): doi:10.1007/s11005-024-01777-0], [Rev. Math. Phys 38, 2461006 (2026): doi:10.1142/S0129055X24610063], [Commun. Math. Phys. 406, 201 (2025): 10.1007/s00220-025-05360-z] and [J. Math. Phys. 67, 022101 (2026): doi:10.1063/5.0311182] including a previously missing extension of the Gallavotti--Cohen (steady state) version of the FT to two-time measurement entropy production in the algebraic setting of quantum dynamical systems. We focus on the open quantum systems, where a small system interacts with several (infinitely extended) reservoirs, and discuss the following approaches: naive, two-time measurement, phase space contraction and ancilla state tomography. We review the connection with modular theory. The latter provides a way to extend the classical notions to the quantum domain. We discuss the quantum version of the principle of regular entropic fluctuations, introduced in the classical context in [Nonlinearity 24, 699 (2011): doi:10.1088/0951-7715/24/3/003]. Finally, we relate the statistical properties of these various notions of entropy production to spectral resonances of quantum transfer operators and apply this approach to a paradigmatic model of non-equilibrium statistical mechanics: the spin-fermion model. The results presented here shed a new light on the nature of entropic fluctuations in quantum statistical mechanics.
We study entropic fluctuations in the Spin–Fermion model describing an N-level quantum system coupled to several independent thermal free Fermi gas reservoirs. We establish the quantum Evans–Searles and Gallavotti–Cohen fluctuation theorems and identify their link with entropic ancilla state tomography and quantum phase space contraction of non-equilibrium steady state. The method of proof involves the spectral resonance theory of quantum transfer operators developed by the authors in previous works.
The present thesis is concerned with the study of a quantum physical system composed of a small particle system (such as a spin chain) and several quantized massless boson fields (as photon gasses or phonon fields) at positive temperature. The setup serves as a simplified model for matter in interaction with thermal "radiation" from different sources. Hereby, questions concerning the dynamical and thermodynamic properties of particle-boson configurations far from thermal equilibrium are in the center of interest. We study a specific situation where the particle system is brought in contact with the boson systems (occasionally referred to as heat reservoirs) where the reservoirs are prepared close to thermal equilibrium states, each at a different temperature. We analyze the interacting time evolution of such an initial configuration and we show thermal relaxation of the system into a stationary state, i.e., we prove the existence of a time invariant state which is the unique limit state of the considered initial configurations evolving in time. As long as the reservoirs have been prepared at different temperatures, this stationary state features thermodynamic characteristics as stationary energy fluxes and a positive entropy production rate which distinguishes it from being a thermal equilibrium at any temperature. Therefore, we refer to it as non-equilibrium stationary state or simply NESS. The physical setup is phrased mathematically in the language of C*-algebras. The thesis gives an extended review of the application of operator algebraic theories to quantum statistical mechanics and introduces in detail the mathematical objects to describe matter in interaction with radiation. The C*-theory is adapted to the concrete setup. The algebraic description of the system is lifted into a Hilbert space framework. The appropriate Hilbert space representation is given by a bosonic Fock space over a suitable L2-space. The first part of the present work is concluded by the derivation of a spectral theory which connects the dynamical and thermodynamic features with spectral properties of a suitable generator, say K, of the time evolution in this Hilbert space setting. That way, the question about thermal relaxation becomes a spectral problem. The operator K is of Pauli-Fierz type. The spectral analysis of the generator K follows. This task is the core part of the work and it employs various kinds of functional analytic techniques. The operator K results from a perturbation of an operator L0 which describes the non-interacting particle-boson system. All spectral considerations are done in a perturbative regime, i.e., we assume that the strength of the coupling is sufficiently small. The extraction of dynamical features of the system from properties of K requires, in particular, the knowledge about the spectrum of K in the nearest vicinity of eigenvalues of the unperturbed operator L0. Since convergent Neumann series expansions only qualify to study the perturbed spectrum in the neighborhood of the unperturbed one on a scale of order of the coupling strength we need to apply a more refined tool, the Feshbach map. This technique allows the analysis of the spectrum on a smaller scale by transferring the analysis to a spectral subspace. The need of spectral information on arbitrary scales requires an iteration of the Feshbach map. This procedure leads to an operator-theoretic renormalization group. The reader is introduced to the Feshbach technique and the renormalization procedure based on it is discussed in full detail. Further, it is explained how the spectral information is extracted from the renormalization group flow. The present dissertation is an extension of two kinds of a recent research contribution by Jakšić and Pillet to a similar physical setup. Firstly, we consider the more delicate situation of bosonic heat reservoirs instead of fermionic ones, and secondly, the system can be studied uniformly for small reservoir temperatures. The adaption of the Feshbach map-based renormalization procedure by Bach, Chen, Fröhlich, and Sigal to concrete spectral problems in quantum statistical mechanics is a further novelty of this work.
The fluctuations of macroscopic observables in quantum systems which are in a nonequilibrium steady state are studied rigorously in the thermodynamic limit. In particular, the nonequilibrium steady state (NESS) of a quantum spin system that is composed of two infinite reservoirs at different temperatures coupled in a bounded region is considered. Under suitable assumptions on the interaction and the asymptotics of the fluctuations, a nonequilibrium central limit theorem for the fluctuations of the NESS expectation value of the empirical average of observables is proven by showing the analyticity of the corresponding moment generating function in a neighbourhood of the origin using the cluster expansion method. Furthermore, the large deviation principle is proven for a class of observables by establishing the existence and differentiability of the corresponding moment generating function.
We derive an exact (classical and quantum) expression for the entropy production of a finite system placed in contact with one or several finite reservoirs, each of which is initially described by a canonical equilibrium distribution. Although the total entropy of system plus reservoirs is conserved, we show that system entropy production is always positive and is a direct measure of system–reservoir correlations and/or entanglements. Using an exactly solvable quantum model, we illustrate our novel interpretation of the Second Law in a microscopically reversible finite-size setting, with strong coupling between the system and the reservoirs. With this model, we also explicitly show the approach of our exact formulation to the standard description of irreversibility in the limit of a large reservoir.