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.
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 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.
Abstract 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 consider and compare several possible extensions of these fluctuation theorems to quantum systems. In addition to the direct two-time measurement approach whose discussion is based on Benoist et al. (Lett Math Phys 114:32, 2024. https://doi.org/10.1007/s11005-024-01777-0 ), we discuss a variant where measurements are performed indirectly on an auxiliary system called ancilla, and which allows to retrieve non-trivial statistical information using ancilla state tomography. We also show that modular theory provides a way to extend the classical notion of phase space contraction rate to the quantum domain, which leads to a third extension of the fluctuation theorems. We further discuss the quantum version of the principle of regular entropic fluctuations, introduced in the classical context in Jakšić et al. (Nonlinearity 24:699, 2011. https://doi.org/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. The obtained results shed a new light on the nature of entropic fluctuations in quantum statistical mechanics.
Nous considerons une chaine de spin quantique XY qui consiste dun parti a gauche, un parti aumilieu, et un parti a droit qui sont initialement en equilibre thermique aux temperatures T_l, T_c, et T_r, respectivement. Les systemes a gauche et a droit sont des reservoirs thermiques inniment etendus et le systeme central est un petit systeme quantique qui lie les deux reservoirs. S'il y a une difference en temperature, la chaleur et l'entropie secoulent dun parti de la chaine vers l'autre. Nous considerons les fonctionnelles Evans-Searles et Gallavotti-Cohen qui decrivent les fluctuations de ce flux quant a l'etat initiale du systeme et l'etat stationnaire hors-equilibre atteint par le systeme dans la limite du temps a l'infini. En plus, nous denissons les statistiques entieres pour la chaine XY et nous considerons la fonctionnelle d'entropie associee, ainsi qu'une classe naturelle des fonctionnelles qui interpole entre la fonctionnelle des statistiques entieres et la quantication directe de la caracterisation variable de la fonctionnelle Evan-Searles qui apparait dans la mecanique statistique classique hors-equilibre. La transformation Jordan-Wigner associe un gaz Fermi libre et une matrice Jacobi a notre chaine XY. En utilisant ce representation nous sommes capable de calculer les fonctionnelles entropiques dans la limite du temps a l'infini comme fonction des donnees de la matrice Jacobi associee. Nous demontrons que les fonctionnelles Gallavotti-Cohen et Evans-Searles sont identiques dans cette limite. En plus, Nous montrons que tous ces fonctionnelles entropiques sont egales dan la limite du temps a l'infini si et seulement si la matrice Jacobi est sans reflexion.
The fluctuation-dissipation theorem (FDT) is a cornerstone of statistical mechanics, linking the linear response of a system to external perturbations with the equilibrium fluctuations of its constituent degrees of freedom. Traditionally formulated for Hermitian quantum systems in thermal equilibrium, its applicability to non-Hermitian quantum systems, which are increasingly relevant in areas such as open quantum systems, PT-symmetric quantum mechanics, and topological phases, remains a complex and largely unresolved challenge. This paper presents a theoretical framework for a universal fluctuation-dissipation theorem applicable to a broad class of non-Hermitian quantum systems. We demonstrate how the presence of non-Hermiticity, characterized by complex energy eigenvalues and non-orthogonal eigenstates, fundamentally modifies the relationship between correlation functions and response functions. Our derivation utilizes a generalized time-evolution operator and incorporates concepts from non-Hermitian spectral theory, revealing a modified FDT that accounts for energy gain or loss inherent in these systems. The resulting theorem provides insights into the interplay between quantum noise and dissipation in environments far from conventional thermal equilibrium, offering a crucial tool for understanding and engineering non-Hermitian quantum devices. We discuss the implications for non-equilibrium steady states and the breakdown of standard notions of temperature and entropy, proposing a path towards experimental verification in platforms like photonic lattices and cold atoms with controlled losses.
Abstract In this brief report, following the recent developments on formulating a quantum analogue of the classical energy equipartition theorem for open systems where the heat bath comprises of independent oscillators, i.e. bosonic degrees of freedom, we present an analogous result for fermionic systems. The most general case where the system is connected to multiple reservoirs is considered and the mean energy in the steady state is expressed as an integral over the reservoir frequencies. Physically this would correspond to summing over the contributions of the bath degrees of freedom to the mean energy of the system over a suitable distribution function ρ ( ω ) dependent on the system parameters. This result holds for nonequilibrium steady states, even in the nonlinear regime far from equilibrium. We also analyze the zero temperature behaviour and low temperature corrections to the mean energy of the system.