arXiv subjects

Claude-Alain Pillet

Publications and source records attributed to Claude-Alain Pillet.

At least 19 recordsLinked to original sources

arXiv2026

Entropy Maximization and Weak Gibbsianity of Quasi-Free Fermionic States

In their 1972 study of approach to equilibrium, Lanford and Robinson showed that gauge-invariant quasi-free states of lattice fermions maximize entropy among all translation-invariant states with a fixed two-point function, and suggested that the maximizer is unique. In subsequent work on this topic, the uniqueness question re-emerged, together with the problem of whether such quasi-free states are weak Gibbs states. We provide a positive answer to both questions within a class of states whose momentum-space two-point function $\widehat C$ satisfies $0<\widehat C(k)<1$ and belongs to the Wiener algebra of the Brillouin zone. The proof reveals that both the entropy maximization principle and weak Gibbsianity follow directly from the thermodynamic formalism for lattice fermions.

math-ph↗

arXiv2026

On entropy production of repeated quantum measurements III. Quantum detailed balance

In light of the dynamical-systems approach to entropy production in repeated quantum measurements, proposed and illustrated in Commun. Math. Phys. 357, 77-123 (2018) [arXiv:1607.00162] and J. Stat. Phys. 182, 44 (2021) [arXiv:2012.03885], we characterize the KMS quantum detailed balance condition for quantum channels via time-reversal invariance and the vanishing of the entropy production for the associated informationally complete quantum instruments.

quant-ph↗

arXiv2026

Miniatures on Open Quantum Systems

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.

math-ph↗

arXiv2026

Fluctuation Theorem and Thermodynamic Formalism

We study the Fluctuation Theorem (FT) for entropy production in chaotic discrete-time dynamical systems on compact metric spaces, and extend it to empirical measures, all continuous potentials, and all weak Gibbs states. In particular, we establish the FT in the phase transition regime. These results hold under minimal chaoticity assumptions (expansiveness and specification) and require no ergodicity conditions. They are also valid for systems that are not necessarily invertible and involutions other than time reversal. Further extensions involve asymptotically additive potential sequences and the corresponding weak Gibbs measures. The generality of these results allows to view the FT as a structural facet of the thermodynamic formalism of dynamical systems.

math-ph↗

arXiv2026

What is a Fluctuation Theorem?

This book provides a modern review of Fluctuation Relations and Fluctuation Theorems in nonequilibrium statistical mechanics. It focuses on the pioneering perspectives of Gallavotti and Cohen, according to which a fluctuation theorem describes the statistics of the deviations of entropy production from its expected value. For time-reversal invariant systems, these fluctuations obey a universal (i.e., model-independent) symmetry called the fluctuation relation. The probabilistic framework introduced in the first part of the book allows for a very general formulation of Fluctuation Relations and Theorems for both deterministic and stochastic dynamical systems. The authors further explore models of physical interest, illustrating this framework by concrete applications. The second part of the book focuses on chaotic dynamics. The formulation of two general Fluctuation Theorems, followed by the detailed study of a concrete example, provides the reader with an understanding of both the theoretical and practical aspects of the subject.

math-ph↗

arXiv2025

Entropic Fluctuation Theorems for the Spin-Fermion Model

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.

math-ph↗

arXiv2024

On the thermodynamic limit of two-times measurement entropy production

We provide a justification, via the thermodynamic limit, of the modular formula for entropy production in two-times measurement proposed in [Benoist, Bruneau, Jakšić, Panati and Pillet: Lett. Math. Phys. 114:32, (2023)]. We consider the cases of open quantum systems in which all thermal reservoirs are either (discrete) quantum spin systems or free Fermi gases.

math-ph↗

arXiv2024

A note on two-times measurement entropy production and modular theory

Recent theoretical investigations of the two-times measurement entropy production (2TMEP) in quantum statistical mechanics have shed a new light on the mathematics and physics of the quantum-mechanical probabilistic rules. Among notable developments are the extensions of entropic fluctuation relations to quantum domain and discovery of a deep link between 2TMEP and modular theory of operator algebras. All these developments concerned the setting where the state of the system at the instant of the first measurement is the same as the state whose entropy production is measured. In this work we consider the case where these two states are different and link this more general 2TEMP to modular theory. The established connection allows us to show that under general ergodicity assumptions the 2TEMP is essentially independent of the choice of the system state at the instant of the first measurement due to a decoherence effect induced by the first measurement. This stability sheds a new light on the concept of quantum entropy production, and, in particular, on possible quantum formulations of the celebrated classical Gallavotti--Cohen Fluctuation Theorem which will be studied in the continuation of this work.

math-ph↗

arXiv2023

A note on adiabatic time evolution and quasi-static processes in translation-invariant quantum systems

We study the slowly varying, non-autonomous quantum dynamics of a translation invariant spin or fermion system on the lattice $\mathbb Z^d$. This system is assumed to be initially in thermal equilibrium, and we consider realizations of quasi-static processes in the adiabatic limit. By combining the Gibbs variational principle with the notion of quantum weak Gibbs states introduced in [Jakšić, Pillet, Tauber, arXiv:2204.00440], we establish a number of general structural results regarding such realizations. In particular, we show that such a quasi-static process is incompatible with the property of approach to equilibrium studied in this previous work.

math-ph↗

arXiv2023

Approach to equilibrium in translation-invariant quantum systems: some structural results

We formulate the problem of approach to equilibrium in algebraic quantum statistical mechanics and study some of its structural aspects, focusing on the relation between the zeroth law of thermodynamics (approach to equilibrium) and the second law (increase of entropy). Our main result is that approach to equilibrium is necessarily accompanied by a strict increase of the specific (mean) energy and entropy. In the course of our analysis, we introduce the concept of quantum weak Gibbs state which is of independent interest.

math-ph↗

arXiv2022

Markovian Repeated Interaction Quantum Systems

We study a class of dynamical semigroups $(\mathbb{L}^n)_{n\in\mathbb{N}}$ that emerge, by a Feynman--Kac type formalism, from a random quantum dynamical system $(\mathcal{L}_{ω_n}\circ\cdots\circ\mathcal{L}_{ω_1}(ρ_{ω_0}))_{n\in\mathbb{N}}$ driven by a Markov chain $(ω_n)_{n\in\mathbb{N}}$. We show that the almost sure large time behavior of the system can be extracted from the large $n$ asymptotics of the semigroup, which is in turn directly related to the spectral properties of the generator $\mathbb{L}$. As a physical application, we consider the case where the $\mathcal{L}_ω$'s are the reduced dynamical maps describing the repeated interactions of a system $\mathcal{S}$ with thermal probes $\mathcal{C}_ω$. We study the full statistics of the entropy in this system and derive a fluctuation theorem for the heat exchanges and the associated linear response formulas.

math-ph↗

arXiv2021

On entropy production of repeated quantum measurements II. Examples

We illustrate the mathematical theory of entropy production in repeated quantum measurement processes developed in a previous work by studying examples of quantum instruments displaying various interesting phenomena and singularities. We emphasize the role of the thermodynamic formalism, and give many examples of quantum instruments whose resulting probability measures on the space of infinite sequences of outcomes (shift space) do not have the (weak) Gibbs property. We also discuss physically relevant examples where the entropy production rate satisfies a large deviation principle but fails to obey the central limit theorem and the fluctuation-dissipation theorem. Throughout the analysis, we explore the connections with other, a priori unrelated topics like functions of Markov chains, hidden Markov models, matrix products and number theory.

math-ph↗

arXiv2019

A Detailed Fluctuation Theorem for Heat Fluxes in Harmonic Networks out of Thermal Equilibrium

We continue the investigation, started in [J. Stat. Phys. 166, 926-1015 (2017)], of a network of harmonic oscillators driven out of thermal equilibrium by heat reservoirs. We study the statistics of the fluctuations of the heat fluxes flowing between the network and the reservoirs in the nonequilibrium steady state and in the large time limit. We prove a large deviation principle for these fluctuations and derive the fluctuation relation satisfied by the associated rate function.

math-ph↗

arXiv2019

Large Deviations and Fluctuation Theorem for Selectively Decoupled Measures on Shift Spaces

We establish the Level-1 and Level-3 Large Deviation Principles (LDPs) for invariant measures on shift spaces over finite alphabets under very general decoupling conditions for which the thermodynamic formalism does not apply. Such decoupling conditions arise naturally in multifractal analysis, in Gibbs states with hard-core interactions, and in the statistics of repeated quantum measurement processes. We also prove the LDP for the entropy production of pairs of such measures and derive the related Fluctuation Relation. The proofs are based on Ruelle-Lanford functions, and the exposition is essentially self-contained.

math-ph↗

arXiv2019

Large deviations and entropy production in viscous fluid flows

We study the motion of a particle in a random time-dependent vector field defined by the 2D Navier-Stokes system with a noise. Under suitable non-degeneracy hypotheses we prove that the empirical measures of the trajectories of the pair (velocity field, particle) satisfy the LDP with a good rate function. Moreover, we show that the law of a unique stationary solution restricted to the particle component possesses a positive smooth density with respect to the Lebesgue measure in any finite time. This allows one to define a natural concept of the entropy production, and to show that its time average is a bounded function of the trajectory. The proofs are based on a new criterion for the validity of the level-3 LDP for Markov processes and an application of a general result on the image of probability measures under smooth maps to the laws associated with the motion of the particle.

math-ph↗

arXiv2017

Energy statistics in open harmonic networks

We relate the large time asymptotics of the energy statistics in open harmonic networks to the variance-gamma distribution and prove a full Large Deviation Principle. We consider both Hamiltonian and stochastic dynamics, the later case including electronic RC networks. We compare our theoretical predictions with the experimental data obtained by Ciliberto et al. [Phys. Rev. Lett. 110, 180601 (2013)].

math-ph↗

arXiv2016

On entropy production of repeated quantum measurements I. General theory

We study entropy production (EP) in processes involving repeated quantum measurements of finite quantum systems. Adopting a dynamical system approach, we develop a thermodynamic formalism for the EP and study fine aspects of irreversibility related to the hypothesis testing of the arrow of time. Under a suitable chaoticity assumption, we establish a Large Deviation Principle and a Fluctuation Theorem for the EP.

math-ph↗

arXiv2016

Full statistics of erasure processes: Isothermal adiabatic theory and a statistical Landauer principle

We study driven finite quantum systems in contact with a thermal reservoir in the regime in which the system changes slowly in comparison to the equilibration time. The associated isothermal adiabatic theorem allows us to control the full statistics of energy transfers in quasi-static processes. Within this approach, we extend Landauer's Principle on the energetic cost of erasure processes to the level of the full statistics and elucidate the nature of the fluctuations breaking Landauer's bound.

math-ph↗