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Vojkan Jakšić

Publications and source records attributed to Vojkan Jakšić.

18 recordsLinked to original sources
arXiv2026

Approach to Equilibrium in Translation-Invariant Quantum Systems: Structural Results II

We continue the development, started in [AHP 25 (2023), 715-749], of a structural theory of approach to equilibrium in translation-invariant quantum spin systems. Dynamical Steady States (DSS) are defined as w*-limit points of the time-averaged evolution of an initial state. We show that if a DSS is an equilibrium state of a sufficiently regular interaction, then the specific energy of this interaction is necessarily a constant of the motion. Two distinct mechanisms lead to this conclusion, one based on dynamical surface energies and the other on conservation of specific relative entropy. We conjecture that these properties hold for a natural class of interactions. For a DSS admitting a physical equilibrium description, the associated equilibrium states are characterized by a dynamical Gibbs variational principle: they are precisely the minimizers of the specific relative entropy from the initial state over the physical equilibrium states. The achieved minimum equals the increase of specific entropy. The structural theory also yields characterizations of approach to thermal equilibrium, based on two model-dependent questions: restoration of a sufficiently local thermodynamic description at large times and classification of the corresponding local constants of motion. Once locality is restored, the constants of motion determine the possible asymptotic equilibrium states. In particular, if finite-range locality is restored and the energy is the only finite-range constant of the motion, then every DSS is a thermal equilibrium state at the temperature fixed by energy conservation. The finite-time dynamical input underlying this theory is established in a companion paper.

math-ph↗
arXiv2026

Dynamical Conservation Laws in Quantum Spin Systems

We study dynamical conservation laws and the propagation of thermodynamic structure in translation-invariant lattice quantum spin systems. Using Lieb-Robinson bounds [CMP 28 (1972), 251-257] in arbitrary dimension, and Araki's bounds [CMP 14 (1969), 120-157] in dimension one, we prove conservation of the specific entropy, specific energy, and specific relative entropy under suitable assumptions on the interactions. The transported local Hamiltonians admit surface energies that vanish per unit volume in the van Hove limit, and local observables remain a core for the transported generators at every finite time. Weak Gibbsianity is preserved for sufficiently small times in arbitrary dimension and for all times in dimension one. These results provide the analytical foundation for the structural theory of approach to equilibrium developed in a companion paper, and establish partial results toward the conjectures formulated there.

math-ph↗
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

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↗
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

Recurrence times, waiting times and universal entropy production estimators

The universal typical-signal estimators of entropy and cross entropy based on the asymptotics of recurrence and waiting times play an important role in information theory. Building on their construction, we introduce and study universal typical-signal estimators of entropy production in the context of nonequilibrium statistical mechanics of one-sided shifts over finite alphabets.

math-ph↗
arXiv2022

On a waiting-time result of Kontoyiannis: mixing or decoupling?

We introduce conditions of lower decoupling to the study of waiting-time estimations of the cross entropy between two mutually independent stationary stochastic processes. Although similar decoupling conditions have been used in the literature on large deviations and statistical mechanics, they appear largely unexplored in information theory. Building on a result of Kontoyiannis, namely Theorem 4 in [Kontoyiannis, J. Theor. Probab., 1998], and replacing the $ψ$-mixing condition in this result with a lower decoupling condition, we considerably extend the validity of waiting-time estimation of cross entropy.

math.PR↗
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

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↗
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↗
arXiv2015

Conductance and absolutely continuous spectrum of 1D samples

We characterize the absolutely continuous spectrum of the one-dimensional Schrödinger operators $h=-Δ+v$ acting on $\ell^2(\mathbb{Z}_+)$ in terms of the limiting behavior of the Landauer-Büttiker and Thouless conductances of the associated finite samples. The finite sample is defined by restricting $h$ to a finite interval $[1,L]\cap\mathbb{Z}_+$ and the conductance refers to the charge current across the sample in the open quantum system obtained by attaching independent electronic reservoirs to the sample ends. Our main result is that the conductances associated to an energy interval $I$ are non-vanishing in the limit $L\to\infty$ iff ${\rm sp}_{\rm ac}(h)\cap I=\emptyset$. We also discuss the relationship between this result and the Schrödinger Conjecture.

math-ph↗
arXiv2010

Entropic Fluctuations in Statistical Mechanics I. Classical Dynamical Systems

Within the abstract framework of dynamical system theory we describe a general approach to the Transient (or Evans-Searles) and Steady State (or Gallavotti-Cohen) Fluctuation Theorems of non-equilibrium statistical mechanics. Our main objective is to display the minimal, model independent mathematical structure at work behind fluctuation theorems. Besides its conceptual simplicity, another advantage of our approach is its natural extension to quantum statistical mechanics which will be presented in a companion paper. We shall discuss several examples including thermostated systems, open Hamiltonian systems, chaotic homeomorphisms of compact metric spaces and Anosov diffeomorphisms.

math-ph↗