Rey_Bellet, Luc
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Professor, Department of Mathematics and Statistics
Last Name
Rey_Bellet
First Name
Luc
Discipline
Mathematics
Other Physics
Statistics and Probability
Other Physics
Statistics and Probability
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Statistical Mechanics, Probability and Analysis
Introduction
Luc Rey-Bellet studies a variety of problems in statistical mechanics (both equilibrium and nonequilibrium). Among them are the physical and mathematical properties of non-equilibrium steady states; the theory of large deviations and its applications to physical systems (billiards and quantum systems); coarse-graining strategies and numerical schemes for lattice spin systems; evolutionary game theory.
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Publication Metadata only Ergodic properties of Markov processes(2006-01) Rey-Bellet, LPublication Metadata only Open classical systems(2006-01) Rey-Bellet, LPublication Metadata only Large deviations in non-uniformly hyperbolic dynamical systems(2008-01) Rey-Bellet, L; Young, LSWe prove large deviation principles for ergodic averages of dynamical systems admitting Markov tower extensions with exponential return times. Our main technical result from which a number of limit theorems are derived is the analyticity of logarithmic moment generating functions. Among the classes of dynamical systems to which our results apply are piecewise hyperbolic diffeomorphisms, dispersing billiards including Lorentz gases, and strange attractors of rank one including Hénon-type attractors.Publication Open Access A NOTE ON THE NON-COMMUTATIVE LAPLACE–VARADHAN INTEGRAL LEMMA(2010-01) De Roeck, W; Maes, C; Netocny, K; Rey-Bellet, LWe continue the study of the free energy of quantum lattice spin systems where to the local Hamiltonian H an arbitrary mean field term is added, a polynomial function of the arithmetic mean of some local observables X and Y that do not necessarily commute. By slightly extending a recent paper by Hiai, Mosonyi, Ohno and Petz [10], we prove in general that the free energy is given by a variational principle over the range of the operators X and Y. As in [10], the result is a non-commutative extension of the Laplace–Varadhan asymptotic formula.Publication Metadata only Coarse-graining schemes and a posteriori error estimates for stochastic lattice systems(2007-01) Katsoulakis, MA; Plechac, P; Rey-Bellet, L; Tsagkarogiannis, DKThe primary objective of this work is to develop coarse-graining schemes for stochastic many-body microscopic models and quantify their effectiveness in terms of a priori and a posteriori error analysis. In this paper we focus on stochastic lattice systems of interacting particles at equilibrium. The proposed algorithms are derived from an initial coarse-grained approximation that is directly computable by Monte Carlo simulations, and the corresponding numerical error is calculated using the specific relative entropy between the exact and approximate coarse-grained equilibrium measures. Subsequently we carry out a cluster expansion around this first – and often inadequate – approximation and obtain more accurate coarse-graining schemes. The cluster expansions yield also sharp a posteriori error estimates for the coarse-grained approximations that can be used for the construction of adaptive coarse-graining methods. We present a number of numerical examples that demonstrate that the coarse-graining schemes developed here allow for accurate predictions of critical behavior and hysteresis in systems with intermediate and long-range interactions. We also present examples where they substantially improve predictions of earlier coarse-graining schemes for short-range interactions.Publication Open Access Fluctuations of the Entropy Production in Anharmonic Chains(2002-01) Rey-Bellet, L; Thomas, LPublication Open Access Deterministic Equations for Stochastic Spatial Evolutionary Games(2010-01) Hwang, SH; Katsoulakis, MA; Rey-Bellet, LIn this paper we investigate the approximation properties of the coarse-graining procedure applied to kinetic Monte Carlo simulations of lattice stochastic dynamics. We provide both analytical and numerical evidence that the hierarchy of the coarse models is built in a systematic way that allows for error control in both transient and long-time simulations. We demonstrate that the numerical accuracy of the CGMC algorithm as an approximation of stochastic lattice spin flip dynamics is of order two in terms of the coarse-graining ratio and that the natural small parameter is the coarse-graining ratio over the range of particle/particle interactions. The error estimate is shown to hold in the weak convergence sense. We employ the derived analytical results to guide CGMC algorithms and we demonstrate a CPU speed-up in demanding computational regimes that involve nucleation, phase transitions and metastability.Publication Open Access Low regularity solutions to a gently stochastic nonlinear wave equation in nonequilibrium statistical mechanics(2005-01) Rey-Bellet, L; Thomas, LEWe consider a system of stochastic partial differential equations modeling heat conduction in a non-linear medium. We show global existence of solutions for the system in Sobolev spaces of low regularity, including spaces with norm beneath the energy norm. For the special case of thermal equilibrium, we also show the existence of an invariant measure (Gibbs state).Publication Open Access Exponential Convergence to Non-Equilibrium Stationary States in Classical Statistical Mechanics(2001-01) Rey-Bellet, L; Thomas, LWe continue the study of a model for heat conduction [6] consisting of a chain of non-linear oscillators coupled to two Hamiltonian heat reservoirs at different temperatures. We establish existence of a Liapunov function for the chain dynamics and use it to show exponentially fast convergence of the dynamics to a unique stationary state. Ingredients of the proof are the reduction of the infinite dimensional dynamics to a finite-dimensional stochastic process as well as a bound on the propagation of energy in chains of anharmonic oscillators.Publication Open Access Entropic fluctuations in statistical mechanics: I. Classical dynamical systems(2011-01) Jakšić, V; Pillet, C-A; Rey-Bellet, LWithin 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. In addition to 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.