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 Ruelle-Lanford functions for quantum spin systems(2010-01) Ogata, Y; Rey-Bellet, LWe prove a large deviation principle for the expectation of macroscopic observables in quantum (and classical) Gibbs states. Our proof is based on Ruelle-Lanford functions [20, 34] and direct subadditivity arguments, as in the classical case [23, 32], instead of relying on G¨artner-Ellis theorem, and cluster expansion or transfer operators as done in the quantum case in [21, 13, 27, 22, 16, 28]. In this approach we recover, expand, and unify quantum (and classical) large deviation results for lattice Gibbs states. In the companion paper [29] we discuss the characterization of rate functions in terms of relative entropies.Publication 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 Ergodic properties of Markov processes(2006-01) Rey-Bellet, LPublication 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 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 Large deviations in quantum lattice systems: One-phase region(2005-01) Lenci, M; Rey-Bellet, LWe give large deviation upper bounds, and discuss lower bounds, for the Gibbs-KMS state of a system of quantum spins or an interacting Fermi gas on the lattice. We cover general interactions and general observables, both in the high temperature regime and in dimension one.Publication COARSE-GRAINING SCHEMES FOR STOCHASTIC LATTICE SYSTEMS WITH SHORT AND LONG-RANGE INTERACTIONS(2010-01) Katsoulakis, MA; Plechac, P; Rey-Bellet, L; Tsagkarogiannis, DWe develop coarse-graining schemes for stochastic many-particle microscopic models with competing short- and long-range interactions on a d-dimensional lattice. We focus on the coarse-graining of equilibrium Gibbs states and using cluster expansions we analyze the corresponding renormalization group map. We quantify the approximation properties of the coarse-grained terms arising from different types of interactions and present a hierarchy of correction terms. We derive semi-analytical numerical schemes that are accompanied with a posteriori error estimates for coarse-grained lattice systems with short and long-range interactions.Publication 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 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 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.