We prove a large deviation principle for the expectation of macroscopic observables in quantum (a... more We 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 and direct subadditivity arguments, as in the classical case, instead of relying on G\"artner-Ellis theorem, and cluster expansion or transfer operators as done in the quantum case. In this approach we recover, expand, and unify
Within the abstract framework of dynamical system theory we describe a general approach to the Tr... more 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
Monte Carlo methods are a popular tool to sample from high-dimensional target distribution and to... more Monte Carlo methods are a popular tool to sample from high-dimensional target distribution and to approximate expectations of observables with respect to a given distribution, which very often is of Gibbs type. Sampling an ergodic Markov process which has the target distribution as its invariant measure can be used to compute approximately such expectations. Often the Markov processes used are time-reversible (i.e., they satisfy detailed balance) but our main goal here is to assess and quantify, in a novel way, how adding an irreversible part to the process can be used to improve the sampling properties. We focus on the diffusion setting (overdamped Langevin equations) and we explore performance criteria based on the large deviations theory for empirical measures. We find that large deviations theory can not only adequately characterize the efficiency of the approximations, but it can also be used as a vehicle to design Markov processes, whose time averages optimally (in the sense of variance reduction) approximates the quantities of interest. We quantify the effect of the added irreversibility on the speed of convergence to the target Gibbs measure and to the asymptotic variance of the resulting estimators for observables. One of our main finding is that adding irreversibility reduces the asymptotic variance of generic observables and we give an explicit characterization of when observables do not see their variances reduced in terms of a nonlinear Poisson equation. Our theoretical results are illustrated and supplemented by numerical simulations.
In recent papers it has been demonstrated that sampling a Gibbs distribution from an appropriate ... more In recent papers it has been demonstrated that sampling a Gibbs distribution from an appropriate time-irreversible Langevin process is, from several points of view, advantageous when compared to sampling from a time-reversible one. Adding an appropriate irreversible drift to the overdamped Langevin equation results in a larger large deviations rate function for the empirical measure of the process, a smaller variance for the long time average of observables of the process, as well as a larger spectral gap. In this work, we concentrate on irreversible Langevin samplers with a drift of increasing intensity. The asymptotic variance is monotonically decreasing with respect to the growth of the drift and we characterize its limiting behavior. For a Gibbs measure whose potential has one or more critical points, adding a large irreversible drift results in a decomposition of the process in a slow and fast component with fast motion along the level sets of the potential and slow motion in the orthogonal direction. This result helps understanding the variance reduction, which can be explained at the process level by the induced fast motion of the process along the level sets of the potential. Correspondingly the limit of the asymptotic variance is the asymptotic variance of the limiting slow motion which is a diffusion process on a graph.
XIVth International Congress on Mathematical Physics, 2006
We describe the ergodic and thermodynamical properties of chains of anharmonic oscillators couple... more We describe the ergodic and thermodynamical properties of chains of anharmonic oscillators coupled, at the boundaries, to heat reservoirs at positive and different temperatures. We discuss existence and uniqueness of stationary states, rate of convergence to stationarity, heat flows and entropy production, Kubo formula and Gallavotti-Cohen fluctuation theorem.
We discuss various aspects of a series of recent works on the nonequilibrium stationary states of... more We discuss various aspects of a series of recent works on the nonequilibrium stationary states of anharmonic crystals coupled to heat reservoirs (see also ). We expose some of the main ideas and techniques and also present some open problems.
We consider a finite chain of nonlinear oscillators coupled at its ends to two infinite heat bath... more We consider a finite chain of nonlinear oscillators coupled at its ends to two infinite heat baths which are at different temperatures. Using our earlier results about the existence of a stationary state, we show rigorously that for arbitrary temperature differences and arbitrary couplings, such a system has a unique stationary state. (This extends our earlier results for small temperature differences.) In all these cases, any initial state will converge (at an unknown rate) to the stationary state. We show that this stationary state continually produces entropy. The rate of entropy production is strictly negative when the temperatures are unequal and is proportional to the mean energy flux through the system.
We recover, expand, and unify quantum (and classical) large deviation results for lattice Gibbs s... more We recover, expand, and unify quantum (and classical) large deviation results for lattice Gibbs states. The main new ingredient in this paper is a control on the overlap of spectral projections for non-commutative observables. Our proof of large deviations is based on Ruelle-Lanford functions which establishes the existence of a rate function directly by subadditivity arguments, as done in the classical case in , instead of relying on Gärtner-Ellis theorem, and cluster expansion or transfer operators as done in the quantum case in . We assume that the Gibbs states are asymptotically decoupled , which controls the dependence of observables localized at different spatial locations. In the companion paper [29], we discuss the characterization of rate functions in terms of relative entropies.
Within the abstract framework of dynamical system theory we describe a general approach to the Tr... more 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.
We analyze the free energy and construct the Gibbs-KMS states for a class of quantum lattice syst... more We analyze the free energy and construct the Gibbs-KMS states for a class of quantum lattice systems, at low temperature and when the interactions are almost diagonal, in a suitable basis. The models we study may have continuous symmetries, our results however apply to intermediate temperatures where discrete symmetries are broken but continuous symmetries are not. Our results are based on quantum Pirogov-Sinai theory and a combination of high and low temperature expansions.
We study the statistical mechanics of a finite-dimensional non-linear Hamiltonian system (a chain... more We study the statistical mechanics of a finite-dimensional non-linear Hamiltonian system (a chain of anharmonic oscillators) coupled to two heat baths (described by wave equations). Assuming that the initial conditions of the heat baths are distributed according to the Gibbs measures at two different temperatures we study the dynamics of the oscillators. Under suitable assumptions on the potential and on the coupling between the chain and the heat baths, we prove the existence of an invariant measure for any temperature difference, i.e., we prove the existence of steady states. Furthermore, if the temperature difference is sufficiently small, we prove that the invariant measure is unique and mixing. In particular, we develop new techniques for proving the existence of invariant measures for random processes on a non-compact phase space. These techniques are based on an extension of the commutator method of Hörmander used in the study of hypoelliptic differential operators.
The primary objective of this work is to develop coarse-graining schemes for stochastic many-body... more The 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. %such as Ising-type models. The proposed algorithms are derived from an initial coarse-grained approximation that is
We prove a large deviation principle for the expectation of macroscopic observables in quantum (a... more We 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 and direct subadditivity arguments, as in the classical case, instead of relying on G\"artner-Ellis theorem, and cluster expansion or transfer operators as done in the quantum case. In this approach we recover, expand, and unify
Within the abstract framework of dynamical system theory we describe a general approach to the Tr... more 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
Monte Carlo methods are a popular tool to sample from high-dimensional target distribution and to... more Monte Carlo methods are a popular tool to sample from high-dimensional target distribution and to approximate expectations of observables with respect to a given distribution, which very often is of Gibbs type. Sampling an ergodic Markov process which has the target distribution as its invariant measure can be used to compute approximately such expectations. Often the Markov processes used are time-reversible (i.e., they satisfy detailed balance) but our main goal here is to assess and quantify, in a novel way, how adding an irreversible part to the process can be used to improve the sampling properties. We focus on the diffusion setting (overdamped Langevin equations) and we explore performance criteria based on the large deviations theory for empirical measures. We find that large deviations theory can not only adequately characterize the efficiency of the approximations, but it can also be used as a vehicle to design Markov processes, whose time averages optimally (in the sense of variance reduction) approximates the quantities of interest. We quantify the effect of the added irreversibility on the speed of convergence to the target Gibbs measure and to the asymptotic variance of the resulting estimators for observables. One of our main finding is that adding irreversibility reduces the asymptotic variance of generic observables and we give an explicit characterization of when observables do not see their variances reduced in terms of a nonlinear Poisson equation. Our theoretical results are illustrated and supplemented by numerical simulations.
In recent papers it has been demonstrated that sampling a Gibbs distribution from an appropriate ... more In recent papers it has been demonstrated that sampling a Gibbs distribution from an appropriate time-irreversible Langevin process is, from several points of view, advantageous when compared to sampling from a time-reversible one. Adding an appropriate irreversible drift to the overdamped Langevin equation results in a larger large deviations rate function for the empirical measure of the process, a smaller variance for the long time average of observables of the process, as well as a larger spectral gap. In this work, we concentrate on irreversible Langevin samplers with a drift of increasing intensity. The asymptotic variance is monotonically decreasing with respect to the growth of the drift and we characterize its limiting behavior. For a Gibbs measure whose potential has one or more critical points, adding a large irreversible drift results in a decomposition of the process in a slow and fast component with fast motion along the level sets of the potential and slow motion in the orthogonal direction. This result helps understanding the variance reduction, which can be explained at the process level by the induced fast motion of the process along the level sets of the potential. Correspondingly the limit of the asymptotic variance is the asymptotic variance of the limiting slow motion which is a diffusion process on a graph.
XIVth International Congress on Mathematical Physics, 2006
We describe the ergodic and thermodynamical properties of chains of anharmonic oscillators couple... more We describe the ergodic and thermodynamical properties of chains of anharmonic oscillators coupled, at the boundaries, to heat reservoirs at positive and different temperatures. We discuss existence and uniqueness of stationary states, rate of convergence to stationarity, heat flows and entropy production, Kubo formula and Gallavotti-Cohen fluctuation theorem.
We discuss various aspects of a series of recent works on the nonequilibrium stationary states of... more We discuss various aspects of a series of recent works on the nonequilibrium stationary states of anharmonic crystals coupled to heat reservoirs (see also ). We expose some of the main ideas and techniques and also present some open problems.
We consider a finite chain of nonlinear oscillators coupled at its ends to two infinite heat bath... more We consider a finite chain of nonlinear oscillators coupled at its ends to two infinite heat baths which are at different temperatures. Using our earlier results about the existence of a stationary state, we show rigorously that for arbitrary temperature differences and arbitrary couplings, such a system has a unique stationary state. (This extends our earlier results for small temperature differences.) In all these cases, any initial state will converge (at an unknown rate) to the stationary state. We show that this stationary state continually produces entropy. The rate of entropy production is strictly negative when the temperatures are unequal and is proportional to the mean energy flux through the system.
We recover, expand, and unify quantum (and classical) large deviation results for lattice Gibbs s... more We recover, expand, and unify quantum (and classical) large deviation results for lattice Gibbs states. The main new ingredient in this paper is a control on the overlap of spectral projections for non-commutative observables. Our proof of large deviations is based on Ruelle-Lanford functions which establishes the existence of a rate function directly by subadditivity arguments, as done in the classical case in , instead of relying on Gärtner-Ellis theorem, and cluster expansion or transfer operators as done in the quantum case in . We assume that the Gibbs states are asymptotically decoupled , which controls the dependence of observables localized at different spatial locations. In the companion paper [29], we discuss the characterization of rate functions in terms of relative entropies.
Within the abstract framework of dynamical system theory we describe a general approach to the Tr... more 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.
We analyze the free energy and construct the Gibbs-KMS states for a class of quantum lattice syst... more We analyze the free energy and construct the Gibbs-KMS states for a class of quantum lattice systems, at low temperature and when the interactions are almost diagonal, in a suitable basis. The models we study may have continuous symmetries, our results however apply to intermediate temperatures where discrete symmetries are broken but continuous symmetries are not. Our results are based on quantum Pirogov-Sinai theory and a combination of high and low temperature expansions.
We study the statistical mechanics of a finite-dimensional non-linear Hamiltonian system (a chain... more We study the statistical mechanics of a finite-dimensional non-linear Hamiltonian system (a chain of anharmonic oscillators) coupled to two heat baths (described by wave equations). Assuming that the initial conditions of the heat baths are distributed according to the Gibbs measures at two different temperatures we study the dynamics of the oscillators. Under suitable assumptions on the potential and on the coupling between the chain and the heat baths, we prove the existence of an invariant measure for any temperature difference, i.e., we prove the existence of steady states. Furthermore, if the temperature difference is sufficiently small, we prove that the invariant measure is unique and mixing. In particular, we develop new techniques for proving the existence of invariant measures for random processes on a non-compact phase space. These techniques are based on an extension of the commutator method of Hörmander used in the study of hypoelliptic differential operators.
The primary objective of this work is to develop coarse-graining schemes for stochastic many-body... more The 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. %such as Ising-type models. The proposed algorithms are derived from an initial coarse-grained approximation that is
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