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23
NonEquilibrium Statistical Mechanics of Anharmonic Chains Coupled to Two Heat Baths at Different Temperatures
, 1999
"... . We study the statistical mechanics of a finitedimensional nonlinear 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 differ ..."
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Cited by 53 (14 self)
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. We study the statistical mechanics of a finitedimensional nonlinear 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 noncompact phase space. These techniques are based on an extension of the commutator method of H ormander used in the study of hypoelliptic differential operators. 1. Intr...
A GallavottiCohen Type Symmetry in the Large Deviation Functional for Stochastic Dynamics
 J. STAT. PHYS
, 1999
"... ..."
Nonequilibrium statistical mechanics of strongly anharmonic chains of oscillators
 Comm. Math. Phys
"... We study the model of a strongly nonlinear chain of particles coupled to two heat baths at different temperatures. Our main result is the existence and uniqueness of a stationary state at all temperatures. This result extends those of Eckmann, Pillet, ReyBellet [EPR99a, EPR99b] to potentials with ..."
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Cited by 44 (11 self)
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We study the model of a strongly nonlinear chain of particles coupled to two heat baths at different temperatures. Our main result is the existence and uniqueness of a stationary state at all temperatures. This result extends those of Eckmann, Pillet, ReyBellet [EPR99a, EPR99b] to potentials with essentially arbitrary growth at infinity. This extension is possible by introducing a stronger version of Hörmander’s theorem for Kolmogorov equations to vector fields with polynomially bounded coefficients on unbounded domains.
L.: Entropy production in nonlinear, thermally driven hamiltonian systems
 J. Stat. Phys
, 1999
"... Abstract. 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 coupling ..."
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Cited by 39 (16 self)
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Abstract. 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. 1.
On a twotemperature problem for wave equation
 Markov Processes and Related Fields 8
, 2002
"... Consider the wave equation with constant or variable coefficients in IR 3. The initial datum is a random function with a finite mean density of energy that also satisfies a Rosenblatt or IbragimovLinniktype mixing condition. The random function converges to different spacehomogeneous processes a ..."
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Cited by 7 (7 self)
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Consider the wave equation with constant or variable coefficients in IR 3. The initial datum is a random function with a finite mean density of energy that also satisfies a Rosenblatt or IbragimovLinniktype mixing condition. The random function converges to different spacehomogeneous processes as x3 → ± ∞ , with the distributions µ ±. We study the distribution µt of the random solution at a time t ∈ IR. The main result is the convergence of µt to a Gaussian translationinvariant measure as t → ∞ that means central limit theorem for the wave equation. The proof is based on the Bernstein ‘roomcorridor ’ argument. The application to the case of the Gibbs measures µ ± = g ± with two different temperatures T ± is given. Limiting mean energy current density formally is − ∞ · (0,0,T+−T−) for the Gibbs measures, and it is finite and equals to −C(0,0,T+−T−) with C> 0 for the convolution with a nontrivial test function. 1
On convergence to equilibrium distribution, II. The wave equation in odd dimensions, with mixing
"... The paper considers the wave equation, with constant or variable coefficients in R n, with odd n \ 3. We study the asymptotics of the distribution mt of the random solution at time t ¥ R as t Q.. It is assumed that the initial measure m0 has zero mean, translationinvariant covariance matrices, and ..."
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Cited by 7 (6 self)
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The paper considers the wave equation, with constant or variable coefficients in R n, with odd n \ 3. We study the asymptotics of the distribution mt of the random solution at time t ¥ R as t Q.. It is assumed that the initial measure m0 has zero mean, translationinvariant covariance matrices, and finite expected energy density. We also assume that m0 satisfies a Rosenblatt or Ibragimov– Linniktype space mixing condition. The main result is the convergence of mt to a Gaussian measure m. as t Q., which gives a Central Limit Theorem (CLT) for the wave equation. The proof for the case of constant coefficients is based on an analysis of longtime asymptotics of the solution in the Fourier representation and Bernstein’s ‘‘roomcorridor’ ’ argument. The case of variable coefficients is treated by using a version of the scattering theory for infinite energy solutions, based on Vainberg’s results on local energy decay.
Fluctuation relations for diffusion process
 Commun. Math. Phys
"... The paper presents a unified approach to different fluctuation relations for classical nonequilibrium dynamics described by diffusion processes. Such relations compare the statistics of fluctuations of the entropy production or work in the original process to the similar statistics in the timerever ..."
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Cited by 7 (3 self)
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The paper presents a unified approach to different fluctuation relations for classical nonequilibrium dynamics described by diffusion processes. Such relations compare the statistics of fluctuations of the entropy production or work in the original process to the similar statistics in the timereversed process. The origin of a variety of fluctuation relations is traced to the use of different time reversals. It is also shown how the application of the presented approach to the tangent process describing the joint evolution of infinitesimally close trajectories of the original process leads to a multiplicative extension of the fluctuation relations. 1
Metastability in Interacting Nonlinear Stochastic Differential Equations II: LargeN Behaviour
, 2006
"... We consider the dynamics of a periodic chain of N coupled overdamped particles under the influence of noise, in the limit of large N. Each particle is subjected to a bistable local potential, to a linear coupling with its nearest neighbours, and to an independent source of white noise. For strong co ..."
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Cited by 6 (3 self)
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We consider the dynamics of a periodic chain of N coupled overdamped particles under the influence of noise, in the limit of large N. Each particle is subjected to a bistable local potential, to a linear coupling with its nearest neighbours, and to an independent source of white noise. For strong coupling (of the order N 2), the system synchronises, in the sense that all oscillators assume almost the same position in their respective local potential most of the time. In a previous paper, we showed that the transition from strong to weak coupling involves a sequence of symmetrybreaking bifurcations of the system’s stationary configurations, and analysed in particular the behaviour for coupling intensities slightly below the synchronisation threshold, for arbitrary N. Here we describe the behaviour for any positive coupling intensity γ of order N 2, provided the particle number N is sufficiently large (as a function of γ/N 2). In particular, we determine the transition time between synchronised states, as well as the shape of the “critical droplet ” to leading order in 1/N. Our techniques involve the control of the exact number of periodic orbits of a nearintegrable twist map, allowing us to give a detailed description of the system’s potential landscape, in which the metastable behaviour is encoded.
Statistical Mechanics of anharmonic lattices
 In Advances in Differential Equations and Mathematical Physics, Contemporary Mathematics 327
, 2003
"... Abstract. We discuss various aspects of a series of recent works on the nonequilibrium stationary states of anharmonic crystals coupled to heat reservoirs (see also [7]). We expose some of the main ideas and techniques and also present some open problems. 1. ..."
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Cited by 5 (1 self)
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Abstract. We discuss various aspects of a series of recent works on the nonequilibrium stationary states of anharmonic crystals coupled to heat reservoirs (see also [7]). We expose some of the main ideas and techniques and also present some open problems. 1.
On the existence of the dynamics for anharmonic quantum oscillator systems
 Rev. Math. Phys
, 2010
"... Abstract. We construct a W ∗dynamical system describing the dynamics of a class of anharmonic quantum oscillator lattice systems in the thermodynamic limit. Our approach is based on recently proved LiebRobinson bounds for such systems on finite lattices [19]. 1. ..."
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Cited by 5 (5 self)
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Abstract. We construct a W ∗dynamical system describing the dynamics of a class of anharmonic quantum oscillator lattice systems in the thermodynamic limit. Our approach is based on recently proved LiebRobinson bounds for such systems on finite lattices [19]. 1.