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44
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
"... ..."
Exponential Convergence to NonEquilibrium Stationary States in Classical Statistical Mechanics
 Comm. Math. Phys
, 2001
"... We continue the study of a model for heat conduction [6] consisting of a chain of nonlinear oscillators coupled to two Hamiltonian heat reservoirs at dierent temperatures. We establish existence of a Liapunov function for the chain dynamics and use it to show exponentially fast convergence of the d ..."
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Cited by 32 (5 self)
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We continue the study of a model for heat conduction [6] consisting of a chain of nonlinear oscillators coupled to two Hamiltonian heat reservoirs at dierent 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 innite dimensional dynamics to a nitedimensional stochastic process as well as a bound on the propagation of energy in chains of anharmonic oscillators. 1 Introduction In its present state, nonequilibrium statistical mechanics is lacking the rm theoretical foundations that equilibrium statistical mechanics has. This is due, perhaps, to the extremely great variety of physical phenomena that nonequilibrium statistical mechanics describes. We will concentrate here on a system which is maintained, by suitable forces, in a state far from equilibrium. In such an idealization, the nonequilibrium phenome...
Fourier’s law for a microscopic model of heat conduction
 J. Stat. Phys
, 2005
"... Abstract. We consider a chain of N harmonic oscillators perturbed by a conservative stochastic dynamics and coupled at the boundaries to two gaussian thermostats at different temperatures. The stochastic perturbation is given by a diffusion process that exchange momentum between nearest neighbor osc ..."
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Cited by 26 (11 self)
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Abstract. We consider a chain of N harmonic oscillators perturbed by a conservative stochastic dynamics and coupled at the boundaries to two gaussian thermostats at different temperatures. The stochastic perturbation is given by a diffusion process that exchange momentum between nearest neighbor oscillators conserving the total kinetic energy. The resulting total dynamics is a degenerate hypoelliptic diffusion with a smooth stationary state. We prove that the stationary state, in the limit as N → ∞, satisfies Fourier’s law and the linear profile for the energy average. 1.
Fourier’s law for a harmonic crystal with selfconsistent stochastic reservoirs
, 2004
"... We consider a ddimensional harmonic crystal in contact with a stochastic Langevin type heat bath at each site. The temperatures of the ‘‘exterior’ ’ left and right heat baths are at specified values TL and TR, respectively, while the temperatures of the ‘‘interior’ ’ baths are chosen selfconsisten ..."
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Cited by 25 (3 self)
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We consider a ddimensional harmonic crystal in contact with a stochastic Langevin type heat bath at each site. The temperatures of the ‘‘exterior’ ’ left and right heat baths are at specified values TL and TR, respectively, while the temperatures of the ‘‘interior’ ’ baths are chosen selfconsistently so that there is no average flux of energy between them and the system in the steady state. We prove that this requirement uniquely fixes the temperatures and the self consistent system has a unique steady state. For the infinite system this state is one of local thermal equilibrium. The corresponding heat current satisfies Fourier’s law with a finite positive thermal conductivity which can also be computed using the Green–Kubo formula. For the harmonic chain (d=1) the conductivity agrees with the expression obtained by Bolsterli, Rich, and Visscher in 1970 who first studied this model. In the other limit, d ± 1, the stationary infinite volume heat conductivity behaves as (add) −1 where ad is the coupling to the intermediate reservoirs. We also analyze the effect of having a nonuniform distribution of the heat bath couplings. These results are proven rigorously by controlling the behavior of the correlations in the thermodynamic limit. KEY WORDS: Fourier’s law; harmonic crystal; nonequilibrium systems; thermodynamic limit; Green–Kubo formula.
THERMAL CONDUCTIVITY FOR A MOMENTUM CONSERVING MODEL
, 2006
"... Abstract. We present here complete mathematical proofs of the results annouced in condmat/0509688. We introduce a model whose thermal conductivity diverges in dimension 1 and 2, while it remains finite in dimension 3. We consider a system of harmonic oscillators perturbed by a stochastic dynamics c ..."
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Cited by 15 (13 self)
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Abstract. We present here complete mathematical proofs of the results annouced in condmat/0509688. We introduce a model whose thermal conductivity diverges in dimension 1 and 2, while it remains finite in dimension 3. We consider a system of harmonic oscillators perturbed by a stochastic dynamics conserving momentum and energy. We compute the finitesize thermal conductivity via GreenKubo formula. In the limit as the size N of the system goes to infinity, conductivity diverges like N in dimension 1 and like ln N in dimension 2. Conductivity remains finite if d ≥ 3 or if a pinning (on site potential) is present. 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
Twotemperature problem for harmonic crystal
 J. Stat. Phys
"... We consider the dynamics of a harmonic crystal in d dimensions with n components, d,n ≥ 1. The initial date is a random function with finite mean density of the energy which also satisfies a Rosenblatt or IbragimovLinniktype mixing condition. The random function converges to different spacehomog ..."
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Cited by 5 (5 self)
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We consider the dynamics of a harmonic crystal in d dimensions with n components, d,n ≥ 1. The initial date is a random function with finite mean density of the energy which also satisfies a Rosenblatt or IbragimovLinniktype mixing condition. The random function converges to different spacehomogeneous processes as xd → ± ∞ , with the distributions µ ±. We study the distribution µt of the solution at time t ∈ IR. The main result is the convergence of µt to a Gaussian translationinvariant measure as t → ∞. The proof is based on the long time asymptotics of the Green function and on Bernstein’s ‘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 is −(0,...,0,C(T+−T−)) with some positive constant C> 0 what corresponds to Second Law. Key words and phrases: harmonic crystal, random initial data, mixing condition, convergence, Gaussian measures, covariance matrices, characteristic functional 1
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.
Thermal Conductivity for a Noisy Disordered Harmonic Chain
, 808
"... Abstract. We consider a ddimensional disordered harmonic chain (DHC) perturbed by an energy conservative noise. We obtain uniform in the volume upper and lower bounds for the thermal conductivity defined through the GreenKubo formula. These bounds indicate a positive finite conductivity. We prove ..."
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Cited by 5 (4 self)
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Abstract. We consider a ddimensional disordered harmonic chain (DHC) perturbed by an energy conservative noise. We obtain uniform in the volume upper and lower bounds for the thermal conductivity defined through the GreenKubo formula. These bounds indicate a positive finite conductivity. We prove also that the infinite volume homogenized GreenKubo formula converges. 1.