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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 72 (17 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...
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 52 (20 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 twotemperature problem for harmonic crystals
 J. Statist. 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 Ibragimov–Linniktype mixing condition. The random function is translationinvariant in x1,. ..."
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Cited by 6 (6 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 Ibragimov–Linniktype mixing condition. The random function is translationinvariant in x1,..., xd−1 and converges to different translationinvariant processes as xd Q±., with the distributions m±. We study the distribution mt of the solution at time t ¥ R. The main result is the convergence of mt to a Gaussian translationinvariant measure as tQ.. 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 m±=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: Harmonic crystal; random initial data; mixing condition; Gaussian measures; covariance matrices; characteristic functional.
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.