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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 ..."
Abstract

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...
Spectral Theory of Thermal Relaxation
 J. Math. Phys
, 1997
"... . We review some results obtained in a recent series of papers on thermal relaxation in classical and quantum dissipative systems. We consider models where a small system S , with a finite number of degrees of freedom, interacts with a large environment R in thermal equilibrium at positive temperatu ..."
Abstract

Cited by 9 (1 self)
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. We review some results obtained in a recent series of papers on thermal relaxation in classical and quantum dissipative systems. We consider models where a small system S , with a finite number of degrees of freedom, interacts with a large environment R in thermal equilibrium at positive temperature T . The zeroth law of thermodynamics postulates that, independently of its initial configuration, the system S approaches a unique stationary state as t !1. By definition, this limiting state is the equilibrium state of S at temperature T . Statistical mechanics further identifies this state with the Gibbs canonical ensemble associated with S . For simple models we prove that the above picture is correct, provided the equilibrium state of the environment R is itself given by its canonical ensemble. In the quantum case we also obtain an exact formula for the thermal relaxation time. Spectral Theory of Thermal Relaxation 2 1. Introduction This paper is an informal outline of the results...