4.4 Article

Weak convergence to equilibrium of statistical ensembles in integrable Hamiltonian systems

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JOURNAL OF MATHEMATICAL PHYSICS
卷 60, 期 5, 页码 -

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AMER INST PHYSICS
DOI: 10.1063/1.5043419

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  1. Office of Science, Office of High Energy Physics, of the U.S. Department of Energy [DE-AC02-05CH11231]

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This article explores the long-time behavior of the bounded orbits associated with an ensemble of initial conditions in a nondegenerate integrable Hamiltonian system, Such systems are inherently nonlinear and subject to highly regular phase space filamentation that can drive the ensemble of orbits toward a stationary state. Describing the statistical ensemble by a probability density on a neighborhood of a family of invariant tori, it is proved that the probability density describing the ensemble at time t converges weakly to an invariant density as t -> infinity. More generally, we provide sufficient conditions for convergence to equilibrium of a multiphase system in action-angle form. These ideas are applied to an illustrative exactly soluble example. This work is relevant for understanding the statistical mechanics of integrable and near-integrable Hamiltonian systems. Published under license by AIP Publishing.

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