Weak convergence to equilibrium of statistical ensembles in integrable Hamiltonian systems
Abstract
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 → ∞. 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. Finally, this work is relevant for understanding the statistical mechanics of integrable and near-integrable Hamiltonian systems
- Publication Date:
- Research Org.:
- Lawrence Berkeley National Laboratory (LBNL), Berkeley, CA (United States)
- Sponsoring Org.:
- USDOE Office of Science (SC), High Energy Physics (HEP)
- OSTI Identifier:
- 1581067
- Alternate Identifier(s):
- OSTI ID: 1511777
- Grant/Contract Number:
- AC02-05CH11231
- Resource Type:
- Accepted Manuscript
- Journal Name:
- Journal of Mathematical Physics
- Additional Journal Information:
- Journal Volume: 60; Journal Issue: 5; Journal ID: ISSN 0022-2488
- Publisher:
- American Institute of Physics (AIP)
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS
Citation Formats
Mitchell, Chad. Weak convergence to equilibrium of statistical ensembles in integrable Hamiltonian systems. United States: N. p., 2019.
Web. doi:10.1063/1.5043419.
Mitchell, Chad. Weak convergence to equilibrium of statistical ensembles in integrable Hamiltonian systems. United States. https://doi.org/10.1063/1.5043419
Mitchell, Chad. Thu .
"Weak convergence to equilibrium of statistical ensembles in integrable Hamiltonian systems". United States. https://doi.org/10.1063/1.5043419. https://www.osti.gov/servlets/purl/1581067.
@article{osti_1581067,
title = {Weak convergence to equilibrium of statistical ensembles in integrable Hamiltonian systems},
author = {Mitchell, Chad},
abstractNote = {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 → ∞. 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. Finally, this work is relevant for understanding the statistical mechanics of integrable and near-integrable Hamiltonian systems},
doi = {10.1063/1.5043419},
journal = {Journal of Mathematical Physics},
number = 5,
volume = 60,
place = {United States},
year = {Thu May 09 00:00:00 EDT 2019},
month = {Thu May 09 00:00:00 EDT 2019}
}
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