The Vlasov equation and the Hamiltonian mean-field model |
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| Institution: | 1. Laboratoire J.-A. Dieudonné, UMR-CNRS 6621, Université de Nice, Parc Valrose, 06108 Nice cedex 02, France;2. Institut Non Linéaire de Nice, UMR-CNRS 6618, 1361 route des Lucioles 06560 Valbonne, France;3. Laboratoire de Physique, UMR-CNRS 5672, ENS Lyon, 46 Allée d’Italie, 69364 Lyon cedex 07, France;4. Dipartimento di Energetica “S. Stecco” and CSDC, Università di Firenze, INFM and INFN, Via S. Marta, 3 I-50139, Firenze, Italy;5. Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, 606-8501 Kyoto, Japan;1. Dipartimento di Física, Politecnico di Torino, Corso Duca degli Abruzzi 24, I-10129 Torino, Italy;2. Dipartimento di Física, Politecnico di Torino, Corso Duca degli Abruzzi 24, I-10129 Torino, Italy;3. INFN and Dipartimento di Física, Università di Cagliari, Cittadella Universitaria, I-09042 Monserrato (CA), Italy;1. Non-A project @ Inria, Parc Scientifique de la Haute Borne, 40 avenue Halley, 59650 Villeneuve d?Ascq, France;2. CRIStAL (UMR-CNRS 9189) Ecole Centrale de Lille, Avenue Paul Langevin, 59651 Villeneuve d?Ascq, France;3. Department of Control Systems and Informatics, University ITMO, 49 avenue Kronverkskiy, 197101 Saint Petersburg, Russia;1. University Lille1, Laboratoire d’Electrotechnique et d’Electronique de Puissance, France;2. University Lille1, Villeneuve d’Ascq Cedex, France;3. Department of Electrical & Computer Engineering of the University of Toronto, Toronto, Canada;4. École Nationale des Arts & Métier Paris-Tech, Lille, France;1. Laboratoire Jean Kuntzmann, Université de Grenoble-Alpes, CNRS, F38041 Grenoble Cedex 9, France;2. Institut Mines-Telecom, Telecom ParisTech, CNRS LTCI, 46 rue Barrault, 75634 Paris Cedex 13, France;3. Department of Mathematics and Statistics, Boston University, Boston, MA 02215, USA;4. Laboratoire Paul Painlevé, UMR 8524 du CNRS, Université Lille 1, 59655 Villeneuve d’Ascq, France;5. Department of Mathematics, Academy of Economical Studies, Bucharest, Romania |
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| Abstract: | We show that the quasi-stationary states of homogeneous (zero magnetization) states observed in the N-particle dynamics of the Hamiltonian mean-field (HMF) model are nothing but Vlasov stable homogeneous states. There is an infinity of Vlasov stable homogeneous states corresponding to different initial momentum distributions. Tsallis q-exponentials in momentum, homogeneous in angle, distribution functions are possible, however, they are not special in any respect, among an infinity of others. All Vlasov stable homogeneous states lose their stability because of finite N effects and, after a relaxation time diverging with a power-law of the number of particles, the system converges to the Boltzmann–Gibbs equilibrium. |
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