Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
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Statistical mechanics of collisionless relaxation in a non-interacting system

Pierre de Buyl

Pierre de Buyl

Center for Nonlinear Phenomena and Complex Systems, Université Libre de Bruxelles (U.L.B.), Code Postal 231, Campus Plaine, B-1050 Brussels, Belgium

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David Mukamel

David Mukamel

Department of Physics of Complex Systems, The Weizmann Institute of Science, Rehovot 76100, Israel

[email protected]

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Stefano Ruffo

Stefano Ruffo

Dipartimento di Energetica ‘Sergio Stecco’, Universitá di Firenze, and INFN, via S. Marta 3, 50139 Firenze, Italia

Laboratoire de Physique, UMR-CNRS 5672, ENS Lyon, 46 Allée d’Italie, 69364 Lyon cedex 07, France

[email protected]

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    We introduce a model of uncoupled pendula, which mimics the dynamical behaviour of the Hamiltonian mean-field (HMF) model. This model has become a paradigm for long-range interactions, such as Coulomb or dipolar forces. As in the HMF model, this simplified integrable model is found to obey the Vlasov equation and to exhibit quasi-stationary states (QSSs), which arise after a ‘collisionless’ relaxation process. Both the magnetization and the single-particle distribution function in these QSSs can be predicted using Lynden-Bell’s theory. The existence of an extra conserved quantity for this model, the energy distribution function, allows us to understand the origin of some discrepancies of the theory with numerical experiments. It also suggests an improvement of Lynden-Bell’s theory, which we fully implement for the zero-field case.

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