Statistical mechanics of collisionless relaxation in a noninteracting system

Abstract

We introduce a model of uncoupled pendula, which mimics the dynamical behavior of the Hamiltonian Mean Field (HMF) model. This model has become a paradigm for long-range interactions, like 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 (QSS), which arise after a "collisionless" relaxation process. Both the magnetization and the single particle distribution function in these QSS 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 us an improvement of Lynden-Bell's theory, which we fully implement for the zero field case.

15 pages, 8 figures, accepted in Philosophical Transactions of the Royal Society A

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• The vmf90 program for the numerical resolution of the Vlasov equation for mean-field systems
• Phase transitions of quasistationary states in the Hamiltonian Mean Field model
• On the effectiveness of mixing in violent relaxation