Non-equilibrium entropy and dynamics in a system with long-range interactions
We extend the core-halo approach of Levin et al (2014 Phys. Rep. 535, 1) for the violent relaxation of long-range interacting system with a waterbag initial condition, in the case of a widely studied Hamiltonian mean field model. The Gibbs entropy maximization principle is considered with the constraints of energy conservation and of coarse-grained Casimir invariants of the Vlasov equation. The core-halo distribution function depends only on the one-particle mean-field energy, as is expected from the Jeans theorem, and depends on a set of parameters which in our approach is completely determined without having to solve an envelope equation for the contour of the initial state, as required in the original approach. We also show that a different ansatz can be used for the core-halo distribution with similar results. This work also reveals a link between a parametric resonance causing the non-equilibrium phase transition in the model, a dynamical property, and a discontinuity of the (non-equilibrium) entropy of the system.