Violent relaxation in the Hamiltonian mean field model: II. Non-equilibrium phase diagrams

TL;DR

This paper develops an approximation scheme to describe violent relaxation in the Hamiltonian Mean Field model, predicting non-equilibrium phase diagrams that match numerical Vlasov simulations, extending previous cold initial condition results.

Contribution

It introduces a generalized approach for violent relaxation in the HMF model applicable to generic initial conditions, improving understanding of non-equilibrium phase diagrams.

Findings

The approximation accurately predicts non-equilibrium phase diagrams.

The approach extends previous models limited to cold initial conditions.

Results agree well with numerical Vlasov equation simulations.

Abstract

A classical long-range-interacting $N$-particle system relaxes to thermal equilibrium on time scales growing with $N$; in the limit $N\to \infty$ such a relaxation time diverges. However, a completely non-collisional relaxation process, known as violent relaxation, takes place on a much shorter time scale independent of $N$ and brings the system towards a non-thermal quasi-stationary state. A finite system will eventually reach thermal equilibrium, while an infinite system will remain trapped in the quasi-stationary state forever. For times smaller than the relaxation time the distribution function of the system obeys the collisionless Boltzmann equation, also known as the Vlasov equation. The Vlasov dynamics is invariant under time reversal so that it does not "naturally" describe a relaxational dynamics. However, as time grows the dynamics affects smaller and smaller scales in phase space, so that observables not depending upon small-scale details appear as relaxed after a short time. Herewith we present an approximation scheme able to describe violent relaxation in a one-dimensional toy-model, the Hamiltonian Mean Field (HMF). The approach described here generalizes the one proposed in G. Giachetti and L. Casetti, J. Stat. Mech.: Theory Exp. 2019, 043201 (2019), that was limited to "cold" initial conditions, to generic initial conditions, allowing us to to predict non-equilibrium phase diagrams that turn out to be in good agreement with those obtained from the numerical integration of the Vlasov equation.