  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.

Statistical Mechanics