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A classical long-range-interacting $N$-particle system relaxes to thermalequilibrium on time scales growing with $N$; in the limit $N\to \infty$ such arelaxation time diverges. However, a completely non-collisional relaxationprocess, known as violent relaxation, takes place on a much shorter time scaleindependent of $N$ and brings the system towards a non-thermal quasi-stationarystate. A finite system will eventually reach thermal equilibrium, while aninfinite system will remain trapped in the quasi-stationary state forever. Fortimes smaller than the relaxation time the distribution function of the systemobeys 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 thedynamics affects smaller and smaller scales in phase space, so that observablesnot depending upon small-scale details appear as relaxed after a short time.Herewith we present an approximation scheme able to describe violent relaxationin a one-dimensional toy-model, the Hamiltonian Mean Field (HMF). The approachdescribed 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 predictnon-equilibrium phase diagrams that turn out to be in good agreement with thoseobtained from the numerical integration of the Vlasov equation.

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