Remarks about the thermostatistical description of the HMF model Part II: Phenomenology of Relaxation Dynamics

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After a general overview of some features of the relaxation dynamics of the Hamiltonian Mean Field model, its equilibrium thermodynamic properties are used to rephrase the out-of-equilibrium regime for energies below the critical point uc=0.75 in terms of an effective dynamical coexistence between a clustered and a gaseous phases, whose existence could be associated to the large relaxation times observed when u₁<u<uc, with u₁=0.5. Starting from the hypothesis that the parametric resonance is the microscopic mechanism allowing the energetic interchange between the particles during the collisional regime, a phenomenological Fokker-Planck equation based on a Langevin equation with a multiplicative noise is proposed in order to describe the collisional relaxation of this system towards its final equilibrium, which supports the following dependence of the collisional relaxation timescale τcr=τ₀N≡√IN/g.