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

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 $u_{c}=0.75$ 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 $u_{1}<u<u_{c}$, with $u_{1}=0.5$. Starting from the hypothesis that the \textit{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 $\tau_{cr}=\tau_{0}N\equiv\sqrt{IN/g}$.