On the dynamical anomalies in the Hamiltonian Mean Field model

Abstract

We study the N-dependence of the thermodynamical variables and the dynamical behavior of the well-known Hamiltonian Mean Field model. Microcanonical analysis revealed a thermodynamic limit which defers from the a priory traditional assumption of the N-dependence of the coupling constant g as 1/N: to tend N?infinity keeping constant E/N^3 and g/N, prescription which guarantees the extensivity of the Boltzmann entropy. The analysis of dynamics leads to approximate the time evolution of the magnetization density m by means of a Langevin equation with multiplicative noise. This equation leads to a Fokker-Planck's equation which is N-independent when the time variable is scaled by the N-dependent time constant T_{mac}=sqrt(IN/g), which represents the characteristic time scale for the dynamical evolution of the macroscopic observables derived from the magnetization density. This results explains the origin of the slow relaxation regimen observed in microcanonical numerical computations of dynamics of this model system. Connection with the system self-similarity is suggested.