Relaxation dynamics of systems with long-range interactions

Systems with long-range interactions (namely, with interaction
potentials decaying slower than $1/r^d$ at large distances $r$ in $d$ dimensions) are rather common, e.g., self-gravitating systems, non-neutral plasmas, dipolar ferroelectrics and ferromagnets, etc. These systems are non-additive, leading to unusual properties, both thermodynamic (e.g., negative microcanonical specific heat, ensemble inequivalence) and dynamic (e.g., slow relaxation, breaking of ergodicity). After a brief review, I will discuss a paradigmatic example, the Hamiltonian Mean-Field model, involving $XY$ spins with
mean-field interactions. For this model, relaxation of some initial
state towards equilibrium proceeds through intermediate
quasistationary states, characterized by a slow variation of
macroscopic observables over time. The life time of these states
diverges with the system size. These states are observed in
deterministic microcanonical evolution within a certain energy
interval. Our recent study suggests that, in the presence of noise,
quasistationary states occur only as a crossover phenomenon, depending on the relative magnitudes of two timescales set by the system size and the level of noise in the evolution. Our proposed scaling form for the relaxation time to equilibrium is verified in simulations by a scheme of piecewise deterministic evolution for the model.