Relaxation dynamics of stochastic long-range interacting systems

TL;DR

This paper investigates how stochastic processes affect the relaxation dynamics of long-range interacting systems, showing that stochasticity limits quasistationary states and alters relaxation times, contrasting with deterministic models.

Contribution

It introduces a generalized HMF model with stochastic collisions, demonstrating that QSS are only transient and that relaxation times do not scale algebraically with system size.

Findings

QSS occur only as a finite-time crossover under stochastic dynamics

Relaxation time does not scale algebraically with system size in stochastic models

Scaling form for relaxation time matches numerical simulations

Abstract

Long-range interacting systems, while relaxing towards equilibrium, may get trapped in nonequilibrium quasistationary states (QSS) for a time which diverges algebraically with the system size. These intriguing non-Boltzmann states have been observed under deterministic Hamiltonian evolution of a paradigmatic system, the Hamiltonian Mean-Field (HMF) model. We study here the robustness of QSS with respect to stochastic processes beyond deterministic dynamics within a microcanonical ensemble. To this end, we generalize the HMF model by allowing for stochastic three-particle collision dynamics in addition to the deterministic ones. By analyzing the resulting Boltzmann equation for the phase space density, we demonstrate that in the presence of stochasticity, QSS occur only as a crossover phenomenon over a finite time determined by the strength of the stochastic process. In particular, we argue that the relaxation time to equilibrium does not scale algebraically with the system size. We propose a scaling form for the relaxation time which is in very good agreement with results of extensive numerical simulations. The broader validity of these results is tested on a different stochastic HMF model involving microcanonical Monte Carlo dynamical moves.