Relaxation times of unstable states in systems with long range interactions

TL;DR

This paper investigates the relaxation times of unstable states in long-range interacting systems, revealing algebraic and logarithmic scaling behaviors in different phases through analytical and numerical methods.

Contribution

It provides a detailed analysis of relaxation times in Hamiltonian models with long-range interactions, highlighting the stability thresholds and scaling laws.

Findings

Stable states have relaxation times increasing algebraically with system size.

Unstable states exhibit relaxation times increasing logarithmically with system size.

Quasistationary states demonstrate robustness in the models studied.

Abstract

We consider several models with long-range interactions evolving via Hamiltonian dynamics. The microcanonical dynamics of the basic Hamiltonian Mean Field (HMF) model and perturbed HMF models with either global anisotropy or an on-site potential are studied both analytically and numerically. We find that in the magnetic phase, the initial zero magnetization state remains stable above a critical energy and is unstable below it. In the dynamically stable state, these models exhibit relaxation time scales that increase algebraically with the number $N$ of particles, indicating the robustness of the quasistationary state seen in previous studies. In the unstable state, the corresponding time scale increases logarithmically in $N$.