Ergodicity Breaking and Parametric Resonances in Systems with Long-Range Interactions

TL;DR

This paper investigates how long-range interactions cause ergodicity breaking and quasi-stationary states, revealing conditions under which systems can be close to ergodic or exhibit chaos due to parametric resonances.

Contribution

It identifies the role of the generalized virial condition in maintaining ergodicity and explains how its violation leads to chaos via parametric resonances in long-range systems.

Findings

Quasi-stationary states can be described by Lynden-Bell statistics under certain conditions.

Violation of the virial condition excites parametric resonances, causing chaos.

Systems do not reach Boltzmann-Gibbs equilibrium in the thermodynamic limit.

Abstract

We explore the mechanism responsible for the ergodicity breaking in systems with long-range forces. In thermodynamic limit such systems do not evolve to the Boltzmann-Gibbs equilibrium, but become trapped in an out-of-equilibrium quasi-stationary-state. Nevertheless, we show that if the initial distribution satisfies a specific constraint - a generalized virial condition - the quasi- stationary-state is very close to ergodic and can be described by Lynden-Bell statistics. On the other hand if the generalized virial condition is violated, parametric resonances are excited, leading to chaos and ergodicity breaking.