Truncated L\'evy Flights and Weak Ergodicity Breaking in the Hamiltonian Mean Field Model

TL;DR

This paper investigates the Hamiltonian mean field model's dynamics, revealing that it exhibits weak ergodicity breaking due to truncated Lévy distributions of sojourn times, with ergodicity restored only after very long times.

Contribution

It introduces a novel analysis of the Hamiltonian mean field model using continuous time random walks and identifies weak ergodicity breaking caused by truncated Lévy distributions.

Findings

Sojourn times follow a Lévy truncated distribution.

Weak ergodicity breaking occurs for long times.

Ergodicity is restored only at very long times for finite systems.

Abstract

The dynamics of the Hamiltonian mean field model is studied in the context of continuous time random walks. We show that the sojourn times in cells in the momentum space are well described by a L\'evy truncated distribution. Consequently the system in weakly non-ergodic for long times that diverge with the number of particles. For a finite number of particles ergodicity is only attained for very long times both at thermodynamical equilibrium and at quasi-stationary out of equilibrium states.