Mixing and Ergodicity in Systems with Long-Range Interactions

TL;DR

This paper develops a new theory for collisionless relaxation in long-range interacting systems, emphasizing local mixing over global ergodicity, and successfully predicts particle distributions and phase transitions in the HMF model.

Contribution

It introduces a theory of local mixing that explains quasi-stationary states in long-range systems, contrasting with traditional ergodic assumptions, and accurately predicts system behavior.

Findings

Accurately predicts particle distribution functions in qSS.

Precisely forecasts phase transitions in the HMF model.

Shows broken global ergodicity in long-range systems.

Abstract

We present a theory of collisionless relaxation in systems with long-range interactions. Contrary to Lynden-Bell's theory of violent relaxation, which assumes global ergodicity and mixing, we show that quasi-stationary states (qSS) observed in these systems exhibit broken global ergodicity. We propose that relaxation towards equilibrium occurs through a process of local mixing, where particles spread over energy shells defined by the manifold to which their trajectories are confined. To demonstrate our theory, we study the Hamiltonian Mean Field (HMF) model, a paradigmatic system with long-range interactions. Our theory accurately predicts the particle distribution functions in qSS observed in molecular dynamics simulations without any adjustable parameters. Additionally, it precisely forecasts the phase transitions observed in the HMF model.