Ergodicity and Central Limit Theorem in Systems with Long-Range Interactions

TL;DR

This paper examines the limitations of ergodicity and the applicability of the central limit theorem in long-range interacting systems, revealing slow convergence to Gaussian distributions and the non-attainment of complete mixing.

Contribution

It demonstrates that quasi-stationary states prevent full ergodic mixing and shows that the generalized central limit theorem leads only to Gaussian attractors in such systems.

Findings

Quasi-stationary states hinder complete ergodic mixing.

Velocity sums converge slowly to Gaussian distributions.

No non-Gaussian attractors exist in the studied models.

Abstract

In this letter we discuss the validity of the ergodicity hypothesis in theories of violent relaxation in long-range interacting systems. We base our reasoning on the Hamiltonian Mean Field model and show that the life-time of quasi-stationary states resulting from the violent relaxation does not allow the system to reach a complete mixed state. We also discuss the applicability of a generalization of the central limit theorem. In this context, we show that no attractor exists in distribution space for the sum of velocities of a particle other than the Gaussian distribution. The long-range nature of the interaction leads in fact to a new instance of sluggish convergence to a Gaussian distribution.