Equilibrium and nonequilibrium properties of systems with long-range interactions

TL;DR

This paper reviews the equilibrium and nonequilibrium behaviors of long-range interacting systems, highlighting phenomena like negative specific heat, quasi-stationary states, and their theoretical explanations, with applications to models like free electron lasers.

Contribution

It provides a comprehensive review of long-range systems, emphasizing the role of Lynden-Bell's entropy in understanding quasi-stationary states and extending statistical mechanics to nonequilibrium.

Findings

Quasi-stationary states' duration increases with system size.

Lynden-Bell's entropy explains relaxation processes.

Potential experimental tests in free electron laser dynamics.

Abstract

We briefly review some equilibrium and nonequilibrium properties of systems with long-range interactions. Such systems, which are characterized by a potential that weakly decays at large distances, have striking properties at equilibrium, like negative specific heat in the microcanonical ensemble, temperature jumps at first order phase transitions, broken ergodicity. Here, we mainly restrict our analysis to mean-field models, where particles globally interact with the same strength. We show that relaxation to equilibrium proceeds through quasi-stationary states whose duration increases with system size. We propose a theoretical explanation, based on Lynden-Bell's entropy, of this intriguing relaxation process. This allows to address problems related to nonequilibrium using an extension of standard equilibrium statistical mechanics. We discuss in some detail the example of the dynamics of the free electron laser, where the existence and features of quasi-stationary states is likely to be tested experimentally in the future. We conclude with some perspectives to study open problems and to find applications of these ideas to dipolar media.