Statistical mechanics of collisionless relaxation in a noninteracting system

TL;DR

This paper introduces a simplified model of uncoupled pendula that mimics long-range interacting systems, demonstrating the emergence of Quasi Stationary States and analyzing their properties using Lynden-Bell's theory, with improvements accounting for additional conserved quantities.

Contribution

The paper presents a new integrable model that captures collisionless relaxation and Quasi Stationary States in long-range systems, extending Lynden-Bell's theory with an additional conserved quantity.

Findings

The model obeys the Vlasov equation and exhibits QSS similar to the HMF model.

Lynden-Bell's theory can predict magnetization and distribution functions in QSS.

Inclusion of the energy distribution function improves theoretical predictions.

Abstract

We introduce a model of uncoupled pendula, which mimics the dynamical behavior of the Hamiltonian Mean Field (HMF) model. This model has become a paradigm for long-range interactions, like Coulomb or dipolar forces. As in the HMF model, this simplified integrable model is found to obey the Vlasov equation and to exhibit Quasi Stationary States (QSS), which arise after a "collisionless" relaxation process. Both the magnetization and the single particle distribution function in these QSS can be predicted using Lynden-Bell's theory. The existence of an extra conserved quantity for this model, the energy distribution function, allows us to understand the origin of some discrepancies of the theory with numerical experiments. It also suggests us an improvement of Lynden-Bell's theory, which we fully implement for the zero field case.