A Hierarchy of Dynamic States of Relaxation

TL;DR

This paper introduces a hierarchy of dynamic relaxed gas sphere solutions derived from coupled Poisson and Liouville equations, revealing new relaxation states and extending classical static models.

Contribution

It develops a hierarchical framework linking static and dynamic gas sphere solutions through approximations of the Liouville equation, including a novel dynamic maximum relaxation state.

Findings

Defines a hierarchy of dynamic relaxed gas spheres

Connects solutions to the Poisson and Liouville equations

Identifies a new form of dynamic maximum relaxation

Abstract

We define a hierarchy of dynamic relaxed gas spheres as solutions of the Poisson equation coupled to a hierarchy of approximations of the Liouville equation leading, when this equation is satisfied, to the well-known isothermal gas spheres in the static case, but also to a new form of dynamic maximum relaxation. The two previous steps of the hierarchy correspond to an increasing degree of local relaxation at the center of the configuration.