Statistical mechanics of self-gravitating systems: mixing as a criterion for indistinguishability

TL;DR

This paper links the phase-space mixing level of self-gravitating systems to particle indistinguishability, refining models of incomplete violent relaxation and deriving a new distribution function with features resembling cuspy density profiles.

Contribution

It introduces a novel association between mixing and indistinguishability, leading to a modified distribution function that better captures system density profiles.

Findings

Derived a distribution function with increased slope at high energies

Applied a correction to Stirling's approximation for low energies

Resembles cuspy density profiles but does not produce sharp cusps

Abstract

We propose an association between the phase-space mixing level of a self-gravitating system and the indistinguishability of its constituents (stars or dark matter particles). This represents a refinement in the study of systems exhibiting incomplete violent relaxation. Within a combinatorial analysis similar to that of Lynden-Bell, we make use of this association to obtain a distribution function that deviates from the Maxwell-Boltzmann distribution, increasing its slope for high energies. Considering the smallness of the occupation numbers for large distances from the center of the system, we apply a correction to Stirling's approximation which increases the distribution slope also for low energies. The distribution function thus obtained presents some resemblance to the "S" shape of distributions associated with cuspy density profiles (as compared to the distribution function obtained from the Einasto profile), although it is not quite able to produce sharp cusps. We also argue how the association between mixing level and indistinguishability can provide a physical meaning to the assumption of particle-permutation symmetry in the N-particle distribution function, when it is used to derive the one-particle Vlasov equation, which raises doubts about the validity of this equation during violent relaxation.