Thermodynamics and collapse of self-gravitating Brownian particles in D dimensions

TL;DR

This paper investigates the thermodynamics and collapse phenomena of self-gravitating Brownian particles across different dimensions, providing analytic density profiles, phase diagrams, and stability analysis in both canonical and microcanonical ensembles.

Contribution

It derives explicit density profiles and stability criteria for self-gravitating Brownian particles in arbitrary dimensions, including exact solutions in D=2 and large D limits.

Findings

Density profile decays as x^{-2} in canonical ensemble.

In microcanonical ensemble, decay exponent is non-trivial and derived.

Exact solutions and asymptotic expansions are provided for large D and D=2.

Abstract

We address the thermodynamics (equilibrium density profiles, phase diagram, instability analysis...) and the collapse of a self-gravitating gas of Brownian particles in D dimensions, in both canonical and microcanonical ensembles. In the canonical ensemble, we derive the analytic form of the density scaling profile which decays as f(x)=x^{-\alpha}, with alpha=2. In the microcanonical ensemble, we show that f decays as f(x)=x^{-\alpha_{max}}, where \alpha_{max} is a non-trivial exponent. We derive exact expansions for alpha_{max} and f in the limit of large D. Finally, we solve the problem in D=2, which displays rather rich and peculiar features.