Anomalous diffusion and collapse of self-gravitating Langevin particles in D dimensions

TL;DR

This paper studies the thermodynamics, equilibrium states, and collapse phenomena of self-gravitating Langevin particles with anomalous diffusion in D-dimensional space, revealing complex stability and phase behaviors relevant to astrophysics and biological systems.

Contribution

It introduces a generalized thermodynamic framework for anomalously diffusing self-gravitating particles and analyzes their stability and collapse using analytical and turning point methods.

Findings

Equilibrium states are described by polytropic distributions.

The stability of these states depends on the dimension D and the polytropic index n.

Collapse solutions are characterized by self-similar dynamics when no equilibrium exists.

Abstract

We address the generalized thermodynamics and the collapse of a system of self-gravitating Langevin particles exhibiting anomalous diffusion in a space of dimension D. The equilibrium states correspond to polytropic distributions. The index n of the polytrope is related to the exponent of anomalous diffusion. We consider a high-friction limit and reduce the problem to the study of the nonlinear Smoluchowski-Poisson system. We show that the associated Lyapunov functional is the Tsallis free energy. We discuss in detail the equilibrium phase diagram of self-gravitating polytropes as a function of D and n and determine their stability by using turning points arguments and analytical methods. When no equilibrium state exists, we investigate self-similar solutions describing the collapse. These results can be relevant for astrophysical systems, two-dimensional vortices and for the chemotaxis of bacterial populations. Above all, this model constitutes a prototypical dynamical model of systems with long-range interactions which possesses a rich structure and which can be studied in great detail.