Instability of a uniformly collapsing cloud of classical and quantum self-gravitating Brownian particles

TL;DR

This paper analyzes the stability and nonlinear evolution of a collapsing cloud of self-gravitating particles, comparing classical and quantum models, and introduces new equations and criteria for structure formation in such systems.

Contribution

It derives a new density contrast evolution equation in a collapsing frame and compares stability criteria with classical approaches, also extending to quantum Bose-Einstein condensates.

Findings

Different stability criteria from Jeans analysis.

Self-similar solutions for nonlinear regime.

Quantum effects modeled by nonlinear mean field equations.

Abstract

We study the growth of perturbations in a uniformly collapsing cloud of self-gravitating Brownian particles. This problem shares analogies with the formation of large-scale structures in a universe experiencing a "big-crunch" or with the formation of stars in a molecular cloud experiencing gravitational collapse. Starting from the barotropic Smoluchowski-Poisson system, we derive a new equation describing the evolution of the density contrast in the comoving (collapsing) frame. This equation can serve as a prototype to study the process of self-organization in complex media with structureless initial conditions. We solve this equation analytically in the linear regime and compare the results with those obtained by using the "Jeans swindle" in a static medium. The stability criteria, as well as the laws for the time evolution of the perturbations, are different. The Jeans criterion is expressed in terms of a critical wavelength $\lambda_J$ while our criterion is expressed in terms of a critical polytropic index $\gamma_{4/3}$. We also study the fragmentation process in the nonlinear regime. We determine the growth of the skewness, the long-wavelength tail of the power spectrum and find a self-similar solution to the nonlinear equations valid for large times. Finally, we consider dissipative self-gravitating Bose-Einstein condensates with short-range interactions and show that, in a strong friction limit, the dissipative Gross-Pitaevskii-Poisson system is equivalent to the quantum barotropic Smoluchowski-Poisson system. This yields a new type of nonlinear mean field Fokker-Planck equations including quantum effects.