Postcollapse dynamics of self-gravitating Brownian particles and bacterial populations

Abstract

We address the postcollapse dynamics of a self-gravitating gas of Brownian particles in D dimensions in both canonical and microcanonical ensembles. In the canonical ensemble, the postcollapse evolution is marked by the formation of a Dirac peak with increasing mass. The density profile outside the peak evolves self-similarly with decreasing central density and increasing core radius. In the microcanonical ensemble, the postcollapse regime is marked by the formation of a "binarylike" structure surrounded by an almost uniform halo with high temperature. These results are consistent with thermodynamical predictions in astrophysics. We also show that the Smoluchowski-Poisson system describing the collapse of self-gravitating Brownian particles in a strong-friction limit is isomorphic to a simplified version of the Keller-Segel equations describing the chemotactic aggregation of bacterial populations. Therefore, our study has direct applications in this biological context.

The Generalized Stochastic Smoluchowski Equation

“…If the free energy possesses several local minima, the choice of the equilibrium state depends on a complicated notion of basin of attraction (for self-gravitating Brownian particles [ 28 , 29 ] the free energy is not bounded from below. In that case, the system can either relax towards a local minimum of free energy F at fixed mass – when it exists – or collapse to a Dirac peak [ 30 ], leading to a divergence of the free energy ). According to the preceding results, we conclude that dynamical stability with respect to the generalized mean field Smoluchowski equation and generalized thermodynamical stability in the canonical ensemble coincide.…”

“…for all perturbations δρ that conserve mass. Using Equation (30), the condition of thermodynamical stability can also be written as…”

“…When several stable solutions exist, their selection depends on a notion of basin of attraction [ 14 ]. The mean field Smoluchowski Equation ( 5 ) appeared in different domains of physics such as electrolytes [ 15 , 16 , 17 , 18 ], systems with long-range interactions [ 14 , 19 , 20 , 21 , 22 , 23 ], bacterial populations undergoing chemotaxis [ 24 ], kinetic theory of adsorbates [ 25 ], superconductors [ 26 , 27 ], self-gravitating Brownian particles [ 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 ], directed self-assembly of nanoparticles [ 44 ], 2D Brownian vortices [ 45 , 46 ], colloids at a fluid interface [ 47 ], nucleation [ 48 ], the Brownian mean field (BMF) model [ 49 ], etc. These models are reviewed in [ 50 ].…”

Generalized Stochastic Fokker-Planck Equations

“…The Smoluchowski-Poisson system has been studied in [16,[58][59][60][61][62]. The screened Smoluchowski-Poisson system and the modified Smoluchowski-Poisson system have been studied in [63,64].…”

Global solutions for a problem modeling the dynamics of a system of self-gravitating Brownian particles

“…The local well-posedness of (1.1)-(1.6) has been investigated in [15] and [10]. Explicit solutions describing the pre-collapse and post-collapse dynamics have been obtained in [7,8,17,18] for a spherically symmetric distribution of matter.…”