Virial theorem and dynamical evolution of self-gravitating Brownian particles and bacterial populations in an unbounded domain

TL;DR

This paper derives a Virial theorem for self-gravitating Brownian particles and bacterial populations in unbounded domains, analyzing diffusion, collapse, and stability across different dimensions and equations of state.

Contribution

It extends the Virial theorem to unbounded domains and arbitrary equations of state, providing new insights into diffusion, collapse, and stability of self-gravitating systems.

Findings

Diffusion above critical temperature is modified by self-gravity.

Collapse at critical temperature shows logarithmic density increase.

Evaporation in higher dimensions is mainly pure diffusion with slight slowdown.

Abstract

We derive the Virial theorem appropriate to the generalized Smoluchowski-Poisson system describing self-gravitating Brownian particles and bacterial populations (chemotaxis). We extend previous works by considering the case of an unbounded domain and an arbitrary equation of state. We use the Virial theorem to study the diffusion (evaporation) of an isothermal Brownian gas above the critical temperature T_c in dimension d=2 and show how the effective diffusion coefficient and the Einstein relation are modified by self-gravity or chemotactic attraction. We also study the collapse at T=T_c and show that the central density increases logarithmically with time instead of exponentially in a bounded domain. Finally, for d>2, we show that the evaporation of the system is essentially a pure diffusion slightly slowed-down by self-gravity. We also study the linear dynamical stability of stationary solutions of the generalized Smoluchowski-Poisson system representing isolated clusters of particles and investigate the influence of the equation of state and of the dimension of space on the dynamical stability of the system. Finally, we propose a general kinetic and hydrodynamic description of self-gravitating Brownian particles and biological populations and recover known models in some particular limits.