Estimate of blow-up and relaxation time for self-gravitating Brownian particles and bacterial populations

TL;DR

This paper derives asymptotic expressions for blow-up and relaxation times of self-gravitating Brownian particles and bacterial populations near critical points, confirming results with numerical solutions and analyzing behavior at criticality.

Contribution

It provides the first explicit asymptotic formulas for blow-up and relaxation times near critical points in these systems, validated by numerical solutions.

Findings

Blow-up time scales as (eta - eta_c)^{-1/2} near criticality.

Derived asymptotic expression for relaxation time above critical temperature.

Obtained large-time density profile expansion at the critical point.

Abstract

We determine an asymptotic expression of the blow-up time t_coll for self-gravitating Brownian particles or bacterial populations (chemotaxis) close to the critical point. We show that t_coll=t_{*}(eta-eta_c)^{-1/2} with t_{*}=0.91767702..., where eta represents the inverse temperature (for Brownian particles) or the mass (for bacterial colonies), and eta_c is the critical value of eta above which the system blows up. This result is in perfect agreement with the numerical solution of the Smoluchowski-Poisson system. We also determine the asymptotic expression of the relaxation time close but above the critical temperature and derive a large time asymptotic expansion for the density profile exactly at the critical point.