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

We determine an exact asymptotic expression of the blow-up time $t_{coll}$ for self-gravitating Brownian particles or bacterial populations (chemotaxis) close to the critical point in $d=3$. We show that $t_{coll}=t_{*}{(\eta -{\eta }_c)}^{-1∕2}$ with $t_{*}=\mathrm{0.917\hspace{0.17em}677\hspace{0.17em}02}\dots$, 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 exact asymptotic expression of the relaxation time close to but above the critical temperature and derive a large time asymptotic expansion for the density profile exactly at the critical point.

Figure 1

We plot $\eta \left(\alpha \right)=T^{-1}\left(\alpha \right)$ where $\alpha ={(4\pi {\rho }_0∕T)}^{1∕2}$ parametrizes the series of equilibria. For $\alpha >{\alpha }_c$, unstable modes with $\lambda >0$ arise, which are solutions of Eq. (34). The minimum temperature $T_c$ corresponds precisely to the point of marginal stability $\lambda =0$ [8].

Figure 2

The integrated mass density is plotted at the critical point.

Figure 3

We plot $\chi \left(\xi \right)$ (lower full curve), $F\left(\xi \right)$ (upper full line), and the eigensolution $f\left(\xi \right)$ of Eq. (57) (dashed line; $f\left(\xi \right)$ has been divided by a factor of 3 for convenience).

Figure 4

We plot $t_{coll}\times {(T_c-T)}^{1∕2}∕T_c$ computed numerically as a function of ${(T_c-T)}^{1∕2}$, which should converge to $t_{*}=\mathrm{0.917\hspace{0.17em}977\hspace{0.17em}02}\dots .$ A quadratic fit in the region ${(T_c-T)}^{1∕2}<0.05$ retrieves the first four digits of $t_{*}$.

Figure 5

At $T=T_c$, and starting from a uniform mass density, we have computed numerically ${\rho }_0-\rho (r=0,t)$, where ${\rho }_0$ is the equilibrium central density. This is compared with the exact and universal large time estimate $c_{\rho }∕t$, with $c_{\rho }={\left(\pi \mu \right)}^{-1}=\mathrm{1.850\hspace{0.17em}338\hspace{0.17em}5}\dots$ (dashed line).