Postcollapse dynamics of self-gravitating Brownian particles and bacterial populations

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.

Figure 1

We plot $N_0\left(t\right)$ for small times (solid line). This is compared to $N_0{\left(t\right)}^{\text{theory}}\times [1+a{(t-t_{\mathit{coll}})}^b]$ (dashed line), where $N_0{\left(t\right)}^{\text{theory}}$ is given by Eq. (81) with $\mu =8.38917147\dots$, and $a\approx 1.7$ and $b\approx 0.33$ are fitting parameters. Note that the validity range of this fit goes well beyond the estimated $t_{*}$ with $t_{*}-t_{\mathit{coll}}\sim T^{D∕2}\sim 0.09$. The bottom inset illustrates the exponential decay of $1-N_0\left(t\right)\sim {\mathrm{e}}^{-\mathrm{\lambda }t}$. The best fit for $\mathrm{\lambda }$ leads to $\mathrm{\lambda }\approx 5.6362$ to be compared to the eigenvalue computed by means of Eq. (86), $\mathrm{\lambda }=5.6361253\dots$. Finally, the top inset illustrates the sensitivity of $N_0\left(t\right)$ to the space discretization, which introduces an effective cutoff (a factor of 4 between each of the three curves). Note the small time scale. Even the curve corresponding to the coarsest discretization becomes indistinguishable from the others for $t>0.448$.

Figure 2

In the postcollapse regime, we plot $\rho (r,t)∕{\rho }_0\left(t\right)$ as a function of the scaling variable $x=r∕r_0\left(t\right)$. A good data collapse is obtained for central residual densities in the range ${10}^3–{10}^6$. This is compared to the numerical scaling function computed from Eq. (79) (dashed line). The inset shows the comparison between this postcollapse scaling function (dashed line) and the scaling function below $t_{\mathit{coll}}$ which has been rescaled to have the same value at $x=0$, preserving the asymptotics: $S\left(x\right)={(3+x^2∕4)}^{-1}$ [see Eq. (36), solid line]. Note that the postcollapse scaling function is flatter near $x=0$, as $S\left(x\right)-1∕3\sim x^3$ (in $D=3$) above $t_{\mathit{coll}}$ instead of $S\left(x\right)-1∕3\sim x^2$, below $t_{\mathit{coll}}$.

Figure 3

The main plot represents $(\mathrm{\lambda }T-\frac{1}{4})T^{-2∕3}$ as a function of $T^{1∕3}$ (line and squares), which should converge to $c_{D=3}=2.33810741\dots$ for $T\to 0$ according to Eq. (101). We find perfect agreement with this value using a quadratic fit (dotted line). Furthermore, this fit shows that the slope at $T=0$ is in fact equal to $-2\pm {2.10}^{-4}$, suggesting that the next term to the expansion of Eq. (101) is $\mathrm{\lambda }=1∕4T+c_3{T}^{-1∕3}-2+\cdots$, in $D=3$. In the inset, we plot $\mathrm{\lambda }$ as a function of $T$ up to $T\approx T_c\approx 0.4$. The small-temperature analytical result of Eq. (101) is in very good agreement with the numerical data up to $T\sim 0.03$, whereas the large-$T$ estimate $\mathrm{\lambda }\left(T\right)=3+\frac{9}{10}{T}^{-1}+\cdots$ is only qualitatively correct in this range of physical temperatures, $T⩽T_c$.

Figure 4

For $T=0.01$ $(\mathrm{\lambda }=35.074198\dots ,\mu =0.31739877\dots )$, we plot $h\left(r\right)$ computed numerically from Eq. (90) (dashed line) and the theoretical expression of Eq. (92), which is valid for $1-r⪢{\mu }^2\sim T^{2∕3}$ (solid line). We also plot the theoretical expression of $z_0$ [dash-dotted line; see Eq. (98)] and the next-order perturbation result [dotted line; see Eq. (99)], which are valid in the region $1-r⪡{\mu }^2\sim T^{2∕3}$. The inset is a blowup of the region close to 1. Note how $h\left(r\right)$ varies by a quantity of order unity as $r$ varies by a quantity of order $T=0.01$. We have chosen a not too small value for $T$ in order to be able to visualize the two scale regimes in a single figure. Both approximations shown in the inset are getting better as $T$ decreases.

Figure 5

For $E=-0.45<E_c\approx -0.335$, we plot $N_0\left(t\right)$, for different values of $h$ decreasing by a factor of 2 for each curve from the top one to the bottom one. It is clear that $N_0\left(t\right)$ decreases as $h$ decreases. The inset shows the corresponding temperature plots. The temperature $T\left(t\right)$ increases as $h$ decreases. Note that in the precollapse regime, the temperature essentially does not depend on $h$. For the situation considered, it starts from $T\left(0\right)=0.1$ and culminates at $T\left(t_{\mathit{coll}}\right)\approx 0.5$.