Critical dynamics of self-gravitating Langevin particles and bacterial populations

We study the critical dynamics of the generalized Smoluchowski-Poisson system (for self-gravitating Langevin particles) or generalized Keller-Segel model (for the chemotaxis of bacterial populations). These models [P. H. Chavanis and C. Sire, Phys. Rev. E 69, 016116 (2004)] are based on generalized stochastic processes leading to the Tsallis statistics. The equilibrium states correspond to polytropic configurations with index $n$ similar to polytropic stars in astrophysics. At the critical index $n_3=d∕(d-2)$ (where $d⩾2$ is the dimension of space), there exists a critical temperature ${\Theta }_c$ (for a given mass) or a critical mass $M_c$ (for a given temperature). For $\Theta >{\Theta }_c$ or $M<M_c$ the system tends to an incomplete polytrope confined by the box (in a bounded domain) or evaporates (in an unbounded domain). For $\Theta <{\Theta }_c$ or $M>M_c$ the system collapses and forms, in a finite time, a Dirac peak containing a finite fraction $M_c$ of the total mass surrounded by a halo. We study these regimes numerically and, when possible, analytically by looking for self-similar or pseudo-self-similar solutions. This study extends the critical dynamics of the ordinary Smoluchowski-Poisson system and Keller-Segel model in $d=2$ corresponding to isothermal configurations with $n_3\to +\infty$. We also stress the analogy between the limiting mass of white dwarf stars (Chandrasekhar’s limit) and the critical mass of bacterial populations in the generalized Keller-Segel model of chemotaxis.

Figure 1

Mass-radius relation for relativistic white dwarf stars at $T=0$ [58]. The radius vanishes for a limiting mass $M_{\mathit{\text{Chandra}}}$ corresponding to the ultrarelativistic limit $\left(R\right)$. The dashed line corresponds to the classical limit $\left(C\right)$.

Figure 2

Mass versus central density for relativistic white dwarf stars at $T=0$. Equilibrium states only exist for $M<M_{\mathit{\text{Chandra}}}$. For $M=M_{\mathit{\text{Chandra}}}$, the density profile is a Dirac peak. For $M>M_{\mathit{\text{Chandra}}}$, the system is expected to collapse and form a neutron star or a black hole. The corresponding density profiles are represented in Fig. 4 of [63].

Figure 3

Mass as a function of the central density for two-dimensional box-confined self-gravitating isothermal spheres with fixed temperature. Equilibrium states exist only for $M⩽M_c$. For $M=M_c$, the density profile is a Dirac peak and for $M>M_c$ the system undergoes gravitational collapse. More precisely, this curve represents $\eta \left(\alpha \right)∕4$ so it also gives the inverse temperature $T_c∕T$ as a function of the density contrast $\mathcal{R}={\rho }_0∕\rho \left(R\right)=\mathcal{R}\left(\alpha \right)$ for a fixed mass. The corresponding density profiles are represented in Fig. 1 of [60].

Figure 4

Series of equilibria for box-confined polytropes with different index (the figure is done for $d=3$). The full lines $(\alpha <{\xi }_1)$ correspond to incomplete polytropes whose profile is arrested by the box and the dashed lines $(\alpha >{\xi }_1)$ correspond to complete polytropes that are self-confined.

Figure 5

Mass as a function of the central density for box-confined self-gravitating polytropic spheres with critical index $n=n_3=3$ in $d=3$. Incomplete polytropes with $\rho \left(R\right)>0$ are represented by a solid line and complete polytropes with $R_{*}⩽R$ are represented by a dashed line. For ${\rho }_0\to +\infty$, the density profile tends to a Dirac peak. Equilibrium states exist only for $M⩽M_c$. For $M>M_c$ the system undergoes gravitational collapse. The curve represents $\eta \left(\alpha \right)∕\left[n_3^{n_3∕(n_3-1)}{\omega }_{n_3}\right]$ so it also gives the inverse temperature ${({\Theta }_c∕\Theta )}^{n_3∕(n_3-1)}$ as a function of the density contrast $\mathcal{R}\left(\alpha \right)$ for a fixed mass.

Figure 6

Density profiles of complete and incomplete polytropes for the critical index $n_3=3$ in $d=3$. We have considered three values of the central density ${\rho }_0=(M_c∕S_d{R}^d{\omega }_{n_3}){\alpha }^d$ corresponding to $\alpha =3<{\xi }_1$ (incomplete polytrope: $R_{*}>R$, $M<M_c$), $\alpha ={\xi }_1=6.89685\dots$ (limit polytrope: $R_{*}=R$, $M=M_c$), and $\alpha =20>{\xi }_1$ (complete polytrope: $R_{*}<R$, $M=M_c$).

Figure 7

Mass as a function of the central density for critical polytropes $n=n_3$ for different dimensions of space. We have plotted $\eta =n_3^{d∕2}{\omega }_{n_3}M∕M_c$ as a function of $\alpha ={({\rho }_0{S}_d{R}^d{\omega }_{n_3}∕M_c)}^{1∕d}$. The maximum mass is reached at the bullet corresponding to $\alpha ={\xi }_1$, $\eta ={\eta }_c=n_3^{d∕2}{\omega }_{n_3}$.

Figure 8

For $d=2$, $n=n_3=+\infty$, and deep into the collapse regime for $T=T_c∕2=1∕8$, we plot the density profile (full line), emphasizing its two components: the core is dominated by the invariant scaling profile (dotted line) given analytically by Eq. (81) containing a mass $M_c=T∕T_c$, and the halo obeys pseudoscaling (dashed line) with an exponent $\alpha \left(t\right)$ tending slowly to $d=2$ as ${\rho }_0\to +\infty$.

Figure 9

In $d=3$ and for $n_3=3$ and $\Theta =0.25>{\Theta }_c$ in a finite box $(R=1)$, we show the density at successive times, illustrating the convergence to the equilibrium density profile (dashed line). The inset illustrates the exponentially fast saturation of the central density for $\Theta >{\Theta }_c$, whereas a slower algebraic saturation is expected right at $\Theta ={\Theta }_c$.

Figure 10

For $d=3$, $n=n_3=3$, and deep into the collapse regime for $\Theta =0.75{\Theta }_c$, we plot the density profile (full line), emphasizing its two components: the core is dominated by the bounded invariant scaling profile (complete polytrope of index $n_3$) containing a mass $M_c={(\Theta ∕{\Theta }_c)}^{3∕2}$ (dotted line), and the halo obeys pseudoscaling (dashed line) with an exponent $\alpha \left(t\right)$ tending slowly to $d=3$ as ${\rho }_0\to +\infty$.

Figure 11

For $\Theta =0.75{\Theta }_c$ (here $d=3$, $n=n_3=3$), we plot ${\rho }_0^{-1}\left(t\right)\frac{d{\rho }_0}{dt}$ (top full line) and ${\widehat{\rho }}_0\left(t\right)$ (bottom full line) as a function of ${\rho }_0\left(t\right)$. Both grow with an effective exponent $\alpha ∕3\approx 0.93$ (dotted lines), which slowly increases and should saturate to unity (the dashed line has slope unity).

Figure 12

For $\Theta =0.75{\Theta }_c$ (here $d=3$, $n=n_3=3$), we plot $f_2^{\left(\alpha \right)}\left(x\right)$ (as defined in the text) as a function of $x=r∕r_0$, for different times for which the central density evolves from ${10}^2$ to ${10}^7$. Pseudoscaling is observed. The envelop of the tails decays with an apparent exponent $\alpha \approx 2.8$ (right dashed line), while the small $x$ behavior is quadratic (left dashed line).

Figure 13

In $d=3$ and $n=5>n_3$, and for a given initial density profile [$M\left(r\right)=r^3∕{({\mathrm{e}}^{-r^2}+r^2)}^{3∕2}$; thick line], we show the collapse dynamics observed at $\Theta =0.15$ (full lines for different times before $t_{\mathit{coll}}$) and the evaporation dynamics observed at $\Theta =1$ (dashed lines for different times). For this particular initial condition, we find ${\Theta }_{*}\approx 0.206$.

Figure 14

In $d=3$ and $n=5>n_3$, we present the evaporation density data collapse at $\Theta =1$. As time proceeds, the effect of gravity becomes negligible, and the scaling profiles converge to the one corresponding to free diffusive evaporation (full line). This is the Barenblatt solution whose invariant profile is a Tsallis distribution of Eq. (114) with index $\gamma$.

Figure 15

In $d=3$ and for $n=n_3=3$, we compare the evaporation profiles at different times for $\Theta =1>{\Theta }_c$, for self-gravitating particles (full lines), to the faster evaporation dynamics when gravity is switched off (dashed lines).

Figure 16

In $d=3$ and for $n=n_3=3$, we compare the scaling profiles for $\Theta =0.21$ near ${\Theta }_c\approx 0.20872$, $\Theta =1$, and $\Theta =100$ (top to bottom full lines; for clarity, the $\Theta =0.21$ profile has been scaled down by a factor 150). For $\Theta ⪢1$, the invariant profile corresponds to the Barenblatt solution (pure anomalous diffusion) which is a Tsallis distribution with index ${\gamma }_{4∕3}=1+1∕n_3$. For $\Theta \to {\Theta }_c$ the invariant profile tends to the profile of a steady polytrope with index $n_3$. For an intermediate temperature $\Theta =1$, we illustrate the perfect observed data collapse by plotting $r_0^d\left(t\right)\rho (r,t)$ as a function of $r∕r_0\left(t\right)$, for $t={1.5}^n(n=0,\dots ,13)$. These 14 curves are indistinguishable from the theoretical scaling profile. In the inset, we illustrate the scaling relation of Eq. (134) obtained for different values of $\epsilon =(\Theta -{\Theta }_c)∕{\Theta }_c\to 0$.