Self-gravitating Brownian systems and bacterial populations with two or more types of particles

We study the thermodynamical properties of a self-gravitating gas with two or more types of particles. Using the method of linear series of equilibria, we determine the structure and stability of statistical equilibrium states in both microcanonical and canonical ensembles. We show how the critical temperature (Jeans instability) and the critical energy (Antonov instability) depend on the relative mass of the particles and on the dimension of space. We then study the dynamical evolution of a multicomponent gas of self-gravitating Brownian particles in the canonical ensemble. Self-similar solutions describing the collapse below the critical temperature are obtained analytically. We find particle segregation, with the scaling profile of the slowest collapsing particles decaying with a nonuniversal exponent that we compute perturbatively in different limits. These results are compared with numerical simulations of the two-species Smoluchowski-Poisson system. Our model of self-attracting Brownian particles also describes the chemotactic aggregation of a multispecies system of bacteria in biology.

Figure 1

The dimensionless density profiles ${\tilde{\rho }}_1\left(\xi \right)=\lambda \mu e^{-\mu \psi }$ and ${\tilde{\rho }}_2\left(\xi \right)=e^{-\psi }$ for $\mu =5$ and for $\lambda =1$ in $d=1$ (a) and $d=3$ (b). The dashed line represents the density of the one component system.

Figure 2

The solution of the two-species Emden equation in the $(u,v)$ plane for $d=1$ (up, a) and in $d=3$ (down, b). The single-species case is represented by the dashed line.

Figure 3

Series of equilibria (caloric curves) for a two-component isothermal gas in $d=1$ (up, a) and in $d=3$ (down, b). We plot the inverse normalized temperature $\eta =GM{m}_2\beta ∕R^{d-2}$ as a function of the normalized energy $\Lambda =-E{R}^{d-2}∕G{M}^2$ for several values of the total mass ratio $\chi =M_1∕M_2$. The dashed curve represents the one component case, i.e., $\chi =0$.

Figure 4

Evolution of the critical normalized inverse temperature ${\eta }_c$ (Jeans temperature) and of the critical normalized energy ${\Lambda }_c$ (Antonov energy) as a function of $\chi$ for $d=3$ and $\mu =5$. We plot with a dashed-dotted line the critical temperature ${\eta }_J$ obtained by using the Jeans swindle (see Appendix for details). Note that this “naive” prediction provides a reasonable fit of the exact critical temperature ${\eta }_c$.

Figure 5

Evolution of the minimum Antonov energy ${\left({\Lambda }_c\right)}_{\min }$ as a function of $\mu$ for $d=3$. In the range considered, it decreases approximately linearly as $-0.19719\mu$. Note that the minimum Antonov energy becomes positive for $\mu \simeq 2.627$ and $\chi \simeq 0.68$. Furthermore, the value of $\chi$ for which ${\Lambda }_c$ is minimum is always close to 0.7 (except for $\mu \to 1$). This is probably related to the fact that ${\left({\Lambda }_c\right)}_{\min }\left(\mu \right)$ is almost linear in this range.

Figure 6

Dimensionless density profile ${\tilde{\rho }}_2\left(\xi \right)=e^{-\psi }$ of the lightest particles enclosed within a box for $d=3$, $\mu =5$, $\chi =1∕9$ and $\alpha =5000$. In the limit $\alpha \to +\infty$, the profile decays as ${\xi }^{-2∕\mu }$ for $1⪡\xi <{\xi }^{\prime }\sim \alpha$. This can be contrasted to the ${\xi }^{-2}$ decay for $\xi \to +\infty$ in an open system with fixed $\lambda$ (see Fig. 1).

Figure 7

The $\lambda \left(\eta \right)$ curves for several values of $\chi =M_1∕M_2$ for $d=3$. These curves are parameterized by $\alpha$.

Figure 8

Dimensionless density profiles ${\tilde{\rho }}_1\left(\xi \right)=\lambda \mu e^{-\mu \psi }$ and ${\tilde{\rho }}_2\left(\xi \right)=e^{-\psi }$ in dimension $d=2$ for $\lambda =1$ and for $\mu =2$ and $\mu =5$. For comparison, the dashed line represents the density of the singe-component system with slope $-4$. The slope of the profile $e^{-\psi }\sim {\xi }^{-\delta }$ depends on $\lambda$ and $\mu$.

Figure 9

The solution of Eq. (29) in the $(u,v)$ plane, where $u$ and $v$ are defined by Eq. (38) for the two-components system at $d=2$. The single-species case is represented by the dashed line. All the curves start at $(u,v)=(d,0)$ with the initial slope given by Eq. (41). For $\xi \to +\infty$, they tend to the terminal point $(0,\delta )$.

Figure 10

An ensemble of caloric curves for different values of $\chi =M_1∕M_2$ in the two-dimensional case. The dashed curve represents the one-component case.

Figure 11

The critical temperature ${\eta }_c$ is plotted versus $\chi$ (a) and $\mu$ (b). The solid lines represent the numerical results. They are in excellent agreement with the theoretical results (76, 70) in the two regimes. The transition between these regimes appears when $\chi ={\chi }_{*}=1∕(\mu -2)$ in Fig. 11a and when $\mu ={\mu }_{*}=2+1∕\chi$ in Fig. 11b.

Figure 12

The normalized density profile ${\tilde{\rho }}_2=e^{-\psi }$ is plotted for $\mu =5>2$ and for two different values of $\chi$ situated from both sides of the critical value ${\chi }_{*}=1∕(\mu -2)=1∕3$. For $\chi =1∕9<{\chi }_{*}$ (regime I), the asymptotic slope of the profile is $-{\eta }_c=-3.6799$ given by Eq. (70). For $\chi =1>{\chi }_{*}$ (regime II), the asymptotic slope in the range $1⪡\xi ⪡{\xi }^{\prime }\sim \alpha$ is $-4∕\mu =-0.8$, see Eq. (71). The final asymptotic slope in the range ${\xi }^{\prime }<\xi <\alpha$ is $-{\eta }_c=-8∕5$ given by Eq. (76).

Figure 13

The critical ratio ${\zeta }_c$ as a function of $\mu$ in $d=3$ is plotted in a log-log scale. This function starts at ${\zeta }_c\left(0\right)=1∕6=(d-2)∕2d$ and diverges at $\mu =2$. Below the critical line, species 1 dominates the collapse and above the critical line, species 2 dominates. In that case, our study can still be used with the transformation $(\zeta ,\mu )\to (1∕\zeta ,1∕\mu )$. This gives a corresponding point located below the dashed curve, corresponding to the function $1∕{\zeta }_c(1∕\mu )$. We show an example illustrating this transformation when $\zeta >{\zeta }_c$. The (×) symbol represents the point (0.65,0.41) and the (+) symbols represent (0.65,0.61) and $(1∕0.65,1∕0.61)$, respectively.

Figure 14

We plot the scaling profiles $S_1$ and $S_2$ for $\mu =0.65$ and for different values of $\zeta$. We take $\zeta =0.41<{\zeta }_c=0.511070$ (dashed lines) and $\zeta =0.61>{\zeta }_c$ (solid lines). This corresponds to the points marked × and + in Fig. 13. For $\zeta >{\zeta }_c$, the exponent ${\alpha }^{\prime }$ is obtained from Eq. (94) using the equivalent point $(1∕\mu ,1∕\zeta )$.

Figure 15

The resolution of the time-dependent equations (85) shows that the evolution is self-similar. We fix for the simulation: $\mu =1$, $d=3$, $T=0.2$, $\zeta =0.5$ and $M_1=M_2=0.5$. For $t\to t_{\mathrm{coll}}$, the rescaled densities converge to the invariant profiles $S_1\left(x\right)$ and $S_2\left(x\right)$ predicted by the theory. The profile of species 1 is the same as in the one-component problem: $S_1\left(x\right)=S_0\left(x\right)\sim x^{-2}$. The profile of species 2 has been obtained by solving Eq. (94) numerically: $S_2\left(x\right)\sim x^{-\alpha }$ with $\alpha =1.66554193$.

Figure 16

The resolution of the time-dependent equations (85) shows that the evolution is self-similar. We fix for the simulation: $\zeta =1$, $d=3$, $T=0.2$, $\mu =2$ and $M_1=M_2=0.5$. For $t\to t_{\mathrm{coll}}$, the rescaled densities converge to the invariant profiles $S_1\left(x\right)$ and $S_2\left(x\right)$ predicted by the theory. The profile of species 1 is the same as in the one-component problem: $S_1\left(x\right)=S_0\left(x\right)\sim x^{-2}$. The profile of species 2 has been obtained by solving Eq. (94) numerically: $S_2\left(x\right)\sim x^{-\alpha }$ with $\alpha =1.35191432$. Note that the large $d$ expansion Eq. (107) leads to $\alpha =4∕3$ in fair agreement with the exact numerical result.

Figure 17

Numerical calculation of ${\alpha }_{\zeta }$ and ${\alpha }_{\mu }$ given by Eqs. (118, 122) as a function of the dimension $d$. The dashed line represents the asymptotic value for $d\to +\infty$. The (∘) symbols represent the large $d$ expansion of ${\alpha }_{\zeta }$ and ${\alpha }_{\mu }$ to order $d^{-4}$ given in Eqs. (121, 124).

Figure 18

The exponent $\alpha$ is plotted for $\mu =1,\zeta ⩽1$ and for $\mu ⩾1,\zeta =1$ in $d=3$. The dashed lines give the different asymptotic behaviors obtained analytically. Finally, the dotted line for $\zeta =1$ corresponds to the result Eq. (107) of the large $d$ expansion, and is in excellent agreement with the exact value of $\alpha$ (the two curves are almost indistinguishable).