Virial theorem and dynamical evolution of self-gravitating Brownian particles in an unbounded domain. I. Overdamped models

We derive the virial theorem appropriate to the generalized Smoluchowski-Poisson (GSP) system describing self-gravitating Brownian particles in an overdamped limit. We extend previous works by considering the case of an unbounded domain and an arbitrary equation of state. We use the virial theorem to study the diffusion (evaporation) of an isothermal Brownian gas above the critical temperature $T_c$ in dimension $d=2$ and show how the effective diffusion coefficient and the Einstein relation are modified by self-gravity. We also study the collapse at $T=T_c$ and show that the central density increases logarithmically with time instead of exponentially in a bounded domain. Finally, for $d>2$, we show that the evaporation of the system is essentially a pure diffusion slightly slowed down by self-gravity. We also study the linear dynamical stability of stationary solutions of the GSP system representing isolated clusters of particles and investigate the influence of the equation of state and of the dimension of space on the dynamical stability of the system.

Figure 1

Evolution of the density profile in $d=2$ for different times $t=4^n$ with $n=0,1,2,3,4$. The initial density profile has been chosen as $\rho (r,t=0)\sim (1-r^2{)}^3$ and the temperature as $T=1$. The solid lines correspond to the evolution of self-gravitating Brownian particles described by Eq. (43). The dashed lines correspond to a purely diffusive evolution without gravitational drift. We see that the effect of gravitational attraction is to reduce the diffusion. This is a correction of order $1$ which leads to a constant shift between the two curves in the log-log plot.

Figure 2

Scaling profiles of the density for different values of the temperature. The upper curve corresponds to $T=0.26$ close to $T_c=1∕4$. The solution of the scaling equation (50) (solid line) is in excellent agreement with the analytical expression (66) (dashed line). The lower curve corresponds to a large value of the temperature $T=10$. The solution of the scaling equation (50) (solid line) is in excellent agreement with the analytical expressions (57, 58) (dashed line) for $T⪢1$. In particular, the measured value of the scaling profile at the origin $g^{\prime }\left(x\right)∕x_{\mid x=0}=g^{″}\left(0\right)=1.03574\dots$ is in good agreement with the expression $1+\frac{1}{T}\ln 2∕2+O\left(T^{-2}\right)$ coming from the theory. Finally, the middle curve corresponds to an intermediate value $T=1$ of the temperature and shows the data collapse for $t=4^{n+1}$ ($n=1,2,3,4$; solid line) of the density profiles obtained by solving the dynamical equation (43) as in Fig. 1. We see that the scaling regime is reached rapidly and that the invariant profile is perfectly described by the scaling equation (50) (dashed line). This figure shows how one goes from a Gaussian profile for $T⪢1$ to a profile generating an algebraic $r^{-4}$ region for $T\to T_c$.

Figure 3

We plot $a\left(t\right)=\pi \rho (0,t)$ as a function of $\ln t$, starting from a profile with $\rho (r,t=0)=\frac{4}{\pi }(1-r^2{)}^3$, for which $I=1∕5$, in an unbounded domain [in fact a box of radius $R=200⪢\sqrt{b\left(t\right)}$]. Although we expect strong corrections to the asymptotic behavior for finite times (of order $\ln \ln t$), we already find that the observed slope is in fact quite close to the prediction $1∕I=5$ (dashed straight line). We compare this to the evolution of the same initial profile in a box of radius $R=2$, which asymptotically diverges as $\exp (5∕2+\sqrt{t∕2})∕4$ (dashed curve). In an unbounded domain, the inset compares the actual density profile (solid line) to our result Eq. (88) (dashed line), which appears undistinguishable even for the moderate value of $t=6$. We have taken the actual measured value of $a=24.82$ but have fitted $b\sim t$ $(b=6.84)$ and $c=1+1∕\ln \left(ab\right)+\cdots$ $(c=1.44)$ as the expressions given in the text are only valid for very large time.

Figure 4

Evolution of the density profile in $d=3$ for different times $t=4^n$ with $n=0,1,2,3,4$ (compare with Fig. 1 for $d=2$). The initial density profile has been chosen as $\rho (r,t=0)\sim (1-r^2{)}^3$, and the temperature is $T=1$. The solid lines correspond to the case of self-gravitating Brownian particles described by Eq. (92). The dashed lines correspond to a purely diffusive evolution without gravitational drift. We see that the effect of gravitational attraction is to reduce the diffusion for small times. However, this is just a correction which vanishes for $t\to +\infty$ so that the curves rapidly become indistinguishable.