Relaxation of a one-dimensional gravitational system

We study the relaxation towards thermodynamical equilibrium of a one-dimensional gravitational system. This model shows a series of critical energies $E_{cn}$ where different equilibria appear and we focus on the homogeneous $(n=0)$, one-peak $(n=\pm 1)$, and two-peak $(n=2)$ states. Using numerical simulations we investigate the relaxation to the stable equilibrium $n=\pm 1$ of this $N$-body system starting from initial conditions defined by equilibria $n=0$ and $n=2$. We find that in a fashion similar to other long-range systems the relaxation involves a fast violent relaxation phase followed by a slow collisional phase as the system goes through a series of quasistationary states. Moreover, in cases where this slow second stage leads to a dynamically unstable configuration (two peaks with a high mass ratio) it is followed by a different sequence, “violent relaxation–slow collisional relaxation.” We obtain an analytical estimate of the relaxation time $t_{2\to \pm 1}$ through the mean escape time of a particle from its potential well in a bistable system. We find that the diffusion and dissipation coefficients satisfy Einstein’s relation and that the relaxation time scales as $N{e}^{1∕T}$ at low temperature, in agreement with numerical simulations.

Figure 1

(Color online) The transition time $t_{0\to \pm 1}$ from the unstable homogeneous state $n=0$ to the stable equilibria $n=\pm 1$ as a function of total energy $E$, below the critical energy $E_{c1}$. At each energy the crosses are the numerical results obtained for 200 realizations of the initial condition $n=0$, for a system of $N=50$ (a) and $N=1000$ particles (b). The solid line is the mean transition time obtained by averaging over these numerical results. The squares which are slightly shifted to the right show the transition times to states $n=\pm 1$ for realizations which happen to first relax to state $n=2$ over a few dynamical times. The dashed curve is the average obtained for these systems.

Figure 2

(Color online) (a) The evolution with time $t$ of the various contributions $K$, $\overline{\Phi }$, and $\overline{V}$ to the total energy $E$ of a system of $N=1000$ particles. We display the curves obtained for a particular realization of the initial condition defined by the unstable equilibrium $n=0$ at energy $E=-2\mid E_{c1}\mid$. The constant curves are the mean-field energy levels of $K$ (solid lines), $\overline{\Phi }$ (dashed lines), and $\overline{V}$ (dotted lines) for the equilibria $n=0$ and $n=\pm 1$. The system starts at levels $n=0$ at $t=0$ and undergoes a transition to levels $n=1$ at $t\simeq 7t_{\mathrm{dyn}}$. (b) The evolution with time of the masses $M_L$ (dashed lines) and $M_R$ (solid lines) located in the left and right parts of the system ($x<L∕2$ and $x>L∕2$). The constant lines show the values $M_L$, $M_R$ of equilibrium $n=1$.

Figure 3

(Color online) Two snapshots of the density distribution $\rho \left(x\right)$ at times $t=8$ and $12t_{\mathrm{dyn}}$. The histogram shows the matter distribution of the $N$-body system over 30 bins. The dashed curves are equilibria $n=0$ (constant density) and $n=1$ (peak at $x=0$).

Figure 4

(Color online) Two snapshots of the velocity distribution $f\left(v\right)$ at times $t=8$ and $12t_{\mathrm{dyn}}$, in units of $v∕v_{\mathrm{dyn}}$ where we defined $v_{\mathrm{dyn}}=L∕t_{\mathrm{dyn}}$. The histogram corresponds to the $N$-body system whereas the dashed curves are equilibria $n=0$ (narrow Gaussian) and $n=1$ (broad Gaussian).

Figure 5

(Color online) The evolution of the energy components $K$, $\overline{\Phi }$, $\overline{V}$, and $E$ (a) and of the masses $M_L$, $M_R$ (b) as in Fig. 2. We show the curves obtained for a particular realization with $N=50$ particles of the initial homogeneous state at $E=-1.3\mid E_{c1}\mid$. The system keeps wandering over states $n=0,\pm 1$ and does not settle into a stable equilibrium.

Figure 6

(Color online) The evolution of the energy components $K$, $\overline{\Phi }$, $\overline{V}$, and $E$ (a) and of the masses $M_L$, $M_R$ (b) as in Fig. 2. We show the curves obtained for a particular realization with $N=1000$ particles of the initial homogeneous state at $E=-1.3\mid E_{c1}\mid$ which only relaxes to equilibrium $n=1$ over $\sim 8000t_{\mathrm{dyn}}$.

Figure 7

Four snapshots of the phase-space distribution $f(x,v)$ at times $t=0$, 6, 2200, and $2300t_{\mathrm{dyn}}$ for a system of $N=1000$ particles. For this particular realization of the initial homogeneous state $n=0$ the system first undergoes a transition to the equilibrium $n=2$ over $6t_{\mathrm{dyn}}$ and only relaxes to the one-peak state $n=1$ at $t\sim 2250t_{\mathrm{dyn}}$.

Figure 8

(Color online) The transition time $t_{2\to \pm 1}$ from the two-peak equilibrium $n=2$ to the one-peak stable equilibria $n=\pm 1$ as a function of total energy $E$. The crosses correspond to various realizations of the initial equilibrium $n=2$ at a given energy, for systems of $N=50$ (a) and $N=500$ particles (b). The solid line is the mean transition time obtained from these realizations whereas the dashed line is the theoretical prediction (63).

Figure 9

(Color online) The mean transition time $t_{2\to \pm 1}$ as a function of the number $N$ of particles. The various lines correspond to total energies $E=-2.73\mid E_{c1}\mid$, $-2.9\mid E_{c1}\mid$, and $-3.1\mid E_{c1}\mid$ from bottom to top. The squares are the numerical results (averaged over many realizations, solid line in Fig. 8) whereas the dashed-lines are the theoretical prediction (63).

Figure 10

(Color online) (a) The evolution with time $t$ of the various contributions $K$, $\overline{\Phi }$, and $\overline{V}$ to the total energy $E$ of a system of $N=500$ particles. We display the curves obtained for a particular realization of the initial condition defined by the unstable equilibrium $n=2$ at energy $E=-2.8\mid E_{c1}\mid$. The constant curves are the mean-field energy levels of $K$ (solid lines), $\overline{\Phi }$ (dashed lines), and $\overline{V}$ (dotted lines) for the equilibria $n=1$ and $n=2$. The system undergoes a transition from levels $n=2$ to levels $n=1$ at $t\sim 4.7\times {10}^4{t}_{\mathrm{dyn}}$. (b) The evolution with time of the masses $M_L$ (dashed lines) and $M_R$ (solid lines) located in the left and right parts of the system ($x<L∕2$ and $x>L∕2$). The constant lines show the values $M_L$, $M_R$ of equilibrium $n=1$. The two linear solid lines starting from $M_L=M_R=M∕2$ at $t=0$ are the theoretical estimate (62) for the mass transfer between both density peaks.

Figure 11

(Color online) Four snapshots of the density distribution $\rho \left(x\right)$ at times $t=0$, 46 500, 46 800, and $80000t_{\mathrm{dyn}}$. The histogram shows the matter distribution of the $N$-body system over 30 bins. The dashed curves are equilibria $n=1$ and $n=2$.

Figure 12

(Color online) Four snapshots of the velocity distribution $f\left(v\right)$ at times $t=0$, 46 500, 46 800, and $80000t_{\mathrm{dyn}}$. The histogram corresponds to the $N$-body system whereas the dashed curves are equilibria $n=2$ (narrow Gaussian) and $n=1$ (broad Gaussian).

Figure 13

Four snapshots of the phase-space distribution $f(x,v)$ at times $t=0$, 46 500, 46 800, and $80000t_{\mathrm{dyn}}$.

Figure 14

Four snapshots of the phase-space distribution $f(x,v)$ at times $t=46770$, 46 774, 46 776, and $46778t_{\mathrm{dyn}}$ around the transition to the one-peak state.