Relaxation of a 1-D gravitational system

We study the relaxation towards thermodynamical equilibrium of a 1-D gravitational system. This OSC model shows a series of critical energies $E_{cn}$ where new 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 quasi-stationary 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 new 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.