We study the relaxation towards thermodynamical equilibrium of a 1-D gravitational system .
This 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 \rightarrow \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 Ne ^ { 1 / T } at low temperature , in agreement with numerical simulations .