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=±1), and two-peak (n=2) states. Using numerical simulations we investigate the relaxation to the stable equilibrium n=±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±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.

©2006 The American Physical Society