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.
Valageas
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Service de Physique Théorique, CEA Saclay, 91191 Gif-sur-Yvette, France.