Relaxation of a 1-D gravitational system

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We study the relaxation towards thermodynamical equilibrium of a 1-D gravitational system. This OSC model shows a series of critical energies Ecn where new 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 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 t₂→ ± 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 e¹/ᵗ at low temperature, in agreement with numerical simulations.