Chaotic itinerancy and thermalization in one-dimensional   self-gravitating systems
  This is the third paper of the series of our studies of the one-dimensional self-gravitating many-body systems. In this paper, we thus study the transition phenomena after the first transition from a quasiequilibrium. We found that irrespective of the initial conditions, the system wanders between many states which yield the same properties of the quasiequilibrium as the water-bag. This itinerancy is most prominent in the time scale $t\sim 10^5 \sim 10^6 t_c$. In the midway between two succeeding quasiequilibria, the system experiences a transient state, where one particle keeps exclusive high energy and its motion decouples with the others. Though the transient state is not the thermally relaxed equilibrium, its distribution quite resembles the isothermal distribution. Thus the macroscopic relaxation we discussed in the previous papers corresponds to the transition from one of quasiequilibria to a transient state. Averaging the behavior over a time scale much longer than the macroscopic relaxation time gives the isothermal distribution. Distribution of lifetime $\tau$ of the transient states yields a power-law distribution of $\tau^{-2}$. The transient state gives a clear example of chaotic itinerancy in conserved dynamical systems. The mechanism of the onset of itinerancy is examined. 
Toshio Tsuchiya, Naoteru Gouda, and Tetsuro Konishi