Relaxation processes in one-dimensional self-gravitating many-body systems

Although one-dimensional self-gravitating N-body systems have been studied for three decades, the nature of relaxation is still unclear. There have been inconsistent results regarding the relaxation time: some initial states seemed to relax in the time scale T∼${\mathit{Nt}}_{\mathit{c}}$, but other states did not relax even after T∼${\mathit{N}}^2$ ${\mathit{t}}_{\mathit{c}}$, where ${\mathit{t}}_{\mathit{c}}$ is the crossing time. The water-bag distribution was believed not to relax after T∼${\mathit{N}}^2$ ${\mathit{t}}_{\mathit{c}}$. In a previous paper, however, [Phys. Rev. E 50, 2607 (1994)] we found that there are two different relaxation times in the water-bag distribution; in the faster relaxation (microscopic relaxation) the equipartition of energy distribution is attained but the macroscopic distribution turns into the isothermal distribution in the later relaxation (macroscopic relaxation). In this paper, we investigate the properties of the two relaxations. We find that the microscopic relaxation time is T∼${\mathit{Nt}}_{\mathit{c}}$, and the macroscopic relaxation has the much longer time scale 4×${10}^4$ ${\mathit{Nt}}_{\mathit{c}}$, thus the water bag does relax. We can see that the inconsistency about the relaxation times is resolved as we see the two different aspects of relaxation. Further, the physical mechanisms of the relaxation are presented. © 1996 The American Physical Society.