Velocity Correlation Function of the One-Dimensional Self-Gravitating System

We develop a kinetic theory for the one-dimensional self-gravitating system (1d-SG system). It is known that the collision term in usual kinetic theory for one-dimensional system is identically zero, when all particles have the same mass. Here we choose a phenomenological approach. Assuming that the system is near equilibrium, we use an approximation that the system is in a thermal bath, called the thermal bath approximation (TBA). With this approximation, velocity correlation function of the 1d-SG system is investigated. The direct numerical simulation shows < v(t)v(0)> similar to t(-alpha) for small N (alpha = 2.0-1.4), where N is the number of sheets. On the other hand, the TBA shows < v(t)v(0)> similar to t(-2) for small N. For large N, the velocity correlation function exhibits slow relaxation. The reason of slow relaxation is explained in terms of the stochastic differential equation derived from the TBA.