Rapid relaxation in a one-dimensional gravitating system

Abstract

The relaxation of one-dimensional gravitating systems has been studied over the past three decades. The numerical efficiency with which these models can be simulated makes them ideal for studying long time evolution of gravitational systems. Much controversy has surrounded the relaxation time for the one-dimensional system of $N$ parallel mass sheets. Early work suggested a relaxation on the order of ${N}^{2}$ characteristic times; however, subsequent simulations did not bear this out. Instead, it has been shown that relaxation, if it occurs at all, takes on the order of ${10}^{7}$ characteristic times. Here we consider the relaxation of a different one-dimensional system consisting of concentric spherical mass shells. Past studies have shown the two shell system has more robust ergodic properties than its planar counterpart, possibly suggesting a more rapid relaxation for the $N$ shell system. We found that the simulation for 4, 16, and 64 shells converges to the density obtained in the Vlasov limit on a much shorter time scale with an upper bound of approximately ${10}^{5}$ characteristic times. This is orders of magnitude smaller than in the sheet system.