On the sensitivity of the N-body problem to small changes in initial conditions

This paper summarizes the results of a considerable number of numerical simulations which investigate the instability of the gravitational N-body problem towards small, "random" changes in the initial positions of the particles. The specific aims are: (1) to follow the growths of the perturbations of individual particles in the simulation, analyzing their statistical properties; (2) to investigate how these statistical properties vary as the number of particles N in the simulation is changed from 16 to 340; and (3) to determine how these properties vary if one changes the 'softening parameter'. Principal conclusions obtained for the simulations considered include the following: (1) The mean e-folding time t-asterisk for the perturbation in the location of some particle is comparable to the dynamical, or crossing, time t(cr) for all N. (2) There is, however, a weak N-dependence, t-asterisk t(cr) decreasing as N increases but tending toward an asymptotic value for large N. (3) The distribution of e-folding times for a given N has a distinctive and comparatively reproducible shape, the reproducibility improving as N grows. (4) Increasing the 'softening parameter' leads to a systematic increase in t-asterisk, but this time scale remains comparable to t(cr) even when the 'softening parameter' becomes so large as to be comparable to the interparticle spacing.