Chaos and the continuum limit in the gravitational N-body problem II. Nonintegrable potentials

Abstract

This paper continues a numerical investigation of orbits evolved in `frozen,' time-independent N-body realisations of smooth time-independent density distributions corresponding to both integrable and nonintegrable potentials, allowing for N as large as 300,000. The principal focus is on distinguishing between, and quantifying, the effects of graininess on initial conditions corresponding, in the continuum limit, to regular and chaotic orbits. Ordinary Lyapunov exponents X do not provide a useful diagnostic for distinguishing between regular and chaotic behaviour. Frozen-N orbits corresponding in the continuum limit to both regular and chaotic characteristics have large positive X even though, for large N, the `regular' frozen-N orbits closely resemble regular characteristics in the smooth potential. Viewed macroscopically both `regular' and `chaotic' frozen-N orbits diverge as a power law in time from smooth orbits with the same initial condition. There is, however, an important difference between `regular' and `chaotic' frozen-N orbits: For regular orbits, the time scale associated with this divergence t_G ~ N^{1/2}t_D, with t_D a characteristic dynamical time; for chaotic orbits t_G ~ (ln N) t_D. At least for N>1000 or so, clear distinctions exist between phase mixing of initially localised orbit ensembles which, in the continuum limit, exhibit regular versus chaotic behaviour. For both regular and chaotic ensembles, finite-N effects are well mimicked, both qualitatively and quantitatively, by energy-conserving white noise with amplitude ~ 1/N. This suggests strongly that earlier investigations of the effects of low amplitude noise on phase space transport in smooth potentials are directly relevant to real physical systems.