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

This paper continues a numerical investigation of the statistical properties of “frozen-$N$ orbits,” i.e., orbits evolved in frozen, time-independent N-body realizations of smooth density distributions $\rho$ corresponding to both integrable and nonintegrable potentials, allowing for ${10}^{2.5}<~N<~{10}^{5.5}.$ The 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 $\chi$ do not provide a useful diagnostic for distinguishing between regular and chaotic behavior. Frozen-$N$ orbits corresponding in the continuum limit to both regular and chaotic characteristics have large positive $\chi$ even though, for large N, the “regular” frozen-$N$ orbits closely resemble regular characteristics in the smooth potential. Alternatively, viewed macroscopically, both regular and “chaotic” frozen-$N$ orbits diverge as a power law in time from smooth orbits with the same initial condition. However, convergence towards the continuum limit is much slower for chaotic orbits. For regular orbits, the time scale associated with this divergence $t_G\sim N^{1/2}{t}_D,$ with $t_D$ a characteristic dynamical, or crossing, time; for chaotic orbits $t_G\sim (\ln {N)t}_D.$ For $N>\mathrm{}{10}^3-{10}^4,$ clear distinctions exist between the phase mixing of initially localized ensembles, which, in the continuum limit, exhibit regular versus chaotic behavior. Regular ensembles evolved in a frozen-$N$ density distribution diverge as a power law in time, albeit more rapidly than ensembles evolved in the smooth distribution. Chaotic ensembles diverge in a fashion that is roughly exponential, albeit at a larger rate than that associated with the exponential divergence of the same ensemble evolved in smooth $\rho .$ For both regular and chaotic ensembles, finite-$N$ effects are well mimicked, both qualitatively and quantitatively, by energy-conserving white noise with amplitude $\eta \propto 1/N.$ 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.