Chaos and the continuum limit in the gravitational N-body problem: Integrable potentials

This paper summarizes a numerical investigation of the statistical properties of orbits evolved in “frozen,” time-independent N-body realizations of smooth, time-independent density distributions corresponding to integrable potentials, allowing for 10^2.5<~N<~10^5.5. Two principal conclusions were reached: (1) In agreement with recent work by Valluri and Merritt, one finds that, in the limit of a nearly “unsoftened” two-body kernel, i.e., V(r)∝(r²+ε²)^-1/2 for ε⃗0, the value of the largest Lyapunov exponent χ does not decrease systematically with increasing N, so that, viewed in terms of the sensitivity of individual orbits to small changes in initial conditions, there is no sense in which chaos “turns off” for large N. However, it is clear that, for any finite ε, χ will tend to zero for sufficiently large N. (2) Even though χ does not decrease for an unsoftened kernel, there is a clear, quantifiable sense in which, as N increases, chaotic orbits in the frozen-N systems remain “close to” integrable characteristics in the smooth potential for progressively longer times. When viewed in configuration or velocity space, or as probed by collisionless invariants like angular momentum, frozen-N orbits typically diverge from smooth potential characteristics as a power law in time, rather than exponentially, on a time scale ∝N^pt_D, with p≈1/2 and t_D a characteristic dynamical, or crossing, time. For the case of angular momentum, the divergence is well approximated by a t^1/2 dependence, so that, when viewed in terms of collisionless invariants, discreteness effects act as a diffusion process that, presumably, can be modeled by nearly white Gaussian noise in the context of a Langevin or Fokker-Planck description. For position and velocity, the divergence is more rapid, characterized by a nearly linear power-law growth, t^q with q≈1, a result that likely reflects the effects of linear phase mixing. The inference that, pointwise, individual N-body orbits can be reasonably approximated by orbits in a smooth potential only for times <N^1/2t_D has potential implications for various resonance phenomena that can act in real self-gravitating systems.