N-body chaos and the continuum limit in numerical simulations of self-gravitating systems, revisited

Abstract

We revisit the rôle of discreteness and chaos in the dynamics of self-gravitating systems by means of N -body simulations with active and frozen potentials, starting from spherically symmetric stationary states and considering the orbits of single particles in a frozen N -body potential as well as the orbits of the system in the full 6N -dimensional phase space. We also consider the intermediate case where a test particle moves in the field generated by N non-interacting particles, which in turn move in a static smooth potential. We investigate the dependence on N and on the softening length of the largest Lyapunov exponent both of single particle orbits and of the full N -body system. For single orbits we also study the dependence on the angular momentum and on the energy. Our results confirm the expectation that orbital properties of single orbits in finite-N systems approach those of orbits in smooth potentials in the continuum limit N → ∞ and that the largest Lyapunov exponent of the full N -body system does decrease with N , for sufficiently large systems with finite softening length. However, single orbits in frozen models and active self-consistent models have different largest Lyapunov exponents and the N -dependence of the values in non-trivial, so that the use of frozen N -body potentials to gain information on large-N systems or on the continuum limit may be misleading in certain cases.

Symplectic coarse graining approach to the dynamics of spherical self-gravitating systems

“…Qualitatively, the trend of σ f k with N is intermediate between a N −1/2 and N −1/3 power-law behaviour, as shown in the figure by the dashed and dotted lines, respectively. Curiously, the same N −α scaling with 1/3 < α < 1/2 has been also reported by Di Cintio & Casetti (2019, 2020b for the maximal Lyapunov exponent and the amplitude of particle-correlations in N −body simulations of collisionless equilibrium models.…”

Discreteness effects, N-body chaos and the onset of radial-orbit instability

“…In this paper we have continued our study on the effect of N −body chaos and discreteness "noise" on the evolution of orbits and instabilities in spherical self-gravitating systems, following Di Cintio & Casetti (2019b). We have investigated the onset of the radial orbit instability in a family of Osipkov-Merritt-Dehnen models for various values of the Friedman-Polyachenko-Shukhman index ξ and different sizes N and logarithmic central density slopes γ.…”

“…In general, using smaller values of the softening length yields larger values of Λmax. For a discussion of how the softening of the gravitational force influences the values attained by the finite time Lyapunov exponents see Goodman et al (1993); El-Zant (2002); El-Zant et al (2019); Di Cintio & Casetti (2019b).…”

Noise, friction and the radial-orbit instability in anisotropic stellar systems: stochastic N-body simulations

“…In Figure 2 we compare the evolution of the axial ratios c/a and b/a and the anisotropy parameters for γ = 1 models with ξ 0 = 4 and 1.8 for additive noise with and without friction. In general, the presence of noise or noise plus friction does not have a significant effect on the onset of ROI for models with a steeper cusp (generally more unstable, as they admit a larger degree of wildly chaotic orbits see Di Cintio & Casetti (2019), Di Cintio & Casetti (2020)) and low values of the initial anisotropy (i.e. ξ 0 = 1.8), while for larger values of ξ 0 , corresponding to a more violent instability, the evolution of the triaxiality and the anisotropy are affected by the presence of noise, with systematically less anisotropic and more "triaxial" end states associated to the presence of larger amounts of noise and friction.…”