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

## TL;DR

This paper investigates how chaos and discreteness affect the dynamics of self-gravitating systems through $N$-body simulations, analyzing the dependence of Lyapunov exponents on system size and potential softening, and highlighting differences between frozen and active models.

## Contribution

It provides a detailed analysis of the continuum limit in $N$-body simulations, revealing non-trivial $N$-dependence of chaos measures and cautioning against using frozen potentials to infer large-$N$ behavior.

## Key findings

- Lyapunov exponents decrease with increasing $N$ for large systems.
- Single orbits in frozen and active models have different chaos characteristics.
- Frozen models may be misleading for understanding large-$N$ systems.

## Abstract

We revisit the r\^{o}le 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 $N \to \infty$ 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.