Relaxation in N-body simulations of spherical systems

TL;DR

This paper empirically measures relaxation rates in N-body simulations of spherical systems, distinguishing between energy diffusion and energy exchange, and finds that relaxation scales with particle number and is influenced by collective oscillations.

Contribution

It provides empirical measurements of relaxation in N-body simulations, highlighting the roles of collective effects and the impact of different force calculation methods.

Findings

Diffusion scales as N^{-1} in inhomogeneous models

Collective effects cause relaxation to scale as N^{-1/2} in uniform spheres

Energy exchange rate between different mass particles is method-independent

Abstract

I present empirical measurements of the rate of relaxation in N-body simulations of stable spherical systems and distinguish two separate types of relaxation: energy diffusion that is largely independent of particle mass, and energy exchange between particles of differing masses. While diffusion is generally regarded as a Fokker-Planck process, it can equivalently be viewed as the consequence of collective oscillations that are driven by shot noise. Empirical diffusion rates scale as N^{-1} in inhomogeneous models, in agreement with Fokker-Planck predictions, but collective effects cause relaxation to scale more nearly as N^{-1/2} in the special case of a uniform sphere. I use four different methods to compute the gravitational field, and a 100-fold range in the numbers of particles in each case. I find the rate at which energy is exchanged between particles of differing masses does not depend at all on the force determination method, but I do find the energy diffusion rate is marginally lower when a field method is used. The relaxation rate in 3D is virtually independent of the method used because it is dominated by distant encounters; any method to estimate the gravitational field that correctly captures the contributions from distant particles must also capture their statistical fluctuations and the collective modes they drive.