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Relaxation in N-body simulations of spherical systems

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.