Numerical study of energy diffusion in King models

The energy diffusion coefficients D_n(E) (n=1,2) for a system of equal mass particles moving self-consistently in an N-body realisation of a King model are computed from the probability per unit time, P(E, Delta E), that a star with initial energy E will undergo an energy change Delta E. In turn, P is computed from the number of times during the simulation that a particle in a state of given energy undergoes a transition to another state. These particle states are defined directly from the time evolution of E by identifying them with the event occuring between two local maxima in the E(t) curve. If one assumes next that energy changes are uncorrelated between different states, one can use diffusion theory to compute D_n(E). The simulations employ N=512, 2048,... , 32768 particles and are performed using an implementation of Aarseth's direct integrator N-body1 on a massively parallel computer. The more than seven million transitions measured in the largest N simulation provide excellent statistics. The numerically determined D(E)'s are compared against their theoretical counterparts which are computed from phase-space averaged rates of energy change due to independent binary encounters. The overall agreement between them is impressive over most of the energy range, notwithstanding the very different type of approximations involved, giving considerable support to the valid usage of these theoretical expressions to simulate dynamical evolution in Fokker-Planck type calculations.