Shadowing-based reliability decay in softened n-body simulations

A shadow of a numerical solution to a chaotic system is an exact solution to the equations of motion that remains close to the numerical solution for a long time. In a collisionless n-body system, we know that particle motion is governed by the global potential rather than by interparticle interactions. As a result, the trajectory of each individual particle in the system is independently shadowable. It is thus meaningful to measure the number of particles that have shadowable trajectories as a function of time. We find that the number of shadowable particles decays exponentially with time as e(-mut) and that for epsilon is an element of [similar to0.2, 1] (in units of the local mean interparticle separation (n) over bar), there is an explicit relationship among the decay constant mu, the time step h of the leapfrog integrator, the softening epsilon, and the number of particles N in the simulation. Thus, given N and epsilon, it is possible to precompute the time step h necessary to achieve a desired fraction of shadowable particles after a given length of simulation time. We demonstrate that a large fraction of particles remain shadowable over similar to100 crossing times even if particles travel up to about 1/3 of the softening length per time step. However, a sharp decrease in the number of shadowable particles occurs if the time step increases to allow particles to travel farther than the softening length in 1 time step or if the softening is decreased below similar to0.2 (n) over bar.