Numerical convergence in solving the Vlasov equation.

When the Vlasov equation is investigated numerically using the method of test particles, the two-body interactions that inevitably arise in the simulation (but are not present in the Vlasov equation itself) drive the collection of test particles toward a state of classical thermal equilibrium. We estimate the relaxation time associated with this thermalization. The Vlasov equation plays a central role in classical (and semiclassical) time-dependent mean field theory, and has been used to model a wide variety of many-body processes, from the gravitational N-body problem [1], to plasma physics [2], to nuclear dynamics [3]. While the content of the Vlasov equation is conceptually simple — interactions among N 1 particles are replaced by a common mean-field potential — solutions are harder to come by, and must in general be sought numerically. This is often accomplished with the test particle method: a swarm of numerical particles is used to simulate a distribution f(r,p, t) in onebody phase space, and the mean-field potential in which these test particles evolve is obtained from this distribution. Thus, while the Vlasov equation replaces a physical many-body problem with the self-consistent evolution of a one-body phase-space distribution, the test particle method in turn replaces the Vlasov equation with a numerical many-body problem. This raises the issue of accuracy: what sort of confidence do we have that the evolution of f(r,p, t) as obtained by the test particle method resembles the true evolution under the Vlasov equation? In particular, since the test particles inevitably interact with one another