Chaos and the continuum limit in charged particle beams

We investigate the validity of the Vlasov-Poisson equations for calculating properties of systems of N charged particles governed by time-independent Hamiltonians. Through numerical experiments we verify that there is a smooth convergence toward a continuum limit as N→∞ and the particle charge q→0 such that the system charge Q=qN remains fixed. However, in real systems N and q are always finite, and the assumption of the continuum limit must be questioned. We demonstrate that Langevin simulations can be used to assess the importance of discreteness effects, i.e., granularity, in systems for which the physical particle number N is too large to enable orbit integrations based on direct summation of interparticle forces. We then consider a beam bunch in thermal equilibrium and apply Langevin techniques to assess whether the continuum limit can be safely applied to this system. In the process we show, especially for systems supporting a sizable population of chaotic orbits that roam globally through phase space, that for the continuum limit to be valid, N must sometimes be surprisingly large. Otherwise the influence of granularity on particle orbits cannot be ignored.