Chaos and the continuum limit in charged particle beams

Abstract

This paper examines discreteness effects in nearly collisionless N-body systems of charged particles interacting via an unscreened r^-2 force, allowing for bulk potentials admitting both regular and chaotic orbits. Both for ensembles and individual orbits, as N increases there is a smooth convergence towards a continuum limit. Discreteness effects are well modeled by Gaussian white noise with relaxation time t_R = const * (N/log L)t_D, with L the Coulomb logarithm and t_D the dynamical time scale. Discreteness effects accelerate emittance growth for initially localised clumps. However, even allowing for discreteness effects one can distinguish between orbits which, in the continuum limit, feel a regular potential, so that emittance grows as a power law in time, and chaotic orbits, where emittance grows exponentially. For sufficiently large N, one can distinguish two different `kinds' of chaos. Short range microchaos, associated with close encounters between charges, is a generic feature, yielding large positive Lyapunov exponents X_N which do not decrease with increasing N even if the bulk potential is integrable. Alternatively, there is the possibility of larger scale macrochaos, characterised by smaller Lyapunov exponents X_S, which is present only if the bulk potential is chaotic. Conventional computations of Lyapunov exponents probe X_N, leading to the oxymoronic conclusion that N-body orbits which look nearly regular and have sharply peaked Fourier spectra are `very chaotic.' However, the `range' of the microchaos, set by the typical interparticle spacing, decreases as N increases, so that, for large N, this microchaos, albeit very strong, is largely irrelevant macroscopically. A more careful numerical analysis allows one to estimate both X_N and X_S.