Dynamics of One-dimensional Self-gravitating Systems Using Hermite-Legendre Polynomials

Abstract

The current paradigm for understanding galaxy formation in the universe de- pends on the existence of self-gravitating collisionless dark matter. Modeling such dark matter systems has been a major focus of astrophysicists, with much of that effort directed at computational techniques. Not surprisingly, a comprehensive understanding of the evolution of these self-gravitating systems still eludes us, since it involves the collective nonlinear dynamics of many-particle systems inter- acting via long-range forces described by the Vlasov equation. As a step towards developing a clearer picture of collisionless self-gravitating relaxation, we analyze the linearized dynamics of isolated one-dimensional systems near thermal equi- librium by expanding their phase space distribution functions f(x,v) in terms of Hermite functions in the velocity variable, and Legendre functions involving the position variable. This approach produces a picture of phase-space evolution in terms of expansion coefficients, rather than spatial and velocity variables. We obtain equations of motion for the expansion coefficients for both test-particle dis- tributions and self-gravitating linear perturbations of thermal equilibrium. This development presents the opportunity to avoid time-consuming N-body simula- tions that are limited by statistical uncertainty and provides a powerful analysis tool for understanding the relaxation to equilibrium.Â