  The current paradigm for understanding galaxy formation in the universe
depends 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 interacting 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 equilibrium 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 distributions and self-gravitating linear perturbations of
thermal equilibrium. N-body simulations of perturbed equilibria are performed
and found to be in excellent agreement with the expansion coefficient approach
over a time duration that depends on the size of the expansion series used.
