Violent relaxation, phase mixing, and gravitational Landau damping

This paper outlines a geometric interpretation of hows generated by the collisionless Boltzmann equation, focusing in particular on the coarse-grained approach toward a time-independent equilibrium. The starting point is the recognition that the collisionless Boltzmann equation is a noncanonical Hamiltonian system with the distribution function f as the fundamental dynamical variable, the mean held energy H[f] playing the role of the Hamiltonian, and the natural arena of physics being Gamma, the infinite-dimensional phase space of distribution functions. Every time-dependent equilibrium f(0) is an energy extremal with respect to all perturbations delta f that preserve the constraints (Casimirs) associated with Liouville's theorem. If the extremal is a local energy minimum,f(0) must be linearly stable, but if it corresponds instead to a saddle point, f(0) may be unstable. If an initial f(t = 0) is sufficiently close to some linearly stable lower energy f(0) its evolution can be visualized as involving linear phase-space oscillations about f(0) which, in many cases, would be expected to exhibit linear Landau damping. If, instead, f(0) is far from any stable extremal, the how will be more complicated, but, in general, one might anticipate that the evolution can be visualized as involving nonlinear oscillations about some lower energy f(0). In this picture, the coarse-grained approach toward equilibrium usually termed violent relaxation is interpreted as nonlinear Landau damping. Evolution of a generic initial f(0) involves a coherent initial excitation delta f(0) = f(0)-f(0), not necessarily small, being converted into incoherent motion associated with nonlinear oscillations about some f(0) which, in general, will exhibit destructive interference. This picture allows for distinctions between regular and chaotic "orbits" in Gamma: stable extremals f(0) all have vanishing Lyapunov exponents, even though "orbits" oscillating about f(0) may well correspond to chaotic trajectories with one or more positive Lyapunov exponents.