Violent Relaxation, Phase Mixing, and Gravitational Landau Damping

This paper outlines a geometric interpretation of flows generated by the collisionless Boltzmann equation, focusing in particular on the coarse-grained approach towards 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 field energy H[f ] playing the role of the Hamiltonian and the natural arena of physics being Γ, the infinite-dimensional phase space of distribution functions. Every time-independent equilibrium f0 is an energy extremal with respect to all perturbations δf that preserve the constraints (Casimirs) associated with Liouville’s Theorem. If the extremal is a local energy minimum, f0 must be linearly stable but, if it corresponds instead to a saddle point, f0 may be unstable. If an initial f(t = 0) is sufficiently close to some linearly stable lower energy f0, its evolution can be visualised as involving linear phase space oscillations about f0 which, in many cases, would be expected to exhibit linear Landau damping. If instead f(0) is far from any stable extremal, the flow will be more complicated but, in general, one might anticipate that the evolution can be visualised as involving nonlinear oscillations about some lower energy f0. In this picture, the coarse-grained approach towards equilibrium usually termed violent relaxation is interpreted as nonlinear Landau damping. Evolution of a generic initial f(0) involves a coherent initial excitation δf(0) ≡ f(0) − f0, not necessarily small, being converted into incoherent motion associated with nonlinear oscillations about some f0 which, in general, will exhibit destructive interference. This picture allows for distinctions between regular and chaotic “orbits” in Γ: Stable extremals f0 all have vanishing Lyapunov exponents, even though “orbits” oscillating about f0 may well correspond to chaotic trajectories with one or more positive Lyapunov exponents.