Coarse-grained collisionless dynamics with long-range interactions
  We present an effective evolution equation for a coarse-grained distribution
function of a long-range-interacting system preserving the symplectic structure
of the non-collisional Boltzmann, or Vlasov, equation. We first derive a
general form of such an equation based on symmetry considerations only. Then,
we explicitly derive the equation for one-dimensional systems, finding that it
has the form predicted on general grounds. Finally, we use such an equation to
predict the dependence of the damping times on the coarse-graining scale and
numerically check it for some one-dimensional models, including the Hamiltonian
Mean Field (HMF) model, a scalar field with quartic interaction, a 1-d
self-gravitating system, and the Self-Gravitating Ring (SGR).
