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 noncollisional Boltzmann, or Vlasov, equation. First, we 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 this 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 model, a scalar field with quartic interaction, a 1-d self-gravitating system, and a self-gravitating ring.