Coarse-grained collisionless dynamics with long-range interactions

TL;DR

This paper derives an effective evolution equation for coarse-grained distribution functions in long-range interacting systems, preserving symplectic structure, and validates it through numerical simulations of various one-dimensional models.

Contribution

It introduces a new coarse-grained evolution equation for long-range systems that maintains the symplectic structure, supported by explicit derivation and numerical validation.

Findings

The derived equation matches predictions for one-dimensional systems.

Damping times depend on the coarse-graining scale as predicted.

Numerical checks confirm the theoretical predictions for multiple models.

Abstract

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).