Topology of Collisionless Relaxation

TL;DR

This study uses molecular dynamics simulations to analyze how collisionless systems with long-range interactions evolve, revealing that initial phase space conditions influence the structure of the final stationary state, especially under certain virial conditions.

Contribution

It demonstrates that for specific initial distributions satisfying a generalized virial condition, the final state can be predicted using Casimir invariants of Vlasov dynamics.

Findings

Final states have compact, simply connected regions if initial distributions are hole-free.

Microscopic holes are confined to outer phase space regions.

Predictive capability for final states under virial conditions using invariants.

Abstract

Using extensive molecular dynamics simulations we explore the fine-grained phase space structure of systems with long-range interactions. We find that if the initial phase space particle distribution has no holes, the final stationary distribution will also contain a compact simply connected region. The microscopic holes created by the filamentation of the initial distribution function are always restricted to the outer regions of the phase space. In general, for complex multilevel distributions it is very difficult to a priori predict the final stationary state without solving the full dynamical evolution. However, we show that for multilevel initial distributions satisfying a generalized virial condition, it is possible to predict the particle distribution in the final stationary state using Casimir invariants of the Vlasov dynamics.