Vlasov versus N-body: the H\'enon sphere

TL;DR

This study compares N-body simulations with a Vlasov solver for Hénon spheres, demonstrating strong agreement in phase-space density evolution despite some convergence challenges in colder systems.

Contribution

It introduces new statistical tools for comparing N-body and Vlasov phase-space densities and provides a detailed validation of N-body methods against Vlasov solutions for spherical systems.

Findings

Excellent agreement between N-body and Vlasov methods in phase-space density.

New statistical measures effectively compare the two approaches.

Convergence issues remain in colder systems even with very large N.

Abstract

We perform a detailed comparison of the phase-space density traced by the particle distribution in Gadget simulations to the result obtained with a spherical Vlasov solver using the splitting algorithm. The systems considered are apodized H\'enon spheres with two values of the virial ratio, R ~ 0.1 and 0.5. After checking that spherical symmetry is well preserved by the N-body simulations, visual and quantitative comparisons are performed. In particular we introduce new statistics, correlators and entropic estimators, based on the likelihood of whether N-body simulations actually trace randomly the Vlasov phase-space density. When taking into account the limits of both the N-body and the Vlasov codes, namely collective effects due to the particle shot noise in the first case and diffusion and possible nonlinear instabilities due to finite resolution of the phase-space grid in the second case, we find a spectacular agreement between both methods, even in regions of phase-space where nontrivial physical instabilities develop. However, in the colder case, R=0.1, it was not possible to prove actual numerical convergence of the N-body results after a number of dynamical times, even with N=10$^8$ particles.