Brownian regime of finite-N corrections to particle motion in the XY hamiltonian mean field model

TL;DR

This paper analyzes the finite-N corrections to particle motion in the XY Hamiltonian mean field model, showing that these corrections behave like Brownian noise over specific time scales, supported by analytical proofs and simulations.

Contribution

It provides a rigorous proof that finite-N effects in the XY-HMF model manifest as Brownian noise, extending understanding of particle dynamics without phase transitions.

Findings

Finite-N corrections behave as independent Brownian noises.

Corrections persist over time scales diverging at least as N^{2/5}.

Simulations confirm analytical predictions.

Abstract

We study the dynamics of the N-particle system evolving in the XY hamiltonian mean field (HMF) model for a repulsive potential, when no phase transition occurs. Starting from a homogeneous distribution, particles evolve in a mean field created by the interaction with all others. This interaction does not change the homogeneous state of the system, and particle motion is approximately ballistic with small corrections. For initial particle data approaching a waterbag, it is explicitly proved that corrections to the ballistic velocities are in the form of independent brownian noises over a time scale diverging not slower than $N^{2/5}$ as $N \to \infty$, which proves the propagation of molecular chaos. Molecular dynamics simulations of the XY-HMF model confirm our analytical findings.