Exploring the thermodynamic limit of Hamiltonian models: convergence to the Vlasov equation

TL;DR

This paper investigates the emergence of Quasi Stationary States in long-range interacting systems, demonstrating that the Vlasov equation accurately describes their behavior and highlighting the importance of particle correlations.

Contribution

It provides a detailed comparison between N-body simulations and the Vlasov model for the Hamiltonian Mean Field system, confirming the Vlasov framework's validity for QSS analysis.

Findings

Vlasov model accurately predicts QSS in the HMF system

Lynden-Bell's theory yields precise analytical predictions

Small scale fluctuations affect Vlasov solutions in certain parameter regions

Abstract

We here discuss the emergence of Quasi Stationary States (QSS), a universal feature of systems with long-range interactions. With reference to the Hamiltonian Mean Field (HMF) model, numerical simulations are performed based on both the original $N$-body setting and the continuum Vlasov model which is supposed to hold in the thermodynamic limit. A detailed comparison unambiguously demonstrates that the Vlasov-wave system provides the correct framework to address the study of QSS. Further, analytical calculations based on Lynden-Bell's theory of violent relaxation are shown to result in accurate predictions. Finally, in specific regions of parameters space, Vlasov numerical solutions are shown to be affected by small scale fluctuations, a finding that points to the need for novel schemes able to account for particles correlations.