Linear response theory for long-range interacting systems in quasistationary states

TL;DR

This paper develops a linear response theory for long-range systems in quasistationary states, analyzing their response to external perturbations via the Vlasov equation, and validates predictions with simulations on the Hamiltonian mean-field model.

Contribution

It introduces an explicit linear response formula for QSSs in long-range systems and applies it to the Hamiltonian mean-field model, providing analytical and numerical insights.

Findings

Response formulas match N-particle simulations for large N

Explicit response for homogeneous QSSs derived

Long-time relaxation to equilibrium observed in simulations

Abstract

Long-range interacting systems, while relaxing to equilibrium, often get trapped in long-lived quasistationary states which have lifetimes that diverge with the system size. In this work, we address the question of how a long-range system in a quasistationary state (QSS) responds to an external perturbation. We consider a long-range system that evolves under deterministic Hamilton dynamics. The perturbation is taken to couple to the canonical coordinates of the individual constituents. Our study is based on analyzing the Vlasov equation for the single-particle phase space distribution. The QSS represents stable stationary solution of the Vlasov equation in the absence of the external perturbation. In the presence of small perturbation, we linearize the perturbed Vlasov equation about the QSS to obtain a formal expression for the response observed in a single-particle dynamical quantity. For a QSS that is homogeneous in the coordinate, we obtain an explicit formula for the response. We apply our analysis to a paradigmatic model, the Hamiltonian mean-field model, that involves particles moving on a circle under Hamilton dynamics. Our prediction for the response of three representative QSSs in this model (the water-bag QSS, the Fermi-Dirac QSS, and the Gaussian QSS) is found to be in good agreement with $N$-particle simulations for large $N$. We also show the long-time relaxation of the water-bag QSS to the Boltzmann-Gibbs equilibrium state.