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

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