  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.
