Linear response theory in the Vlasov equation for homogeneous and for inhomogeneous quasistationary states.

Response to small external forces is investigated in quasistationary states of Hamiltonian systems having long-range interactions. Quasistationary states are recognized as stable stationary solutions to the Vlasov equation, and, hence, the linear response theory to the Vlasov equation is proposed for spatially one-dimensional systems with periodic boundary condition. The proposed theory is applicable both to homogeneous and to inhomogeneous quasistationary states and is demonstrated in the Hamiltonian mean-field model. In the homogeneous case magnetic susceptibility is explicitly obtained, and the Curie-Weiss like law is suggested in a high-energy region. The linear response is also computed in the inhomogeneous case, and resonance absorption is investigated to extract nonequilibrium dynamics in the unforced system. Theoretical predictions are examined by direct numerical simulations of the Vlasov equation.