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

Abstract

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.

Nonlinear response for external field and perturbation in the Vlasov system

“…The leading order is identical to the linear response of homogeneous states [5,6]. One remarkable difference between the present method and the linear response theory is that the latter is based on approximated solutions to the Vlasov equation constructed by perturbation technique.…”

“…We stress the following two observations: (h-i) At the critical temperature T = T c , the slope 2/3 corresponding to the critical exponent δ = 3/2 is successfully observed, and the slope goes to 1 with increasing T as predicted by the linear response theory [5,6]. We note that the slope seems to change smoothly between 2/3 and 1, but it must be 1 in the limit of small h except for T = T c .…”

“…With this assumption g satisfies the normalization condition (6). The assumption A4 eliminates L 1 [g ].…”

“…The exponents γ + and γ − are obtained in Refs. [5,6] and [7] respectively. The exponent β for prepared stable inhomogeneous states is 1/2 in usual case, as we will explain in Sec.…”

“…In the Vlasov system, the linear response theory for the external field has been recently developed [5,6]. This theory gives the asymptotic value of observable associated with the external field if the field is small enough.…”