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Linear response theory for hydrodynamic and kinetic equations with long-range interactions

We apply the linear response theory to systems with long-range interactions described by hydrodynamic equations such as the Euler, Smoluchowski, and damped Euler equations. We analytically determine the response of the system submitted to a pulse and to a step function. We compare these results with those obtained for collisionless systems described by the Vlasov equation. We show that, in the linear regime, the evolution of a collisionless system (Vlasov) with the waterbag distribution is the same as the evolution of a collision-dominated gas without dissipation (Euler). In this analogy, the maximum velocity of the waterbag distribution plays the role of the velocity of sound in the corresponding barotropic gas. When submitted to a step function, these systems exhibit permanent oscillations. Other distributions exhibit Landau damping and relax towards a steady state. We illustrate this behaviour with the Cauchy distribution which can be studied analytically. We apply our results to the HMF model and obtain a generalized Curie-Weiss law for the magnetic susceptibility. Finally, we compare the linear response theory to the initial value problem for the linearized Vlasov equation and report a case of algebraic damping of the initial perturbation.