Eur. Phys. J. D (2014) 68: 329
https://doi.org/10.1140/epjd/e2014-50215-3

Regular Article

General linear response formula for non integrable systems obeying the Vlasov equation^*

¹ CEA – Service de Physique de l’État condensé, Centre d’Études de Saclay, 91191 Gif-sur-Yvette, France
² Dipartimento di Fisica e Astronomia and CSDC, Università degli Studi di Firenze, CNISM and INFN, Via Sansone 1, 50019 Sesto Fiorentino, Italy

^a e-mail: stefano.ruffo@unifi.it

Received: 15 March 2014
Received in final form: 24 July 2014
Published online: 3 November 2014

Abstract

Long-range interacting N-particle systems get trapped into long-living out-of-equilibrium stationary states called quasi-stationary states (QSS). We study here the response to a small external perturbation when such systems are settled into a QSS. In the N → ∞ limit the system is described by the Vlasov equation and QSS are mapped into stable stationary solutions of such equation. We consider this problem in the context of a model that has recently attracted considerable attention, the Hamiltonian mean field (HMF) model. For such a model, stationary inhomogeneous and homogeneous states determine an integrable dynamics in the mean-field effective potential and an action-angle transformation allows one to derive an exact linear response formula. However, such a result would be of limited interest if restricted to the integrable case. In this paper, we show how to derive a general linear response formula which does not use integrability as a requirement. The presence of conservation laws (mass, energy, momentum, etc.) and of further Casimir invariants can be imposed a posteriori. We perform an analysis of the infinite time asymptotics of the response formula for a specific observable, the magnetization in the HMF model, as a result of the application of an external magnetic field, for two stationary stable distributions: the Boltzmann-Gibbs equilibrium distribution and the Fermi-Dirac one. When compared with numerical simulations the predictions of the theory are very good away from the transition energy from inhomogeneous to homogeneous states.