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Linear response theory for long-range interacting systems in quasistationary states

Aurelio Patelli¹, Shamik Gupta², Cesare Nardini^1,2, and Stefano Ruffo^2,3

¹Dipartimento di Fisica ed Astronomia, Università di Firenze and INFN, Via Sansone 1, IT-50019 Sesto Fiorentino, Italy
²Laboratoire de Physique de l’École Normale Supérieure de Lyon, Université de Lyon, CNRS, 46 Allée d’Italie, FR-69364 Lyon cédex 07, France
³Dipartimento di Energetica “Sergio Stecco” and CSDC, Università di Firenze, CNISM and INFN, via S. Marta 3, IT-50139 Firenze, Italy

Phys. Rev. E 85, 021133 – Published 23 February, 2012

DOI: https://doi.org/10.1103/PhysRevE.85.021133

Abstract

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 a 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, which involves particles moving on a circle under Hamiltonian 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.

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$\begin{matrix} \tilde{F} (k, p, t) = \frac{1}{2 π} \int d q F (q, p, t) e^{- i k q}, \end{matrix}$
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Linear response theory for long-range interacting systems in quasistationary states

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