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

Aurelio Patelli, Shamik Gupta, Cesare Nardini, and Stefano Ruffo

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

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.

Received 4 December 2011

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

Authors & Affiliations

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

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Vol. 85, Iss. 2 — February 2012

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Images

Figure 1
$〈 m_{x} 〉 (t)$ vs time $t$ for the (a) water-bag QSS and (b) Fermi-Dirac QSS in the HMF model under the action of the perturbation, Eqs. (52) and (54), with $h = 0.1$ switched on at time $t_{0} = 25$ . (a) The full line in the main plot shows the result of $N$ -particle simulation, while the dashed horizontal line is the theoretical time-averaged value of $〈 m_{x} 〉 (t)$ given in Eq. (73). The system size is $N = 10^{5}$ , while the parameter $p_{0}$ , corresponding to energy $e = 0.7$ , is approximately $1.095$ . In the inset, the numerical result (full line) is compared with the theoretical prediction (72) (dashed line). (b) The full line represents simulation results, while the horizontal dashed line is the theoretical asymptotic value given in Eq. (81). The system size is $N = 10^{5}$ , while $β = 10$ and $μ = 1.2$ , giving energy $e \approx 0.7$ .Reuse & Permissions
Figure 2
$〈 m_{x} 〉 (t)$ vs time $t$ for the Gaussian QSS in the HMF model under the action of the perturbation, Eqs. (52) and (54), with $h = 0.1$ switched on at time $t_{0} = 25$ . The line made of pluses represents the result of $N$ -particle simulation, while the dashed horizontal line is the theoretical asymptotic value given in Eq. (85). The system size is $N = 10^{5}$ , while $β = 0.5$ , so that the energy $e = 1.5$ .Reuse & Permissions
Figure 3
Linear response of a water-bag QSS [(a), (b)], a Fermi-Dirac QSS with $β = 10$ [(c), (d)], and the homogeneous equilibrium state [(e), (f)] for the HMF model under the perturbation, Eqs. (52) and (54), with $h = 0.01$ . All simulation data have been averaged over several thousand realizations of the initial state; for details, see text. In each case, left panel shows the time evolution of the averaged magnetization ${〈 m_{x} 〉}_{ensemble av} (t)$ as obtained from $N$ -particle simulations, and its asymptotic approach either to the time average in Eq. (73) for the water-bag initial state or to ${\bar{m}}_{x}$ given in Eq. (81) for the Fermi-Dirac QSS, or to ${\bar{m}}_{x}$ given in Eq. (85) for the Gaussian QSS. In the right panel, we show the $N$ -particle simulation results for the asymptotic magnetization as a function of energy (the parameter $μ$ in the Fermi-Dirac case). The error bars denote the standard deviation of fluctuations around the asymptotic value. The results compare well with the theoretical predictions. The system size $N$ is 16000 for panels (a)–(d) and 10000 for panels (e) and (f).Reuse & Permissions
Figure 4
Two-step relaxation of the water-bag QSS toward the Boltzmann-Gibbs equilibrium: $〈 m_{x} 〉 (t)$ vs time $t$ for increasing system size from $N = 2000$ to $N = 64 000$ (left to right). Under the perturbation, Eqs. (52) and (54) with $h = 0.01$ , the water-bag initial QSS with $e = 0.65$ relaxes to an intermediate inhomogeneous QSS with $〈 m_{x} 〉 \approx 0.125$ (lower horizontal dash-dotted line) and then to the equilibrium state with $〈 m_{x} 〉 \approx 0.42$ (upper horizontal dashed line). The blue thick lines refer to running averages performed to smooth out local fluctuations.Reuse & Permissions

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