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

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 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, that involves particles moving on a circle under Hamilton 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.

13 pages, 4 figures; v2: typos fixed; v3: small changes, close to the published version

References in corpus (8)

• Statistical mechanics and dynamics of solvable models with long-range interactions
• A review of linear response theory for general differentiable dynamical systems
• Thermodynamics and dynamics of systems with long-range interactions
• Statistical Mechanics of Unbound Two Dimensional Self-Gravitating Systems
• Quasi-stationary states and the range of pair interactions
• Relaxation to thermal equilibrium in the self-gravitating sheet model
• Relaxation times of unstable states in systems with long range interactions
• Quasistationarity in a model of classical spins with long-range interactions

Cited by in corpus (16)

• Nonequilibrium Statistical Mechanics of Systems with Long-Range Interactions: Ubiquity of Core-Halo Distributions
• Nonlinear response for external field and perturbation in the Vlasov system
• Non-mean-field Critical Exponent in a Mean-field Model : Dynamics versus Statistical Mechanics
• Nonstandard transitions in the Kuramoto model: A role of asymmetry in natural frequency distributions
• Strange Scaling and Temporal Evolution of Finite-Size Fluctuation in Thermal Equilibrium
• Attractor non-equilibrium stationary states in perturbed long-range interacting systems
• Collective $1/f$ fluctuation by pseudo-Casimir-invariants
• General linear response formula for non integrable systems obeying the Vlasov equation
• Conditions for predicting quasistationary states by rearrangement formula
• Linear response theory for coupled phase oscillators with general coupling functions
• Non-diagonalizable and non-divergent susceptibility tensor in the Hamiltonian mean-field model with asymmetric momentum distributions
• Generalization of Fluctuation-Dissipation Theorem to Systems with Absorbing States
• Passive viscoelastic response of striated muscles
• Relaxation to equilibrium in models of classical spins with long-range interactions
• A role of asymmetry in linear response of globally coupled oscillator systems
• Nondivergent and negative susceptibilities around critical points of a long-range Hamiltonian system with two order parameters