Generalized maximum entropy approach to quasi-stationary states in long range systems

TL;DR

This paper introduces a generalized maximum entropy method incorporating additional conserved quantities to better describe quasi-stationary states in long-range interacting systems, successfully explaining complex velocity distributions.

Contribution

It develops a generalized Gibbs ensemble approach that accounts for pseudo-conserved moments, improving the theoretical understanding of QSS in long-range systems.

Findings

Accurately predicts multiple bumps in velocity distributions.

Matches simulation results both above and below phase transition.

Resolves previously unexplained features of QSS.

Abstract

Systems with long-range interactions display a short-time relaxation towards Quasi Stationary States (QSS) whose lifetime increases with the system size. In the paradigmatic Hamiltonian Mean-field Model (HMF) out-of-equilibrium phase transitions are predicted and numerically detected which separate homogeneous (zero magnetization) and inhomogeneous (nonzero magnetization) QSS. In the former regime, the velocity distribution presents (at least) two large, symmetric, bumps, which cannot be self-consistently explained by resorting to the conventional Lynden-Bell maximum entropy approach. We propose a generalized maximum entropy scheme which accounts for the pseudo-conservation of additional charges, the even momenta of the single particle distribution. These latter are set to the asymptotic values, as estimated by direct integration of the underlying Vlasov equation, which formally holds in the thermodynamic limit. Methodologically, we operate in the framework of a generalized Gibbs ensemble, as sometimes defined in statistical quantum mechanics, which contains an infinite number of conserved charges. The agreement between theory and simulations is satisfying, both above and below the out of equilibrium transition threshold. A precedently unaccessible feature of the QSS, the multiple bumps in the velocity profile, is resolved by our new approach.