Self-consistent inhomogeneous steady states in Hamiltonian mean field dynamics

TL;DR

This paper introduces a method to identify exact inhomogeneous steady states in Hamiltonian mean field models, revealing that their relaxation times diverge with system size and providing insights into stability and state selection.

Contribution

It proposes a new approach to find inhomogeneous steady states in long-range interacting systems and analyzes their stability and relaxation properties.

Findings

Relaxation time of inhomogeneous states diverges as N^1.

Method applicable to other globally coupled particle models.

Provides a way to evaluate stability limits of steady states.

Abstract

Long-lived quasistationary states, associated with stationary stable solutions of the Vlasov equation, are found in systems with long-range interactions. Studies of the relaxation time in a model of $N$ globally coupled particles moving on a ring, the Hamiltonian Mean Field model (HMF), have shown that it diverges as $N^\gamma$ for large $N$, with $\gamma \simeq 1.7$ for some initial conditions with homogeneously distributed particles. We propose a method for identifying exact inhomogeneous steady states in the thermodynamic limit, based on analysing models of uncoupled particles moving in an external field. For the HMF model, we show numerically that the relaxation time of these states diverges with $N$ with the exponent $\gamma \simeq 1$. The method, applicable to other models with globally coupled particles, also allows an exact evaluation of the stability limit of homogeneous steady states. In some cases it provides a good approximation for the correspondence between the initial condition and the final steady state.