Inhomogeneous Quasi-stationary States in a Mean-field Model with Repulsive Cosine Interactions

TL;DR

This paper investigates the formation and stability of inhomogeneous states in a mean-field model with repulsive cosine interactions, revealing long-lived structures that eventually decay to uniform equilibrium.

Contribution

It provides a detailed explanation of how inhomogeneous quasi-stationary states arise and persist in a repulsive mean-field system, introducing a canonical transformation and a one-particle Hamiltonian model.

Findings

Inhomogeneous structures can be extremely stable for long times.

Decay to uniform equilibrium occurs on very long time scales.

A one-particle Hamiltonian captures the main phenomenology.

Abstract

The system of N particles moving on a circle and interacting via a global repulsive cosine interaction is well known to display spatially inhomogeneous structures of extraordinary stability starting from certain low energy initial conditions. The object of this paper is to show in a detailed manner how these structures arise and to explain their stability. By a convenient canonical transformation we rewrite the Hamiltonian in such a way that fast and slow variables are singled out and the canonical coordinates of a collective mode are naturally introduced. If, initially, enough energy is put in this mode, its decay can be extremely slow. However, both analytical arguments and numerical simulations suggest that these structures eventually decay to the spatially uniform equilibrium state, although this can happen on impressively long time scales. Finally, we heuristically introduce a one-particle time dependent Hamiltonian that well reproduces most of the observed phenomenology.