Slowing down of so-called chaotic states: "Freezing" the initial state

TL;DR

This paper investigates the critical slowing down of chaotic states in XY networks, revealing that characteristic time-scales diverge with system size, following a square root law, and provides analytical explanations for this phenomenon.

Contribution

It introduces a detailed analysis of the divergence of time-scales in chaotic states of XY networks and offers an analytical framework to understand this critical slowing down.

Findings

Time scales diverge as (N)  \,  \, au(N) \u2192  \, ext{as } N ightarrow \u221e

Scaling law  \, au(N) \u223c  \, ext{for different energy densities}

Analytical explanation using thermodynamic equations and eigenvalues of the adjacency matrix

Abstract

The so-called chaotic states that emerge on the model of $XY$ interacting on regular critical range networks are analyzed. Typical time scales are extracted from the time series analysis of the global magnetization. The large spectrum confirms the chaotic nature of the observable, anyhow different peaks in the spectrum allows for typical characteristic time-scales to emerge. We find that these time scales $\tau(N)$ display a critical slowing down, i.e they diverge as $N\rightarrow\infty$. The scaling law is analyzed for different energy densities and the behavior $\tau(N)\sim\sqrt{N}$ is exhibited. This behavior is furthermore explained analytically using the formalism of thermodynamic-equations of the motion and analyzing the eigenvalues of the adjacency matrix.