Higher Order Spectra in Avalanche Models

TL;DR

This paper applies Haar transform-based higher order spectral analysis to mean field avalanche models, verifying theoretical predictions with simulations and experimental data, and discovering new exponents at critical points.

Contribution

It introduces a Haar transform approach for analyzing higher order spectra in avalanche models, providing analytical calculations and experimental validation.

Findings

Analytical expressions for Haar power spectrum and higher order spectra.

Verification of theoretical results with numerical simulations.

Identification of novel exponents in experimental data.

Abstract

Utilizing the Haar transform, we study the higher order spectral properties of mean field avalanche models, whose avalanche dynamics are described by Poisson statistics at a critical point or critical depinning transition. The Haar transform allows us to obtain a time series of noise powers, $H(f_1,t)$, that gives improved time resolution over the Fourier transform. Using $H(f_1,t)$ we analytically calculate the Haar power spectrum, the real 1.5 spectra, the second spectra, and the real cross second spectra in mean field avalanche models. We verify our theoretical results with the numerical results from a simulation of the T=0 mean field nonequilibrium random field Ising model (RFIM). We also extend our higher order spectra calculation to data obtained from a numerical simulation of the T=0 infinite range RFIM for $d=3$, and experimental data obtained from an amorphous alloy, $Fe_{21}Co_{64}B_{15}$. We compare the results and obtain novel exponents.