Evidence of discrete scale invariance in DLA and time-to-failure by canonical averaging

TL;DR

This paper introduces a novel canonical averaging method to detect discrete scale invariance in complex systems like DLA and rupture models, overcoming the limitations of standard ensemble averaging.

Contribution

The paper presents a new averaging scheme that preserves discrete scale invariance signals by aligning realization-dependent scales, enabling clearer detection in physical systems.

Findings

Discrete scale invariance observed in DLA and rupture models.

Canonical averaging enhances detection of hierarchical scales.

Method applicable to various systems with scale invariance signatures.

Abstract

Discrete scale invariance, which corresponds to a partial breaking of the scaling symmetry, is reflected in the existence of a hierarchy of characteristic scales l0, c l0, c^2 l0,... where c is a preferred scaling ratio and l0 a microscopic cut-off. Signatures of discrete scale invariance have recently been found in a variety of systems ranging from rupture, earthquakes, Laplacian growth phenomena, ``animals'' in percolation to financial market crashes. We believe it to be a quite general, albeit subtle phenomenon. Indeed, the practical problem in uncovering an underlying discrete scale invariance is that standard ensemble averaging procedures destroy it as if it was pure noise. This is due to the fact, that while c only depends on the underlying physics, l0 on the contrary is realisation-dependent. Here, we adapt and implement a novel so-called ``canonical'' averaging scheme which re-sets the l0 of different realizations to approximately the same value. The method is based on the determination of a realization-dependent effective critical point obtained from, e.g., a maximum susceptibility criterion. We demonstrate the method on diffusion limited aggregation and a model of rupture.