A universal route from avalanches in mean-field models with random fields to stochastic Poisson branching events

TL;DR

This paper establishes a universal mapping from avalanches in mean-field models to Poisson branching processes, providing a unified framework for understanding avalanche statistics across different models.

Contribution

It introduces an exact mapping of critical and subcritical avalanches in mean-field models to memoryless Poisson branching processes, unifying their analysis.

Findings

Mapping applies to athermal RFIM and fiber bundle models.

Avalanche statistics approach criticality in these models.

Differences lie in field density and interaction evolution.

Abstract

Avalanches in mean-field models can be mapped to memoryless branching processes defining a universality class. We present a reduced expression mapping a broad family of critical and subcriticial avalanches in mean-field models at the thermodynamic limit to rooted trees in a memoryless Poisson branching processes with random occurrence times. We derive the exact mapping for the athermal random field Ising model and the democratic fiber bundle model, where avalanche statistics progress towards criticality, and as an approximation for the self-organized criticality in slip mean-field theory. Avalanche dynamics and statistics in the three models differ only on the evolution of the field density, interaction strength, and the product of both terms determining the branching number.