Scale-free nonlinear conservative cascades and their stationary spectra

TL;DR

This paper presents a unified framework for understanding various complex processes through scale-free nonlinear conservative cascades, classifying them by indices and deriving a formula for their power-law spectra.

Contribution

It introduces a classification scheme for cascades and provides an algebraic formula linking the power-law exponent to cascade indices.

Findings

Power-law steady spectra can arise in diverse processes.

The power-law exponent depends only on three cascade indices.

A simple algebraic formula relates indices to spectral exponents.

Abstract

We show that a variety of complex processes can be viewed from the unified standpoint of scale-free nonlinear conservative cascades. Examples include certain turbulence models, percolation, cluster coagulation (aggregation) and fragmentation, `coarse-grained' forest fire model of self-organized criticality, and scale-free network growth. We classify such cascades by the values of three indices, and show how power-law steady spectra may arise. The power-law exponent is proven to depend only on the values of the three indices by a simple algebraic formula.