Hierarchical Diffusion, Aging and Multifractality

TL;DR

This paper investigates aging processes in hierarchically structured multifractal phase spaces, revealing crossover behaviors in auto-correlation functions and their relation to multifractal exponents.

Contribution

It introduces a model of aging in multifractal phase spaces and characterizes the crossover in auto-correlation functions with scaling laws and exponent relations.

Findings

Auto-correlation functions exhibit crossover from one power law to another.

The crossover follows a simple $t/ w$ scaling law.

Exponents are related to multifractal mass exponents.

Abstract

We study toy aging processes in hierarchically decomposed phase spaces where the equilibrium probability distributions are multifractal. We found that the an auto-correlation function, survival-return probability, shows crossover behavior from a power law $t^{-x}$ in the quasi-equilibrium regime ($t\ll\tw$) to another power law $t^{-\lambda}$ ($\lambda \geq x$) in the off-equilibrium regime ($t\gg\tw$) obeying a simple $t/\tw$ scaling law. The exponents $x$ and $\lambda$ are related with the so called mass exponents which characterize the multifractality.