# Universality Classes of Fluctuation Dynamics in Hierarchical Complex Systems

## TL;DR

This paper introduces a unified framework for modeling the short-term fluctuation statistics of multiscale complex systems, revealing two universality classes characterized by distinct tail behaviors in their distributions.

## Contribution

It develops a hierarchical stochastic model that derives two fundamental classes of fluctuation dynamics, supported by empirical validation in turbulence and financial data.

## Key findings

- Two universality classes identified: power law and stretched exponential tails.
- The model's predictions align well with empirical data from turbulence and markets.
- Distributions are expressed using Meijer G-functions for analytical clarity.

## Abstract

A unified approach is proposed to describe the statistics of the short time dynamics of multiscale complex systems. The probability density function of the relevant time series (signal) is represented as a statistical superposition of a large time-scale distribution weighted by the distribution of certain internal variables that characterize the slowly changing background. The dynamics of the background is formulated as a hierarchical stochastic model whose form is derived from simple physical constraints, which in turn restrict the dynamics to only two possible classes. The probability distributions of both the signal and the background have simple representations in terms of Meijer G-functions. The two universality classes for the background dynamics manifest themselves in the signal distribution as two types of tails: power law and stretched exponential, respectively. A detailed analysis of empirical data from classical turbulence and financial markets shows excellent agreement with the theory.