# Displacement Autocorrelation Functions for Strong Anomalous Diffusion: A Scaling Form, Universal Behavior, and Corrections to Scaling

## TL;DR

This paper introduces an exactly solvable model to analyze strong anomalous diffusion, deriving a universal scaling form for displacement autocorrelation functions and demonstrating its applicability across various complex systems.

## Contribution

The authors present a simple, exactly solvable model and a universal scaling form for autocorrelation functions in strong anomalous diffusion, including corrections to scaling.

## Key findings

- Derived analytical expressions for moments of displacement.
- Established a universal scaling collapse for autocorrelation functions.
- Validated the scaling form across multiple complex systems.

## Abstract

Strong anomalous diffusion is characterized by asymptotic power-law growth of the moments of displacement, with exponents that do not depend linearly on the order of the moment. The exponents concerning small-order moments are dominated by random motion, while higher-order exponents grow by faster trajectories, such as ballistic excursions or "light fronts". Often such a situation is characterized by two linear dependencies of the exponents on their order. Here, we introduce a simple exactly solvable model, the Fly-and-Die (FnD) model, that sheds light on this behavior and on the consequences of light fronts on displacement autocorrelation functions in transport processes. We present analytical expressions for the moments and derive a scaling form that expresses the long-time asymptotics of the autocorrelation function $\langle x(t_1)\,x(t_2)\rangle$ in terms of the dimensionless time difference $(t_2-t_1)/t_1$. The scaling form provides a faithful collapse of numerical data for vastly different systems. This is demonstrated here for the Lorentz gas with infinite horizon, polygonal billiards with finite and infinite horizon, the L\'evy-Lorentz gas, the Slicer Map, and L\'evy walks. Our analysis also captures the system-specific corrections to scaling.