Average shape of fluctuations for subdiffusive walks

TL;DR

This paper analytically derives the average fluctuation shape for subdiffusive processes with broad waiting time distributions, revealing a transition from semicircular to table-like shapes as subdiffusivity increases.

Contribution

It introduces a fractional diffusion approach to analytically characterize fluctuation shapes in subdiffusive walks, highlighting a shape transition not seen in normal diffusion.

Findings

Fluctuation shape becomes table-like with increased subdiffusivity.

Analytical results agree with numerical simulations.

Shape transition contrasts with classical semicircular fluctuation shape.

Abstract

We study the average shape of fluctuations for subdiffusive processes, i.e., processes with uncorrelated increments but where the waiting time distribution has a broad power-law tail. This shape is obtained analytically by means of a fractional diffusion approach. We find that, in contrast with processes where the waiting time between increments has finite variance, the fluctuation shape is no longer a semicircle: it tends to adopt a table-like form as the subdiffusive character of the process increases. The theoretical predictions are compared with numerical simulation results.