Anomalous dynamical scaling determines universal critical singularities

TL;DR

This paper investigates how anomalous diffusion leads to universal critical singularities in the probability density function's scaling behavior, with exact models illustrating phase transition-like phenomena.

Contribution

It introduces a specific decay form in the scaling function that results in universal singularities, linking anomalous diffusion to phase transition analogies.

Findings

Derivation of a decay form implying universal singularities

Exact solutions for continuous time random walks as examples

Identification of singularities analogous to second order phase transitions

Abstract

Anomalous diffusion phenomena occur on length scales spanning from intracellular to astrophysical ranges. A specific form of decay at large argument of the probability density function of rescaled displacement (scaling function) is derived and shown to imply universal singularities in the normalized cumulant generator. Exact calculations for continuous time random walks provide paradigmatic examples connected with singularities of second order phase transitions. In the biased case scaling is restricted to displacements in the drift direction and singularities have no equilibrium analogue.