Singularity of Levy walks in the lifted Pomeau-Manneville map

TL;DR

This paper investigates the relationship between deterministic Pomeau-Manneville maps and stochastic Levy walks, revealing that their equivalence in producing superdiffusion is limited to specific parameter values and is affected by bifurcations and aging effects.

Contribution

It demonstrates that the previously assumed matching between deterministic superdiffusion and Levy walks is only valid for measure-zero parameter sets and explores the impact of bifurcations and aging.

Findings

Matching only holds for measure-zero parameter values.

Bifurcation causes deviations between deterministic and stochastic diffusion.

Aging significantly alters superdiffusive behavior.

Abstract

Since groundbreaking works in the 1980s it is well-known that simple deterministic dynamical systems can display intermittent dynamics and weak chaos leading to anomalous diffusion. A paradigmatic example is the Pomeau-Manneville (PM) map which, suitably lifted onto the whole real line, was shown to generate superdiffusion that can be reproduced by stochastic Levy walks (LWs). Here we report that this matching only holds for parameter values of the PM map that are of Lebesgue measure zero in its two-dimensional parameter space. This is due to a bifurcation scenario that the map exhibits under variation of one parameter. Constraining this parameter to specific singular values at which the map generates superdiffusion by varying the second one, as has been done in previous literature, we find quantitative deviations between deterministic diffusion and diffusion generated by stochastic LWs in a particular range of parameter values, which cannot be cured by simple LW modifcations. We also explore the effect of aging on superdiffusion in the PM map and show that this yields a profound change of the diffusive properties under variation of the aging time, which should be important for experiments. Our findings demonstrate that even in this simplest well-studied setting, a matching of deterministic and stochastic diffusive properties is non-trivial.