Comment on "Non-Normalizable Densities in Strong Anomalous Diffusion: Beyond the Central Limit Theorem"

TL;DR

This paper refutes claims that non-normalizable densities occur in Levy walk diffusion, clarifying that finite-time densities are normalizable and that infinite covariant densities are not physically realizable.

Contribution

It demonstrates that non-normalizable densities in Levy walk diffusion are not physically justified and clarifies the distinction between finite-time densities and infinite covariant densities.

Findings

Finite-time densities are normalizable and useful for moments.

Infinite covariant densities do not correspond to physical reality.

Experimental data are always finite-time, invalidating claims of non-normalizability.

Abstract

A comment on the Letter by A. Rebenshtok, S. Denisov, P. H\"anggi, and E. Barkai, Phys. Rev. Lett., vol. 112, 110601 (2014). It is shown that the recent claims that the particle distributions or densities can become non-normalizable in the case of anomalous Levy walk diffusion cannot be justified. In particular, one can define a normalizable finite-time density which become infinite covariant density (ICD) in the limit of infinite time. The former one can be used to find any generalized moment, while the latter one does not. In this respect, we point out that any decent real or numerical experiment is done in finite time. Hence, ICD does not correspond to any physical reality, contrary to the claims made.