Point Transformations and the Relationships Among Anomalous Diffusion, Normal Diffusion and the Central Limit Theorem

TL;DR

This paper introduces a novel point transformation framework linking anomalous diffusion, normal diffusion, and the Central Limit Theorem, revealing new diffusion equations and distribution behaviors through generalized operators and transformations.

Contribution

It develops a new point transformation approach inspired by quantum mechanics to unify and extend diffusion equations and their relation to the CLT, including novel equations with bi-modal distributions.

Findings

Diffusion equations derived capture known and new diffusion behaviors.

Transformations relate anomalous diffusion to normal diffusion via the CLT.

Experimental diffusion scaling can identify the underlying point transformation.

Abstract

We present new connections among anomalous diffusion (AD), normal diffusion (ND) and the Central Limit Theorem. This is done by defining a point transformation to a new position variable, which we postulate to be Cartesian, motivated by considerations from super-symmetric quantum mechanics. Canonically quantizing in the new position and momentum variables according to Dirac gives rise to generalized negative semi-definite and self-adjoint Laplacian operators. These lead to new generalized Fourier transformations and associated probability distributions, which are form invariant under the corresponding transform. The new Laplacians also lead us to generalized diffusion equations, which imply a connection to the CLT. We show that the derived diffusion equations capture all of the Fractal and Non-Fractal Diffusion equations of O'Shaughnessy and Procaccia. However, we also obtain new equations that cannot (so far as we are able to tell) be expressed as examples of the O'Shaughnessy and Procaccia equations. These equations also possess asymptotics that are related to a CLT but with bi-modal distributions as limits. The results show, in part, that experimentally measuring the diffusion scaling law can determine the point transformation (for monomial point transformations). We also show that AD in the original, physical position is actually ND when viewed in terms of displacements in an appropriately transformed position variable. Finally, we show that there is a new, anomalous diffusion possible for bi-modal probability distributions that also display attractor behavior which is the consequence of an underlying CLT.