TITLE:
Point Transformations and Relationships among Linear Anomalous Diffusion, Normal Diffusion and the Central Limit Theorem
AUTHORS:
Donald Kouri, Nikhil Pandya, Cameron L. Williams, Bernhard G. Bodmann, Jie Yao
KEYWORDS:
Generalized Fourier Analysis, Normal Diffusion, Anomalous Diffusion, Point Transformations, Canonical Quantization, Super Symmetric Quantum Mechanics
JOURNAL NAME:
Applied Mathematics,
Vol.9 No.2,
February
28,
2018
ABSTRACT: We present new connections among linear anomalous diffusion (AD), normal
diffusion (ND) and the Central Limit Theorem (CLT). 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
Anomalous Diffusion equations of O’Shaughnessy and Procaccia. However,
we also obtain new equations that cannot (so far as we can tell) be expressed
as examples of the O’Shaughnessy and Procaccia equations. 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. We
illustrate the ideas both analytically and with a detailed computational example
for a non-trivial choice of point transformation. Finally, we summarize
our results.