Regular and anomalous diffusion I: Foundations

Abstract

Diffusion is a generic term for random motions whose positions become more and more diffuse with time. Diffusion is of major importance in numerous areas of science and engineering, and the research of diffusion is vast and profound. This paper is the first in a stochastic `intro series' to the multidisciplinary field of diffusion. The paper sets off from a basic question: how to quantitatively measure diffusivity? Having answered the basic question, the paper carries on to a follow-up question regarding statistical behaviors of diffusion: what further knowledge can the diffusivity measure provide, and when can it do so? The answers to the follow-up question lead to an assortment of notions and topics including: persistence and anti-persistence; aging and anti-aging; spectral densities, white noise, and their generalizations; the Wiener-Khinchin theorem and its generalizations; and colored noises. Then, observing diffusion from a macro level, the paper culminates with: the universal emergence of power-law diffusivity; the three universal diffusion regimes -- one regular, and two anomalous; and the universal emergence of 1/f noise. The paper is entirely self-contained, and its prerequisites are undergraduate mathematics and statistics.