Super-resolved anomalous diffusion: deciphering the joint distribution of anomalous exponent and diffusion coefficient

TL;DR

This paper develops a universal statistical framework to accurately estimate and analyze the joint distribution of anomalous diffusion parameters,  and D, in heterogeneous media, accounting for finite data effects and experimental variability.

Contribution

It introduces estimators based on time-averaged MSD, derives their joint density using copulas, and provides an exact formula for the D- relation, applicable to any Gaussian process.

Findings

Derived a universal joint density formula for  and D estimators.

Validated the methodology with numerical simulations of fractional Brownian motion.

Quantified estimation accuracy using the Hellinger distance.

Abstract

The molecular motion in heterogeneous media displays anomalous diffusion by the mean-squared displacement $\langle X^2(t) \rangle = 2 D t^\alpha$. Motivated by experiments reporting populations of the anomalous diffusion parameters $\alpha$ and $D$, we aim to disentangle their respective contributions to the observed variability when this last is due to a true population of these parameters and when it arises due to finite-duration recordings. We introduce estimators of the anomalous diffusion parameters on the basis of the time-averaged mean squared displacement and study their statistical properties. By using a copula approach, we derive a formula for the joint density function of their estimations conditioned on their actual values. The methodology introduced is indeed universal, it is valid for any Gaussian process and can be applied to any quadratic time-averaged statistics. We also explain the experimentally reported relation $D\propto\exp(\alpha c_1+c_2)$ for which we provide the exact expression. We finally compare our findings to numerical simulations of the fractional Brownian motion and quantify their accuracy by using the Hellinger distance.