Time-averaged MSD of Brownian motion

TL;DR

This paper derives exact statistical properties of the time-averaged mean-square displacement for Brownian motion, providing insights into its distribution and moments, which are crucial for analyzing single-particle tracking data.

Contribution

It introduces an exact formula for the Laplace transform of the TAMSD distribution for Brownian motion, enabling precise characterization of its statistical behavior.

Findings

Derived the Laplace transform of TAMSD distribution

Calculated the first four cumulant moments of TAMSD

Provided an accurate generalized Gamma distribution approximation

Abstract

We study the statistical properties of the time-averaged mean-square displacements (TAMSD). This is a standard non-local quadratic functional for inferring the diffusion coefficient from an individual random trajectory of a diffusing tracer in single-particle tracking experiments. For Brownian motion, we derive an exact formula for the Laplace transform of the probability density of the TAMSD by mapping the original problem onto chains of coupled harmonic oscillators. From this formula, we deduce the first four cumulant moments of the TAMSD, the asymptotic behavior of the probability density and its accurate approximation by a generalized Gamma distribution.