On the distribution of estimators of diffusion constants for Brownian motion

TL;DR

This paper analyzes the statistical distribution of estimators used to determine diffusion constants from Brownian motion trajectories, providing explicit formulas and numerical validation.

Contribution

It offers new analytical results for the distribution of diffusion constant estimators based on quadratic functionals of Brownian motion, with confirmation via simulations.

Findings

Explicit distribution formulas derived for various estimators

Analytical results validated by numerical simulations

Enhanced understanding of estimator variability in Brownian motion analysis

Abstract

We discuss the distribution of various estimators for extracting the diffusion constant of single Brownian trajectories obtained by fitting the squared displacement of the trajectory. The analysis of the problem can be framed in terms of quadratic functionals of Brownian motion that correspond to the Euclidean path integral for simple Harmonic oscillators with time dependent frequencies. Explicit analytical results are given for the distribution of the diffusion constant estimator in a number of cases and our results are confirmed by numerical simulations.