Optimal fits of diffusion constants from single time data points of Brownian trajectories

TL;DR

This paper investigates methods to accurately estimate the diffusion constant from single Brownian trajectories, proposing weighted functionals that improve precision and can match ensemble averages.

Contribution

It introduces a class of weighted estimators for diffusion constants that enhance accuracy from single trajectories, addressing a key challenge in single particle tracking.

Findings

Weighted functionals can accurately estimate the diffusion constant.

Optimal weight functions yield true ensemble averages.

Precision improves with higher trajectory resolution.

Abstract

Experimental methods based on single particle tracking (SPT) are being increasingly employed in the physical and biological sciences, where nanoscale objects are visualized with high temporal and spatial resolution. SPT can probe interactions between a particle and its environment but the price to be paid is the absence of ensemble averaging and a consequent lack of statistics. Here we address the benchmark question of how to accurately extract the diffusion constant of one single Brownian trajectory. We analyze a class of estimators based on weighted functionals of the square displacement. For a certain choice of the weight function these functionals provide the true ensemble averaged diffusion coefficient, with a precision that increases with the trajectory resolution.