Optimal least-squares estimators of the diffusion constant from a single Brownian trajectory

TL;DR

This paper investigates optimal weighted least-squares estimators for accurately determining the diffusion constant from a single Brownian trajectory, addressing statistical reliability and effects of disorder in high-resolution microscopy data.

Contribution

It identifies ergodic weight functions for least-squares estimators and analyzes how disorder impacts their effectiveness in single-particle tracking.

Findings

Derived ergodic weight functions for estimators

Analyzed the influence of disorder on estimator performance

Provided guidelines for reliable diffusion constant estimation

Abstract

Modern developments in microscopy and image processing are revolutionising areas of physics, chemistry, and biology as nanoscale objects can be tracked with unprecedented accuracy. However, the price paid for having a direct visualisation of a single particle trajectory with high temporal and spatial resolution is a consequent lack of statistics. This naturally calls for reliable analytical tools which will allow one to extract the properties specific to a statistical ensemble from just a single trajectory. In this article we briefly survey different analytical methods currently used to determine the ensemble average diffusion coefficient from single particle data and then focus specifically on weighted least-squares estimators, seeking the weight functions for which such estimators are ergodic. Finally, we address the question of the effects of disorder on such estimators.