Optimal estimates of the diffusion coefficient of a single Brownian trajectory

TL;DR

This paper develops an optimal maximum likelihood estimator for accurately determining the diffusion coefficient from single Brownian trajectories, outperforming traditional methods and analyzing the impact of environmental disorder.

Contribution

It introduces a maximum likelihood approach for estimating diffusion constants from single trajectories and examines disorder effects on estimate distribution.

Findings

Maximum likelihood estimator is more efficient than least squares.

Disorder increases the likelihood of underestimating the diffusion coefficient.

The method improves accuracy in nanoscale particle tracking.

Abstract

Modern developments in microscopy and image processing are revolutionizing areas of physics, chemistry and biology as nanoscale objects can be tracked with unprecedented accuracy. The goal of single particle tracking is to determine the interaction between the particle and its environment. The price paid for having a direct visualization of a single particle is a consequent lack of statistics. Here we address the optimal way of extracting diffusion constants from single trajectories for pure Brownian motion. It is shown that the maximum likelihood estimator is much more efficient than the commonly used least squares estimate. Furthermore we investigate the effect of disorder on the distribution of estimated diffusion constants and show that it increases the probability of observing estimates much smaller than the true (average) value.