Log it: How to fit an active Brownian particle's mean squared displacement with improved parameter estimation

TL;DR

This paper improves parameter estimation for active Brownian particles by fitting the full MSD expression, using bootstrapping for confidence intervals, and employing MLSD for more reliable physical parameter extraction.

Contribution

It introduces a comprehensive fitting approach for ABP MSD data, addressing biases from traditional short-time fits and heteroscedasticity, enhancing accuracy and confidence in parameter estimation.

Findings

Full MSD fitting yields more accurate parameters.

Bootstrapping provides reliable confidence intervals.

MLSD improves physical parameter extraction.

Abstract

The active Brownian particle (ABP) model is widely used to describe the dynamics of active matter systems, such as Janus microswimmers. In particular, the analytical expression for an ABP's mean-squared-displacement (MSD) is useful as it provides a means to describe the essential physics of a self-propelled, spherical Brownian particle. However, the truncated or 'short-time' form of the MSD equation is typically fitted, which can lead to significant problems in parameter estimation. Furthermore, heteroscedasticity and the often statistically dependent observations of an ABP's MSD lead to a situation where standard ordinary least squares (OLS) regression will obtain biased estimates and unreliable confidence intervals. Here, we propose to revert to always fitting the full expression of an ABP's MSD at short timescales, using bootstrapping to construct confidence intervals of the fitted parameters. Additionally, after comparison between different fitting strategies, we propose to extract the physical parameters of an ABP using its mean logarithmic squared displacement (MLSD). These steps improve the estimation of an ABP's physical properties, and provide more reliable confidence intervals, which are critical in the context of a growing interest in the interactions of microswimmers with confining boundaries and the influence on their motion.