Single particle tracking in systems showing anomalous diffusion: the role of weak ergodicity breaking

TL;DR

This paper analyzes how anomalous diffusion in complex systems affects single particle tracking data, focusing on non-ergodic behavior and the statistical properties of time-averaged mean squared displacement for different stochastic processes.

Contribution

It provides a detailed analysis of time-averaged mean squared displacement in anomalous diffusion, including new analytical expressions for velocity correlation functions for key processes.

Findings

Distribution of time-averaged MSD preserves process characteristics even in short series

Analytical expressions for velocity correlation functions are derived

Insights aid in interpreting single particle tracking data in complex systems

Abstract

Anomalous diffusion has been widely observed by single particle tracking microscopy in complex systems such as biological cells. The resulting time series are usually evaluated in terms of time averages. Often anomalous diffusion is connected with non-ergodic behaviour. In such cases the time averages remain random variables and hence irreproducible. Here we present a detailed analysis of the time averaged mean squared displacement for systems governed by anomalous diffusion, considering both unconfined and restricted (corralled) motion. We discuss the behaviour of the time averaged mean squared displacement for two prominent stochastic processes, namely, continuous time random walks and fractional Brownian motion. We also study the distribution of the time averaged mean squared displacement around its ensemble mean, and show that this distribution preserves typical process characteristic even for short time series. Recently, velocity correlation functions were suggested to distinguish between these processes. We here present analytucal expressions for the velocity correlation functions. Knowledge of the results presented here are expected to be relevant for the correct interpretation of single particle trajectory data in complex systems.