Spectral density of individual trajectories of an active Brownian particle

TL;DR

This paper analytically investigates the spectral density of individual trajectories of active Brownian particles, revealing distinctive features that differentiate active from passive motion, and explores finite-time effects and correlations.

Contribution

It provides exact analytical expressions for the spectral density and its variability in active Brownian motion, including finite-time and cross-correlation analyses, advancing understanding of single-trajectory spectral features.

Findings

Spectral density distinguishes active from passive Brownian motion.

Finite observation time affects spectral density and variability.

Cross-correlations between spatial components are characterized.

Abstract

We study analytically the single-trajectory spectral density (STSD) of an active Brownian motion as exhibited, for example, by the dynamics of a chemically-active Janus colloid. We evaluate the standardly-defined spectral density, i.e. the STSD averaged over a statistical ensemble of trajectories in the limit of an infinitely long observation time $T$, and also go beyond the standard analysis by considering the coefficient of variation $\gamma$ of the distribution of the STSD. Moreover, we analyse the finite-$T$ behaviour of the STSD and $\gamma$, determine the cross-correlations between spatial components of the STSD, and address the effects of translational diffusion on the functional forms of spectral densities. The exact expressions that we obtain unveil many distinctive features of active Brownian motion compared to its passive counterpart, which allow to distinguish between these two classes based solely on the spectral content of individual trajectories.