Spectral density of individual trajectories of an active Brownian particle

Alessio Squarcini,Gleb Oshanin,Alexandre Solon

doi:10.1088/1367-2630/ac44e6

Abstract

We study analytically the single-trajectory spectral density (STSD) of an active Brownian motion (BM) 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 γ of the distribution of the STSD. Moreover, we analyse the finite-T behaviour of the STSD and γ, 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 BM compared to its passive counterpart, which allow to distinguish between these two classes based solely on the spectral content of individual trajectories.

Highlights

Active matter encompasses a variety of systems that are driven locally out of equilibrium, at the scale of each constituents
In the other class of models, particles self-propel in a direction that fluctuates because of rotational noise. These active Brownian particles (ABPs) [19] are appropriate to describe, for example, Janus colloidal particles decorated with a catalytic patch which prompts a chemical reaction in the surrounding solution [20, 21]
Let us first note that since the translational and rotational noises are uncorrelated, the variance var SxD(f, T, D) of the process xD(t) naturally decomposes into the variance of the process x(t), which we have studied in the previous Sections, and the variance of the spectral density of a Brownian motion Xt, i.e

Summary

Introduction

Active matter encompasses a variety of systems that are driven locally out of equilibrium, at the scale of each constituents. As can be expected intuitively, on large time scales the ABP follows a standard two-dimensional BM with an effective diffusion coefficient De = v2τR/2 [47] Note that it remains true even if the particle is placed in an external potential, provided the variations of the potential are small on the scale of the persistence length lp = vτR [17]. Neither the form of the standard power spectral density μx(f ), nor its ageing properties or the behaviour of the characteristic parameter γ are known for the much popular ABP model. The paper is organised as follows: in Sec. 2 we first evaluate the exact form of the standard textbook power spectral density of trajectories of an ABP in the limit of an infinite observation time before discussing some features of the finite-T case.

Position correlation functions

Coefficient of variation and cross correlations

Finite-T behaviour of the coefficient of variation

Pearson correlation coefficient

Effects of a translational diffusion

Conclusions