The Countoscope for self-propelled particles

TL;DR

The paper introduces the Countoscope method to analyze particle number fluctuations in out-of-equilibrium self-propelled particles, providing theoretical predictions that match simulations and reveal key physical regimes.

Contribution

It extends the Countoscope approach to active particles, deriving analytical formulas for number correlations and the intermediate scattering function for various active particle models.

Findings

The mean-squared number difference exhibits three distinct time-dependent regimes.

Theoretical predictions align with stochastic simulations across regimes.

Limiting laws in each regime help quantify self-propulsion properties.

Abstract

Particle number fluctuations $N(t)$, measured in virtual observation boxes of an image or a simulation, offer a way to quantify particle dynamics when particle tracking is impractical, such as in high-density systems. While traditionally limited to equilibrium diffusive systems, we extend this approach -- named ``Countoscope'' -- to out-of-equilibrium self-propelled particles: Active Brownian (ABPs), Run and Tumble (RTPs), and Active Ornstein-Uhlenbeck Particles (AOUPs). For AOUPs, we leverage their Gaussian statistics to derive a general formula applicable to any Gaussian system. For ABPs and RTPs, we derive the intermediate scattering function (ISF) -- and thus the correlations of $N(t)$ -- using an exact perturbative expansion over the probability density fields, revealing key physical features of the ISF and of the number correlations. Our theoretical predictions for the mean-squared number difference $\langle \Delta N^2(t) \rangle = \langle (N(t) - N(0))^2 \rangle$ match stochastic simulations and exhibit three time-dependent scaling regimes: diffusive, advective, and long-time enhanced diffusive, reflecting the regimes of the mean squared particle displacement. We further uncover limiting laws in each of these regimes that are useful to quantify self-propulsion properties.