Role of mass fluctuations in the diffusion of clusters of Brownian particles with activity

Abstract

Motivated by the anomalous diffusion observed in clusters of active Brownian particles (ABPs), where the center-of-mass diffusion coefficient scales as $D\sim N^{-1/2}$ with respect to the number $N$ of particles in the cluster, we derive a minimal theoretical framework starting from the single-particle Langevin equations. The model consists of two coupled stochastic equations: one for the cluster center-of-mass trajectory and one for the mass evolution $N(t)$, explicitly accounting for stochastic displacements induced by particle attachment and detachment. We specialize and validate the framework against ABP simulations of isolated clusters in stationary conditions, where $N(t)$ follows a Gaussian process with mean $N_0$, variance $\propto N_0^{\beta }$, and persistence time $\propto N_0^{\kappa }$. Analytical solution of the coupled equations yields the long-time diffusion coefficient as the sum of two contributions: a conventional term $\propto N_0^{-1}$) due to thermal noise plus summation of active forces, and a fluctuation-driven term $\propto N_0^{-\delta }$ with $\delta =2-2/d-\beta +\kappa$, where $d$ is the spatial dimension. We demonstrate that anomalous scaling emerges whenever the second term becomes dominant. The model predicts $D\sim N^{-\alpha }$ with $\alpha =0.63\pm 0.06$, in good quantitative agreement with large-scale ABP simulations.

10 pages, 3 figures