Role of mass fluctuations in the diffusion of clusters of Brownian particles with activity
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 $\alpha=0.63\pm0.06$, in good quantitative agreement with large-scale ABP simulations.