A kinetic model and scaling properties for non-equilibrium clustering of self-propelled particles

We demonstrate that the clustering statistics and the corresponding phase transition to non-equilibrium clustering found in many experiments and simulation studies with self-propelled particles (SPPs) with alignment can be obtained from a simple kinetic model. The key elements of this approach are the scaling of the cluster cross-section with the cluster mass -- characterized by an exponent $\alpha$ -- and the scaling of the cluster perimeter with the cluster mass -- described by an exponent $\beta$. The analysis of the kinetic approach reveals that the SPPs exhibit two phases: i) an individual phase, where the cluster size distribution (CSD) is dominated by an exponential tail that defines a characteristic cluster size, and ii) a collective phase characterized by the presence of non-monotonic CSD with a local maximum at large cluster sizes. At the transition between these two phases the CSD is well described by a power-law with a critical exponent $\gamma$, which is a function of $\alpha$ and $\beta$ only. The critical exponent is found to be in the range $0.8 < \gamma < 1.5$ in line with observations in experiments and simulations.