A kinetic model and scaling properties of 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 by a simple kinetic model. The key elements of this approach are the scaling of the cluster cross-section with cluster size-described by an exponent alpha-and the scaling of the cluster perimeter with cluster size-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 a non-monotonic CSD with a local maximum at large cluster sizes. Through a finite-size study of the kinetic model, we show that the critical point P-c that separates the two phases scales with the system size N as P-c proportional to N-xi, while the CSD p(m), at the critical point Pc, is always a power law such that p(m) proportional to m(-gamma), where m is the cluster size. Our analysis shows that the critical exponents xi and gamma are a function of alpha and beta, and even provides the relationship between them. Furthermore, the kinetic approach suggests that in the thermodynamic limit, a genuine clustering phase transition, in two and three dimensions, requires that alpha = beta. Interestingly, the critical exponent gamma is found to be in the range 0.8 < gamma < 1.5 in line with the observations from experiments and simulations.