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

Abstract

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 α—and the scaling of the cluster perimeter with cluster size—described by an exponent β. 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 Pc that separates the two phases scales with the system size N as Pc∝N−ξ, while the CSD p(m), at the critical point Pc, is always a power law such that p(m)∝m−γ, where m is the cluster size. Our analysis shows that the critical exponents ξ and γ are a function of α and β, 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 α = β. Interestingly, the critical exponent γ is found to be in the range 0.8 < γ < 1.5 in line with the observations from experiments and simulations.