Active matter beyond mean-field: Ring-kinetic theory for self-propelled particles

Abstract

A ring-kinetic theory for Vicsek-style models of self-propelled agents is derived from the exact N-particle evolution equation in phase space. The theory goes beyond mean-field and does not rely on Boltzmann's approximation of molecular chaos. It can handle pre-collisional correlations and cluster formation which both seem important to understand the phase transition to collective motion. We propose a diagrammatic technique to perform a small density expansion of the collision operator and derive the first two equations of the BBGKY-hierarchy. An algorithm is presented that numerically solves the evolution equation for the two-particle correlations on a lattice. Agent-based simulations are performed and informative quantities such as orientational and density correlation functions are compared with those obtained by ring-kinetic theory. Excellent quantitative agreement between simulations and theory is found at not too small noises and mean free paths. This shows that there is parameter ranges in Vicsek-like models where the correlated closure of the BBGKY-hierarchy gives correct and nontrivial results. We calculate the dependence of the orientational correlations on distance in the disordered phase and find that it seems to be consistent with a power law with exponent around -1.8, followed by an exponential decay. General limitations of the kinetic theory and its numerical solution are discussed.

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