Asymptotically exact scattering theory of the Kuramoto-Vicsek model

Abstract

We consider the Kuramoto-Vicsek model of self-propelled particles with velocity-alignment interactions. Starting from the exact $N$-particle Liouville equation, a kinetic equation for the one-particle distribution function is obtained in a self-consistent manner. We show that the usual mean-field assumption of molecular chaos leads to qualitatively wrong predictions such as an infinite coefficient of self-diffusion. Going beyond mean-field and applying the refined assumption of \emph{one-sided molecular chaos} where the two-particle-correlations during binary interactions are explicitly taken into account, we analytically calculate the scattering of particles in the limit of low density and obtain explicit expressions for the dynamical noise of an effective one-particle Langevin-equation and the corresponding self-diffusion. The theory is developed in detail for anti-aligning couplings, where exact analytical results are obtainable. In this calculation, the superposition principle of traditional kinetic theory is modified to handle a system with non-Hamiltonian dynamics involving phase-space compression. The predicted theoretical expressions for the relaxation of hydrodynamic modes and the self-diffusion coefficient are in excellent, quantitative agreement with agent-based simulations. At large particle densities, a given particle is constantly approached and abandoned by different collision partners. Modeling this switching by a random telegraph process and exactly solving a self-consistent integral equation, we obtain explicit expressions for the noise correlations of the effective one-particle Langevin-equation. We also consider the effect of frozen disorder in the particle speeds and show how this can be used to calculate the exact Boltzmann collision operator for positive alignment strengths.