Nonreciprocal antialigning active mixtures: Deriving the exact Boltzmann collision operator

We consider the effect of nonreciprocity in a binary mixture of self-propelled particles with antialigning interactions, where a particle of type A reacts differently to a particle of type B than vice versa. Starting from a well-known microscopic Langevin model for the particles, setting up the corresponding exact N-particle Fokker-Planck equation, and making Boltzmann's assumptions of low density and one-sided molecular chaos, the nonlinear active Boltzmann equation with the exact collision operator is derived. In this derivation, the effect of phase-space compression and the buildup of pair correlations during binary interactions is explicitly taken into account, leading to a theoretical description beyond mean field. This extends previous results for reciprocal interactions, where it was found that orientational order can emerge in a system with purely antialigning interactions. Although the equations of motion are more complex than in the reciprocal system, the theory still leads to analytical expressions and predictions. Comparisons with agent-based simulations show excellent quantitative agreement of the dynamic and static behavior in the low-density and/or small coupling limit.