Non-reciprocal anti-aligning active mixtures: deriving the exact Boltzmann collision operator

TL;DR

This paper derives an exact Boltzmann collision operator for a binary mixture of self-propelled particles with non-reciprocal anti-aligning interactions, extending active matter theory beyond mean-field and validating results with simulations.

Contribution

It provides the first derivation of a non-linear active Boltzmann equation incorporating non-reciprocal interactions and phase-space effects, advancing the theoretical understanding of active mixtures.

Findings

Analytical expressions for non-reciprocal active mixtures

Excellent agreement with agent-based simulations in low-density regimes

Extension of active matter models to include non-reciprocal interactions

Abstract

We consider the effect of non-reciprocity in a binary mixture of self-propelled particles with anti-aligning 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 non-linear active Boltzmann equation with the exact collision operator is derived. In this derivation, the effect of phase-space compression and the build-up 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 anti-aligning 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.