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Pair interactions between active particles need not follow Newton's thirdlaw. In this work we propose a continuum model of pattern formation due tonon-reciprocal interaction between multiple species of scalar active matter.The classical Cahn-Hilliard model is minimally modified by supplementing theequilibrium Ginzburg-Landau dynamics with particle number conserving currentswhich cannot be derived from a free energy, reflecting the microscopicdeparture from action-reaction symmetry. The strength of the asymmetry in theinteraction determines whether the steady state exhibits a macroscopic phaseseparation or a traveling density wave displaying global polar order. Thelatter structure, which is equivalent to an active self-propelled smecticphase, coarsens via annihilation of defects, whereas the former structureundergoes Ostwald ripening. The emergence of traveling density waves, which isa clear signature of broken time-reversal symmetry in this active system, is ageneric feature of any multi-component mixture with microscopic non-reciprocalinteractions.

Scalar Active Mixtures: The Non-Reciprocal Cahn-Hilliard Model