Dynamical patterns of nonreciprocally coupled conserved fields: a unifying model

Multicomponent active systems ubiquitously exhibit non-reciprocal effective interactions between the two species that seemingly evade Newton's third law. Nonreciprocity arises at the mesoscopic level when forces are mediated by a nonequilibrium environment and generates emergent traveling and oscillating patterns. We show that coupled Cahn-Hillard equations provide a unifying model for dynamical pattern formation in non-reciprocally coupled binary systems with conserved fields. We demonstrate that a number of physical systems can be explicitly mapped to this generic form, including mixtures of active and passive Brownian particles, active poroelastic media, mass-conserving reaction-diffusion systems, and bacterial populations with mutual motility control. In all these systems mass conservation is responsible for the Goldstone mode which interacts with the non-reciprocal couplings mediating mass redistribution and driving the onset of traveling states. We identify a generic form of the dispersion relation governing the growth of linear fluctuations that provides the identifying signature for this class of nonreciprocally coupled conservation laws. Finally, the propagation speed of the traveling waves can be read-off from the dispersion relation even deep in the nonlinear regime.