Nonreciprocal reaction-diffusion systems with a Lyapunov functional: traveling wave instabilities that decrease the free energy and oscillatory coarsening

Nonreciprocal interactions in nonequilibrium mixtures have emerged as a paradigmatic route to spatiotemporal patterns in active and driven systems. The onset of these patterns is commonly characterized by a linear stability analysis of a homogeneous state, where nonreciprocity manifests as an asymmetric Jacobian. Here, we reveal two classes of reaction-diffusion systems where the Jacobian is asymmetric, and thus the system is nonreciprocal, yet the free energy remains a Lyapunov functional. In the first class of systems, nonreciprocity is superficial as oscillatory instabilities are precluded and small-wavelength instabilities cannot occur in the absence of long-range forces. More interestingly, in the second class of systems there can be both small-wavelength and oscillatory instabilities which necessarily saturate to a static state that minimizes a free energy. We prove that this is the case for two kinds of nonideal reaction-diffusion systems with thermodynamically-consistent fluxes and reactions. Furthermore, we numerically demonstrate the resulting `"oscillatory coarsening", where instabilities first appear as traveling waves and then phase-separated regimes oscillate in density before saturating to the thermodynamically prescribed binodals. In doing so, we clarify the role of nonreciprocity in reaction-diffusion systems and unveil how free energy-minimizing states can be achieved through a decidedly nonequilibrium mechanism.