Spontaneous oscillations driven by nonreciprocal cross-diffusion

Systems with purely diffusive dynamics are commonly dissipative and evolve to homogeneous steady states. Here, by studying a binary mixture of diffusive particles, we demonstrate that nonreciprocal cross-diffusivities can give rise to a counter-intuitive oscillatory state, in addition to the recently discovered traveling state. Using the Fourier-Galerkin method, we obtain a one-mode approximation that highlights the underlying physics of this nonreciprocity-driven oscillation. We construct a phase diagram which agrees well with the numerical simulations. The model equations we study provide a simplified description of the dynamics of a mixture of purely repulsive active and passive Brownian colloids.