  The formation of dynamical patterns is one of the most striking features of
non-equilibrium physical systems. Recent work has shown that such patterns
arise generically from forces that violate Newton's third law, known as
non-reciprocal interactions. These non-equilibrium phenomena are challenging
for modern theories. Here, we introduce a model mixture of active
(self-propelled) and passive (diffusive) particles with non-reciprocal
effective interactions, which is amenable to exact mathematical analysis. We
exploit state-of-the-art methods to derive exact hydrodynamic equations for the
particle densities. We study the resulting collective behavior, including the
linear stability of homogeneous states and phase coexistence in large systems.
This reveals a novel phase diagram with the spinodal associated with active
phase separation protruding through the associated binodal, heralding the
emergence of dynamical steady states. We analyze these states in the
thermodynamic limit of large system size, showing, for example, that sharp
interfaces may travel at finite velocities, but traveling phase-separated
states are forbidden. The model's mathematical tractability enables precise new
conclusions beyond those available by numerical simulation of particle models
or field theories.
