Separation versus diffusion in a two species system

We consider a finite number of particles that move in $\mathbb Z$ as independent random walks. The particles are of two species that we call $a$ and $b$. The rightmost $a$ particle becomes a $b$ particle at constant rate, while the leftmost $b$ particle becomes $a$ particle at the same rate, independently. We prove that in the hydrodynamic limit the evolution is described by a non linear system of two PDE's with free boundaries.

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