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 nonlinear system of two PDE’s with free boundaries.
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