Hydrodynamics and boundary-induced phase transitions in the $n$-species particle-exchange process

The $n$-species particle-exchange process (PEP($n$)) is an exclusion process in which particles of $n$ different species exchange positions on neighbouring sites with rates chosen such that the invariant measure on the discrete torus is a product measure. We address the large-scale hydrodynamic behaviour of this process which yields a system of $n$ coupled inviscid Burgers equations. This system of conservation laws is shown to admit Riemann invariants for arbitrary $n$ from which explicit solutions of the Riemann problem in terms of shock waves and rarefaction fans are obtained. We also introduce the open PEP($n$), in which particles are exchanged with boundary reservoirs. For a distinguished manifold of boundary rates, we prove that the invariant measure is the same product measure as in the periodic system. The hydrodynamic description in terms of Riemann invariants is used to derive the stationary phase diagram explicitly in terms of microscopic boundary rates. In the generic case, the steady state exhibits $2n+1$ phases, with boundary-induced phase transitions analogous to those of the single-species asymmetric simple exclusion process.