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Summary. We study aspects of the hydrodynamics of one-dimensional totally asym- metric K-exclusion, building on the hydrodynamic limit of Seppalainen (1999). We prove that the weak solution chosen by the particle system is the unique one with maximal current past any fixed location. A uniqueness result is needed because we can prove neither differentiability nor strict concavity of the flux function, so we cannot use the Lax-Oleinik formula or jump conditions to define entropy solutions. Next we prove laws of large numbers for a second class particle in K-exclusion. The macroscopic trajectories of second class particles are characteristics and shocks of the conservation law for the particle density. In particular, we extend toK-exclusion Ferrari's result that the second class particle follows a macroscopic shock in the Rie- mann solution. The technical novelty of the proofs is a variational representation for the position of a second class particle, in the context of the variational coupling method.

Second class particles as microscopic characteristics in totally asymmetric nearest-neighbor $K$-exclusion processes