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

We study aspects of the hydrodynamics of one-dimensional totally asymmetric 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 to K-exclusion Ferrari's result that the second class particle follows a macroscopic shock in the Riemann 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.