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

摘要

We study aspects of the hydrodynamics of one-dimensional totally asymmetricK-exclusion, building on the hydrodynamic limit of Seppalainen (1999). We provethat the weak solution chosen by the particle system is the unique one withmaximal current past any fixed location. A uniqueness result is needed becausewe can prove neither differentiability nor strict concavity of the fluxfunction, so we cannot use the Lax-Oleinik formula or jump conditions to defineentropy solutions. Next we prove laws of large numbers for a second classparticle in K-exclusion. The macroscopic trajectories of second class particlesare characteristics and shocks of the conservation law for the particledensity. In particular, we extend to K-exclusion Ferrari's result that thesecond class particle follows a macroscopic shock in the Riemann solution. Thetechnical novelty of the proofs is a variational representation for theposition of a second class particle, in the context of the variational couplingmethod.