We consider attractive irreducible conservative particle systems on ℤ, without necessarily nearest-neighbor jumps or explicit invariant measures. We prove that for such systems, the hydrodynamic limit under Euler time scaling exists and is given by the entropy solution to some scalar conservation law with Lipschitz-continuous flux. Our approach is a generalization of Bahadoran et al. [Stochastic Process. Appl. 99 (2002) 1–30], from which we relax the assumption that the process has explicit invariant measures.
"Euler hydrodynamics of one-dimensional attractive particle systems." Ann. Probab. 34 (4) 1339 - 1369, July 2006. https://doi.org/10.1214/009117906000000115