One-dimensional hard rod caricatures of hydrodynamics

We give here a rigorous deduction of the "hydrodynamic " equation which holds in the hydrodynamic limit, for a model system of one-dimensional identical hard rods interacting through elastic collisions. The equation should be considered as the analog of the Euler equation of real hydrodynamics. Owing to the degener-acy of the model, it is written in terms of a function g(q, v, /) expressing the density of particles with velocity v at the point q at time /. For this equation we prove an existence and uniqueness theorem in some natural class of functions. Our main result is the proof that if {Pc, c> 0} is a class of initial states which are homogeneous on a scale much less than € ~ ! , and if the corresponding particle densities tend, as c-> 0, in the proper scale, to the initial hydrodynamic density g£q,v \ then, under some general assumptions on the states P € and on go, the particle densities of the evolved states at time e"1/, tend as e-»0 to the unique solution of the hydrodynamic equation with initial condition g0. The proof is completed by exhibiting a large class of initial families {Pc, c> 0} which possess the required properties. KEY WORDS: Nonequilibrium statistical mechanics; hydrodynamic limit; Euler equation; one-dimensional hard rods