One-dimensional hard rod caricature of hydrodynamics

Abstract

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 degeneracy of the model, it is written in terms of a functiong(q, v, t) expressing the density of particles with velocityv at the pointq at timet. For this equation we prove an existence and uniqueness theorem in some natural class of functions. Our main result is the proof that if {∈, ∈ >0} is a class of initial states which are homogeneous on a scale much less than e−1, and if the corresponding particle densities tend, ase→0, in the proper scale, to the initial hydrodynamic densityg o (q,v), then, under some general assumptions on the states ∈− and ong 0, the particle densities of the evolved states at timee −1 t, tend ase→0 to the unique solution of the hydrodynamic equation with initial conditiong 0. The proof is completed by exhibiting a large class of initial families {∈, ∈ >0} which possess the required properties.