Hydrodynamic limits for deterministic particle systems

Establish a rigorous hydrodynamic limit for the deterministic Newtonian particle system with local density, momentum, and energy empirical measures converging to a solution of the corresponding Euler equations.

Background

The notes discuss a deterministic system of interacting particles evolving according to Newtonian dynamics and conserving total mass, momentum, and energy. After rescaling time and space, the formal macroscopic variables are expected to satisfy a system of five conservation laws of Euler type.

The formal derivation requires replacing nonlinear microscopic quantities, including momentum products and interaction-force terms, by averages under infinite-volume local equilibrium measures. The notes explain that obtaining the necessary ergodic theorem for the purely deterministic dynamics is difficult; adding noise can make rigorous local averaging and limiting arguments possible.