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.

References

Our arguments are heuristic; a complete proof remains open and involves hard issues of ergodic theory.

— Some Considerations on the Fluid-Dynamical Limit of Particle Systems  (2609.18550 - Pulvirenti et al., 16 Sep 2026) in Section 2, “How to derive the Euler equations from particle systems”

The rigorous passage to such a limit in general is open!

— Notes on Hydrodynamic Limits and Related Topics  (2608.20252 - Sethuraman, 20 Aug 2026) in Section 2, subsection “Why is it called ‘hydrodynamics’?”