On rigorous Hydrodynamics, Self-diffusion and the Green-Kubo formulae

Abstract

We describe some thoughts and results revolving around the mathematically rigorous derivation of hydrodynamics from deterministic microscopic mechanical model. We recall first the theoretical physics standard on linearized hydrodynamics. In particular we point to the fact that the Green-Kubo formulae may serve as a starting point for a rigorous theory: One looks upon hydrodynamics only from the asymptotics of certain relevant two point equilibrium correlation functions. These describe fluctuations of the locally conserved hydrodynamic fields. We try first to catch the flavour in simple examples of mathematical nature, in particular we observe the well known fact that self-diffusion is part of the theory. Then we elaborate on the physical example of the hard rods system and show how one may understand the bulk diffusion in that system — of the particle density per velocity — as a phenomenon of self-diffusion of a velocity pulse.