On a weak Hydrodynamic Limit theory

To understand mechanical origin of probability in statistical and continuum mechanics, it is useful to study hydrodynamic limit for interacting particles following deterministic Hamiltonian dynamics. Traditional approach on such a program faces many difficulties. One of them is about rigorous justification of canonical type ensembles. This is because that relevant deterministic ergodic theory is still largely out of reach. Another huge barrier is on making rigorous sense out of hyperbolic conservation laws. Such PDEs are used to express F=ma and thermodynamic relations in the continuum.

We examine a new line of thoughts by formulating the hydrodynamic limit program as a multi-scale abstract Hamilton-Jacobi theory in space of probability measures.

This talk will focus on derivation of an isentropic model. Through mass transport calculus, we develop tools to reduce the hydrodynamic problem to known results on finite dimensional weak KAM (Kolmogorov-Arnold-Moser) theory, showing sufficiency of using a weak version of ergodic results on micro-canonical type ensembles, instead of the canonical ones. We will also reply on recent progress of viscosity solution theory for abstract Hamilton-Jacobi equation in space of probability measures (an example of Alexandrov space). Such approach gives a weak and indirect characterization on evolution of the limiting continuum model using generating-function formalism at the level of canonical transformation in calculus of variations. It avoids the use of hyperbolic systems of PDEs, which operates at the level of abstract Euler-Lagrange equations from the action functionals.

All together, these techniques enable us to realize a weaker but rigorous version of the hydrodynamic limit program for some nontrivial cases.

This is a joint work with Toshio Mikami from Tsuda University, Tokyo, Japan.