Rigorous scaling limit for Navier-Stokes and Boltzmann equation

In the now 35 years old survey paper ''Kinetic equations from Hamiltonian dynamics'', Herbert Spohn mentions two important unsolved problems in mathematical physics: On p.571 the hydrodynamic limit, and on p.603 the derivation of the nonlinear Boltzmann equation from quantum mechanics.

Where can I find the current state of affairs? In particular, are there now rigorous derivations of the Navier-Stokes equations and/or the Boltzmann equations from either nonrelativistic many-particle quantum mechanics or quantum field theory? If yes, under which assumptions?

1 Answer

The Boltzmann Equation from Quantum Field Theory

We show from first principles the emergence of classical Boltzmann equations from relativistic nonequilibrium quantum field theory as described by the Kadanoff-Baym equations. Our method applies to a generic quantum field, coupled to a collection of background fields and sources, in a homogeneous and isotropic spacetime. We show that the system follows a generalized Boltzmann equation whenever the WKB approximation holds. The generalized Boltzmann equation, which includes off-shell transport, is valid far from equilibrium and in a time dependent background, such as the expanding universe.