A Poisson map from kinetic theory to hydrodynamics with non-constant entropy

Kinetic theory describes a dilute monatomic gas using distribution function $f(q,p,t)$, the expected phase-space density of particles. The evolves according to collisionless Boltzmann equation in high Knudsen number limit. Fluid dynamics is an alternative description hydrodynamic variables that are functions position and time only. These evolve compressible Euler equations inviscid Both systems noncanonical Hamiltonian systems. Each configuration space infinite-dimensional Poisson manifold, flow generated by functional via bracket. We construct map $\mathcal{J}_1$ from respects brackets on two spaces i.e. map. It maps $p$-integral entropy $f\log f$ density. This belongs family include generalised densities as additional variables. whole can be Taylor expansion further depends formal parameter. If kinetic-theory factors through $\mathcal{J}_1$, exact reduction kinetic fluid possible. not case, but ignoring relative $f$ its local Maxwellian, we approximate resulting reduced generates equations. thus derive approximation theory. also give analogous derivation Euler--Poisson with non-constant entropy, starting Vlasov--Poisson equation.