  Reversible evolution of macroscopic and mesoscopic systems can be
conveniently constructed from two ingredients: an energy functional and a
Poisson bracket. The goal of this paper is to elucidate how the Poisson
brackets can be constructed and what additional features we also gain by the
construction. In particular, the Poisson brackets governing reversible
evolution in one-particle kinetic theory, kinetic theory of binary mixtures,
binary fluid mixtures, classical irreversible thermodynamics and classical
hydrodynamics are derived from Liouville equation. Although the construction is
quite natural, a few examples where it does not work are included (e.g. the
BBGKY hierarchy). Finally, a new infinite grand-canonical hierarchy of Poisson
brackets is proposed, which leads to Poisson brackets expressing non-local
phenomena such as turbulent motion or evolution of polymeric fluids.
Eventually, Lie-Poisson structures standing behind some of the brackets are
identified.
