A hierarchy of Poisson brackets in non-equilibrium thermodynamics

Abstract

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.

GENERIC guide to the multiscale dynamics and thermodynamics

“…A general answer to this question is not known. The interested reader can find some partial results, for example, in [6,7]. Very useful is also the computer assisted verification of the Jacobi identity developed in [8].…”

“…The Poisson bracket expressing the kinematics of the one particle distribution function ( ) r v f , is the Lie-Poisson bracket (see section 2.1) that can be obtained from the Lie group of canonical transformations in the one particle phase space with coordinates ( ) r v , or alternatively by a reduction [6]. The dissipation potential corresponding to the binary collisions is a particular case of the dissipation potential arising in chemical kinetics [24].…”

“…Following [40], we extend now the above contact formulation of classical thermodynamics to the mesoscopic thermodynamics expressed in the fundamental thermodynamic relation (6). We begin by introducing a space…”

Thermodynamic Explanation of Landau Damping by Reduction to Hydrodynamics

“…This Poisson bracket can be derived from the Liouville Poisson bracket by projection or it can be seen as the Lie-Poisson bracket corresponding to symplectic transformations on the one-particle cotangent bundle (see, e.g., [25]). Any real-valued function of the distribution function σ : R → R then generates a Casimir of the bracket,…”

“…Poisson brackets can often be derived from more detailed Poisson brackets by such projections (see, e.g., [25]). …”