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.

Accepted in Physica D, 2016

Cited by in corpus (11)

• Continuum Mechanics and Thermodynamics in the Hamilton and the Godunov-type Formulations
• Hamiltonian and Godunov Structures of the Grad Hierarchy
• Modeling wavefields in saturated elastic porous media based on thermodynamically compatible system theory for multiphase mixtures
• On Hamiltonian continuum mechanics
• Non-convex dissipation potentials in multiscale non-equilibrium thermodynamics
• Gradient and GENERIC evolution towards reduced dynamics
• Ehrenfest regularization of Hamiltonian systems
• Multiscale thermodynamics of charged mixtures
• A case study of non-Fourier heat conduction using Internal Variables and GENERIC
• Landau damping in the multiscale Vlasov theory
• On novel Hamiltonian descriptions of some three-dimensional non-conservative systems