Hamiltonian and Godunov Structures of the Grad Hierarchy

Abstract

The time evolution governed by the Boltzmann kinetic equation is compatible with mechanics and thermodynamics. The former compatibility is mathematically expressed in the Hamiltonian and Godunov structures, the latter in the structure of gradient dynamics guaranteeing the growth of entropy and consequently the approach to equilibrium. We carry all three structures to the Grad reformulation of the Boltzmann equation (to the Grad hierarchy). First, we recognize the structures in the infinite Grad hierarchy and then in several examples of finite hierarchies representing extended hydrodynamic equations. In the context of Grad's hierarchies we also investigate relations between Hamiltonian and Godunov structures.

accepted in Phys.Rev.E

References in corpus (4)

• A hierarchy of Poisson brackets in non-equilibrium thermodynamics
• Externally driven macroscopic systems: Dynamics versus Thermodynamics
• Extra mass flux in fluid mechanics
• Boltzmann-Equation Based Derivation of Balance Laws in Irreversible Thermodynamics

Cited by in corpus (9)

• On Hamiltonian continuum mechanics
• Multiscale Thermodynamics
• On the role of geometry in statistical mechanics and thermodynamics II: Thermodynamic perspective
• Gradient and GENERIC evolution towards reduced dynamics
• Multiscale thermodynamics of charged mixtures
• A case study of non-Fourier heat conduction using Internal Variables and GENERIC
• The Markov Process Admits a Consistent Steady-State Thermodynamic Formalism
• Multiscale Theory
• On Extensions, Lie-Poisson Systems, and Dissipations