Metriplectic Algebra for Dissipative Fluids in Lagrangian Formulation

Massimo Materassi

doi:10.3390/e17031329

Abstract

The dynamics of dissipative fluids in Eulerian variables may be derived from an algebra of Leibniz brackets of observables, the metriplectic algebra, that extends the Poisson algebra of the frictionless limit of the system via a symmetric semidefinite component, encoding dissipative forces. The metriplectic algebra includes the conserved total Hamiltonian H, generating the non-dissipative part of dynamics, and the entropy S of those microscopic degrees of freedom draining energy irreversibly, which generates dissipation. This S is a Casimir invariant of the Poisson algebra to which the metriplectic algebra reduces in the frictionless limit. The role of S is as paramount as that of H, but this fact may be underestimated in the Eulerian formulation because S is not the only Casimir of the symplectic non-canonical part of the algebra. Instead, when the dynamics of the non-ideal fluid is written through the parcel variables of the Lagrangian formulation, the fact that entropy is symplectically invariant clearly appears to be related to its dependence on the microscopic degrees of freedom of the fluid, that are themselves in involution with the position and momentum of the parcel.

Highlights

The history of Theoretical Physics is, to a certain extent, the discovery of symmetries of physical laws, allowing to bypass the necessity of solving the equations of motion explicitly and gaining deep insights about the essence of first principles themselves.The highest achievements of this simplification process are the least action principles [1,2], with the Feynman path integral [3] as their most recent descendant, and the study of invariances [4], the Hamiltonian formalism [2,5] and the Hamilton-Jacobi theory [5−7]
In the context of Hamiltonian mechanics, the dynamics of physical systems appears in the form of algebra of Poisson brackets [8], composing together the physical observables to both represent the motion of the system and the symmetry properties of its dynamics
The generator S of the dissipative dynamics ψ diss has zero Poisson bracket with any other observable depending on ψ

Summary

Introduction

The history of Theoretical Physics is, to a certain extent, the discovery of symmetries of physical laws, allowing to bypass the necessity of solving the equations of motion explicitly and gaining deep insights about the essence of first principles themselves. Enz demonstrated in [21] that, in order for a system to have decreasing energy, non-time-reversible dynamics and phase volume shrinking with time (three conditions defining dissipative dynamics), it is necessary to add to the antisymmetric symplectic product of Hamiltonian systems, a symmetric semidefinite part Such a formalism was originally introduced as mixed-bracket formulation by Enz and Turski, in [22−24], in the kinetic theory of the phase transitions, I order transitions. The fluid entropy has zero Poisson bracket with any other quantity in both formulations, but the expression of the symplectic product in Lagrangian variables (LV) makes it clear that S is not a Casimir invariant due to the parcel relabeling symmetry (that allows the fluid to possess an Eulerian representation at all), but because it encodes degrees of freedom in involution with the parcel position and momentum (and that enter parcel dynamics only through dissipation).

Metriplectic Complete Systems

Casimir Invariants

Lagrangian Formulation for Viscous Fluids

Entropy and the Casimir Invariant Condition

Conclusions