Irreversible dynamics, Onsager-Casimir symmetry, and an application to turbulence

Abstract

Irreversible contributions to the dynamics of nonequilibrium systems can be formulated in terms of dissipative, or irreversible, brackets. We discuss the structure of such irreversible brackets in view of a degeneracy implied by energy conservation, where we consider different types of symmetries of the bracket corresponding to the Onsager and Casimir symmetries of linear irreversible thermodynamics. Slip and turbulence provide important examples of antisymmetric irreversible brackets and offer guidance for the more general modeling of irreversible dynamics without entropy production. Conversely, turbulence modeling could benefit from elucidating thermodynamic structure. The examples suggest constructing antisymmetric irreversible brackets in terms of completely antisymmetric functions of three indices. Irreversible brackets without well-defined symmetry properties can arise for rare events, causing big configurational changes.

Formulation of the relativistic heat equation and the relativistic kinetic Fokker–Planck equations using GENERIC

“…Our approach sheds light on this issue because we show that Eqs. (4) and (5) actually have the same form, see (26) below, and that is why they share the same stationary distribution. Secondly, as being GENERIC systems, (3) and (5) will automatically justify the first and the second laws of thermodynamics.…”

“…Note that this relation also holds true in the classical kinetic Fokker-Planck equation and is known as the fluctuation-dissipation (or Einstein) relation. It is this relation that allows us to write all these systems into a common formulation as in (26). We note that this relation is more general than the one mentioned in Ref.…”

“…(33) is slaved to Eq. (26). In other words, the addition of the excess variable e is simply a mathematical technique, and does not change the original system.…”

“…Let ρ t be a solution of (26). Suppose that V is non-negative and that the initial data ρ 0 satisfy  R 2d H(q, p) ρ 0 (dqdp) < ∞.…”

“…In the original papers [20,21], it was originally introduced in the context of complex fluids with applications to the classical hydrodynamics and to non isothermal kinetic theory of polymeric fluid. Recently it has been applied further to anisotropic inelastic solids [22], to viscoplastic solids [23], to thermoelastic dissipative materials [24], to the soft glassy rheology model [25], to turbulence [26] as well as to the damped Timoshenko and damped Bresse systems [27]. GENERIC in the relativistic setting have been investigated in, e.g., Refs.…”