Metriplectic Algebra for Dissipative Fluids in Lagrangian Formulation

It is known that the dynamics of dissipative fluids in Eulerian variables can be derived from an algebra of Leibniz brackets of observables, the metriplectic algebra, that extends the Poisson algebra of the zero viscosity limit via a symmetric, semidefinite component. This metric bracket generates 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, that generates dissipation. This $S$ is a Casimir of the Poisson algebra to which the metriplectic algebra reduces in the frictionless limit. In the present paper, the metriplectic framework for viscous fluids is re-written in the Lagrangian Formulation, where the system is described through material variables: this is a way to describe the continuum much closer to the discrete system dynamics than the Eulerian fields. Accordingly, the full metriplectic algebra is constructed in material variables, and this will render it possible to apply it in all those cases in which the Lagrangian Formulation is preferred. The role of the entropy $S$ of a metriplectic system is as paramount as that of the Hamiltonian $H$, 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 appears to be more clearly related to its dependence on the microscopic degrees of freedom of the fluid, that do not participate at all to the symplectic canonical part of the algebra (which, indeed, involves and evolves only the macroscopic degrees of freedom of the fluid parcel).


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.
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. An element of the Poisson algebra, namely the Hamiltonian, is the observable that generates the time evolution of any other quantity through the brackets.
This route to the algebrization of dynamics also leads to Dirac's formulation of Quantum Mechanics [9], according to which the algebra of quantum observables is simply a commutation algebra of operators, isomorphic to the Poisson algebra of the respective classical ones. A further progress along this path is the introduction of non-canonical symplectic tensors [10−12], yielding the concept of Lie-Poisson bracket and rendering Casimir invariant quantities important in dynamics. The inclusion of constrained systems in the Hamiltonian framework [13] is another crucial step, allowing for the quantization of gauge systems (including potentially the "fundamental forces of nature", which are indeed gauge forces).
Almost all the benefits of the development just mentioned are, generally speaking, restricted to the Physics of non-dissipative systems: in Lagrangian and Hamiltonian mechanics, as well as in the context of action principles [5,8], only systems undergoing "conservative forces" are treated, while no form of "dissipation" is considered in the fundamental quantum laws, intended as the basic principles of Physics [14]. When dissipative quantum systems are referred to, one typically considers open systems in interaction with some "environment" only partially observed [15], and these are not regarded as "fundamental" (one should however mention the "line of thought", expressed in [16], for instance, in which dissipation is included in the fundamental laws of Quantum Mechanics).
Dissipation, i.e., the presence of interactions rendering the dynamics non-Hamiltonian and non-time-reversible, however appears to be omnipresent in the known universe (especially in the world of complex systems [17], from space plasmas [18] to planetary systems, to life [19]). Dissipation is not only an annoying obstacle to the ambitious abstract dream of finding a Hamiltonian version of as many systems as possible, it rather appears as a constitutive factor of the world [20]: the attempt to bring it under the framework of algebrized dynamics is henceforth expected to give deep insights in our understanding of nature.
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, particularly I order transitions. This mixed-bracket formulation was used in the early 1970s to formulate the dynamics of viscous and heat transporting fluids [25]; then, it was later used extensively in the theory of magnetic systems [26−28], and relativistic plasma [29]. Once a dissipative system is algebrized through the use of mixed-brackets, its quantization may be tried, as described in [21] and [25]: the symplectic part is mapped into the usual quantum commutation algebra, while an anticommutator algebra possibly quantizes the semimetric part giving rise to dissipation [25]. Extensive reviews of mixed-bracket approach are found in [30] and [31].
In the mixed-bracket representation, the system evolution is still generated by the Hamiltonian H that may be either conserved or not, due to the semidefinition of the symmetric part of the bracket. When H is not conserved, however, the formalism does not keep track of the dissipated energy, as the system does not include that part of the environment giving rise to dissipation and draining energy (e.g., a thermal bath, microscopic degrees of freedom or radiation dissipated). A deeper insight about irreversibility of the process may, however, be gained by closing the system with the degrees of freedom that are draining energy. In order to do this, one has to extend the Hamiltonian and include a new observable, namely an entropy.
The dynamics of energetically closed systems relaxing to asymptotic equilibria due to dissipation has been described in by Morrison in [32] via the so-called metriplectic formalism, and this is the approach adopted here. The non-dissipative limit of the system is Hamiltonian, so that there exists some function H and an algebra of Poisson brackets that describes the system in the absence of dissipation. When dissipation is turned on, the Hamiltonian is still constant during the motion, but friction drives the system to an asymptotic equilibrium: this is done introducing the semimetric bracket, and an observable representing the entropy S of the closure of the system.
All the foregoing fruitful attempts to put dissipative systems on the way to algebrization of Physics is represented in [33,34] in terms of Leibniz algebrae.
Quite a few dynamical systems have been reformulated as metriplectic: in [32] the kinetic Vlasov-Poisson approximation of a collisional plasma was described as Hamiltonian in the collisionless limit, while collisional terms are shown to arise from a metric bracket. In [35], a non-ideal fluid described in Eulerian variables (EV) is presented as a non-canonical Hamiltonian system, with the addition of a metric bracket providing the dissipative terms due to the finite viscosity and thermal conductivity. In [36], the non-canonical Hamiltonian dynamics of a free rigid body throughout the space of its angular velocity is enriched by a metric contribution allowing the rotator to relax down to asymptotic equilibria (corresponding to spinning around one of its inertial axes).
In [37], there appeared a specific formulation turning the Boltzmann equation into algebraic dynamics, which violates the antisymmetric-symmetric duality of conservative-dissipative components. The formulation of metriplectic dynamics given in [32,35,36] has been refined in the work of Grmela and Ottinger (see [38] and [39]), extended to more complex system contexts, and renamed as GENERIC (General Equation for the Nonequilibrium Reversible-Irreversible Coupling).
A general review of Poisson and metric brackets to describe energetically isolated or non-isolated systems (referred to as complete and incomplete) may be found in [40]. In his PhD thesis [41], Fish applies the results of [32,35,36] examining metriplectic systems of various types under the point of view of manifold properties, and also giving interesting examples from applied physics and biophysics. In [42] the unity of all those approaches is recognized.
The metriplectic system describing neutral fluids in [35] has been generalized to non-ideal magneto-hydrodynamics in [43], while examples of how to algebrize simple mechanical systems with friction are provided in [44].
In the present paper, the Lagrangian Formulation (LF) of the metriplectic algebra for a viscous fluid is constructed. The symplectic part of the metriplectic system is taken from [35,45,46], while the metric part is an original contribution presented here for the first time as far as the author is aware of, by mapping the metric bracket in EV to its expression in parcel variables.
Even if rather interesting from the viewpoint of mathematical completeness, still the translation of the Eulerian metriplectic algebra to the Lagrangian one can be questioned as to whether it is worth the effort in physical terms, due to its very complicated appearance. Instead, it should be underlined that the symmetry-related role of the fluid entropy appears much clearer in the Lagrangian algebra than in the Eulerian one, not to mention that whenever the use of LF is preferred to that of the Eulerian Formulation (EF), the expressions found here will be applied.
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 simply because it encodes degrees of freedom in involution with the parcel position and momentum (and that enter parcel dynamics only through dissipation).
The paper is organized as follows.
In Section 2 the general framework of metriplectic complete systems is sketched, while in Section 3 we discuss briefly the role of Casimir invariants of the theory with respect to algebra reduction and dissipative processes. Section 4 is dedicated to the key result of this paper: the LF is constructed for viscous fluids, and their metriplectic algebra is formulated in the material variables. With this result in mind, a speculation on the nature of the fluid entropy as a Casimir invariant of the theory is presented in Section 5.
Conclusions are reported in Section 6, where possible applications and future developments of the present research are also sketched.

Metriplectic Complete Systems
In this § we essentially follow the line of reasoning of [32,35,36], and resume those results. Consider an energetically closed system with dissipation, and describe its state as a point ψ moving in a suitable phase space V. Also, refer to its algebra of observables O as a subset of ( ) According to the metriplectic scheme, its dynamics ψ will be expressed as the sum of a referred to as metric bracket. λ is a negative constant parameter (making physical sense only in the correspondence with an asymptotic equilibrium [43]).
The generator S of the dissipative dynamics diss ψ has zero Poisson bracket with any other while the metric bracket (., .) must have H among its null modes ( ) If the evolution of the system works as depending on ψ evolves according to the same rule Due to the conditions (1) and (2), this general rule also implies the first of these equations means that H is constant because it is not altered by dissipation, that just redistributes energy but does not destroy nor creates it; the second condition in (5) states that S asymptotically and monotonically grows during the motion, as a Lyapunov quantity is expected to do in the correspondence with an asymptotic stable state [47]. This also implies that the energy redistribution takes place irreversibly. The conditions (1) and (2), together with the properties of {., .} and (., .) as Leibniz brackets [33], allow for the definition of a total metriplectic generator F = H + λS so that, provided the new bracket is defined, one may simply state for any observable Φ . The new Leibniz structure defined in (6) is the metriplectic bracket, while the metriplectic generator F is sometimes referred to as free energy.

Casimir Invariants
The condition (1) attributes to the metric generator S, whatever it is physically, the algebraic character of Casimir invariant (CI) of the Poisson bracket {., .}, as was recognized in [25]. Now, the metric generator may be a CI for one of the two following reasons. Either, the symplectic bracket {., .} includes derivatives with respect to the variables on which S depends too, and nevertheless admits a non-trivial kernel to which S belongs; this case, we will refer to as C1, is typical for  The free dumped rotator presented in [36], and revisited in [41], is easily recognized to be a C1 case: the square angular momentum is such a CI when the phase space of the rotator is reduced from the 6-dimensional space of angles and their canonical momenta to the 3 R of angular velocities; in this case, the Casimir quantity is the square angular momentum 2 L  , that depends on the same vector L  (the angular momentum) appearing in the symplectic part. Systems with dissipative constants regarded as control parameters depending on an external variable are properly C2 cases (e.g., the Lodka-Volterra, Lorentz and Van Der-Pole systems in [41], or the elementary mechanics dissipative systems reported in [44], where the external variable is the state of a thermal bath).
One should underline that the distinction between C1 and C2 Casimir just stressed is, in a sense, matter of choice of coordinates in the phase space V, since, as some S is in involution with any other quantity, it must be possible to find a description of V with { } .,. completely independent of the arguments of S: in this sense, all the metriplectic systems might be cast into the C2 format. This is indeed the case for the rotator mentioned in [36] and [41]: the .,. , as in the C2 type systems: this reads the original algebra of the rotator as reducible to a C2 one.
Reviewing this line of thought of the fluid dynamical systems is postponed to § 6. Last but not least, deciding whether the Boltzmann entropy playing the role of the metric generator for the Vlasov-Poisson collisional plasma is a C1 or C2 quantity deserves a deeper investigation, involving the fact that Vlasov-Poisson equation results from the truncation of a hierarchy of equations involving many-particle variables [50], the symplectic limit of which has been studied in [51].
The origin of being a CI for the entropy of a viscous fluid is investigated here, writing its metriplectic algebra explicitly in LV.

Lagrangian Formulation for Viscous Fluids
In [34], the viscous fluid equation is described in the Eulerian Formalism (EF), via the fields mass and mass-specific entropy density ( ) t x,  σ . In the non-dissipative limit the dynamics takes a non-canonical Hamiltonian form: the Poisson bracket between two any Greek indices are used for the SO(3)-vector components of v  and of the position x  in the space, and a summation convention holds, so that scalar products in 3 R read used for spatial gradients. The Hamiltonian functional of the system reads where p is the pressure.
Let then viscosity and thermal conductivity be finite. Let the viscosity tensor be of the form , with Λ constant (this Λ is formed by Kronecker tensors and physical "constitutive" constants, see [43]); let the heat flux I  be related to local temperature T as T I α α κ∂ − = : then, the symplectic algebra (7) must be completed by the metric bracket [35] ( ) , , This gives rise to the equations of motion: The symbol β β ∂ ∂ = ∂ 2 has been used.
In the LF, the fluid is subdivided into material parcels labeled by a continuous three-index a  , and the motion and evolution of each a  -th parcel is followed [52]. As far as its motion throughout the space is concerned, the a  -th fluid parcel is described at time t by its position ( ) can be made). Being that the parcel is not purely pointlike, these will be understood as the position and momentum of the center-of-mass of the parcel. Since the parcel is a system of ( ) 23 10 O microscopic particles, it must be equipped also by some variable describing those μDoF: its mass-specific entropy density ( ) t a s ,  is given this role [53]. The fact that the μDoF of the a  -th parcel are all encoded in the thermodynamical variable ( ) t a s ,  suggests that they are treated statistically: in a sense, the metriplectic formalism is the algebrization of a stochastic dynamics in which what remains of the probabilistic noise is the equilibrium thermodynamics of it [23,24,44].
In the optics of [54,55], s encodes the thermodynamics of the relative variables.
The field configuration ( ) is the initial mass density of the a  -th parcel. The mass-specific internal energy density U depends on the density of the parcel, that reads J 0 ρ ρ = because of mass conservation [56], and on its entropy. The dynamics of the nondissipative limit in LV is governed by an apparently canonical Poisson bracket, reading: (in Equation (14) the "dot" means "time derivative along the motion of the parcel", also called Lagrangian, or material, derivative). In order to complete the algebraic dynamics of the non-ideal fluid in LF, the metric part must be produced. The first step is to consider that the equations of motion to be reproduced are the translation of the system (10) in parcel variables: The definition of i A α was already given in (14). The operator μ ∇ is the derivative with respect to μ ζ intended as the differential operator The Eulerian field is the value taken by the corresponding Lagrangian quantity attributed to the parcel that, at that given time, transits at that given point in the space: hence, one should understand Special care must be used to treat the relationship between the functional derivative with respect to any Eulerian field ( ) , do not exactly coincide: . As a result, one may write: All in all, the metric bracket for a viscous fluid in LF reproducing Equations (15) with [ ] s S as a metric generator, reads: The bracket (17) is easily shown to exhibit all the necessary properties for it to be a metric bracket: it is thoroughly symmetric in the Ψ ↔ Φ exchange, while about semidefiniteness one may note that holds, provided the correct "dictionary" is used, so that one may count of the fact that ( ) L Ψ Φ, inherits all the good properties from those demonstrated for ( in [35,43] and references therein. With the finding (17) we have the complete metriplectic algebra of viscous fluid dynamics in the LF, which can be reported as: As anticipated before, the advantage of looking at the metriplectic fluid dynamics in the LF, instead of in the EF, is that a certain subtlety about entropy is clarified, that has to do with the question of it to be a CI of the theory. This clarification comes in Section 5.

Entropy and the Casimir Invariant Condition
Back to what was described in the Introduction, we speculate here on the entropy of fluids [35,43], that appears as in-between the "two ways of being a Casimir" C1 and C2.
On the one hand, this S clearly encodes information on the μDoF of the continuum, while the fluid velocity describes a macroscopic point of view of the system, as it happens in the C2 case. On the other hand, Morrison and Padhye had algebraic reasons to show, in [45,46], that this S belongs to a family of quantities conserved via a "C1 mechanism" (out of the reduction of the algebra (13) that become the physical quantities in the EF, and are invariant under parcel relabeling transformations (RT) [56]). Examining the LF of the fluid, with the RT more clearly readable, the opinion of the author here has become that the viscous fluid may be considered on the same foot as those mentioned in [41] and [44], classifying its S in the C2 case. The RTs, on which the LF to EF reduction is based, are smooth invertible maps ' a a    that leave the Hamiltonian (12) and the Eulerian fields (16) , with 1 C given in (18): this is the set of the quantities that can be constructed in the LF but have not a corresponding Eulerian quantity, because they are not RT-invariant [45,46,56]. The physical difference between S and any 1 C is that S includes only the μDoF responsible for dissipation, while the 1 C s mix them with the parcel's center-of-mass variables ( ) π ζ   , . In few words, only S is expected to play the driving role in dissipation processes.
The physical difference of roles for 1 C and 2 C in (18) persists in the metriplectic algebra of the fluid in the EF. In terms of Eulerian fields those quantities appear as follows with symmetric brackets other than that in (9) remains an open question, also because it is not clear yet how to define the metric bracket "from First Principles" (about this: Turski and his co-authors showed how to construct metriplectic systems out of Lie-Poisson algebrae via Cartan metrics and group theory reasoning [25][26][27][28]25], but this does not seem to be strictly the case here).

Conclusions
A viscous fluid with suitable border conditions relaxes to an asymptotic equilibrium due to the presence of dissipation, while it can be written in a Hamiltonian form in its frictionless limit. This is a perfect system to be put in a metriplectic form according to the formalism in [32,44] and references therein.
Fluids may be represented in EF or in LF, and the metriplectic framework for the EF was already known [35]. Here, the metriplectic algebra in the LF is obtained, adopting the parcel variables as in [52] to describe the fluid in a metriplectic form: the resulting picture is clearer than the one in EF.
The π . Hence, the metric generation of dissipation in this case shows the same mechanism as presented in [44] and in Chapter 8 of [41].
Despite the Lagrangian Formulation leads to equations of motion that are more complicated than the ones in EF, stating the dynamics of a viscous fluid in parcel variables appears crucial in order to describe more transparently mesoscopic coherent structures of matter [18].
In their EF, fluids (and plasmas) appear to be often characterized by modes representing local subsets of the continuum in which the parcels move with macroscopic scale correlations (e.g., in vortices or current structures); collective variables describing such field configurations will probably be better described by adopting parcel variables ( ) , because long range correlation are likely to form well defined patterns "in the a  -space" rather than "in the x  -space", since the " a  -space" is the set 0 D of parcels' identities, where it is possible to keep track of which parcel has interacted with which other one, and hence developed correlation at mesoscopic scales. Forthcoming studies will investigate the application of what obtained here to the LF of vortices [58,59], while a contact with the tetrad formalism, describing parcels of various scales [60−64], will be made. Moreover, the LF of an MHD collisional plasma may be constructed, as an extension of the present study to electromagnetic degrees of freedom. An important point made here is about the distinction between Casimir C1 depending on variables with respect to which derivatives appear in the Poisson bracket, and C2 depending on degrees of freedom that are left untouched by the symplectic bracket. Here the fluid entropy has been clarified to be a C2 type Casimir because it encodes the µDoF of the parcel, while { } L .,. only acts on the parcel's center-of-mass. According to the line of thought that reinterprets the C2 as C1 systems written suitably, exposed in § 3 for the rigid rotator, those cases reported in [41] are expected to be obtainable from "big" symplectic algebae .,. , , , , ... .,. , .,. , , .,. , , has already operated. The author suggests that pursuing an investigation capable of filling in the gap "…" in (21) will give a deep insight into the problem of the origin of irreversibility tout court.