Multiscale thermodynamics of charged mixtures

Fluid mechanics and electrodynamics are two theories of Hamiltonian nature, which are coupled through the Lorentz force. Besides the fields of electric displacement and magnetic field, there are also the fields of polarization and magnetization, which are interacting with both matter and electromagnetic field. We propose a geometrical construction of reversible evolution equations of all the mentioned fields in mutual coupling. Afterwards, dissipation is imposed to particular fields, which are then reduced to the respective constitutive relations playing a role on less detailed levels of description. In summary, we propose multiscale thermodynamics of mixtures of fluids, electrodynamics, polarization and magnetization in mutual interaction.


Introduction
Theoretical electrochemistry aims to describe and predict behavior of chemically reacting systems of charged substances. The modeling methods vary according to the characteristic times and lengths of the observed electrochemical systems. This paper aims to develop a hierarchy of continuum models on different levels of description using the framework of the General Equation for Non-Equilibrium Reversible-Irreversible Coupling (GENERIC) [1,2,3,4].
Let us thus briefly recall GENERIC. Consider an isolated system described by state variables x. The state variables can be for instance position and momentum of a particle, field of probability density on phase space, fields of density and momentum density, electromagnetic fields, etc. Evolution of functionals F (x) of the state variables is then expressed aṡ where the former term on the right hand side stands for a Poisson bracket of the functional and energy while the latter for scalar product of gradient of F and gradient of a dissipation potential. Conjugate variables (derivatives of entropy in the entropic representation) are denoted by x * . The Poisson bracket is antisymmetric, which leads to automatic energy conservation, and satisfies Jacobi identity, which expresses intrinsic compatibility of the reversible evolution. The irreversible term yields a generalized gradient flow driven by gradient of entropy and ensures the second law of thermodynamics. Many successful models in non-equilibrium thermodynamics have been cast in the GENERIC structure (1), and many new thermodynamically consistent models have been obtained by seeking that structure. Our strategy in this paper will be to couple the Poisson bracket of fluid mechanics and Poisson bracket of electrodynamics in vacuum by semidirect product. In other words, we let the electrodynamics be advected by fluid mechanics as in [4]. To go beyond, we add also the field of polarization density and a canonically coupled momentum of polarization. This is important to express the behavior of dipole moment of molecules in interaction with electromagnetic field and the overall motion. Moreover, we add magnetization (the famous Landau & Lifschitz model) advected by fluid mechanics. This way we build a hierarchy of levels of description with appropriate Poisson brackets expressing kinematics on the levels.
Subsequently, we introduce dissipation on the most detailed levels of description, which leads to reduction to less detailed (lower) levels, finishing on the level of mechanical equilibrium, where the evolution is governed by the generalized Poisson-Nernst-Planck equations. We believe that such a complete and geometric picture of continuum thermodynamics of matter coupled with electrodynamics (including polarization and magnetization) was missing in the literature.

Hierarchy of Poisson brackets 2.1 Fluid mechanics of mixtures
Let us start with fluid mechanics, where state variables are fields of density, momentum density and entropy density, x = (ρ, u, s). The Poisson bracket expressing kinematics of fluid mechanics has been long known [5,6,7,8,9]. The Poisson bracket can be easily extended to mixtures with multiple densities, momenta and entropies (i.e. temperatures), see e.g. [10].

Classical fluid mechanics
The Poisson bracket generating one-component compressible fluid mechanics (hydrodynamic Poisson bracket) is where ρ, u and s are density, momentum density and volume entropy density, respectively, see the references above. Once energy, usually is provided, reversible evolution of an arbitrary functional F of the state variables readṡ where integration by parts was used several times 1 . Boundary terms disappear as we assume isolated (e.g. periodic) system. By comparing with the chain rulė we can read the evolution equations for fluid mechanics, the compressible non-isothermal Euler equations for ideal fluids.

Fluid mechanics of mixtures
Consider now a mixture of n species (denoted by Greek indexes from set Ω), each of which is described by its own density, momentum density and entropy density. The Poisson bracket expressing kinematics of state variables x = (ρ α , u α , s α ), α ∈ Ω, is This Poisson bracket can be derived for instance by projection from the Liouville equation [10]. It consists of sum of n Poisson brackets (2), each expressed in terms of variables of mixture component α.
Poisson bracket (7) depends on n momenta and entropies, each for one component of the mixture, which is a rather detailed description allowing for independent motion of the constituents and for different temperatures of them (as in cold plasma, where electrons have different temperature than ions). We are, however, often interested in less detailed description, keeping only densities of the species, the total momentum and total entropy, By letting the arbitrary functional depend only on state variables x = (ρ α , u, s), bracket (7) becomes referred to as the classical mixture hydrodynamic bracket and generating the reversible part of Classical Irreversible Thermodynamics (CIT) [11]. The descriptions of the fluid and fluid mixture dynamic considered in the remainder of the paper will be based on the brackets (2) and (9), respectively. Hence, the description considering the distinguished momenta of species (7) is further avoided.

Electrodynamics in vacuum
The reversible evolution of electromagnetic fields is generated by the canonical Poisson bracket, see [4], where A stands for the vector potential and Y = −D denotes negative of electric displacement field (either in variables (A, Y) or (A, D)). Let us define the magnetic field as In order to express the bracket (10) in terms of magnetic field, the assumed functionals to be dependent only on the curl of A. Bracket (10), transformed in terms of (D, B), see [12], reads as This is the Poisson bracket expressing kinematics of electromagnetic fields D and B. Let us suppose the following energy of the electromagnetic field in vacuum: here ε 0 and µ 0 stand for vacuum permittivity and vacuum permeability, respectively. The evolution equations of the electromagnetic field given by (12) and (11) read where the conjugate fields are actually electric and magnetic intensities, E D = E and E B = H. Using energy (11), the evolution equations can be rewritten as where ε 0 E = D denotes the electric field as well. Applying divergence to (15) gives the following evolution equations: Hence, the usual constraints-Gauß's law for electric and magnetic charge [13]hold true if satisfied by the initial condition, see [14].

Electromagnetic field advected by charged fluids
The purpose of this section is to formulate coupled kinematics of fluids and electromagnetic fields. We employ the theory of semidirect product to find such coupling, and then we perform a transformation unveiling the usual form Lorentz force.

Semidirect product
We have already recalled the Hamiltonian nature of fluid mechanics. There is, however, a finer structure behind, the Lie-Poisson dynamics, where the Poisson bracket is the Lie-Poisson bracket on a Lie algebra dual. Another examples of Lie-Poisson dynamics are rigid body rotation or kinetic theory. In [15] it is explained how to construct new Hamiltonian dynamics by letting one Hamiltonian dynamics be advected by another. Having a Lie algebra dual l * (for instance fluid mechanics), an another Lie algebra dual or cotangent bundle is advected by l * by the construction of semidirect product. One can even think of mutual action of the two Hamiltonian dynamics, which leads to the structure of matched pairs [12], [16]. Here, however, we restrict the discussion only to one-sided action of one Hamiltonian system to another, i.e. to the semidirect product. A general formula for the Poisson bracket of semidirect product of a Lie algebra dual l * and cotangent bundle is the Lie-Poisson bracket on the Lie algebra dual, {F, G} (T * M ) is the canonical Poisson bracket on the cotangent bundle, •, • is a scalar product (usually L 2 , i.e. integration over the domain, or duality in distributions D ′ ), m ∈ l * is the momentum density (element of the Lie algebra dual) and ⊲ is the action of l * on T * M , minus the Lie derivative −L, see Appendix B. Instead of single-component fluids, we can take bracket (9), and after the transformation from (A, Y) to (D, B), as in Sec. 2.2, we obtain Poisson bracket where m denotes total momentum density (of matter and electromagnetic field), Note that the notation introduced in (18)  Bracket (18) expresses kinematics of a CIT mixture and electromagnetic field, with state variables x = (ρ α , m, s, D, B), and was found (for the single species case) in [17,18,19].

Transformation to mass momentum
Let us suppose that each mixture component carries charge e 0 zα mα proportional to mixture density. The free charge density is defined as Poisson bracket (18) can be now transformed by means of relation (19) to the mass momentum u instead of the total momentum m. The calculation was carried out in [12] and [4] and leads to Poisson bracket which is the Poisson bracket expressing evolution of a CIT mixture coupled with electromagnetic field (using the mass momentum u).
The relation between formulation (22) and (18) can be also viewed in terms of surface balances of the electric induction flux and magnetic induction flux for static and moving surfaces as it is shown in [

Gauß's law for electric and magnetic charges
Equations (15) represent the Gauß's law for the electric and magnetic charges in vacuum. Let us now consider the dynamics of coupled matter and electromagnetic field generated by brackets (22) and (18). The evolution equation of magnetic field for bracket (22) reads as and for (18) as As for vacuum, if the initial condition div B(r, t = 0) = 0 is satisfied then neither (23) nor (24) can violate the Gauß's for magnetic charge for further times.
The discussion for the electric charge is more subtle. The evolution equations of D for brackets (22) and (18) are respectively. The divergent part of the first two equations is equal to the charge-weighted sum of the evolution equations of the partial mass densities generated by the respective brackets. Therefore, if is satisfied as an initial condition then again neither (25) nor (26) can violate it.
Casting divergence on equation (26) gives Clearly, the form of equation (28) is the same as for a conserved continuumadvected density (quantity). Therefore, analogously to the previous, if (27) is satisfied as an initial condition it will be satisfied during the evolution. From the geometric point of view the Gauß's laws can be seen as gauge invariance of the respective Lie algebraic structures [14].
We have shown that the dynamics of charged continuum generated by brackets (22) and (18) structurally preserves the Gauß's laws for electric and magnetic charge.

Polarization
The reversible evolution of a charged mixture in electromagnetic field is described by one of the Poisson brackets in the previous section and a choice of energy. But such description does not, in general, capture the intrinsic dipole moments of the molecules, i.e. polarization. An additional bound charge is present due to internal dipole density of the matter on top of the modeled free charge.
Description of the polarization charge depends profoundly on the chosen variables, the time/space scales and the internal structure of the assumed matter. The classical treatment on the macroscopic level, see e.g. [21], resorts to the definition of polarization vector P. The divergence part of P is set equal to the density of polarization charge. The time derivative of P represents current, therefore it is added to the left-hand side of the Ampère's law.
Russakoff in [22] acquired the polarization as a consequence of averaging of microscopic Maxwell's equations with point charges and subsequent expansion with respect to spatially correlated charges. This approach leads, compared to the Purcell's, to a definition of the polarization related magnetization M which is proportional to the averaged relative velocities of the correlated charges. Neither Russakoff nor Purcell discussed the dynamics of the point charges nor the correlated collections of charges, i.e. molecules and ions.

Intrinsic dynamics of polarization
Density of polarization P represents a vector field just as the displacement field D. Advection of the vector field by the fluid mechanics is then expressed by a semidirect product as in Sec. 2.3.1. Similarly as in that Section, we can add (besides P) the conjugate momentum variable (to be denoted by µ). The Poisson bracket expressing advection of the pair (µ, −P) by fluid mechanics is shown in the following section.
The canonical Poisson bracket of state variables µ and P reads which is analogical to (10). Bracket (29) can represent a continuum of elementary dipoles with fixed centres of mass, but changing length and orientation. Indeed, covector field µ can be interpreted as proportional to the relative momentum of a dipole particles, c.f. variable t in (124) and (125). The divergence part of P represents the bound charge density,

Advected cotangent bundle (µ, −P)
When the dipoles are not fixed in space, but advected by a fluid, the interaction is captured by coupling of bracket (29) and fluid dynamics bracket (9) by semidirect product, which is analogical to bracket (130). Note that the total momentum of the coupled system is denoted as µ u. This Poisson bracket expresses kinematics of fluid mechanics advecting the polarization density with its conjugate momentum (relative momentum of the intrinsic dipole charges). Note that bracket (31) may be briefly expressed as using the definition of {F, G} (SP) from (18) and {F, G} (SP) A from (31). The evolution equations given by (31) are as follows, where x † denotes derivative of energy with respect to x (as everywhere in this paper). Later they will be equipped with dissipation of µ triggering subsequent relaxation of polarization.

Reduced variable M
Instead of the conjugate polarization momentum µ, we can choose to work with curl of it, Similarly to the relation of vector potential and magnetic field, considering functionals depending only on M instead of µ turns Poisson bracket (33) to Bracket (35) has identical form as electromagnetic bracket (12), and gives evolution equation structurally similar to (115) due to the presence of curl M † .
Coupling of (35) and (2) given by the semidirect product reads as The form of {F, G} SP , defined in (18), is the same as for coupling of {F, G} (EM) and the fluid mixture due to the form of (35). In this case, the relation of the total momentum, denoted M u, and the mass momentum is related to the total momentum as The evolution equations given by (36) are as follows, The reason to carry out the projection from µ to M was to bring the equations closer to comparable results in literature, e.g. [23]. Contrary to the construction used in (115), M plays role of an independent variable beforehand and its reversible evolution does not depend on the employed entropy principle (nor dissipation) as it is the case in [23]. Moreover, M itself does not appear in the evolution equation for P, the conjugate M † does due to (35). Therefore, M = M † in the context of (115) and (121).

Coupling to the electromagnetic field
Having coupled fluid mechanics with polarization density and its conjugate momentum, let us make the final step -coupling electromagnetic field. Both pairs (µ, P) and (D, B) were coupled to fluid mechanics by semidirect product. Advection of both pairs by fluid mechanics can be thus expressed (using (12), (18), (31)) by Poisson bracket The evolution equations implied by (39) are Bracket (39) can be further projected, using (36) and thus replacing µ with M, as follows, This bracket leads to the following system of equations: Brackets (39) and (41) can be also seen as coupling of (29) and (34) to (18), respectively, using the Lie derivative technique. The total momenta are

Total charge
Maxwell equation (27) was derived by taking divergence of the evolution equation for D. However, the usual form of the equation also contains the bound charge explicitly. Let us thus define the field of electric induction D as For a functional F (D, P) = F (D, P) then holds that For the quadratic energy of the electromagnetic field D 2 /(2ε 0 ), Eq. (45) can be rewritten as which is the usual relation between electric displacement D, electric field E = D † and polarization. Since div D is equal to the free charge density, it also holds that which is the usual form of the Maxwell equation for div D.
Transformation (45) has of course implications on the form of Poisson brackets (31) and (36). Let us assume the functionals in (39) and (41) depend on D instead of D. The former then reads Bracket (50) can be further projected, using (36) and thus replacing µ with M, as follows, The form of (50) and (51) makes it clear that the transformation (45) just slightly alters the evolution equations given in (39) and (42), respectively. Obviously, only the evolution of D and the covector variable to P, either µ or M, are changed.
The altered evolution equations -in the respective cases -of the electric induction and the covector quantity then become, cf. (40), and, cf. (42), respectively. The formulation with D and M are comparable with similar equations known from literature, see [23,Section 3].
The divergent part of (52) evolves as a convected density although the polarization charge density is not conserved due to div µ † . After the conjugate polarization momentum µ has relaxed to zero, bound charge becomes a conserved quantity. In other words, by transformation (45) one recovers the usual from of the Maxwell equation (48). Apparently, the divergent part of (54) evolves as a conserved mass-related density.
In summary, reversible evolution equations for a mixture coupled with electromagnetic fields, polarization and its conjugate momentum were constructed in a geometric way (semidirect products). By further transformation from D to D the usual form of the Gauß law including bound charge is recovered.

Magnetization
The magnetization of matter, see e.g. [21,Sec. 11], is due to the orientation of spins. Perhaps due to the resemblance with dynamics of rigid body rotation, the pioneering model of magnetization by Landau & Lifshitz [24] was based on that dynamics. In the following text we first recall the Hamiltonian formulation of rigid body dynamics and then we let the rigid body dynamics be advected by the fluid (using again the semidirect product theory).

Intrinsic dynamics of magnetization
The configuration manifold of rigid body rotations is the Lie group SO(3). The standard machinery of differential geometry, e.g. [5,25] or [4,Eq. 3.69], concludes that the Lie algebra dual, where angular momentum M seen from the body reference frame plays the role of state variable, is equipped with Poisson bracket where γ is the gyromagnetic ratio. Bracket (56) is also called the spin bracket, see e.g. [26]. The evolution equation implied by this bracket is (by the same procedure as in Sec. 2.1.1) E being energy of the rotation. Derivative of energy with respect to M is the angular velocity ω.

Advection by fluid mechanics
Similarly as in the case of electromagnetic field or polarization, we will now construct the Poisson bracket expressing advection of magnetization and its dynamics by fluid mechanics (using again semidirect product). Now, however, the advected structure is not a cotangent bundle, but a Lie algebra dual. The general formula for semidirect product of two Lie algebra duals, see [12,Eq. 40], then gives Poisson bracket where M m is the new total momentum of the coupled system and M denotes the magnetization. Note that the advected electrodynamics are kept in the bracket for completeness. The right action of velocity field F m on the vector field G m is defined as negative of Lie derivative as usually, Using (59) which is the explicit form of Poisson bracket expressing kinematics of magnetization advected by fluid mechanics. The evolution equation equations implied by (60) are These evolution equations show how magnetization is advected by fluid mechanics, and how such advection affects the fluid motion itself. Moreover, magnetization keeps its intrinsic rigid-body-like dynamics. Finally, note that there is no explicit coupling to the electromagnetic field just as in the original [24] paper. The coupling is achieved implicitly later by letting energy depend on both M and B.
The first line is due to the dynamics of the CIT-mixture. The second line of (62) accounts for the electromagnetism and its coupling to continuum. The third line contains the polarization bracket and its coupling to continuum. The fourth line of (62) is due to the magnetization dynamics and its coupling to continuum. Brackets {·, ·} (SP) , {·, ·} (SP) A and {·, ·} (SP) M were found due to the semidirect product theory. The evolution equations implied by the General bracket are This is the most detailed set of reversible evolution equations expressing evolution of a mixture coupled with electromagnetic field, polarization and its conjugate momentum and magnetization. The bracket (62) can be projected to the levels of description upon which it was built. One can simply evaluate the bracket (62) on a set of functionals independent of a certain variables, see [10]. For instance the projection from µ to M can be seen as evaluation on functionals independent of div µ. In the rest of this paper we enrich the reversible equations for irreversible terms in order to reduce this rather detailed description to the common continuum models coupling matter and electromagnetic field.

Continuum thermodynamics and reductions
After having constructed a hierarchy of Poisson brackets for fluid mechanics of mixtures advecting electrodynamics, polarization and its conjugate momentum and magnetization, let us now enrich that detailed reversible dynamics by dissipative irreversible terms. This allows to see relaxation of fast mesoscopic variables and the effects on dynamics of less detailed variables. For instance we let the conjugate polarization momentum µ relax to recover the standard Single Relaxation Time (SRT) model, which is widely used for comparison with experiments. We also let the magnetization M relax to recover the full Landau & Lifshitz model not only evolving in the laboratory frame, but being advected by the fluid. Finally, we approach the level of mechanical equilibrium, where evolution is governed by generalized Nernst-Planck-Poisson equations. In summary, a comprehensive multiscale thermodynamic construction of fluid mixtures equipped with electrodynamics, polarization and magnetization is provided.

Gradient dynamics
But before adding dissipative terms to the actual evolution equations, let us recall the general framework of gradient dynamics, where irreversible evolution is generated by derivatives of a dissipation potential [27]. Sound statistical arguments for gradient dynamics based on the large deviations principle was found in [28,29,30]. This paragraph closely follows [4, Sec. 4.5, 4.6].

Dissipation potential
Consider a set of state variables x, and let energy, entropy and mass of the system be denoted by E(x), S(x) and M (x), respectively. A dissipation potential Ξ : x * → R is a family of functionals of conjugate variables x * parametrized by x. We require every parametrization Ξ[x * ] = Ξ(x)[x * ] to satisfy: The irreversible evolution of a functional F (x) is then given as Gradient dynamics automatically satisfies the second law of thermodynamics (growth of entropy in isolated systems). This is guaranteed for instance for convex dissipation potentials, but also non-convexity far from the origin (equilibrium) can be taken into account [31]. Moreover, it is in close relation to the method of entropy production maximization [32]. Gradient dynamics plays a key role when formulating dissipation in the GENERIC framework.

Energetic representation of gradient dynamics
The reversible evolution, treated in the preceding sections, has two building blocks, a Poisson bracket and energy. Energy then enters the evolution equations via the conjugate variables to the energy, while entropy being one of the state variables. This is the energetic representation [33]. In contrast, the irreversible evolution is expressed with respect to conjugates to the entropy (64) to keep the gradient structure, i.e. in the entropic representation. Let us now recall the conversion rules between the two representations. Entropy in the entropic representation and energy in the energetic representation are expressed as S(e, ξ) = dr s ((e, ξ)(r)) and E(s, ξ) = dr e ((s, ξ)(r)) , where ξ i are the state variables other than s and e. Assuming that s(e, ξ) and e(s, ξ) are locally algebraic (i.e. not involving any gradients), the transformation rules can be also resolved locally. It holds that δE δs (r) = ∂e ∂s ((s, ξ)(r)).
Hence, the known transformation rules [4] can be applied. Eventually, we have Pointwise application of the transformation rules (68) yields Suppose that Ξ(e * , ξ * ) = Ξ(s † , ξ † ). Then the functional derivatives with respect to the entropic variables can be locally transformed into derivatives with respect to the energetic variables as follows, Conservation of energy requires that δΞ δe * | e * =Se = 0 (recalling the assumption of algebraic dependencies). Therefore, has to be satisfied. This is for instance satisfied for zero-homogeneous functions of ξ † i s † . Gradient dynamics of state variable ξ i at point r is then and irreversible evolution of entropy at point r, s(r) = S, δ r becomes, according to (64), A simple notorious example of the dissipation potential is This is a prototype of dissipation potential, since any general dissipation potential can be approximated by a quadratic one due to the convexity near equilibrium and flatness at equilibrium.

Dynamic maximum entropy principle
The principle of maximum entropy (MaxEnt), where unknown value of a variable is determined by finding the maximum value of entropy subject to constraints given by declared knowledge, has been successfully applied in many fields (information theory, thermodynamics, etc.) [34]. However, in non-equilibrium thermodynamics the problem is not only to find value of a fast variable that has relaxed, but also to find the vector field along which the fast variable evolves, its evolution equation, when only less detailed variables are among the state variables (observables). To this end we recall the method of Dynamic MaxEnt (DynMaxEnt) [35,4]. Let us demonstrate the principle on perhaps the simplest possible example -an isothermal damped particle in potential field. State variables are position and momentum of the particle (q, p). Reversible evolution is given by Hamilton canonical equations (canonical Poisson bracket on the cotangent bundle and energy). Let the irreversible evolution be given by friction velocity v = p † , i.e. using a quadratic dissipation potential. The overall evolution equations are theṅ taking energy as These are the equations for a particle in potential field V (q) moving with friction coefficient ζ > 0. Let us now treat the state and conjugate variables as independent quantities. Motivation for this is provided by contact-geometric formulation of non-equilibrium thermodynamics [36] and [4]. The minimum of energy subject to the knowledge of q is at p = 0, which is the value approached by evolution of p. At this relaxed value the evolution equation for momentum becomes To satisfy this equation, we have solve it, which actually provides a constitutive relation for the conjugate variable p † . Plugging this constitutive relation back into the equation for q, we obtaiṅ This evolution drives position q towards minima of potential V (q). In summary, by relaxing the fast variable p, the originally reversible equation for q becomes irreversible while p being enslaved by q. This procedure can be carried out analogically in the case of continuum thermodynamics as we shall demonstrate below.

Relaxation of conjugate polarization momentum µ
In subsection 2.4 polarization was equipped with conjugate momentum µ. Inspired by the relaxation of the damped particle in subsection 3.2, we shall let the conjugate momentum relax to recover dissipative evolution of polarization.

Polarization relaxation via µ
For convenience we suppose that the energy of the considered system does depend on the magnetic field and magnetization. Therefore, all derivatives of energy w.r.t the aforementioned fields vanish.
Let us moreover assume a dissipation potential quadratic in µ, see (76), Assuming also energy quadratic in µ, the MaxEnt value of µ is zero. Using (63f), the DynMaxEnt relaxation of µ can be formulated as which is the constitutive relation to be plugged into the remaining evolution equations. In particular, equation for polarization becomes Note that the new terms appearing in the right hand side of equation (83) can be seen as generated by dissipation potential Ξ (µ) evaluated at constitutive relation (82), Similar ideas were presented in [37]. This is the dissipation potential generating irreversible evolution of P.
We may also pursue further relaxation of the polarization. Assuming energy quadratic in P, implies the MaxEnt value of P being zero. Equation (83) consequently gives as the Dynamic MaxEnt value of P † . Moreover, if the energy is quadratic in D, then, using (49), we can write: Hence, we have recovered the usual form of Coulomb's law for a linear isotropic dielectric material.

Single Relaxation Time model
Consider again equation (83) with a further assumption of mechanical equilibrium, i.e. m † = 0. Then, for quadratic energy the evolution of polarization (83) becomes which represents a dissipative evolution of polarization subjected to electric field. Let us first analyze the evolution equation by applying harmonic electric field D ε 0 = E 0 exp(iωt). Equation (89) then gives the Single Relaxation Time (SRT) model of polarization, see e.g. [38], provided that P = P 0 exp(iωt). In summary, by letting the conjugate polarization momentum µ relax, a dissipative evolution of polarization is obtained (83). If the mechanical equilibrium is further assumed, this dissipative evolution is compatible with the SRT model widely used for comparison to experiments. Finally, the equilibrium of the dissipative evolution is the linear relation between polarization and electric intensity known from electrostatics.

Polarization relaxation via M
The conjugate momentum to polarization µ can be replaced by its curl, M, as above in Eq. (34), which brings the equations closer to results in literature (see Sec. 4). Suppose that energy of the considered system is independent of the magnetic field and magnetization, so that derivatives of energy w.r.t the aforementioned fields vanish. Quadratic dissipation in ε ijk ∂ j M † k reads Assuming also energy quadratic in M, the MaxEnt value of M is zero. Relaxation of M can be then expressed as, using (55) and (68), In the isothermal case (s † = const.) it follows from (42) that which represents relaxation only of the curl part of the polarization field, keeping the bound charge intact. The Dynamic MaxEnt allows to obtain the value of M † consistent with a given dissipation potential. The impact of such reduction on the polarization will be always introduced through curl M † . Clearly, the divergence part of P cannot be by this affected. Therefore, Dynamic MaxEnt of M cannot, in principle, lead to relation (86).

Relaxation of magnetization
Let us now discuss relaxation of magnetization M inspired by the Landau & Lifshitz model [24]. Consider the following dissipation potential analogical to (91), The derivative of (94) w.
Hence, the irreversible evolution of M due to (94) reads which is compatible with the Landau & Lifshitz model of magnetization once suitable energy is provided, [39,Sec. 3.7].
Having recovered the Landau & Lifshitz model, let us also formulate its generalization version advected by fluid mechanics and interacting with magnetic field. We take quadratic energy derivatives of which energy are This permits us to define the field of magnetic intensity which is the usual relation between H, B and M. This is our motivation of the choice of energy (98). For energy (97) it holds, moreover, that Combining (61b) and (96) then leads to the evolution of the magnetization, using also (97), (98) and 100, Equation (101), supplied with the rest of Eqs. (63f), is the generalized Landau-Lishitz magnetization relaxation model, where magnetization relaxes, interact with electromagnetic field and where it is advected by the fluid.

Electro-diffusion -dissipation of D and ρ α
Poisson bracket (62) for mixtures may be endowed with a weakly non-local electro-diffusion dissipation potential describing an irreversible evolution of the partial densities (friction between components of the mixture and the zero-th species, e.g. solvent), where M αβ is a symmetric, positive definite matrix of binary diffusion coefficients. The dissipation potential conserves both mass and energy.
The irreversible part of the evolution equations can be expressed in energy-conjugate variables, using (68), as Note that the divergent part of (103c) is identical to the sum of (103b) and (103a) weighted by the charge per mass of the respective species. Therefore, the irreversible evolution given by (102) is compatible with Gauß's law given by (27). In other words Note that a dissipation potential introducing dissipation of the partial mass densities (i.e. containing ρ * α ) is required to contain corresponding terms with D * so that an irreversible evolution compatible with the Gauß law, c.f. (104), is introduced via Ξ D * into the evolution equation for D and viceversa. Therefore, if the validity of Gauß's law for the free charge is required then the form of dissipation potential involving D † is thus restricted. For example, the irreversible evolution given by would not preserve the structure of Gauß's law. This is our motivation for suggesting dissipation potential (102).

Generalized Poisson-Nernst-Planck-Stokes
The electro-diffusion due to (102) introduces a dissipative fluxes identical to those of the generalized Planck-Nernst-Poisson systems (gPNP) presented in [23], although, on more detailed level of description. In this paragraph, we would like to discuss the reduction of the description of the mass momenta for the gPNP systems. Let us consider a level of description where µ, M, P are already relaxed, i.e. (18). Moreover, on short enough distances magnetic field effects are usually negligible compared with effects of the electric field, in contrast to long distances, where electric field usually does not play any relevant role due to screening. Let us analyze the former case, paving the way towards electrochemical problems.
Assuming that no magnetic field is present and that its evolution equation (23) be satisfied, it follows that where ϕ plays the role of electric potential. Following (87), energy where ε = ε 0 (1 + χ). Moreover, negligible magnetic field also implies m = u.
The considered level of description consists of (ρ α , u, s, ϕ). Further relaxation of momentum may lead to certain difficulties. Suppose for a moment that energy does not depend on the momentum and so the unknowns are (ρ α , s, ϕ). The momentum equation then reads Clearly, the first two terms of equation (108) form the gradient of the fluid pressure, so that curl of the equation would be: Thus ϕ would have to satisfy an additional equation, so that the unknowns (ρ α , ϕ) would be overdetermined if considered in more than one spatial dimension, see [40,Eq. 16a] or [23,Eq. 7.39].
To avoid this, a dissipation of mass momentum might be introduced. Again, the energy is quadratic in u, see (3), hence, its MaxEnt value is zero, leaving the remainder of the momentum equation as an constitutive equation for the velocity field. For the dissipation potential generating the irreversible part of the Navier-Stokes equations, see e.g. [4,Eqns. 4.74,4.76]. The viscous dissipative terms are then added to the reversible balance of mass momentum (18), with neglected magnetic field. The whole gPNP-Stokes systems eventually reads: The reversible part of partial mass densities evolution are equipped with the generalized Nernst-Planck flux (103a) and (103b). The unknowns of the system the are (ρ α , s, ϕ, v).
In summary, the reduction of momentum in (ρ α , u, s, ϕ) should either consider dissipation in u or give no further relevance to the momentum equation.

Comparison to the Dreyer et al. approach
C. Guhlke, W. Dreyer et al. in [41] published a comprehensive analysis of fluid mixtures coupled with electromagnetic fields, including polarization and magnetization. Their treatment of surfaces as independent thermodynamic systems interacting with the bulk, being beyond the scope of the presented work, have elucidated many electrochemical problems using nonequilibrium thermodynamics, for instance unified theory of the Helmholtz and Stern layers, a derivation of Butler-Volmer equations, or useful asymptotic techniques, see [42,43,23,44,44].
Since our goal is in close relation to that works, let us compare the two approaches in detail. Dreyer et al. assumed that the total charge density can be described as a continuum-advected density giving, formally, the divergent part of D. Additionally, they assumed the free charge density as (21), therefore, they concluded that the polarized charge density is also a continuum-advected density, They formally defined the evolution of the polarization as where M is magnetization and J P i is the dissipative polarization flux and v is the barycentric velocity. The divergent part of (115) is They assume that the total momentum, i.e. the momentum of the electromagnetic field and the mass, and the total energy are conserved. The coupling between the charged fluid and the electromagnetic field is then given by the choice of Lorentz force [23, Eqn. 3.22a]: as a source term in the mass momentum balance and choice of Joule heating: as a source term in the internal energy balance. Symbol J F denotes the dissipative free charge flux equivalent to (103c). E and B stand for the electric and magnetic field, respectively. Eventually, the system of equations is closed with making use of an entropy principle. First, this leads to a reformulation of the polarization evolution equation. Second, an additional evolution equation for the magnetization is found. They read where ρψ denotes the free energy density. The divergence part of the reversible part of (119); i.e. neglecting the dissipative term, e.g. assuming τ P be large, yields Clearly, the closure-supplied additional terms to the evolution of the polarization do not respect the continuum-advected density structure of (114) compared to (115) which was used for the construction of the balance equations. In other words, the reversible evolution suggested by Dreyer et al.
does not conserve the polarization charge density. In summary, the main difference between the treatment of the polarization charge presented by Dreyer et al. and those introduced in subsection 2.4 is the recognition of the co-vector quantities structures (P, µ) and (P, M), respectively. This results to a different coupling to the continuum.
Also the Lorentz force acting on the total charge is different for the two treatments. The total momentum [23, Eqn. 3.23a], cf. (117), reads Hence, the magnetic field acts upon the polarization charge as it would be a free charge density. This we find controversial as the form should stem, in our opinion, from a reduction of more detailed level of description of a charged mixture. On the other hand, this is certainly a shortcoming of the presented treatment as a certain form of interaction between the magnetic field and moving polarization charge should be present. Although, once the magnetic field is relaxed, this coupling does not alter the form of equations.

Summary and Conclusions
In the first section a hierarchy of Poisson brackets describing the reversible dynamics of a charged, polarized and magnetized continua coupled with electromagnetic field has been developed by means of differential geometry. The semidirect product of the fluid mechanics Lie Algebra dual (ρ, u, s) and electromagnetism cotangent bundle (A, −D) results in a reversible electromagneto-hydrodynamics was already known [14,19,12]. Newly, cotangent bundle (µ, −P) describing dynamics of polarization charge is also coupled using the same technique. Finally, the spin dynamics represented by the Lie algebra of SO(3) is coupled to the Lie algebra dual of fluid mechanics giving rise to the reversible dynamics of spin fluid.
The second section is dedicated to the introduction of the irreversible dynamics and reduction of the before-built levels of description using the dissipation potential to formulate the irreversible evolution and the Dynamic Maximization of Entropy (DynMaxEnt) technique to find passage from finer to rougher levels of description. In this manner, the conjugate momentum to the polarization, µ, is relaxed giving rise to dissipation of the polarization itself. The further exploitation of the induced dissipation of P leads to recovering of P = ε 0 χE formula and the Single Relaxation Time model for dielectrics. The dissipation potential for the magnetic moment M was found so that the Landau-Lifshitz model of spin relaxation was recreated. Finally, the electro-diffusion dissipation potential was introduced, leading to a generalized Nernst-Planck-Poisson-Stokes model.
In summary, we present a geometric construction of a hierarchy of Poisson brackets expressing kinematics of fluid mixtures, electrodynamics, polarization and its conjugate momentum and magnetization. Afterwards, dissipation is introduced on detailed levels of description, which are then reduced to less detailed and more common levels.

A Elementary dipole
Two classical charged mass points are described by their positions, r 1 and r 2 , momenta p 1 and p 2 , carrying charge q 1 and q 2 , respectively. The dynamics of the particles is governed by the canonical Poisson bracket: Assume that q = q 1 = −q 2 and consider the following transformation of the variables: Bracket (123) then transforms into:

B Semi-direct product
This Section closely follows [12] and [4,Sec. 3]. Hamiltonian formulation of electromagnetism is given by the co-tangent bundle T * M = V × V * , and canonical Poisson bracket for A ∈ V and Y ∈ V * , interpreting Y as −D.
The semidirect product of the latter and former is, see [12,Eq. 64] given as where m denotes the total momentum of the joined dynamics. The right action of l on T * M is defined as for the co-vector field a. Introducing (129) after some algebra. This bracket can be further projected to the functional dependent on B, see (11), then Bracket {F, G} (EM) A transforms to electromagnetic bracket (12). The remainder of the terms of (130) affected by (131) can be written as The term with second derivatives stemming from by-parts integration cast of the second term on the first line of (130) contains ε imn ∂ i ∂ m F Bn and is, therefore, identically zero. Finally, introducing (132) into (130) gives the sought bracket (18) for one-species continuum.