Covariant Relativistic Non-Equilibrium Thermodynamics of Multi-Component Systems

Non-equilibrium and equilibrium thermodynamics of an interacting component in a relativistic multi-component system is discussed covariantly by exploiting an entropy identity. The special case of the corresponding free component is considered. Equilibrium conditions and especially the multi-component Killing relation of the 4-temperature are discussed. Two axioms characterize the mixture: additivity of the energy momentum tensors and additivity of the 4-entropies of the components generating those of the mixture. The resulting quantities of a single component and of the mixture as a whole, energy, energy flux, momentum flux, stress tensor, entropy, entropy flux, supply and production are derived. Finally, a general relativistic 2-component mixture is discussed with respect to their gravitation generating energy–momentum tensors.


Introduction
The treatment of multi-component systems is often restricted to transport phenomena in chemically reacting systems, that means, the mixture consisting of different components is shortly described by 1-component quantities such as temperature, pressure and energy which are not retraced to the corresponding quantities of the several components of the multi-component system.That is the case in non-relativistic physics [1,2,3] as well as in relativistic physics [4,5,6,7,8].In this paper, the single component as an interacting member of the mixture is investigated.Thus, each component of the mixture is equipped with its own temperature, pressure, energy and mass density which all together generate the corresponding quantities of the mixture.
Considering a multi-component system, three items have to be distinguished: one component as a member of the multi-component system which interacts with all the other components of the system, the same component as a free 1-component system separated from the multi-component system and finally the multi-component system itself as a mixture which is composed of its components.Here, all three items are discussed in a covariant-relativistic framework.For finding out the entropy-flux, -supply, -production and -density, a special tool is used: the entropy identity which constrains the possibility of an arbitrary choice of these quantities [9,10,11,12].Following J. Meixner and J.U. Keller that entropy in non-equilibrium cannot be defined unequivocally [13,14,15,16,17], the entropy identity is only an (well set up) ansatz for constructing a non-equilibrium entropy and further corresponding quantities.This fact in mind, a specific entropy and the corresponding Gibbs and Gibbs-Duhem equations are derived.The definition of the rest mass flux densities, of the energy and momentum balances and of the corresponding balances of the spin tensor are taken into account as contraints in the entropy identity by introducing fields of Lagrange multipliers.The physical dimensions of these factors allow to determine their physical meaning.
Equilibrium is defined by equilibrium conditions which are divided into basic ones given by vanishing entropy-flux, -supply and -production and into supplementary ones such as vanishing diffusion flux, vanishing heat flux and zero rest mass production [11,12].The Killing relation of the 4-temperature concerning equilibrium is shortly discussed.Constitutive equations are out of scope of this paper.
The paper is organized as follows: After this introduction, the kinematics of a multicomponent system is considered in the next two sections for introducing the mass flux and the diffusion flux densities.The energy-momentum tensor is decomposed into its (3+1)-split, and the entanglement of the energy and momentum balances are discussed, follwed by non-equilibrium thermodynamics of an interacting component of the mixture and that of the corresponding free component.The equilibrium of both is considered.Thermodynamics of the mixture starts with two axioms: additivity of the energy momentum tensors and of the 4-entropies of the components resulting in those of the mixture.Entropy, entropy flux, -supply and -production are found out.The paper finishes discussing the gravitation generated by a special general-relativistic 2-component system: one component equipped with a symmetric energy-momentum tensor, the other one with a skew-symmetric energy-momentum tensor.A summary and an appendix are added.

The components
We consider a multi-component system consisting of Z components.The component index A runs from 1 to Z.Each component has its own rest frame B A in which the rest mass density ̺ A is locally defined.These rest mass densities are relativistic invariants and therefore frame independent 1 .
In general, the components have different 4-velocities: u A k , A = 1, 2, ..., Z; k = 1, ..., 4, which all are tensors of first order under Lorentz transformation.We now define the component mass flux density as a 4-tensor of first order and the component mass production term as a scalar Here, (1) 2 is the mass balance equation of the A -component .Consequently, we introduce the basic fields of the components The mass production term has two reasons: an external one by mass supply and one internal one by chemical reactions The external mass supply (ex) Γ A depends on the environment of the system, whereas (in) Γ A is determined by chemical reactions depending on the set of frame-independent stoichiometric equations which are discussed in Appendix 12.3.

The mixture
As each component, also the multi-component system has a mass density ̺ and a 4-velocity u k which are determined by the partial quantities of the components.For deriving ̺ and u k , we apply the nearly self-evident Mixture Axiom: The balance equation of a mixture looks like the balance equation of an one-component system.
Especially here, the mixture axiom is postulated for the balance equations of mass, energymomentum and entropy.According to the mixture axiom, the mass balance of the mixture looks according to (1) with vanishing total mass production, if the mass of the mixture is conserved 2 .Now the question arises: which quantities of the components of the mixture are additive?Obviously, neither the mass densities ̺ A nor the 4-velocities u A k are additive quantities according to their definitions.Consequently, we demand in accordance with the mixture axiom that the mass flux densities are additive 3Setting I: For the present, ̺ and u k are unknown.Of course, they depend on the basic fields of the components (2).Contraction with u k and use of (5) 2,3 results in or in more detail The mass density ̺ and the 4-velocity u k of the mixture are expressed by those of the components according to (7) and (5) 4 .According to (5) 4 , the 4-velocity of the mixture is a weighted mean value of the 4-velocities of the components.For the mass density, we have according to (6) 1 also a with the Kluitenberg factor f A4 weighted mean value of the mass density components [18] f resulting in the entanglement of ̺ and u k which are not independent of each other According to (5) 1 and (1) 2 , we obtain the additivity of the mass production terms

The diffusion flux
From (5) 3 and (8) Introducing the diffusion flux density using (227) 2 we obtain By introducing the projectors we obtain the following properties of the diffusion flux density: According to (17) 2 , the diffusion flux density is that part of the mass flux density which is perpendicular to the 4-velocity of the mixture.The diffusion flux density vanishes in 1-component systems (u 3 The Energy-Momentum Tensor

Free and interacting components
The energy-momentum tensor T Akl of the A -component consists of two parts Here,

0
T Akl is the energy-momentum tensor of the free A -component, that is the case, if there are no interactions between the A -component and the other ones.W Akl B describes the interaction between the B -and the A -component.The interaction between the external environment and the A -component is given by the force density k Al which appears in the energy-momentum balance equation and in the balance equations of energy: Consequently, the interaction of the A -component with the other components of the mixture modifies the energy-momentum tensor of the free A -component.Additionally, its interaction with the environment shows up in the source of the energy-momentum balance.
According to its definition, T Akl is the energy-momentum tensor of the " A -component in the mixture".

(3+1)-split
The (3+1)-split of the energy-momentum tensor of the A -component is The (3+1)-components of the energy-momentum tensor are6 The (3+1)-split of tensors is a usual tool in relativistic continuum physics.The (3+1)components -generated by the split-have physical significance which originally is hidden in the unsplitted tensors.Thus, we generate by (3+1)-splitting the following covariant quantities of the A -component: the energy density e A , the momentum flux density p Al , the energy flux density q Ak , the stress tensor t Akl The symmetric part of the energy-momentum tensor ( 25) is and its anti-symmetric part is The stress tensor is composed of the pressure p A > 0, ∧A, and the viscous tensor π Akl We now consider the physical dimensions of the introduced quantities 7 .According to ( 16) and ( 8) By taking (31), (32) 1 and ( 25) into account we obtain The (3+1)-split (25) of the energy-momentum tensor can be written in a more compact form The energy-momentum tensor (37) is that of the A -component in the mixture, that means as dicussed in sect.3.1, the A -component is not a free system and the (3+1)-splitcomponents e A , q Ak , p Al and t Akl include the internal interaction of the A -component with all the other ones.

Additivity
We now consider the equivalent-system composed of the Z components: that is the mixture which consists of these Z interacting components.Because this interaction is already taken into account by the (3+1)-split-components, the energy-momentum tensors of the components are additive without additional interaction terms.Consequently, the energymomentum tensor T kl of the mixture is Setting II: Multiplication with u l results by use of (8) 1 and (38) 2 in and by multiplication with h m l , (39) results in For an 1-component system (u A k ≡ u k ), we obtain according to (42) g Am = g m = 0 taking f A = f = 0 into account.

(3+1)-components of the mixture
Starting with (38), we obtain According to (37) and (39) 1 , these relations are analogous for the mixture.Consequently, from (40) follows resulting with (43) 1 in the energy density of the mixture From (40) follows resulting with (43) 2 in the energy flux density of the mixture From (41) follows resulting with (44) 1 in the momentum density of the mixture And from (44) 2 follows finally which by taking (38) into account results in the stress tensor and the pressure of the mixture The additivity of the energy-momentum tensors (39) results in (46), (48), (50) and (52), relations which express the (3+1)-components of the energy-momentum tensor of the mixture as those of the components and their velocities e, q k , p k , t kl = F e A , q Ak , p Ak , t Akl , u Ak , ̺(̺ The 4-velocity u k is given by (5) 4 .
The influence of the additivity of the energy-momentum tensors on the balance equations of energy and momentum is investigated in the next section.

Entanglement of Energy and Momentum Balances
If the energy-momentum tensors of the A -component and of the mixture are T Akl and T kl , the energy and momentum balances are according to the mixture axiom by use of ( 23) and (24) energy: The balances (56) 3 and (57) 3 follow from ( 23) and ( 24) by the mixture axiom.Here, Ω A and Ω are the energy supplies, and Ω Am and Ω m the momentum supplies of the A -component and of the mixture.The (3+1)-split of the divergence of the energy-momentum tensor of the A -component results by use of (16) If the component index A is cancelled in (58), we obtain the decomposition of the divergence of the energy-momentum tensor of the mixture.Taking (56) and (57) into account, these divergences can be written as The additivity of the energy-momentum tensors (39) results in the additivity of the force densities8 Taking (23) 2 and (24) 2 into account, we obtain by multiplication of (60) with u m , resp.with h p m ,

Ω =
A Inserting ( 23) and (24), we obtain in more detail As (61) indicates, the additivity of the energy-momentum tensors causes that the supplies of energy and momentum are entangled, expressed by the inequalities Also if the total force density and the total momentum supply are zero, we obtain according to (61) 1,2 A As expected, the supplies of energy and momentum remain entangled in a system of vanishing total force and momentum densities.The entanglement vanishes for such isolated systems for which the force and momentum supplies for all A -components are zero.
5 The Spin Tensor

(3+1)-split
The (3+1)-split of the spin tensor of an A -component is defined by inserting (16) 1 into Introducing the following covariant abreviations (68) results in By ( 69) and (70) are introduced: the spin density s ab , the spin density vector Ξ b , the couple stress s kab and the spin stress Ξ kb .
Analogously to (37) and (38), a more compact form of the spin tensor is Taking ( 69) and (70) into account, we obtain expression which are needed for formulating the entropy identity below.

Additivity
Analogously to Setting II, we introduce the spin tensor of the mixture as the sum of the spin tensors of the A -components.
Setting III: According to the mixture axiom, the spin tensor of the mixture is defined by (75) 1 as spin of an 1-component system resulting from (72) with A ≡ 1 → blank.

Spin balance equation
If there exists an external angular momentum density a spin balance equation of each A -component and of the mixture has to be taken into account According to Setting III, the additivity of the partial angular momenta is valid.
6 Thermodynamics of Interacting Components

The entropy identity
Starting with the (3+1)-split of the entropy 4-vector and the entropy balance equation we have to define the following four quantities in accordance with the balance equations of mass (1) 2 , of energy (23), of momentum (24) and of spin (81) 1 : the entropy density s A , the entropy flux density s Ak , the entropy production σ A and the entropy supply ϕ A .Because there is no unequivocal entropy [15] and consequently, also no unique entropy density, -flux, -production and-supply, we need a tool which helps to restrict the arbitrariness for defining entropies.Such a tool is the entropy identity [10,11] which is generated by adding suitable zeros to the entropy (83) 1 which are related to the balances which are taking into account.These zeros are generated by choosing the following expressions: Consequently, the entropy identity is chosen according to (1), (38) and ( 74) The fields of Lagrange multipliers κ A , λ A , λ A m , Λ A m and Λ A mn are quantities whose physical meaning becomes clear in the course of the exploitation of the entropy identity.
Here, κ A and λ A are scalars, undefined for the present, and for the likewise arbitrary quantities λ A m , Λ A m and Λ A ab , tensors of first and second order.An identification of these Lagrange multipliers is given below after the definitions of entropy flux density, entropy production density and supply in section 6.3.
The entropy identity (85) depends on the balances which are taken into consideration as constraints: the balances of mass, energy, momentum and spin.The electro-magnetic field and quantum fields are included, if the energy-momentum tensor and the spin tensor of these fields are inserted into (85).
Considering the third, the fourth and the fifth row of (85), we obtain that the velocity parts of λ A m , Λ A m and Λ A mn can be set to zero according to (69) and ( 70).The symmetric part of Λ A mn does not contribute to the fifth row of (85) and therefore it is set to zero, too The entropy identity (85) becomes by rearranging This identity transforms into an other one by differentiation and by taking the balance equations of mass (1) 2 , of energy-momentum (22), of spin (81) 1 and of entropy (83) 2 into account.
Here, σ A is the entropy production and ϕ A the entropy supply of the A -component.The identity (88) changes into the entropy production, if s A , s Ak and ϕ A are specified below.
Rearranging the entropy identity results in Now we look for terms of the fifth row of (89) which fit into the first three rows of (89).The shape of these terms is [u Ak ;k scalar/u Ak scalar ;k ] according to the first two rows of (89) and [Ψ Ak ;k (Ψ Ak u A k = 0)] according to the third row.None of the seven terms of the fourth and fifth row of (89) have this shape, but inserting the energy-momentum tensor and the spin tensor into the fifth row of (89) may generate such terms.
The third term of the fifth row of (89) becomes Summing up ( 90) and (91) results in9 Evidently, the term −p A λ A u Ak ;k belongs to the first row of (89).After having inserted the underlined term of (92), the first two rows of (89) become10 Thus, a rearranging of the entropy identity (89) results by taking (93) into account The third term of the fourth row of (94) results in If ( 86) and ( 73) are taken into account, these nine terms are: Rearranging of (96) to (104) results in: (96) and (103): (98), ( 101) and (104): A comparison of (105) and (106) with the first two rows of (94) demonstrates that a term which fits into these rows does not appear in (105) and (106).Thus by taking (93) into account, a rearranging of the entropy identity (89) results in This entropy identity is incomplete: the multi-temperature relaxation is missing which is generated by the different partial temperatures of the components of the mixture.Because of lucidity, the treatment of multi-temperature relaxation is postponed and will be considered below in sect.6.4.In the next section, we now specify s A , s Ak , ϕ A and σ A .
6.2 Exploitation of the entropy identity

Entropy density, Gibbs and Gibbs-Duhem equations
We now define the entropy rest density s A according to the first row of (107) Setting IV: resulting in the specific rest entropy A non-equilibrium state space -which is spanned by the independent variables-contains beside ̺ A , ̺, e A the spin variables Ξ Am and s Amn and additionally p Al which extends the state space in the sense of Extended Thermodynamics11 [19,20].Consequently, we choose the state space [21] The corresponding Gibbs equation according to (109) and ( 110) is Differentiation of (109) results in the Gibbs-Duhem equation by taking (111) into account resulting in Taking ( 113) and (108) into account, the entropy identity (107) becomes The marked terms cancel each other.Taking (12) 2 and (19) 2 into account, we consider This zero contains the diffusion flux which does not appear up to here in the entropy identity (85).That means, the diffusion is missing in (114), and we will not ignore the underbraced terms in (119) 1 , but we insert (119) 3 into (114).Consequently, the entropy identity results in We now specify the entropy flux density s Ak and the entropy supply ϕ A in the next section.

Entropy flux, -supply and -production
According to the first row of (116), we define the entropy flux density Setting V: We now split the entropy identity (116) into the entropy production and the entropy supply.For this end, we need a criterion to distinguish between entropy production and supply.Such a criterion is clear for discrete systems: a local isolation suppresses the entropy supply but not the entropy production.Isolation means: the second row in (116) vanishes, if the A -component is isolated from the exterior of the mixture.Consequently, we define the entropy supply as follows Setting VI: with the result that the entropy identity (116) transfers into the entropy production density by taking (117) and (118) into account As expected, the entropy production is composed of terms which are a product of "forces" and "fluxes" as in the non-relativistic case 12 .The expressions s A , s Ak , ϕ A and σ A contain Lagrange multipliers which are introduced for formulating the entropy identity (85) playing up to here the role of place-holders.Their physical meaning is discussed in the next section.

Fields of Lagrange multipliers
From non-relativistic physics, we know the physical dimensions of the entropy density and the entropy flux density by taking (33) and ( 35) into account According to (108), we have the following equation of physical dimensions Taking (120) 1 and ( 33) into account, we obtain that means, λ A is a reciprocal temperature belonging to the A -component.Therefore, we accept the following Setting VII: with the partial temperature Θ A of the A -component13 and a scalar ν A which is suitably chosen below..According to (117) , we have the following equation of physical dimensions Taking (120) 2 , (12) 2 and (32) 1 into account, we obtain We know from the non-relativistic Gibbs equation that the chemical potentials µ A have the physical dimension of the specific energy e A /̺ A Consequently, we make the following choice by taking (126) into consideration Setting VIII: According to the second term of (118) we have the following equation of physical dimensions that means, λ Ak is proportional to a velocity and at the same time perpendicular to u Ak according to (86) 1 .Consequently, only the velocity u m of the mixture remains for defining λ Ak in accordance with (86) 1 Setting IX: We know from the non-relativistic continuum theory and from (22) 1 and ( 33) the following connection of the physical dimensions 14 From the last term of (71) follows by taking (130) into account From the first term of the third row of (87) follows by use of (83) 1 and (120) 4 and taking (131) into account, we obtain In accordance with (86) 2,3 and analogously to (129), the relations (133) allow the following Setting X: 14 angular momentum = spin density per time Inserting the Lagrange multipliers into the expression of entropy density (108), of entropy flux density (117) and of entropy supply (118), we obtain by use of (69) The entropy production density (119) results by inserting the Lagrange multipliers (123),( 127), ( 129) and ( 134) The underbraced terms result in that means, the spin density does not appear in the entropy production.
The first four terms of the entropy production describe the four classical reasons of irreversibility: diffusion, chemical reactions, heat conduction and internal friction with a by the momentum flux density modified non-equilibrium viscous tensor.The fifth term of (138) 15 is typical for an interacting A -component as a part of the mixture because it contains the 4-velocity of the mixture u m .The same is true for the last two spin terms which vanish for 1-component systems.In any case, all spin terms of the fields (135) to (138) related to entropy vanish with the 4-acceleration.
Up to now, a further phenomenon of irreversibility was not taken into consideration: the multi-temperature relaxation which is discussed in the next section.

Multi-temperature relaxation and the partial temperatures
Because the different components of the mixture have different partial (reciprocal) temperatures λ A , A = 1, 2, ..., Z, a multi-temperature relaxation 16 takes place which is an irreversible phenomenon.Consequently, multi-temperature relaxation has to be taken into account in the entropy identity by adding a suitable zero as done in (85).
A heat transfer H AB between two components of the mixture -A and B-takes place by multi-temperature relaxation, if the corresponding temperatures of the components are different from each other.Consequently, the entropy exchange between these two components is Setting XI: Here H AB is an energy density and G AB an entropy density according to (121) For the A -component, this results according to (140) 2 in The entropy exchange of the A -component according to multi-temperature exchange (143) 1 and (140) 3 has now to be introduced into the entropy identity (107).Because G A has the same physical dimension as s A according to (141) 2 , it fits into the first row of (107).Therefore we add the zero to (107).Taking (140) 3 into account and inserting into (145) we obtain three additional terms which can be directly introduced into the entropy identity without defining an additional Lagrange multiplier.According to (107), the three terms of (147) are attached as follows Introducing these terms as demonstrated in sect.6.1 into the entropy identity (116), the entropy density (135) becomes and the state space ( 110) is extended by and consequently the Gibbs equation ( 111) by (G A /̺) • .The Gibbs-Duhem equation ( 112) is untouched by including the multi-temperature relaxation.
According to (149) the entropy supply (137) results in and the entropy production density (138) becomes by (150) The ten terms of the entropy production density (154) have as already discussed after (138), the following meaning: • by diffusion modified chemical reaction: • multi-temperature relaxation: • four terms describing entropy production by the spin S Akab .

The 4-entropy
We need the 4-entropy of the A -component for describing thermodynamics of a mixture.
Starting with (83) 1 , ( 151) and ( 136), we obtain Rearranging results in The transition from the interacting A -component to the free 1-component system is considered in sect.7 and that to the mixture in sect.8.All quantities introduced up to here are non-equilibrium ones, because we did not consider equilibrium conditions up to now.This will be done in the next section.

Equilibrium conditions
Equilibrium is defined by equilibrium conditions which are divided into basic and supplementary ones [11,12].The basic equilibrium conditions are given by vanishing entropy production, vanishing entropy flux density and vanishing entropy supply18 : A first supplementary equilibrium condition is the vanishing of all diffusion flux densities.According to (12) 1 , we obtain Taking (8) 1 into account, (158) 3 results in Consequently, we have to demand beyond (158) 1 the supplementary equilibrium condition that the mass densities are additive in equilibrium.We obtain according to (8) 2 and ( 14) Taking (158) 2 and (129) into account, (160) 2 yields Further supplementary equilibrium conditions are given by vanishing covariant time derivatives, except that of the four-velocity: • u l eq is in general not zero in equilibrium.Consequently, the time derivatives of all expressions which contain the 4-velocity must be calculated separately, as we will see below.
Another supplementary equilibrium condition is the vanishing of the mass production terms in (10) Thus, we obtain from (1) 2 , (163) 1 and (170) 3 The equilibrium temperature is characterized by vanishing multi-temperature relaxation Often one can find in literature [25] the case of equilibrium of multi-temperature relaxation: although out of equilibrium, only one temperature is considered in multi-component systems.This case is realistic, if the relaxation of multi-temperature relaxation to equilibrium is remarkably faster than that of the other non-equilibrium variables [26].Taking (161) 1 , (26) 2 and (172) into account, the entropy density (151) becomes in equilibrium using the shift of the time derivative Beyond the usual expression for the entropy density in thermostatics 19 , it includes an acceleration dependent spin term.The energy density and the pressure are here defined by the (3+1)-decomposition ( 25) of the energy-momentum tensor.The chemical potential is as well as the temperature introduced as a Lagrange multiplier.Taking (161) 1 , (158) 1 and (172) into account, the entropy flux density (136) vanishes in equilibrium, resulting in using the shift of the time derivative according to (169) and (162).
Finally the entropy supply (153) results in that means, the power exchange caused by the force density and by the angular momentum density vanishes in equilibrium.The entropy production (154) has to vanish in equilibrium according to the basic equilibrium condition (157) 1 .Taking (170) 2 , (158) 1 , (174), (164) 2 , (161) 1 and (172) into account and using (162), (154) results in The third term of the second row of (154) vanishes by shift of the time derivative.In equilibrium, spin terms appear in the vanishing power exchange (175) and in the spin modified internal friction (176).
As demonstrated, equilibrium of an A -component in the mixture is described by three basic equilibrium conditions (157) and six supplementary ones: (158) 1 , (160) 1 , (162), (170) 1,2 and (172).Often, we can find in the literature [27,28] equilibrium conditions which are different from those postulated here.The reason for that is, that entropy production and supply and the entropy flux as starting-points for the basic equilibrium conditions differ from the expressions (135) to (138).Such different equilibrium conditions and their derivations are considered in the next two sections.

Killing relation of the 4-temperature
Starting with (90) 1 , we now consider the following relations Taking ( 178), ( 179) and ( 181) into account, we obtain from (90) 1 Replacing the second row of (154) by the LHS of (182) yields the entropy production of vanishing multi-temperature relaxation and vanishing spin by taking (123) into account without spin: Evident is that is not a sufficient condition for equilibrium because the equilibrium conditions (170) 2 , (158) 1 and (161) 1 are not necessarily satisfied and the entropy production (183) does not vanish.If the energy-momentum tensor is symmetric, (184) results in an expression which as well as ( 184) is not sufficient for equilibrium.Consequently, the Killing relation of the 4-temperature is also not sufficient for equilibrium 20 .
If equilibrium is presupposed, the equilibrium conditions (170) 2 , (158) 1 and (161) 1 are satisfied, the entropy production vanishes and without spin: T Akl eq = T Alk eq : are necessary conditions 21 for equilibrium according to (183), if the spin is ignored.If the spin is taken into account, (187) results by use of the fifth row of (154) in There are different possibilities to satisfy (187) and ( 188) which are discussed in the next section.

The gradient of the 4-temperature
The necessary condition for equilibrium ignoring spin (187) can be differently satisfied generating different types of equilibria.There are three possibilities: If equilibrium exists, one of the following three conditions is valid: 20 a fact which is well-known [12] 21 but as discussed, not sufficient conditions T Akl eq = 1 c 2 e A eq u Ak eq u Al eq − p A eq h Akl eq , (191) c 2 e A eq u Ak eq u Al eq − p A eq h Akl eq , and (187) is valid.(192) Multiplication of (190) 2 with u Al eq results in that means, (190) represents an intensified equilibrium because additionally to the usual equilibrium conditions mentioned in sect.6.6.1, (193) is valid.If (191) is valid, the equilibrium exists in a perfect material whose entropy production is zero.If the considered material is not perfect and if the equilibrium is not intensified, (192) is valid, and the question arises, whether (187) can be valid under these constraints.
To answer this question, we consider (177) to (181) in equilibrium.According to the equilibrium conditions, we obtain Summing up (194) to (198) yields Consequently, (187) is satisfied because each of the three terms vanishes for its own, thus being compatible with (192).If an A -component of a mixture is in equilibrium, two types of equilibria can occur: one in an arbitrary material showing the usual equilibrium conditions and another one which has beyond the the usual equilibrium conditions vanishing temperature gradient and vanishing 4-velocity gradient according to (193).
Evident is that an 1-component system which does not interact with other components is as a special case included in the theory of an A -component in the mixture.This case is discussed in the next section.
7 Special Case: 1-Component System and for shortness, we omit this common index 0. Then the basic fields of an 1-component system are according to (2) rest mass density and 4-velocity: {̺, u k }.

Equilibrium and reversibility
Vanishing entropy production out of equilibrium belongs to reversible processes and vice versa [29].According to (206), is sufficient and necessary for vanishing entropy production in 1-component systems.But concerning equilibrium, (208) is as well as (189) only necessary but not sufficient for it.Thus, all results of sect.6.6.3 change into those of an 1-component system, if the component index A is omitted, eq is replaced by rev, equilibrium is not presupposed and the generated expressions belong to reversible processes and vice versa.Comparing (208) with (189) and ignoring the spin, the derivative of the 4-temperature and the Killing relation of the 4-temperature without spin: (λu l ) rev are rather conditions for reversible processes in 1-component systems because the entropy production is enforced to be zero without existing equilibrium.Independently of the 4temperature, we obtain the well-known fact [30] that according to (206) all processes of perfect materials are reversible in 1-component systems without spin 8 Thermodynamics of a Mixture According to the mixture axiom in sect.2.2, the balance equations of a mixture look like those of an 1-component system.But a mixture as a whole behaves differently from the interacting A -component in the mixture and also differently from an 1-component system which both were discussed in sect.6 and sect.7.Because the interaction between the components is still existing in the mixture, the diffusion fluxes and also the multitemperature relaxation do not vanish as in 1-component systems.Because component indices A do not appear in the description of mixtures, they are summed up in contrast to 1-component systems for which they vanish.The Settings I to III enforce the mixture axiom resulting in mass balance: energy balance: spin balance: The Settings I to XI are concerned with the balance equations ( 211) to (214), with the entropy density, the entropy flux density, the entropy supply, with the Lagrange multipliers and the multi-temperature relaxation.Obviously, we need an additional setting concerning the entropy of the mixture which will be formulated in the next section.

Additivity of 4-entropies 8.1.1 Entropy density and -flux
Like the additivity of the mass flux densities (5) 1 , the energy-momentum tensors (39) 2 and the spin tensors (75) of the A -components, we demand that also of the 4-entropies are additive Setting XII: Consequently, we obtain from (156) by use of (19) 2 and ( 38) According to (84), we obtain the entropy density and the entropy flux density of the mixture by use of ( 13), (8) 1 , ( 17) and ( 42) Taking (40) into consideration, we introduce by comparing with (217) and (218) the Setting XIII: With this setting, the expressions of the entropy density and the entropy flux of the mixture correspond to those which are generated by the additivity of the energy-momentum tensors: (45) to (52).Finally, we obtain the entropy and entropy flux densities of the mixture These expressions of the entropy and entropy flux densites of the mixture are direct results of Setting XII (215).They will be considered in sect.8.2.

Entropy supply and production density
From ( 215) and (83) 2 follows the entropy balance equation of the mixture satisfying the mixture axiom.Accepting the additivity of the entropy supplies of the A -components 25Setting XIV: we obtain from (222) the additivity of the entropy productions of the A -components The entropy supply of the mixture follows from (153), ( 223) and ( 219) The entropy production of the mixture follows from (154), ( 224) and ( 219)

Partial and mixture temperatures
We now consider the positive term in the second row of the entropy density (220) by which a mixture temperature ⋄ Θ can be defined.This mixture temperature is only a formal quantity because it is not evident that a thermometer exists which measures ⋄ Θ: the partial temperatures are internal contact variables [31] measured by thermometers which are selective for the temperature Θ A of the corresponding A -component.Evident is, the measured mixture temperature is a certain mean value of the partial temperatures of the components of the mixture [32,33,34], but this measured mean value may depend on the individual thermometer and may be different from ⋄ Θ, that means, the measured temperature is not unequivocal.Different definitions of the mixture temperature can be found in literature [35].But a unique mixture temperature -independent of thermometer selectivities or arbitrary definitions-is given in the case of multi-temperature relaxation equilibrium (172).This case is often silently presupposed in literature, if only one temperature is used in multi-component non-equilibrium systems.Only this sure case is considered in the sequel.We now introduce the mixture quantities e and q m to the entropy density s and the entropy flux density s m .According to (45) and (47), we obtain Taking ( 227) and (228) into account, (220) and (221) result in Evident is, that partial temperatures of the components appear in all four quantities referring to mixture entropy: entropy density (229), entropy flux density (230), entropy supply (225) and entropy production density (226).These expressions are of a more simple shape, if the mixture is in a multi-temperature equilibrium which is considered in the next section.
The first term in the sum of (231) can be exploited by use of the mean value theorem according to (15) and (8) 1,2 Consequently, the chemical potential of the mixture is and the entropy density of the mixture (231) yields The entropy density (235) of the mixture in multi-temperature equilibrium is similarly constructed, but different from the expression (204) of an 1-component system: there are the energy-, the mass-, the pressure and the spin-term.

Entropy production and -supply
The entropy supply (225) results in The entropy production density (226) becomes The meaning of each individual term of (237) was already discussed with regard to the A -component according to (154).Entropy density (235), entropy flux density (232), entropy supply (236), entropy production density (237) and chemical potential (234) of the mixture are represented by sums of quantities of the A -components.As expected, the (3+1)-components of the energymomentum tensor cannot represent the mentioned thermodynamical quantities because diffusion fluxes, chemical potentials and temperature are not included in the energymomentum tensor.From them, only the energy density e and the energy flux density q m of the mixture appear in entropy and entropy flux densities.
A temperature ⋄ Θ of the mixture can be defined independently of multi-temperature equilibrium.Accordding to (227), 1/ ⋄ Θ is a weighted mean value of the reciprocal partial temperatures of the components arranged with the partial pressures, a construction which seems very special.As already mentioned in sect.8.2, a mixture temperature is not well defined because it depends on the component sensitivity of a thermometer.

Total equilibrium
Evident is that the equilibrium conditions of a mixture follow from those of the A -components which we considered in sect.6.6.1.Consequently, a demand of additional equilibrium conditions for mixtures is not necessary.Presupposing the equilibrium conditions of an A -component (discussed in sect.6.6.1) and multi-temperature equilibrium, we start with the repetition f A eq = 1, w A eq = 0, u Aeq k = u eq k , g Am eq = 0, (ex) Γ A eq = (in) Γ A eq = 0, (238) e eq = A e A eq , q m eq = 0, p m eq = A p Am eq , t jm eq = A t Ajm eq .( 239) Taking ( 238) and (239) into account, the entropy density (235) of the mixture in equilibrium and the entropy flux density (232) result in if the the shifting of the time derivative (169), ( 162), ( 69) and (70) are applied.Interesting is, that the spin terms cancel in equilibrium.

(3+1)-entropy-components and spin
If the spin is taken into consideration 26 , acceleration terms appear in the entropy density and production, (235) and (237), and in the entropy flux density and supply, (232) and (236).The four components (69) and (70) of the spin are differently distributed over the (3+1)-components of the entropy: • the entropy density (151) of an A -component depends on the spin density s Aab and on the spin density vector Ξ Am , whereas the entropy density of the mixture (220) depends on the four spin quantities (69) and (70).In 1-component systems, the entropy density (204) 1 depends only on the spin vector Ξ m .In equilibrium, the entropy density is for all cases independent of the spin, (173) and (240) 1 .
• the entropx flux density (136) of an A -component depends on the couple stress s Akab and on the spin stress Ξ Akm , whereas the entropy flux density of the mixture (232) depends on the four spin quantities (69) and (70).In 1-component systems, the entropy flux density (205) 1 depends only on the spin vector Ξ m .In equilibrium, the entropy flux density (173) and (240) 2 vanishes and induces q Ak eq = 0.
• the entropy supply of an A -component (153) is as well independent of the spin as for the mixture (236) and for an 1-component system (205) 2 .The entropy supply vanishes in equilibrium, and a connection between the force density k Al eq and the angular momentum density m Aab eq is established, ( 175) and (241).
• the entropy production density (154) and (237) does not depend on the spin density s Aab for an A -component and for the mixture, but a dependence upon the three other (3+1)-spin-components exists.In 1-component systems, the entropy production density (206) depends on the spin stress Ξ kb and on the couple stress s kab .The entropy production density vanishes in equilibrium, and a connection between the viscosity tensor π Akl eq and the spin stress and the couple stress is established, ( 176) and (242).9 Balances, Constitutive Equations and the 2 nd Law Up to here, a special material was not taken into account: all considered relations are valid independently of the material which is described by constitutive equations supplementing the balance equations.Especially, the entropy productions (154) of the A -component and (237) of the mixture are not specified for particular materials.There are different possibilities for introducing constitutive equations 27 .Because constitutive equations are not in the center of our considerations, we restrict ourselves on the easiest ansatz which only serves for elucidation of the problem: Balance equations are generally valid for all materials, that means, they cannot be solved without choosing a special material characterized by constitutive equations which inserted into the balance equations transform these into a system of solvable differential equations for the wanted fields.
The entropy production of the A -component (154) is a sum of two-piece products whose factors are so-called "fluxes" and "forces".According to (154), the ten fluxes are and the corresponding ten forces are The entropy production density (154) of an A -component can be written as a scalar product of forces and fluxes a relation which is valid independently of the material in consideration.The material is described by the dependence of the fluxes on the forces, by the constitutive equations which have to be introduced into the expression of the partial entropy production density (246) resulting in the entropy production density of the mixture by ( 224) The inequality is caused by the Second Law which states that the entropy production of the mixture is not negative after having inserted the constitutive equations into the general expression (226).Consequently, the Second Law represents a constraint for the constitutive equations (247) [39], and it makes no sense to take the Second Law into consideration before the constitutive equations are inserted.The entropy production of sub-systems -here the A -components (248) 1 -is not necessarily positive semi-definite.There are different methods for exploiting the dissipation inequality (248) 2 [39,40] which are beyond this paper because special materials are here out of scope.
10 Special Case: General Relativity Theory contain covariant derivatives depending on the geometry of the space-time in which the physical processes occur.Here, the pseudo-Riemannian space of General Relativity Theory (GRT) is chosen as a special case.
In GRT, as a consequence of Einstein's equations the gravitation generating energy-momentum tensor Θ ab has to be symmetric and divergencefree (R ab is the Ricci tensor, g ab the metric, R = R m m ).According to (39) and ( 25), the energy-momentum tensor of the mixture T kl may be neither symmetric nor divergencefree.The same is true for spin divergence S kab ;k .Consequently, both tensors cannot serve as gravitation genrating tensors in Einstein's equations, and the question arises: how can the balance equations (249) 2,3 be incorporated into the general-covariant framework of GRT ?The answer to that question has been proved by the following extended Belinfante/Rosenfeld procedure whose special relativistic version is well known since a long time [41,42,43] a tensor which is symmetric and divergence-free according to (251) and (253) 2 .The general-covariant Belinfante/Rosenfeld procedure transforms by use of the symmetric spin divergence S (ab)c ;c the not necessary divergence-free symmetric part of the energy-momentum tensor T (ab) into a symmetric and divergence-free tensor † Θ ab .Or in other words: the energy-momentum tensor T ab (not necessary symmetric and divergencefree) is tranformed into the mutant † Θ ab (symmetric and divergence-free) 28 .
The decisive step for connecting GRT and GCCT is the following usually used Setting XV: The mutant which is created by the Belinfante/Rosenfeld procedure is the gravitation generating energy-momentum tensor of Einstein's equation (250).According to (249), the mixture (and not single components) determines the geometry.
According to (5) 3 , we obtain and with (8) 1 follows resulting in We obtain from (8) 2 and taking (278) 1 into account As expected, the 4-velocity of the mixture is identical with the uniform component velocities.

Stoichiometric equations
The system of the relativistic stoichiometric equations runs as follows The stoichiometric coefficients ν A α are scalars, and the partial rest mole mass M A 0 is defined using the scalar mole number n A and the mole concentration ζ A of the A -component according to (272) and (273).The stoichiometric coefficients ν A α are determined by the partial rest mole masses M A 0 , A = 1, 2, ..., Z, before and after the αth reaction.The time derivative of the mole number is determined by the reaction velocities Multiplication with M A 0 results by use of (283) in