## Abstract

Recently, the 14 moments model of extended thermodynamics for dense gases and macromolecular fluids has been considered and an exact solution, of the restrictions imposed by the entropy principle and that of Galilean relativity, has been obtained without using Taylor's expansions. Here, we prove the uniqueness of the above solution and exploit other pertinent conditions such as the convexity of the function *h*′ related to the entropy density, the problem of subsystems and the fact that the flux in the conservation law of mass must be the moment of order 1 in the conservation law of momentum. The results present interesting aspects which were not suspected when only approximated solutions of this problem were known.

## 1. Introduction

We analyse here some remarkable properties of a recent extended thermodynamic model for dense gases and macromolecular fluids (Carrisi *et al.* 2009). Some interesting aspects emerge; for example, it is proved that hyperbolicity holds only at the cost of assuming a Lagrange multiplier with three indexes of a higher order than another one with four indexes, so confirming the initial point of the modern theory called consistent order extended thermodynamics (COET). Moreover, it is shown that systems with less moments can be obtained as subsystems only if a physical assumption is used, an assumption which is inherent to the principle of material frame indifference. It is only in this way that the mathematical model becomes consistent.

This model has been obtained through a macroscopic approach, i.e. based on the entropy principle. Another possible formulation of extended thermodynamics for dense gases is based on the kinetic approach; in order to permit a comparison between the two approaches, we now cite some references on the kinetic approach. Enskog (1921) introduced a kinetic theory for dense gases which yields a very good approximation of the behaviour of gases. Later, hydrodynamical-like equations have been derived from the kinetic equation; see, for example, the Chapman–Enskog method (Chapman & Cowling 1970).

Kremer & Rosa (1988) obtained hydrodynamic equations from the local equilibrium distribution function as a kernel linearizing the collision integral in Enskog’s equation; in this way, they were able to derive sound dispersion relations for monoatomic gases by using normal mode analysis. Based on this last paper, Marques & Kremer (1991) obtained linearized hydrodynamic equations involving the second-order terms of the collision integral; in this way, they improved the results previously known in literature and, furthermore, they obtained linearized Burnett equations for monoatomic gases.

Ugawa & Cordero (2007) obtained extended hydrodynamic equations derived from Enskog’s equation by using Grad’s moment expansion method in the bi-dimensional case; among other results, they discussed the nature of a simple one-dimensional heat conduction problem and were able to show that, not too far from equilibrium, the non-equilibrium pressure in this case depends on the density, temperature and heat flux vector.

Coming back to the model (Carrisi *et al.* 2009), we see that the balance equations to describe the 14 moments model of extended thermodynamics for dense gases and macromolecular fluids are
1.1
where the independent variables are *F*, *F*_{i}, *F*_{ij}, *F*_{ill}, *F*_{iill} and are symmetric tensors. *P*_{〈ij〉}, *P*_{ill}, *P*_{iill} are productions and they too are symmetric tensors. The fluxes *G*_{ki}, *G*_{kij}, *G*_{kill}, *G*_{kiill} are constitutive functions and are symmetric over all indexes, except for *k*. The case where also *G*_{ki} is symmetric has already been studied by Carrisi & Pennisi (2008*a*) and also the solution of this eventual condition has been obtained without using expansions around thermodynamical equilibrium. The restrictions imposed by the entropy principle and that of Galilean relativity were firstly studied by Kremer & Beevers (1983) and Kremer (1989), up to second order with respect to equilibrium. In Carrisi *et al.* (2009), we have obtained a non-approximated solution through a non-relativistic limit. We now want to impose the entropy principle for our system (1.1). This is a common routine in extended thermodynamics; readers which are not familiar with this subject may become acquainted with it by reading, for example, the handbook (Müller & Ruggeri 1998). So, we learn that the entropy principle in the present case is equivalent to assuming the existence of potentials *h*′, *ϕ*′_{k} and of the Lagrange multipliers *λ*, *λ*_{i}, *λ*_{ij}, *λ*_{ill}, *λ*_{ppll} such that
1.2
By comparing equation (1.2)_{2} with equation (1.2)_{6}, we obtain the following compatibility condition:
1.3
We now have to impose the Galilean relativity principle; previously, it was imposed through a decomposition of the variables into convective and non-convective parts, with consequent heavy calculations. So, we prefer to apply here the new methodology (Pennisi & Ruggeri 2006), adapted for the present case in Carrisi & Pennisi (2008*b*) and Carrisi *et al.* (2008). In this way, we need no decomposition, but we have only to impose equations (1.2), (1.3) and the following two other conditions:
1.4
After that, we can obtain *λ*_{k} as a function of the remaining Lagrange multipliers, through the implicit function (∂*h*′/∂*λ*_{k})=0; the above-mentioned decomposition will follow as a consequence. Now in Carrisi *et al.* (2009), we have obtained the following solution of equations (1.4):
1.5
with , , , suitable vectors and *X*_{1}–*X*_{8} suitable scalars, whose explicit expressions we do not report for the sake of brevity, but which can be read in Carrisi *et al.* (2009). Moreover, *H*_{0}, *H*_{1}, *H*_{2} and *H*_{3} are arbitrary functions of the scalars *X*_{1}–*X*_{8}.

In the next section, we will prove the uniqueness of this solution. In §3, we will impose the further condition (1.3) and, also for this problem, we will find an exact solution without using expansions. In §4, we will impose the convexity of the function *h*′ which is important in order that our symmetric system is also hyperbolic. We will find interesting results such as the following: although both *λ*_{jpp} and *λ*_{ppll} are zero at equilibrium and the first of these has an index less than the other, it tends to zero faster when the system tends to equilibrium. This fact shows that it is not correct to consider all higher-order moments negligible with respect to the previous ones. This result confirms the starting point of the new theory COET. This is a special approach to rational extended thermodynamics which make use of combinations of moments as fields and to those combinations, called G-moments, may be assigned an order of magnitude in a rational manner. Closure in this theory is an automatic consequence of the assignment of order. As a partial list of papers on this subject, we cite here Müller *et al.* (2003), Barbera (2005), Reitebuch (2005), Müller & Reitebuch (2002), Weiss (2006), Sugiyama & Zhao (2006) and Barbera & Valenti (2006). More precisely, §4 will show that equations (1.5) have to be substituted by
1.6
with *K*_{i} arbitrary functions of *η*_{1}=*X*_{1}, *η*_{i}=(*X*_{i}/*λ*_{ppll}) for *i*=2,…,4 and, moreover, of
1.7
On the other hand, these are compatible with equation (1.5). Obviously, the form (1.6) can be used only if *X*_{1}≠0 on the initial manifold and until that it remains *X*_{1}≠0.

It is interesting to note that the solution (1.6), calculated in *λ*_{ill}=0, becomes
1.8
with *K*_{i} functions of
On the other hand, if we know and , from the above expression of , we obtain *K*_{1}, *K*_{2}, *K*_{3} because they are coefficients of linearly independent vectors. After that, from the above expression of we obtain *K*_{0}; also, their functional dependence is arbitrary because the above expressions of *η*_{1}–*η*_{8} are the most general possible. In other words, if we know the expressions of *ϕ*′^{k} and *h*′ calculated in *λ*_{ill}=0, we will also know them for *λ*_{ill}≠0!

Finally, in §5, the problem of subsystems will be considered and in this case too, we will find unexpected results.

## 2. Uniqueness of the solution (1.5)

In order to prove uniqueness of the solution (1.5), together with equations (1.6) and (1.7) of Carrisi *et al.* (2009), let us begin with the case in which the following two conditions are satisfied:

The vectors

*λ*_{ill},*λ*_{ia}*λ*_{all}, are linearly independent.The four-vectors are linearly independent.

Moreover, the following arguments hold only if continuity is assumed. It is true that weak solutions of our field equations could arise; but at present we are not able to say if our method can also be extended in the presence of weak solutions. This may be the object of future research.

Before proving uniqueness of our solution, we need to consider the following representation theorem: every scalar function of our Lagrange multipliers can be expressed as a function of the scalars of the set This theorem can be proved in a way similar to those used for other representation theorems (Wang 1969; Smith 1971; Pennisi & Trovato 1987; Pennisi 1992), as follows:

It suffices to prove our statement in a particular reference frame and see that in this reference we can obtain the Lagrange multipliers from the knowledge of the scalars in *S*_{1}; so, let us use the frame defined by *λ*_{ill}≡(*λ*_{1ll},0,0), *λ*_{13}=0, *λ*_{1ll}≥0, *λ*_{12}≥0.

— If

*λ*_{1ll}>0,*λ*_{12}>0, we obtain*λ*_{1ll},*λ*_{11},*λ*_{12},*λ*_{22},*λ*_{33},*λ*_{23}, respectively, from*λ*_{all}*λ*_{all},*λ*_{ab}*λ*_{all}*λ*_{bll}, , ,*λ*_{ll}, ; after that, the fourth of these can be expressed as a function of the remaining ones and of through the Hamilton–Kayley theorem.— If

*λ*_{1ll}>0,*λ*_{12}=0, with a rotation around the first axis, we can select the reference where*λ*_{23}=0; after that, we obtain*λ*_{1ll},*λ*_{11},*λ*_{22},*λ*_{33}, respectively, from*λ*_{all}*λ*_{all},*λ*_{ab}*λ*_{all}*λ*_{bll},*λ*_{ll}, .— If

*λ*_{1ll}=0, we may select the reference frame where*λ*_{12}=0,*λ*_{13}=0,*λ*_{23}=0, and obtain*λ*_{11},*λ*_{22},*λ*_{33}from*λ*_{ll}, , .

Until now, we have obtained *λ*_{ill} and *λ*_{ab} as functions of the elements of *S*_{1}; obviously, also *λ*_{ppll} is a function of them. It remains to obtain *λ* and *λ*_{k}. To this end, we note that, from equations (1.6) and (1.7) of Carrisi *et al.* (2009), it follows
2.1
and these are linearly independent for the second hypothesis at the beginning of this section; consequently, the Jacobian determinant, constituted by the derivatives of *X*_{5}–*X*_{8} with respect to *λ* and *λ*_{k}, is non-singular. By using the theorem on implicit functions, it follows that we can obtain *λ* and *λ*_{k} in terms of *X*_{5}–*X*_{8}. This completes the proof of our representation theorem.

But it is obvious that
are invertible functions of , *X*_{1}–*X*_{4} so that also the following representation theorem holds: every scalar function of our Lagrange multipliers can be expressed as a function of the scalars of the set . By applying this theorem, we can now prove our theorem on uniqueness. For the second hypothesis at the beginning of this section, we have that it is possible to obtain the scalar functions *H*_{0}–*H*_{3} such that equations (1.5) hold. For the previous representation theorem, we have that *H*_{i} can be expressed as functions of the elements in *S*. From equations (1.5)_{2} we have that also *h*′ satisfies this property, because the coefficients of *H*_{0}–*H*_{3} are proportional to the elements *X*_{1}–*X*_{4} of *S*.

Let us now impose that equations (1.5) satisfy equations (1.4). To this end, let us use the results of Carrisi *et al.* (2009)
and
where *h*=1,…,8; *r*=0,…,3; *P*_{0}=8, *P*_{1}=1, *P*_{2}=1, *P*_{3}=1, and there is no summation convention over the repeated index *r*. Consequently, by substituting equations (1.5) into equations (1.4), many terms give zero contribution and there remain
and
with *Q*_{1}=*λ*_{ll}, , .

Now, for the first hypothesis at the beginning of this section, the vectors *λ*_{ill}, *λ*_{ia}*λ*_{all}, are linearly independent; consequently, the above relation becomes
This result, for the second hypothesis at the beginning of this section, implies that (∂*H*_{r}/∂*Q*_{s})=0, that is, *H*_{r} does not depend on *Q*_{1}, *Q*_{2}, *Q*_{3}. Consequently, it may depend only on *X*_{1}–*X*_{8}, as we desired to prove.

In this way, we have proved uniqueness only if the conditions (i) and (ii), the beginning of this section, are satisfied. On the other hand, the set in which these conditions are not satisfied is only a submanifold of the domain (for example, condition (i) is violated when *λ*_{ill}=0, or when *λ*_{ia} has an eigenvalue with multiplicity 2 and so on; in all cases, these conditions are violated when the variables satisfy a suitable set of equations or, in other words, in a submanifold); so our result on uniqueness must hold in any case for continuity reasons. This can be clarified better with the following example: if *F*(*x*,*y*) is a continuous function such that
then it follows so that *F*(*x*,*y*)=5 for all values of *x*,*y*.

## 3. The further condition (1.3)

We now want to impose the further condition (1.3); we will see that it can be nicely solved. The solution gives *H*_{0}, *H*_{1}, *H*_{2}, *H*_{3}, in terms of the arbitrary functions *ψ*=*ψ*(*X*_{1},*X*_{2},*X*_{3},*X*_{4},*X*_{5},*Y*_{6},*Y*_{7},*Y*_{8}), *φ*=*φ*(*X*_{1},*X*_{2},*X*_{3},*X*_{4},*Z*_{5}, *X*_{6},*Z*_{7},*Z*_{8}), for *i* going from 1 to 3, for *j*=0,2,3. This solution reads
3.1
where it is understood that the right-hand sides are calculated in
3.2
In order to prove this result, let us start by noting that from equations (1.6) and (1.7) of Carrisi *et al.* (2009), it follows that , , , do not depend on *λ* and, moreover,
3.3
From (3.3)_{9–12} we have also that the coefficients of *H*_{0}, *H*_{1}, *H*_{2}, *H*_{3} in *h*′ do not depend on *λ*; consequently, equation (1.3) becomes
or
This equation, for (3.3)_{13–16} becomes
3.4
To find the solution of these equations, let us distinguish two cases.

### (a) The case X_{1}≠0

We see that equation (3.4) can be written as
3.5
Now, if we add to the second four vectors indicated at the beginning of §2, the first one multiplied by −(1/8)(*X*_{2}/*X*_{1}), to the third one the first one multiplied by −(1/8)(*X*_{3}/*X*_{1}), to the fourth one the first one multiplied by −(1/8)(*X*_{4}/*X*_{1}), those four-vectors transform in
from which it follows that , , are linearly independent. Consequently, equation (3.5) amounts to saying that the coefficients of these three-vectors are zero.

These three equations can be transformed with the following change of functions and of independent variables:
3.6
for *i*=0,…,3. With this change, our equations become

So, it will suffice to define *ψ* from
to obtain, thanks to equations (3.6), the result (3.1), but with *φ*=0, . On the other hand, from equation (1.3), we see that the sum of two solutions is still a solution. Consequently, it will now suffice to prove that equation (3.1) is a solution also with *φ*≠0, , *ψ*=0, *H*_{i}^{*}=0; this will be the result of the following case.

### (b) The case X_{2}≠0

We see that equation (3.4) can be written as
3.7
Now, if we add to the first four vectors indicated at the beginning of §2, the second one multiplied by −8(*X*_{1}/*X*_{2}), to the third one the second one multiplied by −(*X*_{3}/*X*_{2}), to the fourth one the second one multiplied by −(*X*_{4}/*X*_{2}), those four vectors transform in
from which it follows that , , are linearly independent. Consequently, equation (3.7) amounts to saying that the coefficients of these three vectors are zero.

These three equations can be transformed with the following change of functions and of independent variables:
3.8
for *i*=0,…,3. With this change, our equations become
So, it will suffice to define *φ* from
to obtain, thanks to equations (3.8), equations (3.1), but with *ψ*=0, *H*_{i}^{*}=0, as stated before.

## 4. The convexity of *h*′

In order that our system (1.1) be hyperbolic, we now have to impose that the hessian matrix (∂^{2}*h*′/∂*λ*_{A}∂*λ*_{B}) is positive defined, with *λ*_{A} the generic component of the Lagrange multipliers. In other words, the quadratic form *Q*=(∂^{2}*h*′/∂*λ*_{A}∂*λ*_{B})*δλ*_{A}*δλ*_{B} has to be positive definite. Let us exploit this with the potential equation (1.6)_{2} and with equation (1.7); in these expressions, except for replacing *X*_{i} with *X*_{i}/(*X*_{1}) for *i*=5,…,8, the remaining polynomials (1.7) in *X*_{j}/(*X*_{1}) for *j*=2,…,4 have been chosen in order to eliminate from *X*_{i} for *i*=5,…,8 the terms depending only on *λ*_{ab}.

Well, we now want to evaluate this quadratic form *Q* in the reference state, which will be called *C*, where *λ*_{i}=0, *λ*_{ij}=1/3*λ*_{ll}*δ*_{ij}, *λ*_{ill}=0; so, there remain as independent variables *λ*, *λ*_{ll}, *λ*_{ppll}. This is an intermediate state with respect to equilibrium, where we also have *λ*_{ppll}=0. To this end, we need the expressions of our variables up to second order with respect to the state *C*. After some calculations, we find
with *λ*_{〈ab〉}=*λ*_{ab}−(1/3)*λ*_{ll}*δ*_{ij}.

After that, from *h*′=*h*′(*η*_{i}), we find that the expression of *h*′ up to second order with respect to the state *C* is
where the apex * denotes a quantity calculated in the state *C*, so that we have also
Taking into account these intermediate results and the additivity of *Q*=(∂^{2}*h*′/∂*λ*_{A}∂*λ*_{B})*δλ*_{A}*δλ*_{B} and by calculating (∂^{2}*h*′/∂*λ*_{A}∂*λ*_{B}) in the reference state *C*, we find *Q*=*Q*_{1}+*Q*_{2}+*Q*_{3}, with
Consequently, the required convexity holds if
We note that these conditions are continuous in *λ*_{ppll}, so that we may impose them also calculated in *λ*_{ppll}=0; in this way, we will obtain the requested convexity not only in a neighbourhood of the state *C*, but also in a neighbourhood of equilibrium. In other words, one may initially think that if the condition *X*_{1}=*λ*_{ppll}≠0 is not fulfilled, the hyperbolicity of the system is not granted; we have seen here that this is not true, but only if we assume that *λ*_{ill} goes to zero faster than *λ*_{ppll}! The physical meaning of this result is, in our opinion, that it is not correct to consider all higher-order moments negligible with respect to the previous ones and this fact confirms, as stated above, the starting point of COET.

We have performed the same passages also by starting from equation (1.5)_{2}, together with equations (1.6) and (1.7) of Carrisi *et al.* (2009), instead of equations (1.6)_{2} and (1.7); in this way, we have found that *Q* is not positive defined. We conclude that only equation (1.6) is the correct expression to use.

We also note that the results of §3 can be written taking into account the expression (1.6). In particular, we can use the expressions (1.7) to find *X*_{1}–*X*_{8} as functions of *η*_{1}–*η*_{8}. After that, by also using equations (3.2), we can obtain *Y*_{5}–*Y*_{8}, with *Y*_{5}=*X*_{5}, that is
4.1
From *K*_{i}=*η*_{1}*H*_{i} and by defining *ϑ*=*η*_{1}*ψ*, for *i*=1,2,3, we can rewrite equations (3.1). We will limit ourselves to the case *X*_{1}≠0, so that we have *φ*=0, . The result is that the solution gives *K*_{0}, *K*_{1}, *K*_{2}, *K*_{3}, in terms of the arbitrary functions *ϑ*=*ϑ*(*η*_{1},*η*_{2},*η*_{3},*η*_{4},*Y*_{5},*Y*_{6},*Y*_{7},*Y*_{8}), for *i* going from 1 to 3. This solution reads
where it is understood that the right-hand sides are calculated in equation (4.1).

## 5. The subsystems

Other interesting particulars of our solution become manifest when we search the subsystems of equation (1.1).

As example, equations (1.4) calculated in *λ*_{ill}=0, *λ*_{ppll}=0 become the conditions we would have by starting only with equations (1.1)_{1–3}. But equation (1.5)_{2} in *λ*_{ill}=0, *λ*_{ppll}=0 gives *h*′=0 which cannot be accepted for the required convexity. This problem is not avoided either by using equations (1.6) because the consequent solutions do not satisfy the conditions (1.4) calculated for the subsystem. To verify that this is the case, it suffices to note that equation (1.4)_{1} with *η*_{5} instead of *h*′ is satisfied, but if we replace *η*_{5} with its value in *λ*_{ill}=0, *λ*_{ppll}=0, that is, 16*λ*, we see that this satisfies equation (1.4)_{1} no more calculated in *λ*_{ill}=0, *λ*_{ppll}=0! The reason is evident from the fact that *η*_{5} satisfies equation (1.4)_{1}; but if we calculate this equation in *λ*_{ill}=0, we find
or
whose value in *λ*_{ppll}=0 is not a solution of equation (1.4)_{1} calculated in *λ*_{ill}=0, *λ*_{ppll}=0.

An idea may be to redo the passages of §5 of Carrisi *et al.* (2009) but starting from the beginning with *λ*_{ill}=0, *λ*_{ppll}=0, that is, with
and
but in this case, among the scalars there is the one coming from , that is, *λ*_{a}*λ*_{a}−(4/3)*λλ*_{ll} and this, substituted to *h*′ in equation (1.4)_{1} calculated in *λ*_{ill}=0, *λ*_{ppll}=0, does not satisfy it. The same thing can be said if we start from eqn (89) instead of eqn (80), both of Carrisi *et al.* (2009). Furthermore, if we start from eqn (88) instead of eqn (80) (both of Carrisi *et al.* (2009)), we quickly obtain *h*′=0, *ϕ*′^{k}=0. In other words, the subsystem with 10 moments cannot be obtained in any way as a non-relativistic limit.

What about the subsystem with five moments?

If in eqn (9) of Carrisi *et al.* (2009) we substitute *λ*_{ij}=(1/3)*λ*_{ll}*δ*_{ij}, *λ*_{ill}=0, *λ*_{ppll}=0, they become the entropy principle for the system constituted only by equations (1.1)_{1,2} and by the trace of equation (1.1)_{3}, with Lagrange multipliers *λ*, *λ*_{i}, (1/3)*λ*_{ll}, respectively.

Equations (1.4) then become
5.1
With arguments like those described above, the solution of this equation cannot be found from equation (1.5), nor from equation (1.6) calculated in the above values of *λ*_{ij}, *λ*_{ill}, *λ*_{ppll}.

Instead of this, the idea of redoing the passages of §5 of Carrisi *et al.* (2009), but starting from the beginning with *λ*_{ij}=(1/3)*λ*_{ll}*δ*_{ij}, *λ*_{ill}=0, *λ*_{ppll}=0, is successful. In fact, starting from eqn (80) or from eqn (89) (both of Carrisi *et al.* 2009), we find
5.2
where *H*_{0} is a function of *λ*_{ll}, *λ*_{a}*λ*_{a}−(4/3)*λλ*_{ll}. These functions effectively satisfy equations (5.1). More than that, we have that they automatically also satisfy equation (1.3)!

Obviously, we cannot obtain this result by starting from the beginning from eqn (88) of Carrisi *et al.* (2009) because in this case we would quickly obtain *h*′=0, *ϕ*′^{k}=0. On the other hand, if we start from eqn (88) of Carrisi *et al.* (2009) we obtain *λ*^{β}=0 and this is not adequate to describe the relativistic model; the less for its limit!

In other words, we have found that the diagram in figure 1 is not commutative, and this is quite different from the results obtained with expansions around equilibrium, that is, the diagram in figure 2, even if this has been until now proved only for the less restrictive case of ideal gases (Carrisi & Pennisi 2007).

Is there any solution to this problem? Yes. From Pennisi & Ruggeri (2006), Carrisi & Pennisi (2008*b*) and Carrisi *et al.* (2008) we see that *h*′ and *ϕ*′^{k} are not relevant from the physical viewpoint, but the values and which they assume in *λ*_{i} are implicitly defined by (∂*h*′/∂*λ*_{i})=0.

Now, in the five moments case, from equation (5.2)_{1} we find that the equation (∂*h*′/∂*λ*_{i})=0 implicitly defines *λ*_{i}=0. After that, equations (5.2) give
5.3
From the subsystem viewpoint, we firstly find that equations (1.6) and (1.7) together with eqns (6) and (7) of Carrisi *et al.* (2009), calculated in *λ*_{i}=0, *λ*_{ill}=0, *λ*_{ij}=(1/3)*λ*_{ll}*δ*_{ij} become
which remain unchanged when calculated in *λ*_{ppll}=0. These results show us that the functions , calculated starting from equations (1.6) are nothing more than equations (5.3) in the particular case where *K*_{1} depends only on *η*_{6} and, moreover, *K*_{0}=0, *K*_{2}=0, *K*_{3}=0, *K*_{1}=(5/24)*H*_{0}.

In conclusion, we have found that the classical subsystem with five moments contains the classical model with five moments only as a particular case. So, the correct commutative diagram is that in figure 3. If we want to investigate a similar property for the subsystem with 10 moments, we must firstly know the classical model with 10 moments; but, as said above, this cannot be found through a non-relativistic limit. However, we can construct it directly and this will be done in the next subsection.

### (a) The macroscopic model with 10 moments

For the 10 moments model, equations (1.4) become
5.4
From the representation theorems (Wang 1969; Smith 1971; Pennisi & Trovato 1987; Pennisi 1992), we know that every scalar function of *λ*, *λ*_{i}, *λ*_{ij} can be expressed as a function of *λ*, *Q*_{1}, *Q*_{2}, *Q*_{3}, *G*_{0}=*λ*_{i}*λ*_{j}, *G*_{1}=*λ*_{ij}*λ*_{i}*λ*_{j}, *G*_{2}=*λ*_{ia}*λ*_{aj}*λ*_{i}*λ*_{j}.

We also need the Hamilton–Kayley theorem
Taking into account these properties, we have that equation (5.4)_{1} becomes
5.5
because *λ*_{i}, *λ*_{ij}*λ*_{j}, *λ*_{ia}*λ*_{aj}*λ*_{j} are linearly independent.

To solve the system (5.5) we suppose, without loss of generality, that *h*′ is a composite function of
In this way, the system (5.5) becomes (∂*H*/∂*λ*)=0, (∂*H*/∂*G*_{0})=0, (∂*H*/∂*G*_{1})=0 so that the general solution of equation (5.4)_{1} is
5.6
Regarding equation (5.4)_{2}, to avoid passages which may result tedious to read, we will consider only the case with *S*_{3}≠0. We note that at equilibrium, we have *λ*_{ij}=(1/3)*λ*_{ll}*δ*_{ij} from which , so that our hypothesis *S*_{3}≠0 is surely satisfied in a neighbourhood of equilibrium. In this case, a solution of equation (5.4)_{2} is
5.7

and this also satisfies equation (1.3). We also note that this is the only solution of equations (5.4)_{2} and (1.3). In fact, thanks to equation (1.3) we can rewrite equation (5.4)_{2} as
5.8
a solution of this equation is *ϕ*_{k}^{′*}, so that
subtraction of this equation from the previous one gives 0=2*λ*_{ij}(∂/∂*λ*_{j})(*ϕ*′_{k}−*ϕ*_{k}^{′*}); but the matrix *λ*_{ij} in general is not singular, so that (∂/∂*λ*_{j})(*ϕ*′_{k}−*ϕ*_{k}^{′*}), that is, (*ϕ*′_{k}−*ϕ*_{k}^{′*}) is a vectorial function depending only on *λ* and *λ*_{ij}. It follows, from Smith (1971), Wang (1969), Pennisi & Trovato (1987) and Pennisi (1992) that *ϕ*′_{k}−*ϕ*_{k}^{′*}=0, from which uniqueness is proved.

From equation (5.6), we find that the equation (∂*h*′/∂*λ*_{i})=0 implicitly defines *λ*_{i}=0. After that, equation (5.6) gives
which can also be written as
5.9
because *S*_{3} is a function of *Q*_{1},*Q*_{2},*Q*_{3}. Similarly, from equation (5.7) we obtain
5.10
We can now compare equations (5.9) and (5.10) with the corresponding results which can be obtained through the passage to the subsystem. To this end, we firstly find that equations (1.6) and (1.7) together with eqns (6) and (7) of Carrisi *et al.* (2009), calculated in *λ*_{i}=0, *λ*_{ill}=0, become
which remain unchanged when calculated in *λ*_{ppll}=0. These results show us that the functions , calculated starting from equations (1.6) are nothing more than equations (5.9) and (5.10).

Consequently, the commutative diagram is that in figure 4, at least for the case *S*_{3}≠0.

## 6. Conclusions

We retain these results very satisfactorily because they give more insight into the snares hidden in some passages used in extended thermodynamics. So, everything must be handled carefully. In particular, we have proved unicity of the closure of a previous model, where it was found only through a mathematical tool, that is, by taking the non-relativistic limit of a suitable relativistic model. The calculations are complicated, but their elegance bears witness to the goodness of the theory. In the future, we plan to extend this proof also in the presence of weak solutions.

Moreover, we have proved hyperbolicity around thermodynamical equilibrium but at the cost of assuming a Lagrange multiplier with three indexes, of a higher order than another one with four indexes. This is not a defect, because it is similar to a basic concept in COET. We plan, in the future, to investigate this likeness further and, eventually, to re-elaborate the present results but in the context of this new theory.

Finally, we have shown that systems with less moments can be obtained as subsystems only if a physical assumption is used, an assumption which is inherent to the principle of material frame indifference; also, this fact bears witness to the physical consistence of this model. In the future, we also aim to obtain similar results, namely without using Taylor’s expansions, also for the case of ideal gases, where more symmetry conditions are present; at present, we do not know if this is possible, but it would be a very nice thing.

Similar investigations can be tried also for a higher number of moments; this too may be a subject for future research.

## Acknowledgements

We thank two anonymous referees which allowed, with their suggestion, improvements in the presentation of this article.

## Footnotes

- Received October 6, 2009.
- Accepted December 17, 2009.

- © 2010 The Royal Society