Euler’s function

As we shall see later some dissipation potentials are homogeneous Euler s functions. This means for example, that the 0(j,r ) is a homogeneous Euler s function of degree k when every scalar of Ae R the following relation is valid ... 

It can be proved [i ], that the dissipation potential T(X,r ) dual-pair of 0(j,rj) is also homogeneous Euler s-function, like... 

So the entropy production is homogeneous Euler s function of degree... 

In summary, the previously given form of the minimal principle of entropy production leads to a class of generalized Onsager constitutive theory, which is also direct generalization of the linear Onsager s theory having their dissipation potentials as homogeneous Euler s functions. 

The above parts show the minimum principle for vector processes in the frame of the generalized Onsager constitutive theory by the directions of Onsager s last dissip>ation of energy principle. We had seen above that in case of source-free balances, this principle is equivalent with the principle of minimal entropy production. The equivalence of the two theorems in the frame of the linear constitutive theory was proven by Gyarmati [2] first. Furthermore, we showed that in case when the principle of minimal entropy production is used for the determination of the possible forms of constitutive equations, the results are similar to the linear theory in the frame of the Onsager s constitutive theory, where the dissipation potentials are homogeneous Euler s functions. 

No similar conditions exist for RGEP and the flux part of RGEP in generalized Onsager constitutive theory. Also, the assumptions of this theory, contrary to the linear constitutive theory, do not guarantee the stabile extremum of the GEP at a thermodynamic steady-state. It is well known that in the linear constitutive theory the dissipation potentials are second order homogeneous Euler s functions. This is a central prop>erty of the potentials, which guarantees that the two p>arts of the GEP are proportional to the GEP and are equal with... 

The Glansdorff-Prigogine general evolution criterion involves the minimum of global entropy production in such a constitutive theory where the potentials are homogeneous Euler s functions. We show below the strictly convex property of dissipation potentials guarantee the minimum, and the function... 

We show that these types of reactions could be described by a dissip>ation potential which are homogeneous Euler s function of degree 3 of the concentration of chemical components. 

The GEP of the system in stationer case could be minimal then and only then when the dissipation potentials of the subsystems are homogeneous Euler s functions with identical degree. 

In this state the GEP of the system is minimal, because both the dissipation potentials are homogeneous Euler s functions of degree 2. 

A dissipative system could be energy converter, when not all the terms of the entropy production are positive, their sum is positive definite only. The problem will be studied in a dissifjative system having two thermodynamic forces and currents and in case when the dissipation potentials are homogeneous Euler s function of degree k. In this case the currents and forces are connected with the following constitutive equations (see Appendix 7.6.)... 

Fig. 9. The characteristic functions of the non-linear energy converters. P is the output power, r is the efficacy, is the dissipation and E is the ecologic function (the difference between the output power and the dissipation), q is the coupling parameter, and m is the power of the universal Euler s function in (297). These functions are not given on a common scale, they are on certain self-scales to show their behavior near the optimum state.

From this take into account Appendix 7.7., follows that the entropy production, as function of fluxes, is homogeneous Eulers s function of degree X. Because of this property for the flux dissipation potential we obtain... 

Thus the flux potential is homogeneous Euler s function of degree X as well. The flux potential in (301) is strict convex, thus 00in (308). Substitute the flux potential to the Ziegler s principle (306) then after some algebra we obtain the generalized Onsager s last dissipation of energy principle (302), i.e. 

Using the following relation of the homogeneous Euler s function potentials and of the GEP... 

Consequently, again we obtained the result that the sum of the dissipation potentials is minimal in stationer state and to the extremum of GEP is requested the identical degrees of Euler s function dissipation potentials. This means also, that using dissipation potentials has advantage in the thermodynamics of steady-state systems. 

A direct consequence of the possibility to represent the homogeneous Euler s functions by their partial derivative functions is that the thermodynamic currents and dissipation... 

The representation theorem is a simple consequence of the definition of the homogeneous Euler s functions ... 

The proof that the partial derivative functions are one degree lower homogeneous Euler s functions could be shown as below. When the function is ... 

At the end we proved the equivalence of the Ziegler s maximum entropy production principle with the generalized Onsager s last dissipation energy principle in cases when the dissip>ation potentials are homogeneous Euler s functions. 

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