Entropy, generalized minimum production principle

This demonstration of generalized minimum production principle has been given by Callen. This theorem is very general and seems to have wide applicability. However, the proof of this theorem is so closely connected with the reciprocity relations that some restriction will appear in applying it to practical problems, whereas the principle of minimum entropy production in the macroscopic description holds even when the reciprocal relations are not used. 

It is also easy to show that the dissipation functions 0 and W do not exist far from equilibrium, since djo and dxO are not total differentials. Neither do they in general possess integrating multipliers. Therefore the variational principle of Onsager -i cannot be used, but as we shall see it is possible to generalize Prigogine s principle of minimum entropy production in the stationary state to be valid outside near-equilibrium states in the sense that we can construct a function which is minimized in the stationary state and which near equilibrium reduces to the entropy production. 

Although the principle given in Eqs. 159 and 160 is more general than the principle of minimum entropy production, this cannot be used as variational principle. To consider this situation in more detail let us write Eq. 159 in a discrete form ... 

As the principle of minimum entropy production has wide range of applicability, it seems useful to derive this principle by the statistical-mechanical method. This approach will show that the minimum entropy production principle holds in a microscopic description of the system. However, since it is rather complicated to deal with general cases, we shall restrict ourselves to the case of a simple system. Following Klein and Meijer we shall consider a system consisting of two identical chambers, I and II. which contain a total number of N molecules of an ideal... 

Rather lately, an attempt to generalize this principle to a nonlinear irreversible process has been made by Klein, based on a statistical method for a simplified model. He demonstrated that the minimum production properties are also a useful approximation criterion for the stationary state even when the latter is very far from equilibrium. According to a simple example of an irreversible process given in his calculation, it has been found that the entropy production does not decrease monotonically, but passes through its minimum en route to the stationary state. We have already mentioned these results in connection with the variational principle in nonlinear irreversible processes. 

Evans and Baranyai [51, 52] have explored what they describe as a nonlinear generalization of Prigogine s principle of minimum entropy production. In their theory the rate of (first) entropy production is equated to the rate of phase space compression. Since phase space is incompressible under Hamilton s equations of motion, which all real systems obey, the compression of phase space that occurs in nonequilibrium molecular dynamics (NEMD) simulations is purely an artifact of the non-Hamiltonian equations of motion that arise in implementing the Evans-Hoover thermostat [53, 54]. (See Section VIIIC for a critical discussion of the NEMD method.) While the NEMD method is a valid simulation approach in the linear regime, the phase space compression induced by the thermostat awaits physical interpretation even if it does turn out to be related to the rate of first entropy production, then the hurdle posed by Question (3) remains to be surmounted. 

In an important paper (TNC.l), they offered for the first time an extension of nonequilibrium thermodynamics to nonlinear transport laws. As could be expected, the situation was by no means as simple as in the linear domain. The authors were hoping to find a variational principle generalizing the principle of minimum entropy production. It soon became obvious that such a principle cannot exist in the nonlinear domain. They succeeded, however, to derive a half-principle They decomposed the differential of the entropy production (1) as follows ... 

The intrinsic tendency of the system to decrease its entropy production as far as possible is known as the principle of minimum entropy production. It defines a general direction of spontaneous evolutions in thermodynamic systems which are not in equilibrium states. The proof of this principle depends on the validity of On-... 

Returning now to the general case, (7.50) can be considered as a generalized evolution criterion for all real processes. This criterion includes the principle of minimum entropy production in the linear range. An evolution criterion, however, can immediately be retranslated into a stability criterion if for all variations... 

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. 

We shall now present several examples to illustrate the general validity of the principle of minimum entropy production. 

The principle of minimum entropy production can easily be demonstrated for more complex chemical systems. Example 1 can be generalized to an arbitrary number of intermediates. 

In the linear regime we saw that the stationary states are those in which the total entropy production P = a dV reaches a minimum. This criterion also assured the stability of the stationary state. In the far-from-equilibrium nonlinear regime there is no such general principle for determining the state of the system. 

In 1987 we were concerned with the validity of the so-called principle of minimum entropy production rate [4,5]. In the first article we showed by expansion of the entropy production the general invalidity of the principle. Once the entropy production rate is expanded in the affinity, the deviation from equilibrium, then two operations are required (1). the differentiation of the entropy production rate with respect to temperature and (2). the termination of the series expansion in the affinity to simulate the requirement close to equilibrium . The problem arises with the fact that these two operations do not commute. Only if operation 2 proceeds 1, an incorrect procedure, then the dissipation shows an extremum at a stationary state. Only the incorrect procedure leads to a principle . If operation 1 proceeds 2 then the dissipation has no extremum at a stationary state, the same result as obtained in Sects. 12.2 and 12.3 without any approximations. 

Although, as we have seen, the principle of minimum entropy production can be generalized to be valid for a fairly large class of thermodynamically nonlinear systems, the most general result still appears to be the differential principle of Glansdorff and Prigogine, which states that... 

The principle of minimum entropy production has been generalized to a continuous system by Mazur . Although his generalization has also been made for the case of electric current, we shall confine ourselves to the systems in which the thermal conduction and the diffusion take place together with chemical reactions. 

The principle of minimum entropy production holds in the macroscopic description in which the entropy is considered to be a function of the diagonal density matrix, as seen from Klein and Meijer s theory. An attempt has been made by Callen to generalize this principle for the cases where the contribution of the off-diagonal elements of the density matrix to the entropy cannot be neglected. 

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