Principle of Minimum Entropy

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 ... 

In the linear nonequilibrium thermodynamics theory, the stability of stationary states is associated with Prigogine s principle of minimum entropy production. Prigogine s principle is restricted to stationary states close to global thermodynamic equilibrium where the entropy production serves as a Lyapunov function. The principle is not applicable to the stability of continuous reaction systems involving stable and unstable steady states far from global equilibrium. 

The stability of transport and rate systems is studied either by nonequilibrium thermodynamics or by conventional rate theory. In the latter, the analysis is based on Poincare s variational equations and Lyapunov functions. We may investigate the stability of a steady state by analyzing the response of a reaction system to small disturbances around the stationary state variables. The disturbed quantities are replaced by linear combinations of their undisturbed stationary values. In nonequilibrium thermodynamics theory, the stability of stationary states is associated with Progogine s principle of minimum entropy production. Stable states are characterized by the lowest value of the entropy production in irreversible processes. The applicability of Prigogine s principle of minimum entropy production is restricted to stationary states close to global thermodynamic equilibrium. It is not applicable to the stability of continuous reaction systems involving stable and unstable steady states far from global equilibrium. The steady-state deviation of entropy production serves as a Lyapunov function. 

This law may also provide a basis for process optimization analysis of engineering devices involving simultaneous transport phenomena and chemical reactions by use of the principle of minimum entropy production. 

Prigogines principle of minimum entropy production, proved ... 

The entropy production s, being a positive definite, equation (5) gives the condition of a minimum. It is a mathematical form of Prigogine s principle of minimum entropy production according to which at the steady state, all the flows corresponding to the unrestricted forces vanish. 

Recently, the criterion of nonbreaking film flow was thermodynamically substantiated with the aid of Prigogine s principle of minimum entropy production including the case of a double film flow [88],... 

The principle of minimum entropy is restricted to linear phenomenological equations obeying the Onsager relations. In equilibrium thermodynamics, systems tend to maximize the entropy or minimize the free energy. 

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... 

Richardson IW (1969) On the Principle of Minimum Entropy Production. Biophysical... 

For the case of coupled forces and flows, the principle of minimum entropy production can be demonstrated as follows. Consider a system with two forces and flows that are coupled. For notational convenience, we shall represent the total entropy production per unit time by P. Therefore,... 

We shall now show how this condition may also be obtained using the principle of minimum entropy production. The entropy production per unit volume for this system (which we assume is homogeneous) is... 

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. 

Examples 2, 4 and 5 illustrate a common feature implied by the principle of minimum entropy production (Fig. 17.4) in a series of coupled systems, entropy production is extremized when the flows are equal. In a chemical reaction it was the velocity Vk for heat conduction it was the heat flow Jq, for an electric circuit it is the electric current Ik-... 

This shows, once again, that a perturbation in the total entropy production P from its stationary-state value monotonically decreases to its stationary state value, in accordance with the principle of minimum entropy production. A simple proof of (18.2.3) is given in Appendix 18.1. 

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. 

It is useful to think of this principle of minimum entropy production in a more geometrical way. The equation... 

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, 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... 

In the above-mentioned variational principle the intensive variables are kept constant as in the case of Onsager s least dissipation of energy and contrary to the case of the principle of minimum entropy production, which will be dealt with in the latter part of this chapter. Further, it is easily seen that the variational principle Eq. 84 is closely connected with the principle... 

We can show that the above conclusion is true in the case of chemical reactions. These results give full importance to the physical implications of the principle of minimum entropy production. As is well known, the entropy is a maximum at the equilibrium state, and it increases monotonically with time as demonstrated by Boltzmann s and Gibbs / -theorems. The present theorem states that the entropy production is not only a minimum at the stationary state, but also that it decreases monotonically until it attains the stationary value. 

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. 

This implies that Eq. 145 and hence the stationary state are obtained from the variational principle Eq. 152 as shown by Mazur. In other words, the stationary state is characterized as the state of minimum entropy production as in the case of a discontinuous S5 tem. Thus we have demonstrated the principle of minimum entropy production is stiU valid for continuous systems. The extremum is a minimum because of the positivedefiniteness of the entropy production. 

At first sight, the principle of minimum entropy production, especially in the case of a continuous system, seems to have some intimate connection with the principle of least dissipation of energy. However, these two principles are of rather different characters. In the case of least dissipation of energy the flows are varied by keeping the intensive variables at every point constant, while the forces and flows are varied at the same time in accordance with the phenomenological relations under the prescribed boundary values in the case of the principle of minimum production of entropy. 

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... 

This implies that the time rate of entropy production decreases monotonically with time until it attains to the stationary state. Then we can conclude that the stationary state is the state to minimize the entropy production and thus the principle of minimum entropy production has been demonstrated by the statistical method. This is a certainly a extension of Pauli s H-theorem to nonequilibrium stationary state. Furthermore, we can directly see from Eq. 183 that the entropy production does decrease unless aU the occupation probabilities are time-independent. It should be noted that this derivation of the principle does not explicitly depend on the thermod3mamic relations such as the reciprocity and the Gibbs relations. 

In the present section we wish to show that the principle of minimum entropy production is applicable also to the stationary state in a magnetic resonance experiment, in which a collection of spin magnetic moments are subject to a circularly polarized magnetic field perpendicular to a constant magnetic field. Application to this type of phenomenon is done by a reinterpretation and modification of Klein and Meijer s method. This has been done by Klein and he has shown that Overhauser processes of producing nuclear polarization satisfy the principle of minimum... 

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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