Principle of minimum production

V. PRINCIPLE OF MINIMUM PRODUCTION OF ENTROPY A. Discontinuous Systems... 

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. 

In the previous section, the principle of minimum production of entropy is extended to a continuous system, but this extension is legitimated only when the phenomenological coefficients of Eq. 19 are entirely independent of any of intensive variables. This condition is rather restrictive for the practical purpose. 

This implies that the entropy production is a minimum when parameters E, P, and P assume their stationary state values, and hence that the principle of minimum production of entropy holds in a magnetic resonance experiment. 

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. 

The first question posed in the introduction, Question (3), makes the point that one cannot have a theory for the nonequilibrium state based on the first entropy or its rate of production. It ought to be clear that the steady state, which corresponds to the most likely flux, x(x, i), gives neither the maximum nor the minimum of Eq. (61), the rate of first entropy production. From that equation, the extreme rates of first entropy production occur when x = oo. Theories that invoke the Principle of Minimum Dissipation, [10-12, 32] or the Principle of... 

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],... 

In accordance with the general conservation principle of minimum intervention, the main objective is to conserve the glass, and not to recover transparency, through removal of corrosion products and deposits. Only in exceptional circumstances, therefore, may weathering layers be removed to increase the transparency of the glass or to support its interpretation. In any case, damage to the hydrated layer must be avoided this layer is considered to be the skin of the glass, which protects it from further attack. 

Many transition metals and their oxides are quite active for the oxidation of olefins to CO2. From the standpoint of mechanism these oxidations have much of potential interest. According to the principle of minimum atomic rearrangement in a reaction step, a considerable number of steps must be involved in the conversion of an olefin such as CjHg to CO2 and H2O. We know little of these steps. With most of the C02-forming catalysts, very small amounts of intermediate products are observed, even at low conversions. Furthermore, it has been demonstrated that certain possible intermediates, such as acids, are not rapidly oxidized over many of these catalysts, and therefore paths involving these as intermediates can be eliminated. 

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. 

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