Minimum entropy production principle

Example 3.19 Minimum energy dissipation in heat conduction Use the minimum entropy production principle to derive the relation for nonstationary heat conduction in an isotropic solid rod. 

The above equation shows that the steady state is less dissipative for a specified duty in a finite time. This conclusion is in line with the minimum entropy production principle of Prigogine. 

Jaynes ET The minimum entropy production principle, Annu Rev Phys Chem 31 579-601, 1980. 

VI. Statistical Theory of the Minimum Entropy Production Principle. 307... 

VI. STATISTICAL THEORY OF THE MINIMUM ENTROPY PRODUCTION PRINCIPLE... 

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... 

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 ... 

Another consequence of Eq. (4.91) is that if we arrange the n subsystems in time instead of in space, then the collection of subsystems constitutes the reaction path of a batch reactor where Vk is the volume of subsystem k. For a specified conversion and time, we should minimize the sum of Jk(AGk/T)Vk. This minimization leads to results similar to Eq. (4.91), and supports the principle of equipartition of forces. Hence, for a given total conversion and reaction time, minimum entropy production results when the driving force A GIT is equal in all n time intervals. Similarly, the conversion is maximum for a given entropy production and reaction time when the driving forces are uniform. 

The first summation term on the right is the minimum entropy production corresponding to the stationary state. The second sum on the right is zero, according to the Onsager reciprocal relations and the Prigogine principle. Therefore, we have... 

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. 

The extended principle of Le Chdtelier and Prigogine s theorem thus leads to the conclusion that if k out of n forces X1( Xj . X are maintained at fixed values by means of external constraints the system will ultimately reach a state of minimum entropy production that is truly stationary this will be termed a steady-state condition of order k. 

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 book is divided into four parts. Part One, which consists of six chapters, deals with basic principles and concepts of non-equilibrium thermodynamics along with discussion of experimental studies related to test and limitation of formalism. Chapter 2 deals with theoretical foundations involving theoretical estimation of entropy production for open system, identification of fluxes and forces and development of steady-state relations using Onsager reciprocity relation. Steady state in the linear range is characterized by minimum entropy production. Under these circumstances, fluctuations regress exactly as in thermodynamics equilibrium. 

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. 

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