Minimum entropy production theorem

Beyond the domain of validity of the minimum entropy production theorem (i.e., far from equilibrium), a new type of order may arise. The stability of the thermodynamic branch is no longer automatically ensured by the relations (8). Nevertheless it can be shown that even then, with fixed boundary conditions, nonequilibrium systems always obey to the inequality1... 

This is called the minimum entropy production theorem (1, 2). 

The minimum entropy production theorem dictates that, for a system near equilibrium to achieve a steady state, the entropy production must attain the least possible value compatible with the boundary conditions. Near equilibrium, if the steady state is perturbed by a small fluctuation (8), the stability of the steady state is assured if the time derivative of entropy production (P) is less than or equal to zero. This may be expressed mathematically as dPIdt 0. When this condition pertains, the system will develop a mechanism to damp the fluctuation and return to the initial state. The minimum entropy production theorem, however, may be viewed as providing an evolution criterion since it implies that a physical system open to fluxes will evolve until it reaches a steady state which is characterized by a minimal rate of dissipation of energy. Because a system on the thermodynamic branch is governed by the Onsager reciprocity relations and the theorem of minimum entropy production, it cannot evolve. Yet as a system is driven further away from equilibrium, an instability of the thermodynamic branch can occur and new structures can arise through the formation of dissipative structures which requires the constant dissipation of energy. 

Under those conditions P behaves as a Lagrangian in mechanics. Furthermore, as P is a nonnegative function for any positive value of the concentrations X,, by a theorem due to Lyapounov, the asymptotic stability of nonequilibrium steady states is ensured (theorem of minimum entropy production.1-23 These steady states are thus characterized by a minimum level of the dissipation in the linear domain of nonequilibrium thermodynamics the systems tend to states approaching equilibrium as much as their constraints permit. Although entropy may be lower than at equilibrium, the equilibrium type of order still prevails. The steady states belong to what has been called the thermodynamic branch, as it contains the equilibrium state as a particular case. 

Horiuti Nakamura [75] have considered the possibility that there may be more than one set of stoicheiometric numbers leading to an overall reaction between the species s s. They find that the number of such reaction routes is the row nullity of the matrix [a ]. Nakamura has also considered the affinity of the overall reaction near equilibrium [79] and with Yamazaki [20] has related this to the theorem of minimum entropy production. 

It should also be noted that the Prigogine theorem on the minimum entropy production is applicable to the circuit as a whole and for its individual branches (open subsystems). Actually, the maximum amount of entropy is formed in the environment owing to heat transfer to it from the hydraulic circuit. In the circuit itself the energy imparted to the fluid is entirely spent on its motion along the branches, i.e., on performance of effective work, and the entropy production at given conditions of interaction with the environment takes its minimal value equal to zero. The minimality of AS/ was shown in (Gorban et al., 2001, 2006). 

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. 

Evolution toward steady-state. Glansdorff-Prigogine general evolution criterion. The GlansdorfT - Prigogine theorem of the minimum entropy production... 

Figure 17.1 A simple thermal gradient maintained by a constant flow of heat. In the stationary state, the entropy current Js,out — diS/dt + The stationary state can be obtained either as a solution of the Fourier equation for heat conduction or by using the theorem of minimum entropy production. Both lead to a temperature T(x) that is a linear function of the position x...

Demonstrate the theorem of minimum entropy production for an arbitrary number of constrained and unconstrained tliermodynamic forces. 

The analysis begins with a basic theorem For this system, minimum entropy production implies a constant rate of entropy production on each branch of the cycle. In the limit of a slow process, this rate is the same for all branches of the cycle. A second theorem follows Suppose the total cycle time is t, that the entropy produced on the ith branch of the cycle is <7, and that k is the maximum of the heat conductances on all branches. Then a lower bound for the entropy production, per cycle, is... 

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. 

Before concluding this section, we should mention the boundary conditions given by Eqs. 146 and 147. These conditions correspond clearly to Eqs. 124 and 126, respectively, in the case of a discontinuous system. Let us now consider the case where the values of the intensive variables are left free at a certain portion of the boundary surface Q. Then let us determine the boundary condition so as to get the minimum entropy production compatible with the boundary condition at the remaining portion of the boundary. The general theorem of the calculus of variation states that the entropy production becomes a minimum when the flows across vanish, i.e., J n = n = 0 at Qi- This imphes... 

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. 

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. 

Utilizing increasing dissipation or specific entropy production as the parameter for following evolution, this theory provides not only for evolution but also for the acceleration of evolution noted by de Duve (1974). In this thermodynamic scheme, the creation of new order leads to an increase in entropy production, while maintenance steps appear to accord with the theorem of minimum entropy production. 

It might be thought that some principle concerning entropy production, such as the minimum principle invoked in irreversible thermodynamics, might distinguish between equivalent sets of reactions, but the next theorem shows that this is not so. 

A hierarchical classification of processes in respect to their characteristic times underlies, in fact, the theorem on the minimum of the rate of entropy production as well. Let us consider two conjugate processes described by the Onsager equations ... 

Generally, a chemical reaction has a nonlinear relation between the rate and the driving force AIT. Chemical reactors are often designed to operate at the maximum rate of reaction. An alternative is a reactor operated with minimum useful work lost. The lost work per unit time in a chemical reactor is given by the Gouy—Stodola theorem, and is obtained by integrating the entropy production rate over the reactor volume ... 

Also, this contains the entropy production, and was introduced first by Onsager and Machlup in their work about non-equilibrium fluctuation theory [i ]. With this we can formulate the minimum theorem of the generalized Onsager constitutive theory the OM-function is the non-negative function of the fluxes, forces and intensive parameters. It only becomes zero, which is its minimum, when the material equations of the generalized Onsager constitutive theory are satisfied. The OM-function has crucial importance, because it contains all the important constitutive properties of the linear and generalized constitutive theories these follow from the necessary conditions of the minimum of OM-function ... 

Due to the Lagrangian of the functional (99) is the sum of the dissipation potentials, which is equal to the entropy production in case of every real steady-state physical processes, this extremum theorem involves the minimum principle of global entropy production (MPGEP). The physical meaning of MPGEP needs a clarification. Consider the variations of the fluxes and of the intensive parameters as fluctuations of the system around their stationer state values. When these fluctuations are small, the fluctuation of the global entropy production of the system is equal to its first approximations and it has a form... 

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. 

This is the theorem of minimal entropy production (Prigogine (1947), Desoer ). With respect to the variation of AX the entropy production is at a minimum. We can see that this is indeed a minimum by reducing the forces AXjt ( = 1,..., m) to zero, i.e. we approach the equilibrium state where diAS/dt = 0. We can now reverse... 

In accord with the theorem of minimal entropy production we see that the entropy production (following a perturbation) continuously decreases reaching a minimum in the steady state. 

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