Prigogine theorem

We can specially show that the main principles of nonequilibrium thermodynamics (the Onsager relations, the Prigogine theorem, symmetry principle) and other theories of motion (for example, theory of dynamic systems, synergetics, thermodynamic analysis of chemical kinetics) are observed in the MEIS-based equilibrium modeling. In order to do that, we will derive these statements from the principles of equilibrium thermodynamics. 

First of all relying directly on the second law we will try to give the interpretation of the Prigogine theorem. Taking into account that the traditional variables of equilibrium thermodynamics are the parameters of state and, wishing to reveal the formalized relations between both thermodynamics, let us consider two situations sequentially (1) when some parameters of interaction that hinder the attainment of final equilibrium between the open subsystem and other parts of the isolated system that contains this subsystem are set (2) when flows are taken constant for the flow exchange between the open subsystem and the environment. It is obvious that both situations can be reduced to the case of fixing individual forces which is normally considered in the nonequilibrium thermodynamics. 

Probably the presented equilibrium interpretation of the Prigogine theorem cannot be considered as its strict or general proof. At the same time this interpretation reveals the possibilities to automatically observe the principle of the least entropy production at equilibrium modeling of a wider spectrum of physicochemical processes. 

From a satisfactory, to a certain extent, explanation based on the second law of the Prigogine theorem we can pass to an absolutely macroscopic explanation of the Onsager reciprocal relations by changing the order of proofs accepted in the nonequilibrium thermodynamics (in the nonequilibrium thermodynamics the Prigogine theorem is derived from the Onsager relations). 

Hence, the extremum L(x) is the point of minimum. Thus, the problem of entropy maximization is transformed into the problem of heat minimization and the Kirchhoff and Prigogine theorems result from the extension of the second law to the passive isothermal circuits. The graphical interpretation of problem (21) is given in Figure 3b. 

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 capabilities of MEIS and the models of kinetics and nonequilibrium thermodynamics were compared based on the theoretical analysis and concrete examples. The main MEIS advantage was shown to consist in simplicity of initial assumptions on the equilibrium of modeled processes, their possible description by using the autonomous differential equations and the monotonicity of characteristic thermodynamic functions. Simplicity of the assumptions and universality of the applied principles of equilibrium and extremality lead to the lack of need in special formalized descriptions that automatically satisfy the Gibbs phase rule, the Prigogine theorem, the Curie principle, and some other factors comparative simplicity of the applied mathematical apparatus (differential equations are replaced by algebraic and transcendent ones) and easiness of initial information preparation possibility of sufficiently complete consideration of specific features of the modeled phenomena. 

The assumption of linear steady-state and, hence, the application of linear thermodynamics methods allows the application of the Prigogine theorem, then one obtains the Boltzmann-like distribution of subsystems (micropores with their walls) in energy ... 

In such system the rate of increase of entropy Prigogine theorem. Therefore, the entropy of the system is maximum otherwise, an eventual rising of entropy would cause new fluxes increasing system entropy So is maximum, and the Prigogine theorem is equivalent to the principle of maximum of entropy for the system. 

This conclusion is a principal statement of the I. Prigogine theorem (1947, the Nobel Prize winner in 1977). It also is essential in view of the positively determined Rayleigh Onsager function that the spontaneous evolution of the system to its stationary state can be accompanied by only a monotonous decrease in P and, as a result, in that is,... 

It follows from the Prigogine theorem that in cases where the system exists near its thermodynamic equilibrium, any deviation from the system stationary state due to a disturbance of some internal parameters causes an increase in the rate of entropy production. Simultaneously, the spontaneous evolution of the system will make the entropy production rate decreasing again to its minimal value. Hence, the stationary state of an open system nearly its equilibrium is stable. It is obvious here that the stability condi tion of the stationary state is inequality 8P > 0 at the appearance of any disturbance (fluctuation) of those internal parameters whose values are determined by the condition of the system stationarity. 

Why is the Prigogine theorem on the rate of entropy production sig nificant to the field of chemistry, and what are the conditions of its applicability ... 

Express the relationship between the chemical potentials and con centrations of the reaction intermediates Aj in the stationary mode of the process. Write the expression for the rate of entropy production. Formulate the Prigogine theorem on the rate of entropy production in the stationary state for the case of the given system. To what extent is this theorem applicable for the given system at the temperature 1200 K if the affinity of stepwise reaction Rj + R2 <— P equals 2 kJ/mol 50 kJ/mol ... 

When a reactive system is far from its thermodynamic equilibrium, corol laries of the Prigogine theorem, which were derived for the case of the linear nonequilibrium thermodynamics, cannot be applied to analysis directly. Nevertheless, tools of thermodynamics of nonequilibrium pro cesses allow the deduction of some important conclusions on properties of the system, even though strongly nonequilibrium, including in some cases on the stability of stationary states of complex stepwise processes. For several particular cases, theorems similar to the Prigogine theorem can be proved, too. 

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

---