Evolution criterion

TNC. 12. P. Glansdorff and I. Prigogine, On a general evolution criterion in macroscopic physics, Physica 30, 351-374 (1964). 

This is the so-called evolution criterion. The equality sign applies only to steady states. Two cases may then arise ... 

In order to extract the maximal energy out of the available foodstuff oxidative phosphorylation should operate at the state of optimal efficiency in vivo. Since a zero as well as an infinite load conductance both lead to a zero efficiency state, obviously there must be a finite value of the load conductance permitting the operation of the energy converter at optimal efficiency. For linear thermodynamic systems like the one given in equations (1) and (2) the theorem of minimal entropy production at steady state constitutes a general evolution criterion as well as a stability criterion.3 Therefore, the value of the load conductance permitting optimal efficiency of oxidative phosphorylation can be calculated by minimizing the entropy production of the system (oxidative phosphorylation with an attached load)... 

An evolution criterion can be obtained from the rate of change of volumetric entropy production P = fXJXdV > 0 as follows ... 

Let us say that the system has only one independent variable Y for example, the concentration (thermodynamic rush) of some intermediate. In this case, the evolution criterion d P < 0 can be expressed in the form of total differential (3.4)... 

While inequality 35 is the evolution criterion for systems near equilibrium, the reverse inequality, together with Equation 37, can be regarded as a generalized stability condition ... 

Frequently the stability criterion of Glandorff and Prigogine is expressed as an evolution criterion. By interpreting the variations and 6c. in (7.24) as variations during a real time evolution of the system we can formulate the assumption of a positive definite capacitance matrix equivalently as... 

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... 

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

The Glansdorff-Prigogine general evolution criterion, in case of time independent boundary conditions, which are valid during the evolution of the thermodynamic steady state, can be formulated as... 

According to the Glansdorff-Prigogine general evolutions criterion (192), from the above equation we get the necessary conditions of the minimum of the entropy production at the thermodynamics steady state... 

The Glansdorff-Prigogine general evolution criterion involves the minimum of global entropy production in such a constitutive theory where the potentials are homogeneous Euler s functions. We show below the strictly convex property of dissipation potentials guarantee the minimum, and the function... 

We see that we now have two inequalities, P>0 and d P<0. The second inequality is an important evolution criterion. Let us indicate briefly two consequences. If only one concentration, say X, is involved in the evolution, JpP = v X) dA/dX)dX = dW. The variable W, thus defined, is then a kinetic potential . But this is rather an exceptional case. The interesting consequence is that time-independent constraints may lead to states which are not stationary, states that oscillate in time. We shall see examples of such systems in Chapter 19, but let us consider here a simple example of a far-from-equilibrium chemical system where the dependence of velocities on affinities are antisymmetric i.e., vi = IA2,V2 = —lAi (Onsager s relations are not valid for systems far from equilibrium). The derivative dpP/dt in this case becomes ... 

I had a similar experience in 1954 when I spoke, probably for the first time, about the possibility of oscillating reactions. At that time, I had published a short paper with Radu Balescu on the possibility that far from equilibrium we could have chemical oscillations, in contrast with what happens near equilibrium. This work was connected with involvment in the so-called "universal evolution criterion", derived with Paul Glansdorff. My lecture of 1954 had no more success than the one of 1946. The chemists were very skeptical about the possibility of chemical oscillations and in addition, said an outstanding chemist, even if it would be possible, what should be the interest The interest of chemical kinetics was at that time the discovery of well-defined mechanisms, and specially of potential energy surfaces, which one could then connect with quantum mechanical calculations. The appearance of chemical oscillations or other exotic phenomena seemed to him to be of no interest in the direction in which chemical kinetics was traditionally engaged. All this has changed, but to some extent the situation of chemistry in respect to physics remains under the shadow of this distrust of time. 

Remark There exists a general relation, the so called evolution criterion, valid also beyond the linear regime, dx < 0. If we apply this to Eq. (7.73) we conclude that... 

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. 

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