Onsager reciprocity principles

THL.3. 1. Prigogine, Remarque sur le principe de reciprocite d Onsager et le couplage des reactions chimiques (Remarks on Onsager s reciprocity principle and the coupling of chemical reactions). Bull. Cl. Set Acad. Roy. Belg. 32, 30-35 (1946). 

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 simplest case, the reciprocation of two processes 1 and 2 is writ ten in accordance with the Onsager s principle as... 

The Onsager reciprocal relations are not satisfied in open strongly non equilibrium systems. As a result, the assumption on minimization of the entropy production rate is not substantiated. Therefore, the universal criterion of the system that is evolution far from equilibrium should be a generalization of the principle of the minimized entropy production rate in specific terms of nonlinear thermodynamics. 

Equation 3.3.7 expresses the Onsager reciprocal relations (ORR), named after Lars Onsager who first established the principles of irreversible thermodynamics (Onsager, 1931). The ORR have been the subject of many journal papers receiving support as well as criticism, the latter from, in particular, Coleman and Truesdell (1960) and Truesdell (1969). We shall assume the validity of the ORR in the development that follows. 

It is possible for more than two forces to couple. There exists a criterion which allows one to deduce a priori the number of effective couplings. This is Curie s principle of symmetry. The principle states that a macroscopic phenomenon in the system never has more elements of symmetry than the cause that produces it. For example, the chemical affinity (which is a scalar quantity) can never cause a vectorial heat flux and the corresponding coupling coefficient disappears. A coupling is possible only between phenomenon which have the same tensor symmetry. Thus Onsager reciprocity relation is not valid for a situation when the fluxes have different tensorial character. 

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. 

Onsager reciprocity relation is based on (i) principle of microscopic reversibility, (ii) fluctuation theory and (iii) the assumption that decay of fluctuations follows ordinary macroscopic laws. We give below a brief account of its derivation. 

According to Onsager s reciprocity principle, for the steady-state condition of very small volume elements and close to equilibrium, we can write for the inverse-indexed phenomenological coefficients... 

This equation shows that the phenomenological coefficients are related to the reaction rates at equilibrium. The principle of detailed balance or microscopic reversibility is incorporated into /rB = 7 )3 = rf3,eq, and hence the Onsager reciprocal relations are valid. 

We should also mention that the normal solution of the Boltzmann equation discussed here, together with the //-theorem discussed in the previous section, can be used to provide a derivation of the principles of nonequilibrium thermodynamics. For mixtures, one can show that the various diffusion coefficients that occur in the Navier-Stokes equations can be expressed in a form where Onsager reciprocal relations are satisfied. However, both for mixtures and for pure gases the relation between the normal solution and irreversible thermodynamics only holds if one does not go beyond in the -expansion of the distribution function. ... 

ONSAGER RECIPROCAL RELATIONS AND THE SYMMETRY PRINCIPLE in which... 

In this section we shall look at the meaning of linear phenomenological laws in the context of chemical reactions. In a formalism in which the principle of detailed balance or microscopic reversibility is incorporated through the condition that forward rates of every elementary step balance the corresponding reverse rate, the Onsager reciprocity is implicit. No additional relations can be derived for the reaction rates if it is assumed that at equilibrium each elementary step is balanced by its reverse. Therefore, the main task in this section will be to relate the Onsager coefficients Ly and the experimentally measured reaction rates. In our formalism the Onsager reciprocal relations will be automatically valid. 

Being a useful theoretical instrument, the phenomenological approach, however, cannot clarify the molecular mechanisms of chemical processes. The famous Onsager reciprocal relationships were derived in their general form from the principle of the local reversibility of physical processes. For chemical reactions, some useful flux-force relationships can be derived if we know the mechanism of the reactions considered. In this way, in principle, it becomes possible to express the phenomenological Onsager proportionality coefficients through the kinetic constants of elementary steps. 

Unlike other branches of physics, thermodynamics in its standard postulation approach [272] does not provide direct numerical predictions. For example, it does not evaluate the specific heat or compressibility of a system, instead, it predicts that apparently unrelated quantities are equal, such as (1 A"XdQ/dP)T = - (dV/dT)P or that two coupled irreversible processes satisfy the Onsager reciprocity theorem (L 2 L2O under a linear optimization [153]. Recent development in both the many-body and field theories towards the interpretation of phase transitions and the general theory of symmetry can provide another plausible attitude applicable to a new conceptual basis of thermodynamics, in the middle of Seventies Cullen suggested that thermodynamics is the study of those properties of macroscopic matter that follows from the symmetry properties of physical laws, mediated through the statistics of large systems [273], It is an expedient happenstance that a conventional simple systems , often exemplified in elementary thermodynamics, have one prototype of each of the three characteristic classes of thermodynamic coordinates, i.e., (i) coordinates conserved by the continuous space-time symmetries (internal energy, U), (ii) coordinates conserved by other symmetry principles (mole number, N) and (iii) non-conserved (so called broken ) symmetry coordinates (volume, V). 

Some of the variational principles which are to be described in the present article, are very closely connected with Onsager s reciprocity relationsAlthough there have been various methods of derivation of this theorem, we shall follow the traditional method of derivation by Onsager and Casimir. This is based on the consideration of fluctuations in an aged system, and this method is also connected with the derivation of Onsager s principle of least dissipation of energy. ... 

It has been shown by Mazur and Overbeek that the validity of this relation is of a very general character and a direct consequence of Onsager s principle of reciprocity of irreversible phenomena ... 

---