Onsager reciprocity relation nonequilibrium thermodynamics

What has been done so far is to take experimental laws and express them in the form of phenomenological equations, i.e., Eqs. (6.300) and (6.301). Just as the phenomenological equations describing the equilibrium properties of material systems constitute the subject matter of equilibrium thermodynamics, the above phenomenological equations describing the flow properties fall within the purview of nonequilibrium thermodynamics. In this latter subject, the Onsager reciprocity relation occupies a fundamental place (see Section 4.5.7). 

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). 

Far from thermodynamic equiHbrium we find nonfinear interdependence of thermodynamic fluxes and forces. In this case, the Onsager reciprocal relations are generally not satisfied, and the formafism developed in Chapter 2 is not fuUy applicable for analysis of the state of open systems. Analysis of systems that are far from thermodynamic equilibrium is the subject of nonlinear nonequilibrium thermodynamics. 

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. ... 

In general, the diagonal elements of a positive definite matrix must be positive. In addition, a necessary and sufficient condition for a matrix Lg to be positive definite is that its determinant and all the determinants of lower dimension obtained by deleting one or more rows and columns must be positive. Thus, according to the Second Law, the proper coefficients L k should be positive the cross coefficients, (i 7 k), can have either sign. Furthermore, as we shall see in the next section, the elements Ljk also obey the Onsager reciprocal relations Ljk = Lkj. The positivity of entropy production and the Onsager relations form the foundation for linear nonequilibrium thermodynamics. 

Thus nonequilibrium thermodynamics gives a unified theory of irreversible processes. Onsager reciprocal relations are general, valid for all systems in which linear phenomenological laws apply. 

Perhaps the best starting point in a review of the nonequilibrium field, and certainly the work that most directly influenced the present theory, is Onsager s celebrated 1931 paper on the reciprocal relations [10]. This showed that the symmetry of the linear hydrodynamic transport matrix was a consequence of the time reversibility of Hamilton s equations of motion. This is an early example of the overlap between macroscopic thermodynamics and microscopic statistical mechanics. The consequences of time reversibility play an essential role in the present nonequilibrium theory, and in various fluctuation and work theorems to be discussed shortly. 

The coupling of two different electrokinetic ratios (Estr/p and V/I) through Equation (65) is an illustration of a very general law of reciprocity due to L. Onsager (Nobel Prize, 1968). The general theory of the Onsager relations, of which Equation (65) is an example, is an important topic in nonequilibrium thermodynamics. 

It is true that in his own work De Donder did not pursue the consequences of nonequilibrium very far. We have to wait until the basic discovery of Onsager s reciprocity relations in 1931 and till the work of Eckart, Meixner, and many others in the 1940s and the 1950s to see thermodynamics of nonequilibrium processes take shape and be integrated into common knowledge. 

Fortunately, several simplifications can be made (Nye, 1957). Transport phenomena, for example, are processes whereby systems transition from a state of nonequilibrium to a state of equilibrium. Thus, they fall within the realm of irreversible or nonequilibrium thermodynamics. Onsager s theorem, which is central to nonequilibrium thermodynamics, dictates that as a consequence of time-reversible symmetry, the off-diagonal elements of a transport property tensor are symmetrical (i.e., xy = X/,-). This is known as a reciprocal relation. The Norwegian physical chemist Lars Onsager (1903-1976) was awarded the 1968 Nobel Prize in Chemistry for reciprocal relations. Thus, the tensor above can be rewritten as... 

Reciprocal relations have been the first results in the thermodynamics of irreversible processes to indicate that this was not some ill-defined no-man s-land but a worthwhile subject of study whose fertility could be compared with that of equilibrium thermodynamics. Equilibrium thermodynamics was an achievement of the nineteenth century, nonequilibrium thermodynamics was developed in the twentieth century, and Onsager s relations mark a crucial point in the shift of interest away from equilibrium toward nonequilibrium. 

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