Reciprocity relations, onsager

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. 

We consider the fluctuations in two variables aft) and aj t- -r) where r is the time interval. The fluctuation in the average value of the product of the two variables aft)-aj t- -r) and [a (f)aj(t + t)] would differ only by temporal order i.e. by the substitution of t -t. Now using this concept of microscopic reversibility, we obtain following relation 

Equation (2.22) involves the concept of microscopic reversibility. If we assume that decay of fluctuations follows ordinary macroscopic laws, we can write Eq. (2.18) in a form where Jf = 8a /8r and is the force. Substituting Eq. (2.18) into Eq. (2.22) we obtain 

In principle, the results based on fluctuation theory and principle of microscopic reversibility would only be true for systems close to equilibrium. This will also be true for the assumption of linear relation between fluxes and forces. One has to understand also the serious limitation of phenomenological linear laws. The condition is that the magnitude of t should satisfy the inequality, 

The adiabatically insulated system and in the absence of external magnetic field can be characterized by a number of independent parameters. These parameters can be of two types  

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Inserting the definition of G gives the celebrated Onsager reciprocal relations [4, 5]... 

These are the famous Onsager reciprocal relations. Thus there is symmetiy in the ability of a potential to create a flux/, and of the ability of a potential to create a flux The reciprocal relations arc experimentally verifiable connections between effects which superficially might appear to be independent. 

For the case of independent forces and fluxes, the Onsager reciprocal relation holds ... 

Onsager Reciprocity Relations and Generalizations to Nonlinear Response... 

Onsager reciprocity relations as well as the Green-Kubo and Einstein formulas for these coefficients ... 

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). 

The Onsager reciprocity relation, when applied to the present context, predicts that the cross coefficients a2 and a, which determine the rate of flow ofliquid due to the applied electric field and the current passing due to a hydrostatic pressure difference, respectively, are equal, Le.,... 

But v/X is the electro-osmotic velocity of the fluid per unit of electric field, i.e., the electro-osmotic mobility. It is interesting to note that both the electro-osmotic mobility v/X = a2 and the streaming-current coefficient j/AP = a3 have beat proved to be equal to each other and to ltfZ/4nr. This only means that the Onsager reciprocity relation has been shown to be consistent with a simple model of some electrokinetic phenomena. 

Equations (13.44) are the celebrated Onsager reciprocity relations [L. Onsager. Reciprocal relations in irreversible processes. Phys. Rev. 37, 405-26 38, 2265-79 (1931)] that relate off-diagonal Fj 7 and Ft Jj coupling and serve as a cornerstone of modem transport theory. They are derived here without reference to statistical mechanical or fluctuation assumptions. 

These symmetry relations are called the Onsager reciprocal relations. Their meaning is best illustrated by reference to the following problem. Suppose the interaction Hamiltonian is... 

The coefficients Ly are called the coupling coefficients and are assumed to obey the so-called Onsager reciprocity relations Ly = Lji. 

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). 

L. Onsager. Reciprocal relations in irreversible processes. I. Phys. Rev.,... 

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