Onsager Reciprocal Relations and the Symmetry Principle

Phenomenological laws and the cross effects between the flows were independently studied and, until the formalism presented here was developed in the 1930s, there was no unified theory of all the cross effects. Relating the entropy production to the phenomenological laws is the first step in developing a unified theory. For conditions under which the linear phenomenological laws (16.1.2) are valid, entropy production (16.1.1) takes the quadratic form 

Here F. can be positive or negative. A matrix Ljk that satisfies the condition (16.1.8) is said to be positive definite. The properties of positiye definite matrices are well characterized. For example, a two-dimensional matrix Ly is positive definite only when the following conditions are satisfied (exc. 16.1)  

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. 

That reciprocal relations noticed by William Thomson (Lord Kelvin) and others even during the last century. The early explanations of the reciprocal relations were based on thermodynamic reasoning that was not on a firm footing. For this reason. 

Lars Onsager (1903-1976) (Reproduced, by permission, from the Emilio Segre Visual Archives of the American Institute of Physics) 

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

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