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I am currently studying non-equilibrium thermodynamics in the linear regime, and I am wondering why we define the constitutive equations (i.e: the relationship between flux and force) in a basis that is not the eigenbasis of the phenomenologic coefficient matrix. That is to say, if we have two fluxes such that:

$$\begin{cases}J_1=L_{11}X_1+L_{12}X_2 \\ J_2=L_{21}X_1+L_{22}X_2\end{cases}$$

If the system satisfies Onsager's principle: $L_{12}=L_{21}$, which is often the case, at least in my exercises, then I would find it more natural to work in the basis of eigenvectors of the matrix $(L_{ij})$, which I would know to be symmetric and therefore have 2 eigenvalues, and their corresponding eigenvectors would be orthogonal. These new forces and new fluxes would then be independent from one another, just like one decouples two oscillators into their normal modes without giving it much thought, because it makes the problem simpler.

If we have $\lambda_1,\lambda_2$ eigenvalues of $L$, then there exist two thermodynamic forces such that: $L\tilde X_1=\lambda_1\tilde X_1$, $L\tilde X_2=\lambda_2\tilde X_2$, so the constitutive equations are now:

$$\begin{cases}\tilde J_1=\lambda_1\tilde X_1 \\ \tilde J_2=\lambda_2\tilde X_2 \end{cases}$$

If we were interested in knowing the original fluxes, we could then undo the change of variables to work again in the original basis and that would be all. Why don't we do this and prefer to work with the coupled equations?

Lagrangiano
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1 Answers1

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You can find essentially this analysis in [1]. That one may write a pair of simultaneous transport equations such as $$\begin{cases}\tilde J_1=\lambda_1\tilde X_1 \\ \tilde J_2=\lambda_2\tilde X_2 \end{cases} \tag{1}$$ is called Kelvin's (Thomson's) principle of separability and Kelvin used this principle to derive his famous equations on thermolectricity. It is, in fact, a nontrivial assumption and its validity was much debated for over seventy years when Onsager offered his principle of microscopic reversibility that quickly replaced Kelvin's for the former supposedly being a more fundamental principle, ie., one based in statistical mechanics. Despite the almost exclusive reference to microscopic reversibility in modern literature there has always been a minority supporting Kelvin's approach and I had a similar question in this forum.

Here is a quote from [2]

Although Thomson's approach has been criticized in the literature as being incorrect, the only valid objection, as pointed out before, is the arbitrariness involved in his treatment, namely, the thermoelectric phenomenon being separated into a heat flux by itself and an electron flux which carries energy. The separability principle is the philosophical part of his approach; that is, any complicated process can be separated into individual independent processes. The actual separation, as shown previously can be made very general and has no arbitrariness in it. Specifically, the thermoelectric phenomenon can be separated into a heat flux which carries electrons (phonon drag) and an electron flux which carries energy.

[1] K. Pitzer:"IRREVERSIBLE THERMODYNAMICS," Pure and Applied Chemistry v.2, 1-2 https://doi.org/10.1351/pac196102010207

[2] J.C.M. Li: "Thermodynamics for Nonequilibrium Systems. The Principle of Macroscopic Separability and the Thermokinetic Potential," JOURNAL OF APPLIED PHYSICS, v.33, #2 FEBRUARY, 1962, https://doi.org/10.1063/1.1702476

hyportnex
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