Kinetic Interpretation of the Kelvin Relations

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Abstract
The derivation of the Kelvin relations for the reversible heat development and the thermo-electric force in crystals is shown to depend only on the assumption of the principle of detailed balancing. Previous treatments of this subject have been less general. The work of Nordheim is valid only for isotropic materials, and that of Ehrenfest and Rutgers, though applicable to crystals, depends on the assumption of elastic collisions between the electrons and the atoms of the metal. In both treatments the electrons are assumed to be completely free and to be described by an energy function which is spherically symmetrical. All of these restricting assumptions are removed. The solution of the integral equation describing the distribution function of the electrons is obtained by a general method which depends only on the assumption of detailed balancing. This principle gives the integral equation an essentially symmetrical kernel, as a consequence of which a formal solution may be obtained. One may use this solution to derive the electric and heat current equations and to prove certain reciprocal relations from which the Kelvin equations follow. The proof of the existence of these relations, involving as it does only the assumption of detailed balancing, may be considered as a substantiation for the case of conduction in metals of certain very general considerations of Onsager.

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