Onsager's Theorem from Thermodynamics

The Onsager theorem pertains to symmetries inherent in two-dimensional current flow, through either a conducting or a semiconducting medium in the presence of an externally applied magnetic field oriented perpendicular to the current-carrying plane. Although the theorem can be derived on a statistical basis, as done originally by Onsager, in many courses dealing with thermoelectricity, it is necessary to state the theorem without proof, rather than take the time to develop the background properly. Therefore, it is of considerable pedagogical interest to have a derivation of the theorem based upon thermodynamics only. To that end, the following discussion has been found useful.