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At a recently held conference in Paris on the 200th anniversary of the publication of Sadi Carnot's treatise "Réflexion sur la puissance motrice du feu" I was asked to present and assess Constantin Carathéodory's approach to thermodynamics, first published in 1909 [1].
I will give an informal version of this talk focusing on the logical structure of Carathéodory's argument divided into his theorem, his axiom, and his derivation of standard equilibrium thermodynamics, from the outcome of his theorem.
Carathéodory's theorem is a mathematical result on a condition for integrability of 1-forms. I will not try to present the proof, as a pedagogical and accessible presentation with a survey of the earlier mathematical literature (mostly 19th century) appeared recently [2]. Carathéodory's axiom is simply the statement that the conditions of his theorem are satisfied in macroscopic thermodynamics, where the 1-forms of interest are heat and work. I will on the other hand outline Carathéodory's derivation of thermodynamics where he does not need to evoke the Carnot cycle.
One reason why Carathéodory's approach did not catch on in physics, with some exceptions, is the diffuculty of his theorem, particularly when it was first presented. Another reason is that Landsberg and Van Kampen showed in the 1960ies that Carathéodory's axiom is a simple consequence of Kelvin's statement of the Second Law. I will present Van Kampen's version of the argument [3]. I will end by some speculation on settings where Carathéodory's approach perhaps nevertheless could find a niche.
[1] Constantin Carathéodory, “Untersuchungen über die Grundlagen der Thermodynamik,” [Examination of the foundations of thermodynamics], Math. Ann. 67 355-386 (1909)
[2] Pedro F. Da Silva Júnior, “On the integrability of Pfaffian forms on RN“ [arXiv:2110.08337]
[3] U.M. Titulaer and N.G. Van Kampen, “On the deduction of Caratheodory’s axiom from Kelvin’s principle”, Physica 3 1 1029-1032 (1965)