9: The Carathéodory Principle
- Page ID
- 32047
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)
The formulation of the second law from thermodynamics used the concept of heat engines, at least indirectly. But the law is very general and one could ask whether there is another formulation which does not invoke heat engines but leads to the notion of absolute temperature and the principle that entropy cannot spontaneously decrease. Such a version of the second law is obtained in an axiomatization of thermodynamics due to C. Carathéodory.
- 9.1: Mathematical Preliminaries
- We will start with a theorem on differential forms which is needed to formulate Carathéodory’s version of the second law.
- 9.2: Carathéodory Statement of the Second Law
- In the neighborhood of any equilibrium state of a physical system with any number of thermodynamic coordinates, there exist states which are inaccessible by adiabatic processes. The adiabatic processes can be quite general, not necessarily quasi-static. This leads immediately to the notion of absolute temperature and entropy. This has been discussed in a concise and elegant manner in Chandrasekhar’s book on stellar structure. We briefly repeat his argument for completeness.