Carathéodory's principle

Remarks

The principle is a form of the second law due to Constantin Carathéodory. It is used with a theorem of Carathéodory on Pfaffian forms. Let $\omega$ be a smooth differential form of degree $1$ on an open set of $\R^n$. If every neighbourhood of every point contains points that cannot be joined to it by curves along which $\omega = 0$, then locally $\omega = \lambda \, d\sigma$ for functions $\lambda$ and $\sigma$. Applied to the heat form $\delta Q_{\mathrm{rev}}$ of a simple system, this gives $\delta Q_{\mathrm{rev}} = T \, dS$, and so the existence of temperature and entropy. The states with the same entropy form a hypersurface in the state space, and quasistatic adiabatic processes stay on it. States of lower entropy near a given state are then not adiabatically accessible from it.

Going deeper

For a student

Carathéodory's approach avoids heat engines and cycles. It replaces them by a geometric statement about the space of states. The mathematics behind it is the integrability of Pfaffian forms, which is related to the Frobenius theorem.