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From: "Ellis D. Cooper" <xtalv1@netropolis.net>
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Subject: Categories in thermodynamics
Date: Fri, 16 Nov 2007 11:17:56 -0500
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I would like to know whether a category theoretic "rationalization" of the
mathematical theory of thermodynamics published by Elliot H. Lieb
and Jakob Yngvason has been published or at least undertaken. Their axioms
add structure to a preorder:

\begin{description}
\item [(A1) Reflexivity.] $X \stackrel {A}{\sim} X$.
\item [(A2) Transitivity.] $X \prec Y$ and $Y \prec Z$ imply $X \prec Z$.
\item [(A3) Consistency.] $X \prec X'$ and $Y \prec Y'$ imply $(X,Y)
\prec (X',Y')$.
\item [(A4) Scaling Invariance.] If $X \prec Y$, then $tX \prec tY$
for all $t>0$.
\item [(A5) Splitting and recombination.] For $0 < t < 1$,
\begin{center}$X\stackrel {A}{\sim} (tX,(1-t)X)$.\end{center}
\item [(A6) Stability.] If, for some pair of states, $X$ and $Y$,
\begin{center} $(X,\epsilon Z_0) \prec (Y, \epsilon Z_1)$\end{center}
\noindent holds for a sequence of $\epsilon$'s tending to zero and
some some states $Z_0, Z_1$, then $X \prec Y$.
\item [(CH) Comparison hypothesis.] For any two states $X$ and $Y$
in the same state space, either $X \prec Y$ or $Y \prec X$.
\end{description}

REFERENCES
Elliot H. Lieb, Jakob Yngvason, "A guide to entropy and the second
law of thermodynamics," Notices of the AMS, May, 1998, pp. 571-581.

Elliot H. Lieb, Jakob Yngvason, "The physics and mathematics of the second
law of thermodynamics," Physics Reports, Volume 310, Issue 1, March 1999,
pp. 1-96. (This has the proofs.)

Elliot H. Lieb, Jakob Yngvason, "A Fresh look at entropy and the second
law of thermodynamics," Physics Today, April 2000, pp. 32-37. (See also text
only preprint at http://www.esi.ac.at/preprints/ESI-Preprints.html .)

Respectfully yours,
Ellis D. Cooper