Elliott H. Lieb and Jakob Yngvason
The Mathematics of the Second Law of Thermodynamics
(200K, Postscript)

ABSTRACT.  The essence of the second law is the `entropy principle' which states that 
adiabatic processes can be quantified by an entropy function on the space 
of all equilibrium states, whose increase is a necessary and sufficient 
condition for such a process to occur. It is one of the few really 
fundamental physical laws (in the sense that no deviation, however tiny, 
is permitted) and its consequences are far reaching. Since the entropy 
principle is independent of models, statistical mechanical or otherwise, 
it ought to be derivable from a few logical principles without recourse 
to Carnot cycles, ideal gases and other assumptions about such things as 
`heat', `hot' and `cold', `temperature', `reversible processes', etc., 
as is usually done. The well known formula of statistical mechanics, 
$S = -\sum p \, \log p$, is irrelevant for this problem. In this paper 
the foundations of the subject and the construction of entropy from a 
few simple axioms will be presented. Finally, we consider some open 
problems and directions for further study.