 00332 Elliott H. Lieb and Jakob Yngvason
 The Mathematics of the Second Law of Thermodynamics
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Sep 2, 00

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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.
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