Elliott H. Lieb, Jakob Yngvason
The Mathematical Structure of the Second Law of Thermodynamics
Preprint series: ESI preprints

MSC:

80A05 Foundations

80A10 Classical thermodynamics

Abstract: The essence of the second law of classical thermodynamics
is the `entropy principle' which asserts the existence of an additive
and extensive entropy function, $S$, that is defined for all
equilibrium states of thermodynamic systems
and whose increase characterizes the possible state changes under
adiabatic conditions. 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. This principle is independent of
models, statistical mechanical or otherwise, and can be understood
without recourse to Carnot cycles, ideal gases and other assumptions
about such things as `heat', `temperature', `reversible processes',
etc., as is usually done. Also the well known formula of statistical
mechanics, $S = -\sum p \log p$, is not needed for the derivation of
the entropy principle.

This contribution is partly a summary of our joint work (Physics
Reports, Vol.\ 310, 1--96 (1999)) where the existence and uniqueness
of $S$ is proved to be a consequence of certain basic properties of the
relation of adiabatic accessibility among equilibrium states. We also
present some open problems and suggest directions for further study.