https://doi.org/10.1007/s100510170202
Hamiltonian structure of thermodynamics with gauge
1
CEA/Saclay, Service de Physique Théorique, Orme des Merisiers, 91191 Gif-sur-Yvette Cedex, France
2
Centre de recherches ELF, Chemin du Canal, BP 22, 69360 Solaize, France
Corresponding author: a balian@spht.saclay.cea.fr
Received:
5
July
2000
Revised:
2
March
2001
Published online: 15 May 2001
Denoting by the set of extensive variables
which characterize the state of a thermodynamic system, we write the
associated intensive variables
the partial derivatives of the
entropy
in the form
where p0 behaves as a gauge factor. When regarded as independent,
the variables
define a
space
having a canonical symplectic structure where they appear
as conjugate. A thermodynamic system is represented by a n+1-dimensional
gauge-invariant Lagrangian submanifold
of
Any
thermodynamic process, even dissipative, taking place on
is
represented by a Hamiltonian trajectory in
governed by a
Hamiltonian function which is zero on
A mapping between the
equations of state of different systems is likewise represented by a
canonical transformation in
Moreover a Riemannian metric
arises naturally from statistical mechanics for any thermodynamic system,
with the differentials
as contravariant components of an
infinitesimal shift and the
's as covariant ones. Illustrative
examples are given.
PACS: 05.70.Ce – Thermodynamic functions and equations of state / 05.70.Ln – Nonequilibrium and irreversible thermodynamics / 64.10.+h – General theory of equations of state and phase equilibria
© EDP Sciences, Società Italiana di Fisica, Springer-Verlag, 2001