Hamiltonian structure of thermodynamics with gauge

Denoting by q(i) (i = 1, ..., n) the set of extensive variables which characterize the state of a thermodynamic system, we write the associated intensive variables gamma (i),, the partial derivatives of the entropy S = S (q(1), ..., q(n)) equivalent to q(0), in the form gamma (i) = -p(i) / p(0) where p(0) behaves as a gauge factor. When regarded as independent, the variables q(i), p(i) (i = 0, ..., n) define a space T having a canonical symplectic structure where they appear as conjugate. A thermodynamic system is represented by a n + 1-dimensional gauge-invariant Lagrangian submanifold M of T. Any thermodynamic process, even dissipative, taking place on M is represented by a Hamiltonian trajectory in T, governed by a Hamiltonian function which is zero on M A mapping between the equations of state of different systems is likewise represented by a canonical transformation in T. Moreover a Riemannian metric arises naturally from statistical mechanics for any thermodynamic system, with the differentials dq(i) as contravariant components of an infinitesimal shift and the dp(i)'s as covariant ones. Illustrative examples are given.