Hamiltonian structure of thermodynamics with gauge

Abstract

The state of a thermodynamic system being characterized by its set of extensive variables $q^i(i=1,...,n),$ we write the associated intensive variables ${\gamma }_i,$ the partial derivatives of the entropy $S(q^1,...,q^n)\equiv q_0,$ 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 $\mathbb{T}$ having a canonical symplectic structure where they appear as conjugate. A thermodynamic system is represented by a $n+1$-dimensional gauge-invariant Lagrangian submanifold $\mathbb{M}$ of $\mathbb{T}.$ Any thermodynamic process, even dissipative, taking place on $\mathbb{M}$ is represented by a Hamiltonian trajectory in $\mathbb{T},$ governed by a Hamiltonian function which is zero on $\mathbb{M}.$ A mapping between the equations of state of different systems is likewise represented by a canonical transformation in $\mathbb{T}.$ Moreover a natural Riemannian metric exists for any physical system, with the $q^i$'s as contravariant variables, the $p_i$'s as covariant ones. Illustrative examples are given.

Proofs corrections latex vali.tex, 1 file, 28 pages [SPhT-T00/099], submitted to Eur. Phys. J. B

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