Hamiltonian structure of thermodynamics with gauge

Abstract:

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 γ _i, the partial derivatives of the entropy S = S q ¹,..., q ⁿ ≡q ₀, in the form γ _i = - p _i/p ₀ where p₀ behaves as a gauge factor. When regarded as independent, the variables q ⁱ, p _i i = 0,..., n 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 dq ⁱ as contravariant components of an infinitesimal shift and the dp _i's as covariant ones. Illustrative examples are given.