Hamiltonian structure of thermodynamics with gauge

摘要

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 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.