On the Contact Geometry and the Poisson Geometry of the Ideal Gas

Abstract

Abstract:We elaborate on existing notions of contact geometry and Poisson geometry as applied to the classical ideal gas. Specifically, we observe that it is possible to describe its dynamics using a 3-dimensional contact submanifold of the standard 5-dimensional contact manifold used in the literature. This reflects the fact that the internal energy of the ideal gas depends exclusively on its temperature. We also present a Poisson algebra of thermodynamic operators for a quantum-like description of the classical ideal gas. The central element of this Poisson algebra is proportional to Boltzmann's constant. A Hilbert space of states is identified and a system of wave equations governing the wavefunction is found. Expectation values for the operators representing pressure, volume and temperature are found to satisfy the classical equations of state.

On the van der Waals Gas, Contact Geometry and the Toda Chain

“…when a = 0 and b = 0, systems (10) and (11) correctly reduce to the corresponding system of PDEs for the ideal gas, obtained in Ref. [5]. One readily verifies that integration of the systems (10) and (11) lead back to the fundamental Equation 5we started off with.…”

“…The function W(z) coincides with the potential function of the Toda chain; we have already encountered it in Ref. [5] in the context of the ideal gas. Since the latter has a = 0, which causes the change of variables (17) to be singular, one must proceed differently in this case.…”

“…In the limit when the gas is ideal, the momentum p y vanishes identically [5], and the physics is described in terms of the three-dimensional contact submanifold N spanned by x, p x and U.…”

“…In Ref. [5], we have called Equation (4) a partial differential equation of state (PDE of state for short). It plays a role analogous to that played by the Hamilton-Jacobi equation in classical mechanics [2,6,7].…”

Contact geometry and thermodynamics of black holes in AdS spacetimes

“…If f (V, T, ....) = 0 be an equation of state for system then expressing it in terms of derivatives of the thermodynamic potential V = ∂H/∂P and T = ∂H/∂S we get: f (∂H/∂P, ∂H/∂S, ....) = 0 . This is known as the PDE of state [64] or a Hamilton-Jacobi equation [65]. Here the thermodynamic potential assumes the role of the Hamilton's principal function.…”