On the contact geometry and the Poisson geometry of the ideal gas

TL;DR

This paper explores the contact and Poisson geometric structures underlying the classical ideal gas, revealing new geometric descriptions and a quantum-like algebraic framework that connects thermodynamics with quantum concepts.

Contribution

It introduces a novel 3D contact submanifold model for the ideal gas dynamics and develops a Poisson algebra with a quantum-like structure, linking thermodynamics and quantum mechanics.

Findings

The ideal gas dynamics can be modeled on a 3D contact submanifold.

A Poisson algebra of thermodynamic operators is constructed with Boltzmann's constant as a key element.

Wave equations for the ideal gas are derived, with expectation values matching classical equations of state.

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.