q-Legendre Transformation: Partition Functions and Quantization of the Boltzmann Constant

TL;DR

This paper introduces a q-analogue of the Legendre transformation linking quantum mechanics and thermodynamics, incorporating non-commutative constants and an infinite algebraic structure to extend classical thermodynamic concepts.

Contribution

It develops a novel q-Legendre transformation framework using non-commutative algebra and extends the Boltzmann constant into an infinite generator algebra.

Findings

Formulation of a q-analogue of the Legendre transformation.

Introduction of non-commutative Planck-Boltzmann constants.

Proposal to replace the Boltzmann constant with an infinite algebraic structure.

Abstract

In this paper we construct a q-analogue of the Legendre transformation, where q is a matrix of formal variables defining the phase space braidings between the coordinates and momenta (the extensive and intensive thermodynamic observables). Our approach is based on an analogy between the semiclassical wave functions in quantum mechanics and the quasithermodynamic partition functions in statistical physics. The basic idea is to go from the q-Hamilton-Jacobi equation in mechanics to the q-Legendre transformation in thermodynamics. It is shown, that this requires a non-commutative analogue of the Planck-Boltzmann constants (hbar and k_B) to be introduced back into the classical formulae. Being applied to statistical physics, this naturally leads to an idea to go further and to replace the Boltzmann constant with an infinite collection of generators of the so-called epoch\'e (bracketing) algebra. The latter is an infinite dimensional noncommutative algebra recently introduced in our previous work, which can be perceived as an infinite sequence of "deformations of deformations" of the Weyl algebra. The generators mentioned are naturally indexed by planar binary leaf-labelled trees in such a way, that the trees with a single leaf correspond to the observables of the limiting thermodynamic system.