One-parameter Darboux transformations in Thermodynamics

TL;DR

This paper explores Darboux transformations applied to quantum oscillator thermodynamics, revealing superpositions of boson and fermion excitations and proposing a generalized fluctuation-dissipation theorem.

Contribution

It introduces a novel application of Darboux transformations to thermodynamic actions, linking quantum oscillator actions with superpositions of excitations and extending fundamental thermodynamic relations.

Findings

Darboux transformations relate to superpositions of boson and fermion states.

A generalized fluctuation-dissipation theorem is proposed.

Thermodynamic actions satisfy Riccati equations.

Abstract

The quantum oscillator thermodynamic actions are the conjugate intensive parameters for the frequency in any frequency changing process. These oscillator actions fulfill simple Riccati equations. Interesting Darboux transformations of the fundamental Planck and pure vacuum actions are discussed here in some detail. It is shown that the one-parameter "Darboux-Transformed-Thermodynamics" refers to superpositions of boson and fermion excitations of positive and negative absolute temperature, respectively. A Darboux generalization of the fluctuation-dissipation theorem is also briefly sketched