One-parameter Darboux transformations in Thermodynamics

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. Using the Planck energy distribution of a quantum oscillator one can define an action function fP (x) = UP ω = h̄ 2 + h̄ exp(h̄x)− 1 = h̄ 2 coth (