Thermodynamic implications of some unusual quantum theories

TL;DR

This paper explores how certain quantum algebra deformations affect the thermodynamics of macroscopic systems, revealing unusual high-temperature behaviors, negative pressures, and potential violations of thermodynamic laws with implications for early universe evolution.

Contribution

It analyzes the thermodynamic consequences of Kempf-Mangano-Mann deformations, highlighting novel behaviors and potential violations of thermodynamic principles in macroscopic systems.

Findings

Minimal length uncertainty predicts new high-temperature behavior.

Minimal momentum uncertainty leads to unusual large-volume features.

Negative pressures and entropy violations suggest thermodynamic law challenges.

Abstract

Various deformations of the position-momentum algebras operators have been proposed. Their implications for single systems like the hydrogen atom or the harmonic oscillator have been addressed. In this paper we investigate the consequences of some of these algebras for macroscopic systems. The key point of our analysis lies in the fact that the modification of the Heisenberg uncertainty relations present in these theories changes the volume of the elementary cell in the hamiltonian phase space and so the measure needed to compute partition functions. The thermodynamics of a non interacting gas are studied for two members of the Kempf-Mangano-Mann (K.M.M.) deformations. It is shown that the theory which exhibits a minimal uncertainty in length predicts a new behavior at high temperature while the one with a minimal uncertainty in momentum displays unusual features for huge volumes. In the second model negative pressures are obtained and mixing two different gases does not necessarily increase the entropy . This suggests a possible violation of the second law of thermodynamics. Potential consequences of these models in the evolution of the early universe are briefly discussed. Constructing the Einstein model of a solid for the q deformed oscillator, we find that the subset of eigenstates whose energies are bounded from above leads to a divergent partition function.