The Free Quon Gas Suffers Gibbs' Paradox

TL;DR

This paper investigates the statistical mechanics of q-commutation relation particles, revealing that free gas partition functions are q-independent and exhibit Gibbs' paradox, indicating non-extensive thermodynamics.

Contribution

It demonstrates that free gases with q-commutation relations have partition functions independent of q and display Gibbs' paradox, challenging their thermodynamic extensivity.

Findings

Partition functions are q-independent for -1<q<1.

Gibbs' paradox appears without the 1/N! correction.

The system does not exhibit extensive thermodynamic properties.

Abstract

We consider the Statistical Mechanics of systems of particles satisfying the $q$-commutation relations recently proposed by Greenberg and others. We show that although the commutation relations approach Bose (resp.\ Fermi) relations for $q\to1$ (resp.\ $q\to-1$), the partition functions of free gases are independent of $q$ in the range $-1<q<1$. The partition functions exhibit Gibbs' Paradox in the same way as a classical gas without a correction factor $1/N!$ for the statistical weight of the $N$-particle phase space, i.e.\ the Statistical Mechanics does not describe a material for which entropy, free energy, and particle number are extensive thermodynamical quantities.