Exclusion Statistics Transformation and Ensemble Equivalence Tested From a Different Perspective

TL;DR

This paper extends a method to transform fermionic systems into equivalent bosonic systems, demonstrating their thermodynamic equivalence through detailed numerical verification of particle distributions.

Contribution

It generalizes a previous transformation method to establish thermodynamic equivalence between fermionic and bosonic systems with different exclusion statistics.

Findings

Fermionic and bosonic systems have identical entropies and thermodynamic properties.

Numerical calculations confirm the equivalence of particle distributions in both representations.

The method enables calculation of system properties in either fermionic or bosonic form.

Abstract

We generalize the method introduced in J. Phys. A: Math. Gen. 35, 7255 (2002) based on the concept of thermodynamic equivalence and we transform a Fermi system of general density of states into a thermodynamically equivalent Bose system. This consists of mapping configurations of fermions from the original system onto configurations of bosons, the initial and final configurations having the same energy above the many-body ground state energy. In this way we obtain two systems of particles of different exclusion statistics, but which have the same entropies--and therefore identical canonical thermodynamic properties. This method enables one in general to calculate the system properties in either of the bosonic and fermionic representations. We check the method here in microscopic detail by calculating the equilibrium particle distributions in the two representations using the entropy maximization at fixed particle number and fixed ``fermionic'' and ``bosonic'' energies, respectively. Analytical calculations seem difficult to do, but we check the results numerically and we find them equal within the numerical accuracy.