Density of states for the Unitary Fermi gas and the Schwarzschild black hole

TL;DR

This paper introduces a numerical method to calculate the density of states from thermodynamic potentials and applies it to both the unitary Fermi gas and Schwarzschild black hole, revealing their thermodynamic properties.

Contribution

It proposes a simple numerical approach to derive the density of states from Helmholtz free energy, applicable to complex quantum and gravitational systems.

Findings

Density of states for the unitary Fermi gas calculated.

Analysis of the density of states for Schwarzschild black hole.

Demonstration of the method's effectiveness in different systems.

Abstract

The density of states of a quantum system can be calculated from its definition but, in some cases, this approach is quite cumbersome. Alternatively, the density of states can be deduced from the microcanonical entropy or from the canonical partition function. After discussing the relationship among these procedures, we suggest a simple numerical method, which is equivalent in the thermodynamic limit to perform a Legendre transformation, to obtain the density of states from the Helmholtz free energy. We apply this method to determine the many-body density of states of the unitary Fermi gas, a very dilute system of identical fermions interacting with divergent scattering length. The unitary Fermi gas is highy symmetric due to the absence of any internal scale except for the average distance between two particles and, for this reason, its equation of state is called universal. In the last part of the paper, by using the same thermodynamical techniques, we review some properties of} the density of states of a Schwarzschild black hole, which shares with the unitary Fermi gas the problem of finding the density of states directly from its definition.