# Microcanonical analysis of Boltzmann and Gibbs Entropies in trapped cold atomic gases

## TL;DR

This paper compares Boltzmann and Gibbs entropy definitions in a trapped fermionic gas, highlighting differences in temperature predictions and clarifying the implications for systems with bounded energy spectra.

## Contribution

It provides a detailed microcanonical analysis of entropy definitions in a quantum system, emphasizing their differences and physical implications.

## Key findings

- Gibbs temperature aligns more closely with grand canonical temperature for small particle numbers.
- Boltzmann entropy can suggest negative temperatures, which are not realized in this system.
- Significant differences exist between the two entropy definitions in bounded quantum systems.

## Abstract

We analyze a gas of noninteracting fermions confined to a one-dimensional harmonic oscillator potential, with the aim of distinguishing between two proposed definitions of the thermodynamic entropy in the microcanonical ensemble, namely the standard Boltzmann entropy and the Gibbs (or volume) entropy. The distinction between these two definitions is crucial for systems with an upper bound on allowed energy levels, where the Boltzmann definition can lead to the notion of negative absolute temperature. Although negative temperatures do not exist for the system of fermions studied here, we still find a significant difference between the Boltzmann and Gibbs entropies, and between the corresponding temperatures with the Gibbs temperature being closer (for small particle number) to the temperature based on a grand canonical picture.