  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.
