  A question that is currently highly debated is whether the microcanonical
entropy should be expressed as the logarithm of the phase volume (volume
entropy, also known as the Gibbs entropy) or as the logarithm of the density of
states (surface entropy, also known as the Boltzmann entropy). Rather than
postulating them and investigating the consequence of each definition, as is
customary, here we adopt a bottom-up approach and construct the entropy
expression within the microcanonical formalism upon two fundamental
thermodynamic pillars: (i) The second law of thermodynamics as formulated for
quasi-static processes: $\delta Q/T$ is an exact differential, and (ii) the law
of ideal gases: $PV=k_B NT$. The first pillar implies that entropy must be some
function of the phase volume $\Omega$. The second pillar singles out the
logarithmic function among all possible functions. Hence the construction leads
uniquely to the expression $S= k_B \ln \Omega$, that is the volume entropy. As
a consequence any entropy expression other than that of Gibbs, e.g., the
Boltzmann entropy, can lead to inconsistencies with the two thermodynamic
pillars. We illustrate this with the prototypical example of a macroscopic
collection of non-interacting spins in a magnetic field, and show that the
Boltzmann entropy severely fails to predict the magnetization, even in the
thermodynamic limit. The uniqueness of the Gibbs entropy, as well as the
demonstrated potential harm of the Boltzmann entropy, provide compelling
reasons for discarding the latter at once.
