Construction of microcanonical entropy on thermodynamic pillars

TL;DR

This paper derives the microcanonical entropy from fundamental thermodynamic principles, demonstrating that the Gibbs volume entropy is uniquely consistent, while the Boltzmann surface entropy can lead to contradictions, especially in predicting magnetization.

Contribution

The paper provides a bottom-up derivation of the microcanonical entropy, establishing the Gibbs volume entropy as the unique form consistent with thermodynamic pillars.

Findings

Gibbs entropy uniquely satisfies thermodynamic principles

Boltzmann entropy fails to predict magnetization accurately

Volume entropy aligns with the second law and ideal gas law

Abstract

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.