[Paper Review] Resolving the debate about proposed expressions for the classical entropy

Robert H. Swendsen|arXiv (Cornell University)|Feb 19, 2017

Advanced Thermodynamics and Statistical Mechanics24 references3 citations

TL;DR

This paper resolves the long-standing debate over classical entropy by showing that the grand canonical ensemble provides the only thermodynamically consistent expression, validating negative temperatures and correcting flaws in Boltzmann and Gibbs entropies. While the Boltzmann entropy remains an excellent approximation in most cases, the grand canonical formulation uniquely satisfies all thermodynamic requirements, including correct behavior for decreasing densities of states and first-order phase transitions.

ABSTRACT

Despite well over a century of effort, the proper expression for the classical entropy in statistical mechanics remains a subject of debate. The Boltzmann entropy (calculated from a surface in phase space) has been criticized as not being an adiabatic invariant. It has been suggested that the Gibbs entropy (volume in phase space) is correct, which would forbid the concept of negative temperatures. An apparently innocuous assumption turns out to be responsible for much of the controversy, namely, that the energy $E$ and the number of particles $N$ are given exactly. The true distributions are known to be extremely narrow (of order $1/\sqrt{N}$), so that it is surprising that this is a problem. The canonical and grand canonical ensembles provide alternative expressions for the entropy that satisfy all requirements. The consequences are that negative temperatures are thermodynamically valid, the validity of the Gibbs entropy is limited to increasing densities of states, and the completely correct expression for the entropy is given by the grand canonical formulation. The Boltzmann entropy is shown to provide an excellent approximation in almost all cases.

Motivation & Objective

• To resolve the ongoing controversy over the correct expression for classical entropy in statistical mechanics.
• To identify the root cause of the debate, which stems from the unrealistic assumption that energy and particle number are exactly known (microcanonical ensemble).
• To demonstrate that the canonical and grand canonical ensembles provide thermodynamically consistent alternatives with no fundamental flaws.
• To establish that negative temperatures are thermodynamically valid, contrary to claims based on the Gibbs entropy.
• To show that the grand canonical entropy is the only formulation that correctly describes systems with decreasing densities of states and first-order phase transitions.

Proposed method

• Analyzes the thermodynamic consistency of four entropy definitions: Boltzmann, Gibbs, canonical, and grand canonical, using classical statistical mechanics.
• Identifies the microcanonical assumption (exact E and N) as the source of inconsistencies in the Boltzmann and Gibbs entropies, particularly for systems with decreasing densities of states.
• Applies the canonical and grand canonical ensembles to model systems with finite but macroscopic particle numbers (N > 10^12), ensuring fluctuations are experimentally negligible.
• Evaluates entropy behavior in systems with monotonically increasing and decreasing densities of states, comparing predictions for temperature, energy, and phase transitions.
• Uses Stirling’s approximation and the double-tangent construction to assess the validity of the Boltzmann entropy in phase transitions.
• Compares the thermodynamic stability and extensivity of each entropy formulation, particularly focusing on the N-dependence and 1/N corrections.

Experimental results

Research questions

• RQ1 Why do the Boltzmann and Gibbs entropies fail to describe systems with decreasing densities of states consistently?
• RQ2 Can negative temperatures be thermodynamically valid, and what entropy formulation supports this?
• RQ3 What is the correct expression for classical entropy that satisfies all thermodynamic stability conditions, including for first-order phase transitions?
• RQ4 Why is the microcanonical assumption (exact E and N) problematic despite small corrections of order 1/N?
• RQ5 How do the canonical and grand canonical ensembles resolve the inconsistencies present in the Boltzmann and Gibbs formulations?

Key findings

• The grand canonical entropy is the only formulation that satisfies all thermodynamic requirements, including correct behavior for systems with decreasing densities of states.
• The Gibbs entropy fails to predict negative temperatures and incorrectly predicts energy transfer between systems of different sizes in thermal contact when the density of states decreases.
• The Boltzmann entropy is an excellent approximation in most cases, but fails during first-order phase transitions due to broad energy distributions.
• The canonical and grand canonical entropies correctly describe first-order phase transitions, while the Boltzmann and Gibbs entropies do not.
• The grand canonical formulation ensures exact extensivity of entropy for independent-particle models, unlike the other three definitions.
• The 1/N and ln(N)/N errors in the Boltzmann and Gibbs entropies are negligible in macroscopic systems but are sufficient to invalidate thermodynamic consistency in principle, justifying the need for a better formulation.