Gibbs Vs Boltzmann Entropies: A Comparison And Special Relativity's Impact.pdf

A Clarification by E. T. Jaynes

This 1964 article by E.T. Jaynes tackles a long-standing debate surrounding the validity and interpretation of two key entropy expressions in statistical mechanics: those proposed by Gibbs and Boltzmann.

Key Findings:

1. Gibbs Entropy is Correct: The author reaffirms that the Gibbs H-function (or Gibbs entropy) correctly represents entropy as defined in phenomenological thermodynamics.

2. Boltzmann Entropy Inaccurate: The Boltzmann H-function, while elegant, yields an inaccurate "entropy" when interparticle forces are significant. This error grows with system size.

3. Relating Boltzmann Entropy to Gibbs: Boltzmann's other entropy definition (S = k log W) is shown to be consistent with the Gibbs H and derivable from it.

4. Boltzmann H Theorem and Second Law: The article debunks the notion that Boltzmann's H theorem proves the second law for dilute gases. It emphasizes that H itself doesn't relate to experimental entropy measurements.

5. Dynamic Invariance of Gibbs Entropy: The dynamic invariance of the Gibbs H provides a simple proof of the second law for all systems, not just dilute gases.

6. Anthropic Nature of Entropy: Jaynes stresses the "anthropomorphic" nature of entropy - its reliance on both statistical and phenomenological interpretations reflecting our experience with macroscopic systems.

Implications:

The article clarifies fundamental concepts in statistical mechanics, highlighting:

The distinction between mathematical expressions for entropy and their experimental realization.
The need to connect theoretical entropy calculations to observable macroscopic quantities.
The limitations of using coarse-graining as a proof of the second law.
The dynamic invariance of the Gibbs H as a robust foundation for understanding entropy in diverse systems.