I was reading Goldstein's paper Boltzmann's Approach to Statistical Mechanics and he makes the following statement:
The Second Law is concerned with the thermodynamic entropy, and this is given by Boltzmann’s entropy (1), not by the Gibbs entropy (2).
On the next page he states:
Certainly contributing to the tendency to identify the thermodynamic entropy with the Gibbs entropy is the fact that for systems in equilibrium the Gibbs entropy agrees with Boltzmann’s entropy.
However a paper by Jaynes showed that the Gibbs and Boltzmann entropy do not agree for a system in equilibrium and in fact the disagreement is proportional to the inter-particle interactions (in general $S_B \geq S_G$). It is made somewhat confusing by the fact that I'm not sure Jaynes and Goldstein are using exactly the same definition of the Boltzmann entropy. But what Goldstein does make clear is that the Boltzmann entropy is calculated from the 1-particle distribution function, as does Jaynes. This must necessarily exclude inter-particle correlations and so I think is subject to Jaynes' critique.
Does anyone know how Goldstein's statement can be justified?