You probably know that s=k*log(W) is carved on Boltzmann's tombstone. Did you know that he was never able to precisely describe W, at least not to the satisfaction of Albert Einstein, who chided him for not being able to do so? As for a breakthrough, I have always appreciated how Keenan, Keyes, Hill, and Moore (1969) demonstrated that all other thermodynamic properties can be derived through partial derivatives of the Helmholtz Free Energy. This fact was also utilized by Haar, Gallagher, and Kell (1984) plus a few more since then. I am also a big proponent of Maxwell's Criterion (see graph), which is misunderstood and even denied by some authors who are ignorant of its derivation.