One of the classic distinctions between young Boltzmann and old Boltzmann was his view on entropy. Young Boltzmann had his H-theorem where a mechanical quantity H was supposed to represent entropy. This quantity fluctuated even in equilibrium. Old Boltzmann had his $k \ln W$ macrostate entropy which strictly did not fluctuate. Afterwards we had Gibbs/Shannon who generalized the old-Boltzmann point of view, and we arrive at the modern common definition of entropy: $-k\sum_i P_i \ln P_i$. This too does not fluctuate.
Yet even still today there seems to remain two different points of view on entropy that I see people discussing:
- One group says that entropy fluctuates, and the second law of thermodynamics is "only true on average". Entropy is somehow a mechanical variable representing disorder (?). Example: fluctuation theorem. Statistical mechanics is described as: systems want to move the most likely "high entropy microstate" but sometimes they don't.
- Another group maintains the second law as perfect in all situations, and says entropy does not fluctuate. Entropy strictly is a measure of a probability distribution and cannot spontaneously decrease due to the conservation of distinction in mechanics (unitarity, and all that).
I think I understand the second point of view. Yet, I do not understand the first point of view. I do not understand how this fluctuating entropy variable is defined, nor what it is useful for. I am tempted to say these people are just confused but perhaps I am missing something. Any hints?