# Is there any useful sense in which entropy fluctuates?

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:

1. 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.
2. 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?

• The first point of view follows from Loschmidt's paradox which points out that the time-reversal symmetry of the laws of physics should not be able to give rise to a quantity that is guaranteed to always decrease with time. Apr 8, 2016 at 14:20
• I am currently reading this paper: 1.Bartolotta, A., Carroll, S. M., Leichenauer, S. & Pollack, J. Bayesian second law of thermodynamics. Physical Review E 94, (2016). It says that the old Boltzmann view with its fluctuation theorem "has garnered considerable attention in recent years" and gives references. It then goes on to explain how also the modern Gibbs/Shannon formulation allows fluctuations in the entropy when observations are taken into account properly. So perhaps that paper gives some answers to your question. Jun 10, 2017 at 16:41

In Gibbs's view of thermodynamics, entropy is a static property of a stochastic phase space and is given by the functional $$-\sum_i p_i \log p_i$$ where $$p_i$$ is the equilibrium probability of outcome $$i$$. For example, the entropy of a fair die $$(p_i=1/6, i=1\cdots 6)$$ is $$-\sum_{i=1}^6 \left(\frac{1}{6}\right)\log\frac{1}{6} = \log 6.$$ In this context to say that entropy fluctuates is to say that the probability distribution itself fluctuates.