Is there any useful sense in which entropy fluctuates?

Question

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?

Answer 1

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.

But can probability distributions fluctuate?

Actually they can, but there is a but. In the Bayesian view the equilibrium distribution is the most probable distribution among all distributions that satisfy the macroscopic constraints of the system. Indeed, we obtain this distribution by maximizing the entropy functional under the given constraints. It turns out that the most probable distribution is overwhelmingly more probable than any other. In this sense, the probability to observe any distribution other than the equilibrium distribution is infinitesimal when the size of the system increases.

A very good discussion of Gibbs's entropy versus (young) Boltzmann's is given in this paper by Jaynes.

Answer 2

The first argument regarding fluctuating entropy and fluctuation theorems is valid for nonequilibrium systems. When a dissipative system is driven out of equilibrium by some time dependent protocol followed by equilibration, one gets a value of work done, heat dissipated and entropy produced. When the cycle is repeated numerous times one gets a different value of work, heat and entropy production in each run. In other words one talks about trajectory to trajectory fluctuations in the three quantities. Then one takes an ensemble average to calculate distribution of work done, heat dissipated and entropy produced and establishes fluctuation theorems for Nonequilibrium processes in the long time limit. I guess this answers the fluctuating entropy part. Keep in mind that its trajectory based or path based. So one uses path integral techniques for such calculations.