Having read the definitions, I’m pretty sure that a system with Poincare recurrence does not have a meaningful Entropy value (or E is absolutely constant with time). You cannot get any useful work out of the system, or within the system. The speed distribution of particles never changes. Our ability to predict the position and momentum of the particles is constant. Is there some other definition of Entropy that actually shows fluctuations like you’re describing?
Having read the definitions, I’m pretty sure that a system with Poincare recurrence does not have a meaningful Entropy value (or E is absolutely constant with time). You cannot get any useful work out of the system, or within the system. The speed distribution of particles never changes. Our ability to predict the position and momentum of the particles is constant. Is there some other definition of Entropy that actually shows fluctuations like you’re describing?
The ideal gas does have a mathematical definition of entropy, Boltzmann used it in the statistical derivation of the second law:
https://en.wikipedia.org/wiki/Entropy_(statistical_thermodynamics)
Here is an account of Boltzmann work and the first objections to his conclusions:
https://plato.stanford.edu/entries/statphys-Boltzmann/