“What mechanism exists to cause the particles to vary in speed (given the magical non-deforming non-reactive box we are containing things in)?”
The system is a compact deterministc dynamical system and Poincarè recurrence applies: it will return infinitely many times close to any low entropic state it was before. Since the particles are only 3 the time needed for the return is small.
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 Poincarè recurrence theorem doesn’t imply that. It doesn’t imply the system is ergodic, and it only applies to “almost all” states (the exceptions are guaranteed to have measure zero, but then again, so is the set of all numbers anyone will ever specifically think about). In any case, the entropy doesn’t change at all because it’s a property of an abstraction.
An ideal gas in a box is an egodic system. The Poincarè recurrence theorem states that a volume preserving dynamical system (i.e. any conservative system in classical physics) returns infinitely often in any neighbourhood (as small as you want) of any point of the phase space.
“What mechanism exists to cause the particles to vary in speed (given the magical non-deforming non-reactive box we are containing things in)?”
The system is a compact deterministc dynamical system and Poincarè recurrence applies: it will return infinitely many times close to any low entropic state it was before. Since the particles are only 3 the time needed for the return is small.
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/
The Poincarè recurrence theorem doesn’t imply that. It doesn’t imply the system is ergodic, and it only applies to “almost all” states (the exceptions are guaranteed to have measure zero, but then again, so is the set of all numbers anyone will ever specifically think about). In any case, the entropy doesn’t change at all because it’s a property of an abstraction.
An ideal gas in a box is an egodic system. The Poincarè recurrence theorem states that a volume preserving dynamical system (i.e. any conservative system in classical physics) returns infinitely often in any neighbourhood (as small as you want) of any point of the phase space.