I wrote about this at
https://r6.ca/blog/20100802T005835Z.html.
Basically entropy is a function of descriptions of state rather than states themselves. So when you talk about jar of gas having a certain volume and pressure, there is a set of states that satisfy that description and the logarithm of the volume of that set of states (in phase space) is the entropy of that description.
Liouville’s theorem says that the volume of a subset of phase space is preserved under time evolution of the system. So if you want to have a machine that reliably transform from one description to another description, then the volume of the latter description must be at least as large as the initial description. That is the second law of thermodynamics.
(This are a bit more complicated than I state above because you have to work with differentials, but the basic idea is there).
The Poincaré recurrence theorem says individual states eventually almost reoccur, but this doesn't apply to a volume of states. The volume of states usually gets smeared out throughout phase space.
Like if you start the Lorenz equations on a square of points instead of an individual point, it gets smeared out over the entire Lorenz attractor. And while individual points in that square nearly reoccur, the entire square does not. And thus you cannot build a machine to reliably reduce entropy because you cannot predict when your reoccurance will reoccur (even if you can compute an upper bound on how long it will take) since you never know precisely the initial conditions to begin with (you start somewhere in that initial square).
Basically entropy is a function of descriptions of state rather than states themselves. So when you talk about jar of gas having a certain volume and pressure, there is a set of states that satisfy that description and the logarithm of the volume of that set of states (in phase space) is the entropy of that description.
Liouville’s theorem says that the volume of a subset of phase space is preserved under time evolution of the system. So if you want to have a machine that reliably transform from one description to another description, then the volume of the latter description must be at least as large as the initial description. That is the second law of thermodynamics.
(This are a bit more complicated than I state above because you have to work with differentials, but the basic idea is there).
The Poincaré recurrence theorem says individual states eventually almost reoccur, but this doesn't apply to a volume of states. The volume of states usually gets smeared out throughout phase space.
Like if you start the Lorenz equations on a square of points instead of an individual point, it gets smeared out over the entire Lorenz attractor. And while individual points in that square nearly reoccur, the entire square does not. And thus you cannot build a machine to reliably reduce entropy because you cannot predict when your reoccurance will reoccur (even if you can compute an upper bound on how long it will take) since you never know precisely the initial conditions to begin with (you start somewhere in that initial square).