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#### etotheipi

Prof. Tong used the Liouville theorem to prove the Poincaré recurrence theorem in his notes, that given an initial point ##P## in phase space, for any neighbourhood ##D_0## of ##P## there exists a point ##P' \in D_0## that will return to ##D_0## in a finite time.

To illustrate the theorem, he says that if you put a bunch of molecules of gas in a corner of a room, then eventually they'll end up back where they started [granted, the recurrence time exceeds the universe lifetime!].

The question at the end of the section is "Where's your second law of thermodynamics now?!". I can't figure out how its consistent, so I wondered if someone could explain? Thanks!

To illustrate the theorem, he says that if you put a bunch of molecules of gas in a corner of a room, then eventually they'll end up back where they started [granted, the recurrence time exceeds the universe lifetime!].

The question at the end of the section is "Where's your second law of thermodynamics now?!". I can't figure out how its consistent, so I wondered if someone could explain? Thanks!

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