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Hello, so I was watching Susskind's lectures on Statistical Mechanics:

He explained Liouville's Theorem

The only way I could understand Susskind's second point is if Liouville's Theorem is actually a probability statement, much like the 2nd law of Statistical Mechanics: it isn't true, just very very probable: in this way, convergence is possible, just unlikely (if so, what are the prerequisites for the theorem?)

Thank you,

mr. vodka

He explained Liouville's Theorem

*qualitatively*in the following two ways:**No Merging:**Two trajectories in phase space will never merge; this seems obvious using the time-symmetry and determinism in classical mechanics, because when you reverse time everything should still be deterministic, yet if there were merging it wouldn't be.**No Limit-Merging:**Called "practically just as unpleasant if it would be true", namely that trajectories also won't converge toward each other

*diverge*(taking the system as only one ball (read: particle)), but it is also reversible (actually, for Statistical Mechanics, I think it's fair to say*everything is reversible if you have enough info*about the microstate? Here the microstate just 'coincides' with the macrostate if you get my point). Travelling down the trajectories in reverse, we're seeing a convergence. (Okay the example isn't perfect because convergence implies going on forever, yet my time only goes back till the instant I placed the ball on the table, but this doesn't seem to be a main issue here.)The only way I could understand Susskind's second point is if Liouville's Theorem is actually a probability statement, much like the 2nd law of Statistical Mechanics: it isn't true, just very very probable: in this way, convergence is possible, just unlikely (if so, what are the prerequisites for the theorem?)

Thank you,

mr. vodka

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