Hello, so I was watching Susskind's lectures on Statistical Mechanics:(adsbygoogle = window.adsbygoogle || []).push({});

He explained Liouville's Theoremqualitativelyin the following two ways:

Now I don't get this last one; I understand what he says, but I don't see why it should be true. For example think of reversible chaos: a ball on a snookertable will show chaos, meaning initially close points in phase space will

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 otherdiverge(taking the system as only one ball (read: particle)), but it is also reversible (actually, for Statistical Mechanics, I think it's fair to sayeverything is reversible if you have enough infoabout 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

**Physics Forums | Science Articles, Homework Help, Discussion**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Statistical Mechanics: Confused about Liouville's Theorem (Susskind, Time, Chaos)

**Physics Forums | Science Articles, Homework Help, Discussion**