Exploring Liouville's Theorem with Susskind's Lectures on Statistical Mechanics

In summary, Susskind explains Liouville theorem as being a probability statement, and that it's not true, but very very probable.
  • #1
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Hello, so I was watching Susskind's lectures on Statistical Mechanics:

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
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 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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  • #2
Oh, seems an interesting link and I have to check it out.

I once heard for Liouville you also consider the development of volume in phase space and this volume should not converge to zero. I guess for many mechanical problems it stay even constant? And of course the total phase volume should be limited.

Maybe someone else can put this more mathematically correct.
 
  • #3
But in the above described case the density seems to increase. How am I misinterpreting things? All help is welcome!
 
  • #4
mr. vodka said:
Hello, so I was watching Susskind's lectures on Statistical Mechanics:

He explained Liouville's Theorem qualitatively in the following two ways:


Sorry I can't help you, but I would love to see Susskind's explanation of the Liouville theorem. This video lecture series on Stat Mech is very long...can you tell me which video has the part about Liouville?

Thanks...
 
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  • #5
part one, about after 40 min
 

1. What is Liouville's Theorem?

Liouville's Theorem is a fundamental principle in classical mechanics that states that the phase space (a mathematical space that represents all possible states of a system) of a closed system is conserved over time. This means that the distribution of points in the phase space remains constant as the system evolves. In other words, the volume of the phase space remains the same over time.

2. How does Susskind's Lectures on Statistical Mechanics relate to Liouville's Theorem?

Susskind's Lectures on Statistical Mechanics explores the statistical behavior of systems with a large number of particles, which is governed by the laws of classical mechanics. Liouville's Theorem is a fundamental principle in classical mechanics that plays a crucial role in understanding the behavior of these systems.

3. What can we learn from exploring Liouville's Theorem with Susskind's Lectures on Statistical Mechanics?

By exploring Liouville's Theorem with Susskind's Lectures on Statistical Mechanics, we can gain a deeper understanding of the statistical behavior of classical systems with a large number of particles. This can help us make predictions and analyze the behavior of complex systems in various fields, such as physics, chemistry, and engineering.

4. How is Liouville's Theorem applied in real-world situations?

Liouville's Theorem has many practical applications, including in fields such as thermodynamics, fluid mechanics, and celestial mechanics. For example, it is used to analyze the motion of particles in a gas, the behavior of fluids in pipes, and the orbits of planets around the sun.

5. Are there any limitations to Liouville's Theorem?

While Liouville's Theorem is a powerful tool in classical mechanics, it does have its limitations. It assumes that the system is closed and that all forces involved are conservative. It also does not take into account quantum effects, which are necessary for understanding the behavior of systems at very small scales.

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