Liouville & Entropy: Solving the Controversy

[Gerenuk]

Next to all recent Entropy thread I'd also like to have a question solved.

What's the solution to the controversy between the second law of thermodynamics, and Liouville's theorem that for conservative systems (as a gas should be?!) every state should be reached at some point? So eventually after an extraordinary long time all molecules would also gather in the corner.

 

[Oudeis Eimi]

I don't think there is a controversy. The strict "thermodynamic" result is
only valid for an infinite number of particles (thus, no fluctuations). Real
systems do fluctuate though; all particles gathering in one place would be
a BIG fluctuation, thus very low probability, thus something that would only
happen after a very, very long time.

 

[Gerenuk]

OK, that's also my favourite interpretation.

So the second law is rather a statistical result and in an extremely long period of time the second law could be arbitrarily violated?!

Any objections from someone else?

 

[fluidistic]

OK, that's also my favourite interpretation.

So the second law is rather a statistical result and in an extremely long period of time the second law could be arbitrarily violated?!

Any objections from someone else?

You're right. I've posted a few questions of the same kind. For instance, see https://www.physicsforums.com/showthread.php?t=319633.
See the fluctuation theorem. There was a paper about an experiment that showed entropy decreases macroscopically for a few seconds... The paper was accessible from wikipedia. The Second Law is not a "fundamental law".
I just found something related to the article: http://www.newscientist.com/article/dn2572-second-law-of-thermodynamics-broken.html.

 

[sardar]

The controversy between the second law of thermodynamics and Liouville's theorem has been a longstanding debate in the field of thermodynamics. The second law of thermodynamics states that the total entropy of a closed system will always increase over time, while Liouville's theorem states that for conservative systems, every state is eventually reached at some point.

One possible solution to this controversy is to consider the concept of time scales. The second law of thermodynamics is based on macroscopic observations and applies to systems that are observed over relatively short time scales. On the other hand, Liouville's theorem is based on microscopic observations and applies to systems observed over much longer time scales.

Therefore, it is possible that the apparent contradiction between the two principles arises because they are operating on different time scales. In other words, while the second law of thermodynamics may hold true on a macroscopic level, Liouville's theorem may still hold true on a microscopic level, but over a much longer time scale.

Another potential solution is to consider the limitations of Liouville's theorem. While it holds true for conservative systems, it does not take into account the effects of external forces or interactions. In the case of a gas, these external forces could include pressure, temperature, or other environmental factors that may prevent all molecules from gathering in one corner.

It is also important to note that the second law of thermodynamics is a statistical law, meaning that it applies to a large number of particles and not necessarily to individual particles. This means that while it may seem unlikely for all molecules to gather in one corner, it is still possible for a small number of molecules to do so, while the majority of molecules remain evenly distributed.

In conclusion, there is no clear-cut solution to the controversy between the second law of thermodynamics and Liouville's theorem. However, by considering the concept of time scales, the limitations of Liouville's theorem, and the statistical nature of the second law of thermodynamics, we can gain a better understanding of how these principles may coexist and complement each other in explaining the behavior of thermodynamic systems.