Understanding Poincare's recurrence theorem

[JD_PM]

The Poincare's recurrence theorem :

This theorem implies the following:

Suppose a container is divided in two by a wall. Half of it contains particles and the other none. If you were to remove the wall and wait a very very long time, the particles would eventually be found in the same half of the container.

To me this statement is counter-intuitive. I would expect the particles to jiggle around forever.

Why am I wrong?

I have read the proof but I would rather discuss the theorem to understand it.

 

[Orodruin]

Summary:: I want to understand the idea of Poincare's recurrence theorem

To me this statement is counter-intuitive. I would expect the particles to jiggle around forever.

They ”jiggle around” randomly forever. This means that they will eventually come back to the original state (or very close to it) by pure chance. The time for this to happen can be orders of magnitude longer than the age of the Universe.

 

[JD_PM]

Thanks for your answer.

This means that they will eventually come back to the original state (or very close to it) by pure chance.

Doesn't this mean that the second law of Thermodynamics breaks down? Theoretically, if all particles eventually come back to the original state means that $\Delta S =0$. This is actually what shocks me about this theorem.

The time for this to happen can be orders of magnitude longer than the age of the Universe.

Actually my book justifies that the second law is not violated because very long time means $10^{20}$ years.

To me that is not a good reason because, in the end we end up with $\Delta S =0$ so we still have the same issue.

By the way, how is this time calculated?

 

[Orodruin]

Doesn't this mean that the second law of Thermodynamics breaks down?

No. Thermodynamics is about the statistical properties of the system and the resulting macro states. It is not describing the fully deterministic situation. Looking at statistical mechanics, you have the different statistical ensembles that will allow you to derive the thermodynamic properties of the system.

By the way, how is this time calculated?

This depends on how close you want to get to the original state in phase space. Take a characteristic time to move out of that volume and multiply by the ratio of the total phase space volume accessible and the phase space volume you want to move back to. This gives you an upper estimate.

 

[atyy]

Doesn't this mean that the second law of Thermodynamics breaks down?

The laws of thermodynamics are not fundamental, and they can break down in theory, but these are so rare that there is not any breakdown in practice.

 

[JD_PM]

The laws of thermodynamics are not fundamental

What do you mean saying 'not fundamental'?