[guga42k]

If somebody needs to build an intuition about entropy he could think about simple problem.

You are given insulated cylinder with a barrier in the middle. Left side of the cylinder filled with ideal gas A, and the right side filled with gas B. If given a particle one can distinguish A from B. The pressure and temperature on both sides are the same. Then you remove the barrier and gases mix. Question: how much work you need to do to revert the system into the original state? Hint: the work is equal to entropy difference between two states.

More generally, if you have proper insulated system and leave it be for a while. All of sudden you will have to do some work to come back to the original state despite energy conservation law holds.

[ithkuil]

If you need to do work in order to revert to the previous state, does it imply you can extract work when going to the first to the second state?

Given the scenario you just laid out it seems no work can be extracted just by letting mix two substances that are at the same temperature and pressure. But there is something about it that doesn't quite add up to my intuition of symmetry and conservation laws. Could you please elaborate more on that?

[Lichtso]

I think you can very well extract work from having a membrane and selectively let one substance mix into the other but not the other in the first [0]. It is called Osmosis [1].

[0]: https://en.wikipedia.org/wiki/Semipermeable_membrane [1]: https://en.wikipedia.org/wiki/Osmosis

[ithkuil]

I guess what's confusing me in this scenario is that we're not saying that the two halves of the cylinder contain particles with different properties (e.g. different velocities) but only that we can "tell them apart" as if they were coloured differently, but otherwise behaving in exactly the same way.

The former scenario is famously the setting for Maxwell's daemon. I was assuming this scenario is something else.

I'm confused because on one hand I can see that it requires work to reorder the particles once they have been shuffled around. On the other hand I don't see how one could extract work while they get shuffled around if they all have the same momenta.

Perhaps the answer is that we cannot have a system where the microscopic entities are at the same indistinguishable but also distinguishable. Perhaps if they had different "colours' it means they do interact differently with the environment? I'm still confused frankly

[FiatLuxDave]

Historically this caused a lot of confusion, so you aren't the only one. You may find E. Jaynes' classic paper on the Gibbs mixing paradox helpful, especially from page 7 onwards:

https://www.mdpi.org/lin/entropy/cgibbs.pdf

[Lichtso]

> only that we can "tell them apart"

That is irrelevant, we don't need to be able to tell them apart, the membrane needs to be able to. Besides that, they can be completely identical.

> I'm confused because on one hand I can see that it requires work to reorder the particles once they have been shuffled around. On the other hand I don't see how one could extract work while they get shuffled around if they all have the same momenta.

In the illustration with the cylinder from Wikipedia you can see that the level of the one fluid (which the other fluid is selected into) rises. It performs work against gravity and builds up potential energy / increases the pressure. You can harvest that.

> The former scenario is famously the setting for Maxwell's daemon. I was assuming this scenario is something else.

In Maxwell's daemon you start with a substance which is already mixed and separate it into its components. That requires work and is the exact opposite of what is happening here. In fact it is called reverse Osmosis [0]. Osmosis gives you pressure which you can harvest, so reverse osmosis needs pressure back to operate. That completes the cycle.

[0] https://en.wikipedia.org/wiki/Reverse_osmosis

[jimmySixDOF]

>That is irrelevant, we don't need to be able to tell them apart

There is some speculation that intelligent behavior evolved as a response to entropy as a way to exert control over future events in the environment.

https://www.santafe.edu/news-center/news/dedeo-intelligent-b...

[guga42k]

Indeed you can extract work from this system. But because of energy conservation it will result to temperature drop. In case if you want to revert the system to its original state you will have to reheat it (return previously extracted work back) AND also spend some work to reorder particles (reduce entropy).

[guga42k]

>If you need to do work in order to revert to the previous state, does it imply you can extract work when going to the first to the second state?

Nope. The work comes from the system coming from ordered state into unordered. Why the problem above is good for intuition because you can work out how to reverse the state. You invent semi-magical barrier which is fully transparent for particles A and reflects particles B, then you start to push such barrier from left to right up to the middle, compressing gas B (and making work!) and leave left part with gas A only, then repeat similar exercise on the right side.

>Given the scenario you just laid out it seems no work can be extracted just by letting mix two substances that are at the same temperature and pressure. But there is something about it that doesn't quite add up to my intuition of symmetry and conservation laws. Could you please elaborate more on that?

As far as I understand this asymmetry was the exact reason why entropy was introduced. Then later explained by Boltzmann via a measure of number of microscopic states.

Naturally second law of thermodynamics forbids perpetual engines.

[kuchenbecker]

Enter Maxwell's demon as a completely valid solution to this problem showing you can decrease entropy within that system (but you need to exclude the demon from the system).