[javajosh]

Entropy only made sense when I learned it from the perspective of statistical thermodynamics. It's a very programmerly understanding, IMHO, and it's quite intuitive. EXCEPT that the language used is ridiculous: grand canonical ensemble indeed! Anyway, the idea that a system can be in some number of specific states, and that equilibrium is that unique situation where the number of possible specific states is at its maximum, really spoke to me.

[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).

[dekhn]

I took a stat thermo class and it was basically all about entropy, which was expresed as ln W- the log of the number of ways (permutations) that a system can be ordered, which gives a convenient denominator when calculating the probability of a specific permutation. Here's the professor's book, which was still only in latex form when we took the class: https://www.amazon.com/Molecular-Driving-Forces-Statistical-...

[javajosh]

yes, there are lots of quantitative details. I wanted to emphasize the key qualitative concept, from which the others can derive. In a similar way you can derive all of special relativity, and approach an intuition about the strangeness of spacetime, starting with only two ideas: the laws of physics are the same in all reference frames; the speed of light is constant. I prefer to start there and derive e.g. Lorentz factors than start with the mathy stuff.

[kergonath]

> starting with only two ideas: the laws of physics are the same in all reference frames; the speed of light is constant.

Isn’t this redundant, though? The constant velocity for light in a vacuum comes directly from the laws of (classical) electromagnetism in the form of Maxwell’s equations. So “the laws of Physics are the same in all reference frames” implies “Maxwell’s equations are valid in all reference frames”, which in turn implies “the velocity of light in vacuum is the same in all reference frames”. That’s what I understood reading Einstein himself.

I think it’s much stronger that way. Otherwise we get to why light should be a special case, which is difficult to defend. The constant velocity of light (in vacuum) being an unavoidable consequence of the laws of Physics makes it much stronger.

> I prefer to start there and derive e.g. Lorentz factors than start with the mathy stuff.

That’s how Einstein himself explained it (with trains and stuff, but still) and it makes a lot of sense to me. Much more than the professor who did start with the Lorentz transform and then lost everyone after 20 minutes of maths.

[staunton]

> So “the laws of Physics are the same in all reference frames” implies “Maxwell’s equations are valid in all reference frames”, which in turn implies “the velocity of light in vacuum is the same in all reference frames”

From the point of view of physicists before Einstein, this forces you to decide between Newtonian physics and Maxwell's theory, because the reference frames that are "equivalent" are irreconcilably different for those. The "irreconcilable" part is subtle and not obvious. Maxwell's theory "won", but it was the newer theory while Newton's was very well established. The contemporary physicists' efforts to reconcile the two using an "ether" were completely reasonable from their point of view. (And actually, you can't even completely exclude the existence of an ether, as some ether theories are consistent with the standard model to a reasonably high accuracy. What kills them is Occam's razor)

[kergonath]

I know. What I am saying is that the laws of physics being the same implies the velocity of light being the same. Of course it conflicts with Newton’s laws and Galilean transformations when changing reference frames. But what he demonstrated was that the apparent conflict could be solved in a logically consistent way, once you do away with the concept of universal time. From that point, Newton’s laws do not need to be invariant, because they were demonstrably incomplete, in contrast to Maxwell’s, which were verified as far as they could be with the experimental setups of the time.

Sure, postulating the existence of the ether was reasonable in a way at one point in time, I am not saying otherwise. But by 1905 it was on very shaky grounds, with no experimental result to support it. Saying that the theory can be tweaked to reproduce reality is not very useful: all theories can. What a theory needs to be verified is to predict things that the other established theories do not. And on that front, the ether theory is about as powerful as my pet theory that elementary particles are moved by tiny demons that we cannot see (I would make a joke about string theory but it’s way more serious than the ether one).

[javajosh]

>That’s how Einstein himself explained it

Yes, I think we're violently agreeing then. The original 1905 paper is extremely readable.

[contravariant]

That intuition is still a bit shallow though. I don't mean that in a bad way, some intuition is better than none at all. However if you start to dig you'll find out the terminology goes out of wack.

Note that you're describing equilibrium as a unique situation where the number of possible states is at a maximum. Now how can a situation be unique if it has the maximum number of possible states? Clearly the situation is as far from unique as it can be.

To resolve the contradiction requires distinguishing between features of the probability distribution and features of a random sample (i.e. a possible state) and also needs an explanation how it even makes sense to view a deterministic physical system (leave quantum mechanics for now) as a random variable.

The theory that links everything together is ergodic theory, which has a couple of handy theorems. One is that for a certain kind of dynamical system the average over time and the average over the 'possible states' agree. Such a system can also be assigned an entropy. It even suggests that generally a system will be found around states with a probability close to 2^-entropy (this is not absolutely always true ..but close enough for physicists)

Now what does such a system look like? Well we need a state space (easy) and a measure on it which is constant as the system evolves (i.e. we can pick a region in the state space, evolve it and its volume will stay constant). The last part is tricky, but as it turns out classical mechanics gives us the phase space and the canonical volume on it (basically the standard notion of volume) which fit the bill. This gives a probability distribution on the state space and an entropy equal to log(volume in phase space), which matches the definitions in statistical physics but also gives a solid foundation for some of the seemingly arbitrary choices.

So there you have it, that's why a system can have a probability distribution attached to it, despite being deterministic, why 'high entropy states' are common, and why physical systems have a uniform distribution (and therefore an entropy which is the log of the number of states).

This also explains physicists got away with using a uniform distribution without worrying about which variables they used. By pure 'coincidence' the standard choice of variables that physicists use have this incredibly nice property that makes everything work out. I'm not sure if this is too well known so it might be worth abusing this to 'prove' a perpetuum mobile is possible to stop people using uniform distributions without due deliberation.

[passion__desire]

Entropy is mathematical force to be honest.

[Solvency]

Huh? "Force"?

[esafak]

concept

[danq__]

No. A force is the correct characterization. Entropy is indeed a force with a singular direction.

Can you explain why does entropy perpetually increase?

Additionally explain what happens if we reverse time, does your intuition of entropy still make sense? Likely no, because there is a vector directionality to entropy which makes it comparable to a "force".