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I had a discussion yesterday, and I found my understanding of entropy lacking. I know that entropy is not exactly the same as the degree of disorder in a system, but I also know that outside of specifically designed experiments, it's quite difficult to find examples where the two are not intimately linked.

A hydroelectric power plant works by taking water from one basin and pouring it out into another. If these two basins are the same and placed at the same height, but one has a higher water level, then we see that after outputting electric power, the end state is a bit more disordered than the initial state (it goes from one basin having all the water and the other basin having no water to both the basins having some water).

However, if the basins are smaller, but one is filled with water and placed higher than the other, which is empty, so that you may empty the top one completely of water and fill the bottom one entirely, then the degree of disorder seems to be much closer.

In both of these two examples, if I were to just move the water without passing it through a generator, then the disorder has increased because the potential energy of the water has become chaotic movement and heat within the water, so it's not nearly as "available to do useful work", to quote the tag. If the generator is used, the water slows down, but you make electric power which is spent to make something that eventually turns into heat somewhere else.

On the other hand, in the second example we still have one basin full of water and one basin with no water after using it all, so that contribution to the total disorder seems to disappear without the generator noticing any difference.

How come the generator doesn't notice any difference? Have I misunderstood something, and the state "the water is in the lower basin" actually has higher disorder than "the water is in the upper basin"? Or is this one of the places where I have to discard the "entropy = disorder" notion that popular science loves so much?

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  • $\begingroup$ Can you quantify any of this? There might be an interesting question here, but I don't see why your perception of the "disorder" of the two different states should hold true. You have two basins each with a volume $V$ of water in them... and afterwards you have one basin with a volume $2V$ of water in it. Or you have volumes $V+x$, $V-x$ before and $V$, $V$ afterwards. So what? $\endgroup$ – user12029 Nov 12 '16 at 1:15
  • $\begingroup$ @NeuroFuzzy The first example goes form $2V,0$ to $V, V$, say. To me the latter seems like a more "mixed", or disordered state. The second example, however, goes from $V, 0$ to $0, V$, which to me seems like the degree of order hasn't changed at all. My point is that if this is true, then the generator should feel the difference: in one case the entropy / disorder changes more than in the other. But I see no reason why it should feel any difference, since the same amout of water comes through with the same force, and it runs off just as easily. $\endgroup$ – Arthur Nov 12 '16 at 1:21
  • $\begingroup$ Well, popular press notions of entropy are crap, so... $\endgroup$ – Jon Custer Nov 12 '16 at 1:48
  • $\begingroup$ @JonCuster It's not just popular press, though. It's basically anything that's not graduate physics, or possibly undergraduate thermodynamics / statistical mechanics. Also, in my opinion, among the simplifications done by popular press, equating entropy with disorder is not their worst sin. $\endgroup$ – Arthur Nov 12 '16 at 1:52
  • $\begingroup$ I pretty much guarantee that if you read the Wikipedia article on entropy, you will end up mentally more disordered than when you started it : en.wikipedia.org/wiki/Entropy_(order_and_disorder) $\endgroup$ – user108787 Nov 12 '16 at 2:05
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So, entropy additively measures our uncertainty about the microscopic state of a system given whatever we can macroscopically measure about it. You can kind of call that "disorder" if you want, but that's a bit of an imprecise term, useful in the most common case but not 100% universal. It's no huge problem that it's described this way.

What's really missing is the idea of ensembles, which are admittedly hard to explain to a public audience! We have to mostly say the word "closed system" a lot and make people feel like they don't really understand something which they thought they understood very simply.

So the basic idea is that if you have interactions they tend to multiply our uncertainties about the microscopic state of the system. The belief is that if you could really isolate the system from the rest of the world for long enough, there would just be a random choice of actual microscopic state from the total set of microscopic states allowed by the conserved parameters of the system. In such a case one expects the "biggest" macroscopic state (the most uncertain because it contains the most compatible microscopic states) to be the natural resting place of the system. To keep the system totally isolated it is generally sufficient to fix all of its number of particles, fix some container that it is in, and then forbid it from exchanging energy with anything outside the container (or the container itself, which is why the container is fixed; otherwise you can interact with the walls and give them kinetic energy by pushing on them). This, then, is the familiar form of the 2nd law, that the system transitions from the less-likely macrostates to the more-likely macrostates, and not vice versa. The term for this way of analyzing a system is the "microcanonical ensemble" of the system.

Suppose the system partitions into two subsystems $A,~B$ such that every conviguration of the joint system $c \in C_{A+B}$ is actually just a pair of configurations $c = (c_A,~c_B)$ for the subsystems, so $C_{A+B} = C_A \times C_B$ (the configuration-space of $A+B$ is a Cartesian product of the subsystems' configuration spaces). Define a macroscopic state of any of these as a subset of the configuration space; then we have $|c| = |c_A|~|c_B|$ and so we take a logarithm to make this measure additive, $s = s_A + s_B.$ This is a definition of the entropy. Now notice that if there is any conserved quantity $Q$ in the system as a whole, there is some parameter which dictates whether it spontaneously flows from one subsytem to the other under the above randomization: given a $\delta Q$ going from $B$ to $A$ we find that the total entropy change is $$\delta s = \frac{ds_A}{dQ} \delta Q - \frac{ds_B}{dQ} \delta Q$$and this is positive, negative, or zero based on these absorption-tendencies $ds/dQ.$

Now imagine that one of these systems is much much bigger than the one that you're interested in, you can imagine that receiving a $\delta Q$, which matters a lot for your system you're studying, doesn't affect the bigger system much at all, so its $ds_A/dQ$ stays constant as $Q$ varies. These are the other "ensembles," where the system is sharing stuff with the rest of the world.

Now that you understand, these things make a bit more sense. Suppose for example that you have two reservoirs of water at different heights and water flows from high to low. What happens when it hits the bottom is that usually it splashes upward, but not as high as the original. Eventually as it sloshes around it comes to a resting state. In this state the average motion of the particles is higher: you have heated up the system, increasing the number of momentum configurations the particles can have. Eventually all of the water will flow down to the bottom and will be maximally jittery. If the basins are even then it will just equalize the level of water on both sides and otherwise it's the same deal.

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