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Say I put a bunch of powerful square magnets on a nearly frictionless table in a disordered fashion. The second law of thermodynamics states that the system shall spontaneously get more disordered, yet the magnets will attract each other and form a chain (typically), thereby increasing the order of the system-and, seemingly, decreasing its entropy.

It would seem to me that the system is closed and the lattice is indeed the equilibrium state. Therefore, I suspect that by attracting each other, the magnets increase their own entropy by a larger amount than the decrease in entropy caused by the lattice formation. Is it true? If yes, what are the thermodynamics of magnets responsible for this? Is there a microscopic explanation?

Thanks!

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If the distance of those magnets is big enough, there will be no magnetic interaction above static friction, nothing will happen at all. For the rest: without giving the lateral magnetisation of those magnets and the relative oriantation of the magnets the question makes no sense. – Georg Dec 6 '11 at 12:40
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The question makes complete sense, this is a thought experiment, and no specific details are required except for the fact that these magnets are within attraction distance and the static friction is very small (much smaller than the magnetic force). – Greg Dec 6 '11 at 12:54

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The magnets will indeed attract each other. This attraction will put them in motion, and they will head towards each other, converting electromagnetic energy into kinetic energy. Then they will collide, and loose their kinetic energies in the collision, finally coming to rest in a more ordered, low-energy state.

In terms of energy, the outcome of the experiment is that you have converted electromagnetic energy into heat: the heat released in the collisions. This conversion creates far more entropy than the entropy lost by arranging the magnets in a more ordered fashion.

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I don’t completely disagree with genneth’s answer, but I find the term “damping” to be too generic to provide a physical understanding of what is really going on. It’s not the friction on the table (as it’s assumed to be nearly frictionless), and it certainly is not the electromagnetic emission that damp most of the energy. The real culprits are the collisions between the magnets. – Edgar Bonet Dec 8 '11 at 16:09
I can consider a small enough perturbation to the equilibrium state (the lattice), which is more disordered, and they will move back to the equilibrium state without any collisions! So the real culprit cannot be collisions. – Chris Gerig Dec 9 '11 at 10:14
@Chris Gerig: Sure, but then you are talking about a different thought experiment. – Edgar Bonet Dec 9 '11 at 13:24
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It's actually a nice example of why the 2nd law is useful: if you go around trying to account in microscopic detail the balance of things like energy and entropy, then you can easily go wrong. I used to have a student who would do this all the time; I don't know if he ever learnt the lesson...

In this specific case, you neglected damping --- without it, you would never come to an equilibrium and the 2nd law does not apply. With damping, you necessarily dissipate heat, and that loss will more than make up for the macroscopic ordering. This dissipation can be either mechanical friction, or (as an example of why you would be wrong about this situation being "closed") electro-magnetic --- oscillating dipoles emit radiation!

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Excellent answer! Took me a while to figure why damping was an argument, but it is indeed the case. At first I was thinking only of the mechanical friction with the ground. – Greg Dec 6 '11 at 14:26
So You can explain how this attraction by "square" magnets works? – Georg Dec 6 '11 at 17:01
that does not matter, if you had ever played with free-standing permanent magnets (regardless of their shape), you would have noticed that their original orientation does not matter: the repulsive state is an unstable equilibrium point, and whenever they are close enough to repel each other, they align and attract each other. Anyway, that's not the point of the question, as genneth could easily understand. If you are really interested in that question, I am sure you will be able to set up a very nice experiment: apexmagnets.com – Greg Dec 6 '11 at 21:05
So You do not know the lateral magnetisation of that magnets? And BTW when You try to set up a meaningful "thought" experiment, do not waste time by using some physical effect You are not really shure of. Just demand some attaction between the units, and demand fluid friction for their movement. Nobody needs some "real" implementation for a thought experiment. – Georg Dec 6 '11 at 22:12
Good point indeed, but that's stackexchange.com, not Applied Physics Letter, and you're more than welcome to update my question if it does not reach your own quality requirements. – Greg Dec 6 '11 at 23:15
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Consider this for an outrageous answer,

since it is impossible to have an isolated system, hence the magnetic field of the bar magnets in the scattered state interacts with the system around it and will cause some increase in entropy,

Now the magnets align to form a larger and stronger magnet which interacts with its surroundings in a more agressive way compared to the previous case, hence increasing the entropy of the system,

DAmping is the major factor which is increasing disorder here, but this factor will also contribute so small , yet non zero amount to the entire entropy increase.

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