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As far as I interpret it, the law of ever increasing entropy states that "a system will always move towards the most disordered state, never in the other direction".

Now, I understand why it would be virtually impossible for a system to decrease it's entropy, just as it is virtually impossible for me to solve a Rubik's cube by making random twists. However the (ever so small) probability remains.

Why does this law underpin so much of modern physics? Why is a theory that breaks this law useless, and why was Maxwell's demon such a problem? Does this law not just describe what is most likely to happen in complex systems, not what has to happen in all systems?

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Hannesh, you are correct that the second law of thermodynamics only describes what is most likely to happen in macroscopic systems, rather than what has to happen. It is true that a system may spontaneously decrease its entropy over some time period, with a small but non-zero probability. However, the probability of this happening over and over again tends to zero over long times, so is completely impossible in the limit of very long times.

This is quite different from Maxwell's demon. Maxwell's demon was a significant problem because it seemed that an intelligent being (or more generally any computer) capable of making very precise measurements could continuously decrease the entropy of, say, a box containing gas molecules. For anyone who doesn't know the problem, this entropy decrease could be produced via a partitioning wall with a small window that the demon can open or close with negligible work input. The demon allows only fast-moving molecules to pass one way, and slow-moving ones the other way. This effectively causes heat to flow from a cold body of gas on one side of the partition to a hot body of gas on the other side. Since this demon could be a macroscopic system, you then have a closed thermodynamical system that can deterministically decrease its entropy to as little as possible, and maintain it there for as long as it likes. This is a clear violation of the second law, because the system does not ever tend to thermodynamic equilibrium.

The resolution, as you may know, is that the demon has to temporarily store information about the gas particles' positions and velocities in order to perform its fiendish work. If the demon is not infinite, then it must eventually delete this information to make room for more, so it can continue decreasing the entropy of the gas. Deleting this information increases the entropy of the system by just enough to counteract the cooling action of the demon, by Landauer's principle. This was first shown by Charles Bennett, I believe. The point is that even though living beings may appear to temporarily decrease the entropy of the universe, the second law always catches up with you in the end.

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To add to this answer: although the time scales we deal with in reality are not infinite, one can easily show that the probabilities of fluctuations on the macroscopic scale (i.e. decreases in entropy of the order $1\,\,JK^{-1}$) are of the order $e^{-1/k_B} \approx 10^{-10^{23}}$, which is unimaginably minute. Moreover, since the probabilities can be calculated precisely, this explains why thermodynamics is still important on scales small enough that the fluctuations are significant. –  Nathaniel Oct 28 '12 at 23:54
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That's a great answer, thanks –  Hannesh Oct 29 '12 at 9:12
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actually it's a misconception to say that entropy is disorder , Entropy is not ‘disorder’ and entropy change is not ‘from order to disorder’. Entropy measures the dispersal of energy: how much energy is spread out in a process,or how widely spread out it becomes – at a specific temperature. In chemistry the energy that entropy measures is ‘motional energy’, the translational, vibrational and rotational energy of molecules and, if applicable, phase change energy. (Bond energy is only involved when chemical reactions occur.) Let us dispense with at least one popular myth: "Entropy is disorder" is a common enough assertion, but commonality does not make it right. Entropy is not "disorder", although the two can be related to one another. For a good lesson on the traps and pitfalls of trying to assert what entropy is, see Insight into entropy by Daniel F. Styer,American Journal of Physics 68(12): 1090-1096 (December 2000). Styer uses liquid crystals to illustrate examples of increased entropy accompanying increased "order", quite impossible in the entropy is disorder worldview. And also keep in mind that "order" is a subjective term, and as such it is subject to the whims of interpretation.

read this for further clarification http://entropysite.oxy.edu/cracked_crutch.html

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