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On the whole, the entropy of the universe is always increasing. There are far more possible states of "high" entropy than there are of "low" entropy. The example I've seen most often is an egg$-$ there is only one way for an egg to be whole (the low-entropy state), but many possible high-entropy states.

Now, this is only a "statistical" law, however, and, as I understand it, it's not impossible (only incredibly unlikely) for a metaphorical egg to "unbreak".

My question, then, is: have we ever observed a natural example of a "spontaneous" reduction in entropy? Are there any physical processes which work to reduce entropy (and, if so, is there a more concrete way to define entropy? Obviously, a cracked egg is intuitively a higher entropy state than an "uncracked" egg... but what to we really mean by entropy, which is distinct from disorder)?

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    $\begingroup$ Are you talking about the entropy of the entire universe or the entropy of a system? $\endgroup$
    – Mark_Bell
    Apr 11 at 23:04
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    $\begingroup$ Any time a crystal solidifies out of the liquid the entropy decreases. $\endgroup$
    – Jon Custer
    Apr 11 at 23:36
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    $\begingroup$ There are extremely high number of situations every day in which the entropy decreases, even in biological system. There are open systems everywhere. Instead if the question is about the universe as a whole, this is another thing. $\endgroup$
    – Mark_Bell
    Apr 11 at 23:42
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    $\begingroup$ seems to me that living things themselves are an entropy reducing machine for their system, continuously increasing the entropy of the environment $\endgroup$
    – anna v
    Apr 12 at 3:07
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    $\begingroup$ Small nitpick, but I would guess there is more than one way for an egg to be whole, just far less than for it to be broken. $\endgroup$
    – gardenhead
    Apr 12 at 19:43
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Yes, one can observe a decrease in entropy of an isolated system.

The statistics of these observation are quantified by the fluctuation theorem. The logic of it is based on what you (the OP) suggest: since statistical mechanics is about statistical laws, one would expect there to be fluctuations.

The first paragraph of wikipedia states this well, so I'll just copy it with some emphasis added:

While the second law of thermodynamics predicts that the entropy of an isolated system should tend to increase until it reaches equilibrium, it became apparent after the discovery of statistical mechanics that the second law is only a statistical one, suggesting that there should always be some nonzero probability that the entropy of an isolated system might spontaneously decrease; the fluctuation theorem precisely quantifies this probability.

The paper in the first note (pdf) there gives access to a PRL with an experimental observation of a decrease in entropy in an isolated system. Of course, these observations were done in very small systems; as the systems get larger, the chance of observing such fluctuations decreases.

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    $\begingroup$ I'll add that Maxwell's Demon suggests that if you were to create a system tries to "capture" spontaneous entropy reductions resulting from the fluctuation theorem and thus yield net reductions in entropy, the infrastructure of the system itself is conjectured to require enough entropy to counterbalance the otherwise net reduction. You can have reductions in entropy with some small probability, but they cannot be harnessed to yield net negative entropy in any useful way $\endgroup$ Apr 12 at 2:59
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    $\begingroup$ "but they cannot be harnessed to yield net negative entropy in any useful way", if we accept entropy as purely statistical, that assertion is kind of a dogma. Maybe it is possible, and maybe there is nothing but technology limitation preventing Maxwell's Demon from working. $\endgroup$
    – lvella
    Apr 12 at 13:23
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    $\begingroup$ @lvella It's not dogma, it's a theoretical limit, not a technological one. And it doesn't say that some machine couldn't extract some small amount negative entropy for some super small amount of time, it just means that on the whole, any such machine would increase entropy in the long run rather than decrease it. $\endgroup$ Apr 12 at 13:35
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    $\begingroup$ @Shufflepants The question is precisely if, in theory, entropy can decrease. The answer is "yes". Why then can't Maxwell's Demon harness it? Well, because in the older theory of thermodynamics the answer was "no", so we don't want the Demon to break it. That is not a very compelling argument. $\endgroup$
    – lvella
    Apr 12 at 14:04
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    $\begingroup$ @lvella Think of it like gambling. Suppose you can pay $1 and 1/10 of the time you win back your dollar and an extra penny, 9/10 of the time you get nothing. It's theoretically possible to gain money playing this game in the short term, but with those odds, in the long run, you'll lose money almost certainly. The statistical argument against the demon is the same. It's not that we don't want the demon to break it, it's that it's been mathematically proven that for any machine real or theoretical, the expected value of entropy produced will always be positive on a long enough time scale. $\endgroup$ Apr 12 at 14:26
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My question, then, is: have we ever observed a natural example of a "spontaneous" reduction in entropy?

Yes, of course, but as @Mark_Bell pointed out, it depends on if you are talking about the entropy of the system only (or the surroundings only) or the total entropy change (system + surroundings).

If heat spontaneously transfers from the system to the surroundings due to a finite difference in temperature between them, the entropy of the system decreases, but the entropy of the surroundings increases by a greater amount for total entropy change (system plus surroundings = change in entropy of the universe) of greater than zero.

Hope this helps.

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Quoting from this directly related post,

According to the fluctuation theorem the second law of thermodynamics is a statistical law. Violations at the micro scale, therefore, certainly have a non-zero probability.

We can calculate the entropy using the Boltzmann formula $S=k_B\log N$, where $N$ is the number of states. From the initial state $i$ to the the broken final state $f$, the change in entropy is $\Delta S=k_B\log N_f/N_i$. Therefore, $N_f=N_ie^{\Delta S/k_B}$ so that the probability of the transition from the state $i$ to the state $f$ is calculated as $p_{i\to f}=N_f/(N_f+N_i)\approx 1$, because there are several more possible micro-states in the final configuration $p_{f\to i}=1-p_{i\to f}\approx\exp(-\Delta S/k_B)$. Although the increase of entropy is a high number, the Boltzmann constant $0<k_B$ adds $10^{23}$ to the exponent, so that the value of the probability is extremely low.

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