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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)?

Qmechanic
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    Are you talking about the entropy of the entire universe or the entropy of a system? – Mark_Bell Apr 11 '21 at 23:04
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    For the last, in-bracket question: entropy is (proportional to the logarithm of) the number of ways in which you can obtain a particular arrangement, see Boltzmann's definition of entropy. For me, this has always been clearer than the poorly-defined concept of "disorder". – Mauro Giliberti Apr 11 '21 at 23:05
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    Any time a crystal solidifies out of the liquid the entropy decreases. – Jon Custer Apr 11 '21 at 23:36
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    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. – Mark_Bell Apr 11 '21 at 23:42
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    seems to me that living things themselves are an entropy reducing machine for their system, continuously increasing the entropy of the environment – anna v Apr 12 '21 at 03:07
  • You're alive, aren't you? – J... Apr 12 '21 at 11:36
  • Related: Can a broken egg spontaneously reassemble itself (as in the video)? – jng224 Apr 12 '21 at 19:15
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    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. – gardenhead Apr 12 '21 at 19:43
  • If you add work to a system, you can reduce the entropy of the system. When I comb my hair, I add work, and my hair's collective entropy is reduced. There are lots of physical processes which work to reduce entropy by adding work to the system. Not every system is completely irreversible like a dropped egg is. – Flydog57 Apr 13 '21 at 00:43

3 Answers3

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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.

tom10
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    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 – TheEnvironmentalist Apr 12 '21 at 02:59
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    "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. – lvella Apr 12 '21 at 13:23
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    @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. – Shufflepants Apr 12 '21 at 13:35
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    @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. – lvella Apr 12 '21 at 14:04
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    @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. – Shufflepants Apr 12 '21 at 14:26
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    Kudos to your answer. I'm adding my upvote to the others. My answer was strictly from the perspective of continuum (classical) thermodynamics since I am admittedly not conversant in statistical thermodynamics (I never advanced to that level). – Bob D Apr 12 '21 at 21:26
  • @Shufflepants I disagree. When you say "long enough time" that may be true on time scales we can comprehend (e.g 10E100 years) but if you wait long enough for something like e.g. Poincaré recurrence, by definition you have the possiblity to do new work - Maxwell's Demon in this case is a no-op, simply the recurrence of a previous state is sufficient for new (and possibly repeated) work to be done. – Michael Apr 13 '21 at 01:57
  • @Michael You misunderstand "long enough time". It's not about possibility, it's about expected value. The expected entropy by any machine is positive. It is true that you can randomly, briefly get a very small decrease in entropy. The longer you want to keep that entropy decrease and the larger the decrease in entropy you want, the more unlikely it is. Yes, if you wait long enough, in theory you could get arbitrarily large decreases in local entropy, but in the rest of the universe you will get unimaginably larger increases in entropy in the same time scale. It'll never lead to a net decrease. – Shufflepants Apr 13 '21 at 14:45
  • @BobD: Thanks! I feel more like a journalist than a physicist, just mentioning someone else's interesting ideas. The ideas are fairly new, btw, worked out around 1999, so while fully within in the framework of classic stat mech, they do provide something new and show that the 2nd law doesn't work as assumed in small systems. (The comments about the 2nd law being true for large systems and long times are correct but sort of miss the point, as the fluctuation theorem is about small systems, eg, molecular motors, where these fluctuations can be relevant and do work.) – tom10 Apr 14 '21 at 15:11
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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.

Bob D
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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.

kbakshi314
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