# Chance of objects going against greater entropy

My books uses the argument that the multiplicities of a few macrostates in a macroscopic object take up an extraordinarily large share of all possible microstates, such that even over the entire lifetime of the universe, if each microstate had an equal chance of being accessed, big fluctuations away from equilibrium would never occur.

My question to this statistical proof is that, in the real world, is there really an infinitesimal but nonzero chance that macroscopic systems could access some of the more unlikely macrostates (e.g. heat transfer from a cold object to a hot object)?

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I think you are essentially asking whether the ergodic hypothesis (en.wikipedia.org/wiki/Ergodic_hypothesis) is true. I'm not competent to comment, but you might try this book chapter by Thorne and Blandford, especially pages 20 - 27. (pma.caltech.edu/Courses/ph136/yr2008/0803.1.K.pdf). – Mark Eichenlaub Nov 11 '10 at 6:41
This is a question about the validity of equilibrium stat mech. In that framework, of course, big fluctuations away from equilibrium and therefore interesting events (such as life) never happen. Its a good thing then that most interesting systems in the real world are far from equilibrium! – user346 Feb 13 '11 at 7:11

I guess so - I mean, as far as I know, there's no law of physics that strictly prohibits those "exotic" states from being realized. As long as the state exists and can be reached by some path from the "center" of the state space where the likely states are, there should be a nonzero (not even infinitesimal, really) probability of accessing it. But for a typical system, that probability is really, really, really small. So small that it's impossible to intuitively comprehend just how unlikely such an event is.

The thing is, a lot of people aren't used to dealing with even moderately large or small numbers. If you confront them with a probability like $10^{-10^{23}}$, they often fail to put the smallness of that value in perspective, and instead focus on the fact that it's not strictly equal to zero. From there they may start coming up with all sorts of nonsensical ideas about walking through walls and spontaneous combustion (the weird kind) and the like. So physicists usually find it easier to just say the probability is zero - and in fact, for any purpose other than a rigorous mathematical proof, it might as well be.

(Sorry about the rant, I know most people are actually relatively sensible about these things, but it bothers me that the crazy ones seem to get all the attention despite being wrong.)

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I just want to add that such a small probability is much smaller than the probability that we get the model wrong, by any measure you can reasonably assume. In other words, if someone tries to seriously compute the probability of seeing what @wrongusername calls an "exotic macrostate", this probability would be dominated by the small possibility that physics has been totally wrong over the last two centuries. – Frédéric Grosshans Nov 16 '10 at 13:53

You can get a quantitative estimation of the relative probability of 2 macrostates of different entropies $S$ from the $S=k\ln\Omega$ formula, where $k=1.38\times 10^{-23}\ \mathrm{J}/\mathrm{K}$ is the Boltzmann constant.

We have $P\propto\Omega\propto e^{-S/k}$ . This means that $\frac{P_1}{P_2}=e^{-\frac{S_1-S_2}{k}}$.The presence of $k$ as a denominator of the exponent ensures that the probability is small as soon as the entropy difference is bigger than a few times $k$, and completely negligible when the entropy difference is as small as $1000k\sim10^{-20}$ J/K. When you allow $S_1-S_2$ to take any reasonable macroscopic value , the you have the insanely small probabilities David Zaslavsky spoke about.

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Entropy is mathematically defined on distributions of a variable: in physics only the entropy of microscopic degrees of freedom is considered (except for the obligatory section present in virtually each textbook of statistical mechanics: the sharpness of the 'spreading' of a macroscopic variable). Where as the entropy of microscopic variables are observed to increase with time, the entropy of the distribution of a macroscopic variable tends to decrease with time i.e. become sharper. In the context of dynamical systems (although unstable equilibrium points may exist where the entropy actually increases over time)...

So in the context of physics entropy [implicitly of microscopic degrees of freedom] tends to increase with time while entropy of macroscopic variables tend to decrease with time (for example consider the horizontal position of an ensemble of marbles you have randomly thrown -original entropy in horizontal x position is high- in a parabolic potential, after a while each marble comes to rest in the bottom - so final entropy in horizontal x position is low since the distribution after a some time t of the positions in each ensemble becomes more sharply peaked around the bottom of the pit).

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 or in short: microscopic entropy increases, macroscopic entropy decreases. BTW since life rests on making things more certain (i.e. the probability of having 2 legs should be much higher than other amounts of legs) macroscopic decrease of entropy is necessary, and hence microscopic increase of entropy is necessary... – propaganda Jan 25 '12 at 11:20 Wait... I am pretty sure that entropy always increases, particularly at macroscopic levels. And entropy is essentially a logarithmic measure of the multiplicity, not a measure of the uncertainty in quantum mechanics. – wrongusername Jan 25 '12 at 20:52 The logarithmic measure of multiplicity is but one of many equivalent definitions of entropy: when multiplicities are unavailable S[X]=-\int P(X=x)*ln(P(X=x)) dx. Apply to microscopic probability distribution and you get traditional entropy which tends to increase over time, but calculate entropy on macroscopic probability distribution and this will tend to decrease over time... – propaganda Jan 26 '12 at 3:02 (his question doesnt seem to refer to probability of strange events where wavefunction is small but nonzero in quantum mechanics, but to small probabilities in statistical mechanics) – propaganda Jan 26 '12 at 3:04 Oh shoot, I'm sorry then. I saw you talking about variables getting sharper, and linking entropy to sharpness, which I haven't heard of being used in stat mech before, only in QM. Any further recommended readings on macroscopic entropy decrease? – wrongusername Jan 26 '12 at 14:55
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