Chance of objects going against greater entropy

Question

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)?

Answer 1

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.)

Answer 2

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.

Answer 3

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).