If for an isolated system in thermodynamic equilibrium all the accessible microstates are equally probable, why do the gas molecules in an isolated container, never accumulate at one corner of the box or say, on the right half of the box? I emphasize that my question is about the validity of the postulate of equal a priori probabilities.
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asked Jun 22, 2020 at 2:03
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2 Answers
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In principle, if you observed the isolated gas-filled container for long enough, you would occasionally see a state in which all gas molecules were congregated on the right side of the box. But you would likely have to wait for an exceedingly long time, well in excess of the expected life-time of the visible universe, before it happened for the first time. That is because the number of possible states with no obvious grouping of the molecules vastly exceeds the number of possible states in which the molecules are observably grouped.
As a very simple approximation, consider a box containing a very diffuse gas consisting 10^9 molecules. Take the reasonable approximation that the probability of each molecule being on the left or right side of the box is independent of the positions of the other molecules. Each molecule will be on the right side approximately 50% of the time, so the probability that all the molecules are simultaneously to the right side will be 0.5^(10^9), or about one chance in 10^(10^8). That is an exceedingly tiny probability.
As a more graspable example - if you properly shuffle a pack of cards, then every ordering of cards is equally likely, including the one where the cards are all ordered by rank and suite. But the chance that you will spontaneously sort them into that specific order is very small - one in 52!. Of the remaining 52!-1 possible sortings, most of them will appear 'random'.
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There are way more microstates where the gas particles aren't in the corner than there are microstates where the gas is in the corner. So if you're talking the probability of "in the corner" vs "not in the corner" they latter is going to win out.
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$\begingroup$ But all microstates are equally probable by the postulate. $\endgroup$ Commented Jun 22, 2020 at 2:30
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2$\begingroup$ Yea, but "not in the corner" isn't a single microstate. There are plenty of different microstates that fit that description. Just think of all of the different arrangements of the gas particles that still leave it "not in the corner". $\endgroup$– roshokaCommented Jun 22, 2020 at 2:31