In textbooks on statistical mechanics, Boltzmann’s claim that a system converges on the most probable macrostate is often attributed to the Law of Large Numbers, which is commonly defined as the tendency of the number of outcomes (such as the number of heads in a series of N coin flips) to converge on the population average (N/2 for the binomial distribution). However, this is actually the definition of the Law of Averages, as pointed out in the article Wikipedia: Law of averages:
Another application of the law of averages is a belief that a sample’s behavior must line up with the expected value based on population statistics. For example, suppose a fair coin is flipped 100 times. Using the law of averages, one might predict that there will be 50 heads and 50 tails. While this is the single most likely outcome, there is only an 8% chance of it occurring according to… the Binomial Distribution.
To get an intuitive sense of this, imagine flipping a coin 1 million times. Would you expect to get exactly 500,000 heads with a high probability? Furthermore, since the true probability of achieving the most probable outcome is vanishingly small, the probability of each of the other 999,999 outcomes is also infinitesimal.
However, the correct definition of the Law of Large Numbers is that the sample average (the number of heads divided by the number of coin flips) H/N, converges on the expected value of a single coin flip 1/2, not that the sample outcome (the number of heads H), converges toward the population mean N/2. On the other hand, the variance in the distribution of outcomes increases in fixed proportion to the number of molecules, so the probability of an outcome further than a given distance from the population mean increases with N.
A corollary to the idea that thermodynamic equilibrium is defined by the most probable macrostate is the concept of uniformity. While the distribution of observable outcomes of the microstates contained in the most probable macrostate becomes more uniform as the number of molecules n is increased, the proportion of these observable outcomes relative to all possible outcomes decreases as n is increased. The microstates included in the most probable macrostate cannot be considered representative of the system, since they consist of only a tiny portion of the total microstates for large values of N.
Revised 6/6/23