In textbooks on statistical mechanics, Boltzmann’s claim that the 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 (1/2). 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, not the sample outcome, converges toward the population expected outcome. On the other hand, the variance increases in fixed proportion to the number of molecules, so the probability of an outcome further than a given distance from the mean increases with N.
A corollary to the idea that equilibrium is defined by the most probable macrostate is the concept of uniformity. All of the microstates corresponding to the most probable macrostate are assumed to be physically indistinguishable since the individual molecules are assumed to be physically interchangeable. This is the essence of Boltzmann’s definition of macrostate. A microstate represents a specific physical allocation of space or energy among the molecules, and this allocation is isomorphic to the probability distribution of the macrostates. While the average allocation of microstates to the most probable macrostate will be the same as the expected value of the probability distribution, 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