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Bill Dreiss

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For example, the number of molecules found in one half of a box of ideal gas is given by the binomial distribution. However, it can easily be seen that if the number of molecules in one half of the box is within ± 0.6745

Furthermore, by his definition of equilibrium, the probability of being in the most probable macrostate is approximately √(2/π

How can these discrepancies be resolved?

*Lectures on Gas Theory*(Dover Books on Physics) (p. 74), Boltzmann states “In nature, the tendency of transformations is always to go from less probable to more probable states”, by which he means what are now called macrostates. Thus he claims that an ideal gas almost always evolves to the most probable macrostate, which he defines as equilibrium.For example, the number of molecules found in one half of a box of ideal gas is given by the binomial distribution. However, it can easily be seen that if the number of molecules in one half of the box is within ± 0.6745

*σ*of the mean, the probability of the next macrostate being greater than the current macrostate is less than ½, contrary to Boltzmann’s statement.Furthermore, by his definition of equilibrium, the probability of being in the most probable macrostate is approximately √(2/π

*N*) for large*N*(where*N*is the total number of molecules in the gas). This means that the probability of being in equilibrium becomes vanishingly small for large systems, contradicting the common understanding of the meaning of equilibrium.How can these discrepancies be resolved?

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