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Thermal Equilibrium

Eq. (9) establishes the criterion for the MOST PROBABLE CONFIGURATION -- i.e. the value of  $\hat{U}_1$  for which the combined systems have the maximum total entropy, the maximum total number of accessible states and the highest probability. This also defines the condition of THERMAL EQUILIBRIUM between the two systems - that is, if   $U_1 = \hat{U}_1$,  any flow of energy from   ${\cal S}_1$  to   ${\cal S}_2$  or back will lower the number of accessible states and will therefore be less likely than the configuration^15.15 with   $U_1 = \hat{U}_1$. Therefore if we leave the systems alone and come back later, we will be most likely to find them in the "configuration" with  $\hat{U}_1$  in system   ${\cal S}_1$ and   $(U - \hat{U}_1)$  in system   ${\cal S}_2$.

This seems like a pretty weak statement. Nothing certain, just a bias in favour of  $\hat{U}_1$  over other possible values of  U₁  all the way from zero to  U. That is true. STATISTICAL MECHANICS has nothing whatever to say about what will happen, only about what is likely to happen - and how likely! However, when the numbers of particles involved become very large (and in Physics they do become very large), the fractional width of the binomial distribution [Eq. (2)] becomes very narrow, which translates into a probability distribution that is incredibly sharply peaked at  $\hat{U}_1$. As long as energy conservation is not violated, there is nothing but luck to prevent all the air molecules in this room from vacating the region around my head until I expire from asphyxiation. However, I trust my luck in this. A quotation from Boltzmann confirms that I am in distinguished company:

"One should not imagine that two gases in a 0.1 liter container, initially unmixed, will mix, then again after a few days separate, then mix again, and so forth. On the contrary, one finds . . . that not until a time enormously long compared to 10¹⁰¹⁰ years will there be any noticeable unmixing of the gases. One may recognize that this is practically equivalent to never . . . . "
-- L. Boltzmann