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In the Wikipedia article on Maxwell–Boltzmann statistics, there is a point in the derivation that stumps me. When I get to where

$\displaystyle W=N!\prod\frac{g^{N_i}}{N_i!}$

is quoted as a count of the number of possible ways particles can be found in each state, I am told we want to pick the $N_i$ in such a way as to maximize $W$. Why is it important to maximize $W$?

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up vote 4 down vote accepted

That's the main assumption behind a lot of equilibrium statistical mechanics, i.e. the thermal equilibrium macrostate of a system is the one that can be realized in the largest possible amount of different microstates under some general restrictions on total energy and total amount of particles.

Or in other words, the thermal equilibrium state is assumed to be the one with the largest phase space volume. If you look at the total phase space of your system and start dividing it up into volumes with different macroparamaters (i.e. different distributions of the particles over energy levels) you'll find out that there are small cells (for instance all particles in the highest energy level) and larger cells. The largest one is the one the system is most likely to be in most of the time and therefore it is natural to assume that it is that one that corresponds to thermal equilibrium.

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