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64 STATISTICAL PHYSICS

just described. Therefore, the number of microstates (or complexions, as Boltz-
mann called them) corresponding to the partition Eq. 4.4 is given by

Boltzmann took w to be proportional to the probability of the distribution speci-

fied by (n^,n 2 ,. ..). This will be called his second definition of probability.

For later purposes I need to mention a further development, one not due to

Boltzmann. The number of microstates w is now called a fine-grained probability.

For the purpose of analyzing general macroscopic properties of systems, it is very

important to use a contracted description, which leads to the so-called coarse-

grained probability,* a concept that goes back to Gibbs. The procedure is as fol-

lows. Divide n space into cells co 1 ,(o 2 ,. .. such that a particle in COA has the mean

energy EA. Partition the TV particles such that there are NA particles in WA:

The set (NA,EA) defines a coarse-grained state. For the special case of the ideal

gas model, it follows from Eq. 4.5 that the volume W in F space corresponding

to the partition of Eqs. 4.6 and 4.7 is given by

where W is the so-called coarse-grained probability. The state of equilibrium cor-
responds to the maximum Wm.ut of W considered as a function of 7VA and subject
to the constraints imposed by Eqs. 4.6 and 4.7. Thus the Maxwell-Boltzmann
distribution follows** from the extremal conditions

"The names fine-grained and coarse-grained density (feine und grobe Dichte) were introduced by
the Ehrenfests [E45].

**For the classical ideal gas, one can get the Maxwell-Boltzmann distribution directly from Eqs.
4.4 and 4.5; that is just what Boltzmann himself did.

The entropy in equilibrium, £„,, is given by (see Eq. 4.3)