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# Maxwell-Boltzmann Statistics

For the purpose of comparison, it is instructive to consider the purely classical case of Maxwell-Boltzmann statistics. The partition function is written (8.46)

where the sum is over all distinct states of the gas, and the particles are treated as distinguishable. For given values of there are (8.47)

possible ways in which distinguishable particles can be put into individual quantum states such that there are particles in state 1, particles in state 2, et cetera. Each of these possible arrangements corresponds to a distinct state for the whole gas. Hence, Equation (8.46) can be written (8.48)

where the sum is over all values of for each , subject to the constraint that (8.49)

Now, Equation (8.48) can be written (8.50)

which, by virtue of Equation (8.49), is just the result of expanding a polynomial. In fact, (8.51)

or (8.52)

Note that the argument of the logarithm is simply the single-particle partition function

Equations (8.20) and (8.52) can be combined to give (8.53)

This is known as the Maxwell-Boltzmann distribution. It is, of course, just the result obtained by applying the canonical distribution to a single particle. (See Chapter 7.) The previous expression can also be written in the form (8.54)

where (8.55)

The Bose-Einstein, Maxwell-Boltzmann, and Fermi-Dirac distributions are illustrated in Figure 8.1.    Next: Quantum Statistics in Classical Up: Quantum Statistics Previous: Bose-Einstein Statistics
Richard Fitzpatrick 2016-01-25