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

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

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

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

(8.54) |

where

(8.55) |

The Bose-Einstein, Maxwell-Boltzmann, and Fermi-Dirac distributions are illustrated in Figure 8.1.