Consider the expression (8.21). For the case of massive bosons, the numbers assume all values for each , subject to the constraint that . Performing explicitly the sum over , this expression reduces to

(8.41) |

where is the partition function for particles distributed over all quantum states, excluding state , according to Bose-Einstein statistics [cf., Equation (8.23)]. Using Equation (8.28), and the approximation (8.29), the previous equation reduces to

(8.42) |

Note that this expression is identical to (8.35), except that is replaced by . Hence, an analogous calculation to that outlined in the previous section yields

This is called the

Equations (8.20) and (8.30) can be integrated to give

(8.45) |

where use has been made of Equation (8.43).

Note that photon statistics correspond to the special case of Bose-Einstein statistics in which the parameter takes the value zero, and the constraint (8.44) does not apply.