Photon Statistics

Up to now, we have assumed that the number of particles, $N$ , contained in a given system is a fixed number. This is a reasonable assumption if the particles possess non-zero mass, because we are not generally considering relativistic systems in this course (i.e., we are assuming that the particle energies are much less than their rest-mass energies). However, this assumption breaks down for the case of photons, which are zero-mass bosons. In fact, photons enclosed in a container of volume $V$ , maintained at temperature $T$ , can readily be absorbed or emitted by the walls. Thus, for the special case of a gas of photons, there is no requirement that limits the total number of particles.

It follows, from the previous discussion, that photons obey a simplified form of Bose-Einstein statistics in which there is an unspecified total number of particles. This type of statistics is called photon statistics.

Consider the expression (8.21). For the case of photons, the numbers $n_1, n_2,\cdots$ assume all values $n_r=0,1,2,\cdots$ for each $r$ , without any further restriction. It follows that the sums $\sum^{(s)}$ in the numerator and denominator are identical and, therefore, cancel. Hence, Equation (8.21) reduces to

$$\bar{n}_s = \frac{\sum_{n_s} n_s {\rm e}^{-\beta n_s \epsilon_s}} {\sum_{n_s} {\rm e}^{-\beta n_s \epsilon_s}}.$$ (8.35)

However, the previous expression can be rewritten

$$\bar{n}_s =-\frac{1}{\beta}\frac{\partial}{\partial \epsilon_s}\left(\ln\sum_{n_s} {\rm e}^{-\beta n_s \epsilon_s}\right).$$ (8.36)

Now, the sum on the right-hand side of the previous equation is an infinite geometric series, which can easily be evaluated. In fact,

$$\sum_{n_s=0,\infty} {\rm e}^{-\beta n_s \epsilon_s} = 1 + {\rm ... ...e}^{-2 \beta \epsilon_s} + \cdots = \frac{1}{1-{\rm e}^{-\beta \epsilon_s}}.$$ (8.37)

(See Exercise 1.) Thus, Equation (8.36) gives

$$\bar{n}_s =\frac{1}{\beta} \frac{\partial}{\partial\epsilon_s} \l... ...ilon_s}) =\frac{{\rm e}^{-\beta \epsilon_s}} {1-{\rm e}^{-\beta \epsilon_s}},$$ (8.38)

or

$$\bar{n}_s = \frac{1}{{\rm e}^{ \beta \epsilon_s}-1}.$$ (8.39)

This is known as the Planck distribution, after the German physicist Max Planck who first proposed it in 1900 on purely empirical grounds.

Equation (8.20) can be integrated to give

$$\ln Z = -\sum_r \ln\left(1-{\rm e}^{-\beta \epsilon_r}\right),$$ (8.40)

where use has been made of Equation (8.39).