Thermodynamics: Stats

For τ σ , the occupation of these low frequency states is very high, approaching ∞ . This poses no problem however because it is the density of photons per frequency space that is physically important, and so there are very few frequencies that have this high occupancy.

For τ σ , the occupation of the high frequency states is near zero.

When we are summing over quantum states, only non-negative quantum numbers are allowed. Writing the integral alone sums over all 8 quadrants in n-space, and so we divide by 8 to get the right answer.

We saw that the energy goes as τ ⁴ , and we can recall that the temperature can be defined as with appropriate variables held constant. The only way to satisfy such a requirement is to have σ go as τ ³ .