The Boltzmann Factor and the Partition Function

For thermal equilibrium at T>0, a microstate of energy E has weight exp(-E/(k_B T)); k_B is Boltzmann's constant. For two levels with one state each and gap D>0, upper/lower = exp(-D/(k_B T)). If k_B T is much smaller than D, nearly all particles occupy the lower level. With degeneracies g_upper and g_lower, multiply the ratio by g_upper/g_lower.

The partition function Z sums exp(-E/(k_B T)) over all microstates. A level with g states contributes g times its factor. At T>0, E=0 has factor 1 and E>0 has factor below 1. Sum weights, not energies. For infinitely many nondegenerate levels E_n=nD, n=0,1,... and D>0, successive weights have ratio r=exp(-D/(k_B T)). At T>0, 0<r<1, so the geometric-series rule gives Z=1/(1-r).

A state's probability is its weight divided by Z. All state probabilities must sum to 1. For example, weights 1 and 1/3 give Z=4/3, so the probabilities are 3/4 and 1/4. Weights may be rescaled together for probability calculations, but distinguish those relative weights from the raw factors used to define Z.

For two fixed levels with one state each, k_B T much larger than the gap makes their populations nearly equal. Each individual state has probability approaching 1/2. Unequal level degeneracies change this limit: level populations approach the ratio of their numbers of states.

Heating changes the weights because the fixed gap divided by k_B T gets smaller. Thus exp(-D/(k_B T)) approaches 1. The gap need not shrink, and the states need not disappear. Equal high-temperature populations apply to a finite set of equally degenerate levels, not automatically to an infinite spectrum.

Leaving a positive-weight state out of Z gives an incomplete denominator. Dividing retained weights by that denominator describes a restricted set, not the full system. To find the full probabilities and mean energy, include every allowed state. Only a stated approximation can justify neglecting very small weights.