We derived in Section 1 the Boltzmann factor by allowing the temperature to change. If we generalize and allow the number of particles to change as well, we can obtain a ratio of probabilities:
The structure of the above is very similar to what we encountered before. Ignoring the new term returns the familiar form from before. Any term of the form e(Nμ-)/τ is called a Gibbs factor.
So far we only have a mechanism for relative probabilities. To obtain absolute probabilities, we need to have a sum that adds up all of the Gibbs factors. This sum is called the Gibbs sum and is given by:
Note that in taking the sum we begin with N = 0, which has its own corresponding energies , and sum over all the states with N = 0. Then we move to N = 1, and so on.
We can calculate the absolute probability of occupation of a state now using the Gibbs factor and the Gibbs sum just as we did with the Boltzmann factor and sum earlier.