Question

What are the Boltzmann factors?

Answer 1

A general ratio of the population of states can be written in statistical mechanics as:

`N_i/N = (g_i e^(-betaepsilon_i))/(q) = (g_i e^(-betaepsilon_i))/(sum_i g_i e^(-betaepsilon_i))`

where:

• `g_i` is the degeneracy of state `i` with energy `epsilon_i`.
• `beta = 1/(k_BT)` is a constant containing the Boltzmann constant and temperature.
• `N_i` is the number of particles in state `i` and `N` is the total number of particles.

If we then consider a single state relative to energy zero, we have two states such that:

`N_1/N_0 = N_1/N cdot N/N_0`

`= (g_1 e^(-betaepsilon_1))/cancel(g_0 e^(-betaepsilon_0) + g_1 e^(-betaepsilon_1)) cdot cancel(g_0 e^(-betaepsilon_0) + g_1 e^(-betaepsilon_1))/(g_0 e^(-betaepsilon_0))`

Since the `N`'s cancel out, you can see that this can be extended to any number of states. If we then let energy zero be `epsilon_0 = 0`, then:

`(N_i)/(N_0) = (g_i)/(g_0) e^(-betaepsilon_i)`

Thus, the population of state `bbi` with some energy higher than energy zero is given by `e^(-betaepsilon_i) = e^(-epsilon_1//k_BT)`, weighted by the ratio of the degeneracies `g_i` and `g_0`.

We call `e^(-epsilon_i//k_BT)` the Boltzmann factor for state `i`.