# Boltzmann distribution and the most probable configuration

According to boltzmann distribution, $$\frac{N_i}{ N_j}=e^{-\beta(\epsilon_i - \epsilon_j)}$$

Where $N_x$ is number of particles in the $x^{th}$ state in the most probable configuration. $\epsilon$ is energy of that state and $\beta$ is temperature dependent constant.

So, from this we can infer that a higher energy state will always have lesser number of particles than a lower one in the most probable configuration, right?

(For convenience, let us assume that every energy level has only one state.)

My issue is, couple small examples I take seem to violate this principle.

For instance, take the case where you have 5 particles with total energy $5\epsilon$in a system with energy levels $0,\epsilon, 2\epsilon...$. You chalk out a table to find weights of all possible configuration.

Image from Atkins solutions

Now, the most probable configurations are {2,2,0,1,0...} and {2,1,2,0,0...}. But both of these cannot be the most probable configuration if Boltzmann's law is true.

Where am I going wrong?