# Ratio of population in two energy levels using Maxwell-Boltzmann distribution

To calculate the ratio of population in two energy levels using the Maxwell-Boltzmann distribution, you need to compare the probabilities of the two energy levels. The ratio of populations can be determined by dividing the probability of being in one energy level by the probability of being in the other energy level.

The probability density function (PDF) of the Maxwell-Boltzmann distribution in terms of energy is given by:

$$f(E) = \sqrt{\frac{2}{\pi}} \frac{\sqrt{E}}{kT^{3/2}} \exp\left(-\frac{E}{kT}\right).$$

Let's consider two energy levels, $$E_1$$ and $$E_2,$$ with $$E_2>E_1.$$ The ratio of populations between these two levels, denoted as $$R$$, can be calculated as follows:

$$R = \frac{f(E_2)}{f(E_1)} = \frac{\sqrt{\frac{2}{\pi}} \frac{\sqrt{E_2}}{kT^{3/2}} \exp\left(-\frac{E_2}{kT}\right)}{\sqrt{\frac{2}{\pi}} \frac{\sqrt{E_1}}{kT^{3/2}} \exp\left(-\frac{E_1}{kT}\right)}$$

Simplifying this expression, we can cancel out common factors: $$R = \sqrt{\frac{E_2}{E_1}} \exp\left(\frac{E_1 - E_2}{kT}\right)$$ This equation gives you the ratio of population between the two energy levels $$E_1$$ and $$E_2$$ based on the Maxwell-Boltzmann distribution.

My question is that why do the factor of $$\sqrt{\frac{E_2}{E_1}}$$always neglected during calculations?

You give the Maxwell-Boltzmann distribution normalized for a single energy level. In a situation with multiple states, the correct normalization is the partition function: $$Z(\beta) = \sum_n \exp(-\beta E_n)$$ with $$\beta = 1/kT$$. The measure of population in state $$E$$ is then $$f(E) = \frac{1}{Z(\beta)} \exp(-\beta E)$$ With two energy levels, the ratio is $$f(E_2)/f(E_1) = \exp(-\beta(E_2 - E_1))$$ where the normalization (partition function) divides out, which is why you don't see it included.
The question really is about the difference between these two ratios, $$R = \sqrt{\frac{E_2}{E_1}} \exp\left(\frac{E_1 - E_2}{kT}\right)$$ and $$R' = \exp\left(\frac{E_1 - E_2}{kT}\right)$$ The following cartoon may help. Here microstates, shown as dots, are grouped by their energy.
• $$R$$ is the ratio of populations with energy $$E_2$$ relative to a population with energy $$E_1$$. This is represented by the ratio of the areas in this cartoon, if by area we mean the number of states contained in each population.
• $$R'$$ is the ratio of the probability of specific state $$s_B$$, whose energy happens to be $$E_2$$, to the probability of state $$s_A$$, whose energy happens to be $$E_1$$. I say "happens to be" to emphasize that we first pick two states whose probabilities we want to compare, then we look at the energy of the states in order to calculate the ratio of their probabilities.
In a simulation for example we start with some state $$A$$, make a small change, say by moving a particle to a new position to form a new state $$B$$, and ask "what is the probability to accept the new state in the simulation?" To answer this question we use the ratio $$R'$$.