Occupation numbers in the classical limit¶
This notebook can be used to study the occupation numbers for Fermi-Dirac (FD) and Bose-Einstein (BE) statistics in the classical limit, to show they become what is expected classically.
We have
$$ \bar n_s = \frac{1}{e^{\beta(\varepsilon_s – \mu)} \pm 1}, $$where the plus sign is for FD particles and the minus sign is for BE particles. The classical limit is valid when
$$ \beta(\varepsilon_s – \mu) \gg 1, $$so by considering the occupation numbers for large energies, we see that
$$ \bar n_s \to \frac{N e^{\beta \varepsilon_s}}{\zeta}, $$the MB distribution.
Go to the Occupation Numbers section, and click “Run all” in the Cell menu above (if you are using CoLab, this is in the Runtime menu). After you have run the notebook once, you will see an interactive window which allows you to see the occupation numbers (blue dots correspond to FD, red dashes are BE, and the solid black curve is MB) as you change the energy range.
Things to try out:
Change, using the slider, the range of energy to see the figures become the in the high energy limit.
Look at the functions used and modify them for different values of $\beta$ and $\mu$, and construct your own figures to compare.
Create a function to compare the partition functions in the three different statistics, to see they agree in the high energy limit.
Functions¶
In this section, we have imported libraries that we need in order to simulate these outcomes. If you’re only interested in playing around with the results, just “run” the entire notebook and move to the next section. If you’re interested in coding and wish to modify this, have at it!