Fermi–Dirac State Occupation

Worked Example

Consider a state with an energy offset of exactly 1 unit above the chemical potential, and a thermal scale set to 0.5. This allows us to calculate the exact probability of the state being occupied by a fermion at thermal equilibrium under these specific conditions.

delta = 1 theta = 0.5 occupation = 1 / (exp(1 / 0.5) + 1) occupation = 1 / (exp(2) + 1) occupation = 1 / (7.389056 + 1) = 0.1192029

Experiment

Let's observe how the state occupation probability changes as we adjust the system's thermal scale and the state's relative energy offset.

1. Launch the Fermi–Dirac State Occupation module.
2. Select Thermal scale to 0 and observe the sharp step function for occupation at E=0.
3. Increase Thermal scale to 2 and watch the step broaden as thermal fluctuations allow higher energy states to be occupied.
4. Adjust the Energy offset to negative values and observe the occupation approaching 1 (filled state).

Limitations

This module visualizes the occupation probability of a single isolated state. It does not integrate over a density of states to find the total particle number, nor does it dynamically adjust the chemical potential to conserve total particle number as temperature changes. Therefore, it is most applicable when discussing a system where the chemical potential is fixed by a much larger reservoir.

References

The concepts demonstrated in this module are standard in statistical mechanics.

• Fermi-Dirac distribution derivation and thermodynamic properties