Example 7.7: Fermi–Dirac distribution from detailed balancing Assuming Pauli’s principle, energy conservation and the principle of microscopic reversibility, also known as the principle of detailed balancing, (i.e. that the number of reactions 1 + 2 → 3 + 4 is the same as the number of reactions 3 + 4 → 1 + 2), show that the probability for a fermion to occupy state i is fi =

1 1 + eα+βi

, (7.71)

where α and β are constants. [Hint: Consider first the probability of reactions 1 + 2  3 + 4, with 1 − fi denoting the probability that no particle occupies state i]. What do you conclude for reactions 1 + 2 + 3  4 +5 + 6?