yields the average particle number in the individual state |ui〉.
2-a. Fermion distribution function
As the occupation number only takes the values 0 and 1, the first bracket in expression (17) is equal to [e–β(ei – μ)]; as for the other modes (k ≠ i) contribution, in the second bracket, it has already been computed when we determined the partition function. We therefore obtain:
(18)
Multiplying both the numerator and denominator by 1 + e–β (ei – μ) allows reconstructing the function Z in the numerator, and, after simplification by Z, we get:
We find again the Fermi-Dirac distribution function
(20)
This distribution function gives the average population of each individual state |ui〉 with energy e; its value is always less than 1, as expected for fermions.
The average value at thermal equilibrium of any one-particle operator is now readily computed by using (19) in relation (15).
2-b. Boson distribution function
The mode j = i contribution can be expressed as:
(21)
We then get:
(22)
which, using (11), amounts to:
where the Bose-Einstein distribution function
This distribution function gives the average population of the individual state |ui〉 with energy e. The only constraint of this population, for bosons, is to be positive. The chemical potential is always less than the lowest individual energy ek. In case this energy is zero, μ must always be negative. This avoids any divergence of the function
Hence for bosons, the average value of any one-particle operator is obtained by inserting (23) into relation (15).
2-c. Common expression
We define the function fβ as equal to either the function
where the number η is defined as:
(26)
2-d. Characteristics of Fermi-Dirac and Bose-Einstein distributions
We already gave in Complement CXIV (Figure 3) the form of the Fermi-Dirac distribution. Figure 1 shows both the variations of this distribution and the Bose-Einstein distribution. For the sake of comparison, it also includes the variations of the classical Boltzmann distribution:
which takes on intermediate values between the two quantum distributions. For a non-interacting gas contained in a box with periodic boundary conditions, the lowest possible energy e is zero and all the others are positive. Exponential eβ (ei – μ) is therefore always greater than e–βμ. We are now going to distinguish several cases, starting with the most negative values for the chemical potential.
1 (i) For a negative value of βμ with a modulus large compared to 1 (i.e. for μ ≪ —kBT, which corresponds to the right-hand side of the figure), the exponential in the denominator of (25) is always much larger than 1 (whatever the energy e), and the distribution reduces to the classical Boltzmann distribution (27). Bosons and fermions have practically the same distribution; the gas is said to be “non-degenerate”.
2 (ii) For a fermion system, the chemical potential has no upper boundary, but the population of an individual state can never exceed 1. If μ is positive, with μ ≫ kBT:– for low values of the energy, the factor 1 is much larger than the exponential term; the population of each individual state is almost equal to 1, its maximum value.– if the energy ei increases to values of the order μ the population decreases and when ei ≫ μ it becomes practically equal to the value predicted by the Boltzmann exponential (27).
Most of the particles occupy, however, the individual states having an energy less or comparable to μ, whose population is close to 1. The fermion system is said to be “degenerate”.
