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Fermi-Dirac and Bose-Einstein Distribution
Adrian Down
October 19, 2005
1 Fermi-Dirac distribution
1.1 Derivation
1.1.1 Review: Fermions
Recall that fermions have half-integer spin and therefore are subject to the Pauli exclusion principle.
1.1.2 Setup
For definiteness, consider electrons in a box. The results are completely general. The possible wave functions are found by solving the Schr¨odinger equa- tion. Each energy level can support only two states: spin up or spin down. Consider a particular state as the system and all other states in the box as the reservoir, as we did in the case of localized spins previously. We then have the case of a reservoir and a system with energy l. Since the system is a particular state, it can only contain N = 0 or N = 1 particles. The number of particles in the reservoir is then N 0 or N 0 − 1. We consider a Grand Canonical ensemble of such systems, and we are interested in the ensemble average.
1.1.3 Average occupancy
As always, the first step is to write down the partition function. There are only two possible states, and thus two terms in summation.
Z = (^) ︸︷︷︸e^0 N =0,=
1 ·μ−l ︸ ︷︷ ︸τ N =1,=l We want to know the average occupancy of this mode. Because there can be only particle in the system, the average occupancy is equal to the probability that the state is occupied. The probability is given by the Gibbs’s Factor,
〈N (l)〉 =
1 · e
1 ·μ−l τ 1 + e
1 ·μ−l τ
The conventional notation is
〈N (l)〉 = f () l →
f () =
e
−μ τ (^) + 1
This important result is the Fermi-Dirac distribution. It gives the probability of a state with energy and chemical potential μ being occupied in a system populated by fermions.
Note. • “f ” is used as the symbol for the distribution because it is the ’‘Fermi” distribution.
- The energy is understood to be a function of all relevant quantum numbers, so the subscript l is implied.
1.2 Interpretation
1.2.1 Effect of the chemical potential
To see the effect of the chemical potential, evaluate f () at τ = 0. There are two cases
< μ : e
−τ μ → e−∞^ ⇒ f () = 1 > μ : e
−μ τ (^) → e+∞^ ⇒ f () = 0
the state being occupied.
1 − f (μ + δ) =
e
δ τ (^) + 1 − 1 e δτ
=
e
δ τ e
δτ
e−^
δ τ (^) + 1 = f (μ − δ)
2 Bose-Einstein distribution
2.1 Derivation
2.1.1 Review: Bosons
Bosons have integer spin, and are not subject to the exclusion principle. We will see that the key difference between the boson and fermion distributions is the behavior of the chemical potential.
2.1.2 Grand partition function
Consider a similar reservoir setup as before. There are N 0 − N particles in the reservoir and N in the particular state. As always, begin with the partition function. The first two terms are the same, since they correspond to 0 and 1 particles, respectively, but there are now many more terms.
Z =
n=
e
n(μ−l) τ
Recast the sum as
∑
n=
xN^ where x = e−μ−τ
We must have that x < 1, or else the series is divergent. This imposes the condition that
μ − τ
< 0 ⇒ μ <
This condition results in extremely different behavior than the case of fermions, where where all of the states were at energies below μ.
Note. We assumed that the summation was infinite, but this is not really the case. If the bose-einstein system were small and the reservoir were not infinite, we would need a more careful treatment of the system. The error in the upper limit of the summation is essentially negligible for large systems.
From usual facts about geometric series,
Z =
∑^ (∞)
n=
xn^ =
1 − x
2.1.3 Average occupancy
The average occupancy is given by
〈N (l)〉 =
N xN ∑ xN
Use the usual trick of taking derivatives,
〈N (l)〉 =
x (^) dxd
xN ∑ xN^
x (^) dxd
1 −x
1 1 −x =
x (1 − x)^2
· (1 − x) =
x 1 − x
The Bose-Einstein distribution is
N () =
e τδ − 1
Note. • The only difference with the Fermi distribution is the sign of the 1 in the denominator.
- N ( = μ) → ∞ ⇒ the state of = μ is not possible, in agreement with our previous results.
