Chapter 1

Fermi-Dirac Statistics

1.1 Introduction
In quantum mechanics, particles are classified into two categories: fermions and bosons.
While both types of particles are indistinguishable, they exhibit different behaviors under
particle exchange. In this article, we explore the distinction between fermions and bosons
using examples of two-particle systems.

1.1.1 Fermions
Fermions are particles with half-integer spin, such as electrons. They obey the Pauli
exclusion principle, which states that no two fermions can occupy the same quantum state
simultaneously. As a result, the wavefunction describing a system of fermions must be
antisymmetric under particle exchange. Consider a two-fermion system with quantum
states |n1 , s1 ⟩ and |n2 , s2 ⟩. The antisymmetrized wavefunction for the fermions is given by
a Slater determinant:
1
Ψfermions (x1 , x2 ) = √ [ψn1 (x1 )ψn2 (x2 ) − ψn1 (x2 )ψn2 (x1 )] (1.1)
2

This ensures that the wavefunction changes sign upon interchanging the positions of the
two fermions, satisfying the requirement of antisymmetry.

1.1.2 Bosons
Bosons, on the other hand, are particles with integer spin, such as photons. Unlike fermions,
bosons do not obey the Pauli exclusion principle and can occupy the same quantum state
simultaneously. The wavefunction describing a system of bosons must be symmetric under
particle exchange. For a two-boson system, the symmetric wavefunction could be:

1
Ψbosons (x1 , x2 ) = √ [ψn1 (x1 )ψn2 (x2 ) + ψn1 (x2 )ψn2 (x1 )] (1.2)
2

In this case, interchanging the positions of the two photons does not change the overall
sign of the wavefunction, reflecting the symmetric exchange statistics of bosons.
2 CHAPTER 1. FERMI-DIRAC STATISTICS

1.1.3 Differences between bosons and fermions

The table 1.1.3 illustrates the difference between bosons and fermions
Property Fermions Bosons
Spin Half-integer ( 12 , 32 ) Integer (0, 1, 2)
Pauli Exclusion Princi- Obey (cannot share the Do not obey (can occupy
ple same quantum state) the same state)
Quantum Statistics Fermi-Dirac distribution Bose-Einstein distribution
Behavior at Low T Form a degenerate Form a Bose-Einstein
Fermi gas condensate
Example Particles Electrons, protons, neu- Photons, gluons, W/Z
trons, quarks bosons, Higgs boson
Role in Physics Matter particles Force carriers

1.2 Types of Fermi Gases

1.2.1 (a) Ideal Fermi Gas
Example: Free Electron Gas in Metals
Theoretical Justification:
An ideal Fermi gas is a system of fermions that do not interact with each other except
through the Pauli exclusion principle. The electron gas in metals is a good example
because:
ˆ Electrons in metals move in a periodic potential but, to a first approximation, can
be considered free particles.
ˆ The only constraint is Fermi-Dirac statistics, which determines the energy distri-
bution of electrons.
ˆ The Sommerfeld model describes this system well, assuming no significant electron-
electron interactions.

Practical Example:
ˆ Conduction electrons in metals like Copper (Cu), Silver (Ag), and Gold
(Au) follow ideal Fermi gas behavior at moderate temperatures.

1.2.2 (b) Degenerate Fermi Gas

Example: Electrons in White Dwarfs
Theoretical Justification:
A degenerate Fermi gas occurs when the thermal energy is much smaller than the Fermi
energy (kB T ≪ EF ). This means that:
ˆ Almost all states below the Fermi level are filled.
ˆ The gas pressure comes from quantum degeneracy pressure, not thermal effects.
1.2. TYPES OF FERMI GASES 3

Practical Example:

ˆ White dwarfs are stellar remnants where electrons are highly degenerate, support-
ing the star against gravity.

ˆ The Chandrasekhar limit (∼ 1.4 solar masses) describes the balance between
degeneracy pressure and gravity.

1.2.3 (c) Non-Degenerate Fermi Gas

Example: Electrons in a Metal at High Temperatures
Theoretical Justification:

A non-degenerate Fermi gas occurs when the temperature is high compared to the
Fermi temperature (kB T ≫ EF ), meaning:

ˆ The occupation probability follows a classical Maxwell-Boltzmann distribution

rather than a step function.

ˆ The gas behaves more like an ideal classical gas with minor quantum corrections.

Practical Example:

ˆ At very high temperatures (e.g., in metals at temperatures above 100,000 K), the
electrons begin behaving classically rather than as a degenerate quantum sys-
tem.

1.2.4 (d) Weakly Non-Degenerate Fermi Gas

Example: Free Electron Gas in Metals at Moderate Temperature
Theoretical Justification:

A weakly non-degenerate Fermi gas occurs when the temperature is slightly above
the Fermi temperature (kB T ≈ EF ), leading to:

ˆ A mix of quantum and classical behaviors.

ˆ The occupation function starts deviating from a sharp step at ϵF , but quantum effects
still dominate.

Practical Example:

ˆ Semiconductors with moderate doping: At certain temperatures, the conduc-

tion electrons in Silicon (Si) or Germanium (Ge) behave as a weakly non-
degenerate gas.
4 CHAPTER 1. FERMI-DIRAC STATISTICS

1.2.5 (e) Strongly Non-Degenerate Fermi Gas

Example: Electrons in a Plasma at Very High Temperatures
Theoretical Justification:
A strongly non-degenerate Fermi gas occurs at extremely high temperatures (kB T ≫
EF ), where:

ˆ The gas behaves almost like an ideal classical gas.

ˆ The Fermi-Dirac distribution approaches the Maxwell-Boltzmann form.

Practical Example:
ˆ Electrons in a hot plasma (e.g., in the Sun’s corona or fusion reactors)
behave as a strongly non-degenerate gas because their thermal energy dominates
over quantum degeneracy effects.

1.2.6 (f ) Relativistic Fermi Gas

Example: Electrons in Neutron Star Crust
Theoretical Justification:
A relativistic Fermi gas occurs when the particle velocity approaches the speed
of light, meaning:

ˆ The energy of particles follows the relativistic dispersion relation:

p
E = p2 c2 + m2 c4

ˆ The equation of state is different from a non-relativistic Fermi gas.

Practical Example:
ˆ Electrons in the crust of a neutron star are relativistic due to the extreme
density (ρ ∼ 1010 g/cm3 ).

1.2.7 (g) Extreme Relativistic Fermi Gas

Example: Neutrino Gas in the Early Universe
Theoretical Justification:
An extreme relativistic Fermi gas occurs when the mass of the particles is negligible
compared to their momentum, so:
E ≈ pc.

ˆ The equation of state for such a gas becomes radiation-like (P ∝ ρ4/3 ).

ˆ This regime applies to particles with very high energy or very low mass.
1.3. FERMI DIRAC DISTRIBUTION 5

Practical Example:
ˆ Neutrino gas in the early universe: At temperatures much higher than the
neutrino mass (∼eV range), neutrinos behave as an extreme relativistic Fermi
gas, influencing cosmic evolution.

ˆ Electron gas in neutron stars: In the most compact neutron stars, electrons
become ultra-relativistic, contributing significantly to pressure.

1.2.8 Summary Table of Examples

Type of Fermi Gas Theoretical Example Practical Example

Ideal Fermi Gas Free electron gas (Som- Conduction electrons in
merfeld model) metals
Degenerate Fermi Gas Zero-temperature Fermi Electrons in white dwarfs
gas
Non-Degenerate Fermi High-temperature Fermi Electrons in hot metals
Gas gas
Weakly Non- Intermediate kB T ≈ EF Semiconductors (moder-
Degenerate Fermi ate doping)
Gas
Strongly Non- kB T ≫ EF Plasma electrons in fusion
Degenerate Fermi reactors
Gas p
Relativistic Fermi Gas v ≈ c, E ≈ p2 c2 + m2 c4 Electrons in neutron star
crust
Extreme Relativistic E ≈ pc, m ≈ 0 Neutrino gas in the early
Fermi Gas universe

1.3 Fermi Dirac distribution

The Fermi-Dirac distribution function gives the probability that a quantum state of energy
ϵ is occupied by a fermion (a particle obeying the Pauli exclusion principle, like electrons,
neutrons, or protons). It is given by:

1
f (ϵ) =
eβ(ϵ−µ) + 1
where:

ˆ β= 1
kB T (inverse temperature)

ˆ µ is the chemical potential

ˆ kB is Boltzmann’s constant

ˆ T is the temperature

ˆ ϵ is the energy of the state

6 CHAPTER 1. FERMI-DIRAC STATISTICS

1.3.1 Derivation Using the Grand Canonical Ensemble

In the grand canonical ensemble, the probability of a system being in a particular quantum
state depends on the total energy and number of particles. The probability that a quantum
state with energy ϵ is occupied by n fermions is given by the Boltzmann factor:

P (n) ∝ e−β(nϵ−nµ)

where n can only be 0 or 1 due to the Pauli exclusion principle (since fermions cannot
occupy the same state more than once). Thus, the probability of a state being empty
(n = 0) is:
1 1 1
P (0) = e−β(0·ϵ−0·µ) = = −β(ϵ−µ)
Z Z 1+e
And the probability of the state being occupied (n = 1) is:

1 −β(ϵ−µ) e−β(ϵ−µ)
P (1) = e = .
Z 1 + e−β(ϵ−µ)
Since f (ϵ) represents the average occupation number of the state (since n = 0 or n = 1,
the mean value is just P (1)), we obtain:

e−β(ϵ−µ)
f (ϵ) = P (1) = .
1 + e−β(ϵ−µ)

Dividing numerator and denominator by e−β(ϵ−µ) , we simplify:

1
f (ϵ) = .
eβ(ϵ−µ) + 1
This is the famous Fermi-Dirac distribution function.

ˆ If ϵ ≪ µ (low-energy states, well below the Fermi level), then eβ(ϵ−µ) ≈ 0, so f (ϵ) ≈ 1
(state is almost always occupied).

ˆ If ϵ ≫ µ (high-energy states), then eβ(ϵ−µ) ≫ 1, so f (ϵ) ≈ 0 (state is almost always

empty).

ˆ At ϵ = µ, we get f (ϵ) = 21 , meaning states at the chemical potential are half-occupied.

1.3.2 Derivation using entropy maximisation

For a system of fermions, each quantum state i can be either occupied (ni = 1) or empty
(ni = 0) due to the Pauli exclusion principle. The probability that a state is occupied is
fi , and the probability that it is empty is 1 − fi . The entropy of the system is given by
the Shannon entropy formula:
X
S = −kB [fi ln fi + (1 − fi ) ln(1 − fi )] (1.3)
i

where:

ˆ fi is the probability that quantum state i is occupied,

1.3. FERMI DIRAC DISTRIBUTION 7

ˆ 1 − fi is the probability that it is unoccupied,

ˆ kB is Boltzmann’s constant.

Our objective is to maximize S subject to constraints on the total energy and total number
of particles in the system. We must enforce two physical constraints: The total number of
particles is given by: X
N= fi (1.4)
i

The total energy of the system is: X

U= fi ϵi (1.5)
i

To incorporate these constraints in our maximization problem, we employ Lagrange mul-

tipliers α (for the particle number) and β (for the energy constraint). The function to
maximize is: X X
F =S−α fi − β f i ϵi (1.6)
i i

To maximize F, we take the derivative with respect to fi and set it equal to zero:

d
[−kB (fi ln fi + (1 − fi ) ln(1 − fi )) − αfi − βfi ϵi ] = 0 (1.7)
dfi

Differentiating each term:

−kB [ln fi + 1 − (ln(1 − fi ) + 1)] − α − βϵi = 0 (1.8)

Simplifying:
fi
−kB ln = α + βϵi (1.9)
1 − fi
Rearranging:
fi
= e−(α+βϵi )/kB (1.10)
1 − fi
1
We define β = kB T and µ = − α
β , which gives:

fi
= e−β(ϵi −µ) (1.11)
1 − fi

Solving for fi :
1
fi = (1.12)
eβ(ϵi −µ) + 1
Thus, we have derived the Fermi-Dirac distribution function:
1
f (ϵ) = (1.13)
e(ϵ−µ)/kB T + 1
This derivation demonstrates several important physical aspects:

ˆ The Fermi-Dirac distribution naturally emerges from the principle of entropy maxi-
mization.
8 CHAPTER 1. FERMI-DIRAC STATISTICS

ˆ The Pauli exclusion principle is inherently reflected in the form of the function, as
f (ϵ) never exceeds 1.

ˆ At very low temperatures (T → 0), the distribution approaches a step function:

– States with ϵ < µ are fully occupied (f (ϵ) ≈ 1)

– States with ϵ > µ are empty (f (ϵ) ≈ 0)

ˆ The chemical potential µ corresponds to the Fermi energy at absolute zero temper-
ature.

ˆ At high temperatures, the distribution approaches the classical Maxwell-Boltzmann

distribution.

1.4 Fugacity
Fugacity (z) is a generalized activity that represents an “effective pressure” or “corrected
chemical potential” of a gas, particularly in the context of real gases or quantum gases. It
is defined as: µ
z = eβµ = e kB T (1.14)
where:

ˆ µ is the chemical potential,

ˆ β= 1
kB T is the inverse temperature,

ˆ kB is Boltzmann’s constant,

ˆ T is the temperature.

1.5 Fugacity in a Fermi Gas

For an ideal Fermi gas, fugacity helps describe the statistical distribution of particles.
Since fermions obey Fermi-Dirac statistics, the mean occupation number for a quantum
state of energy ϵ is:
1 1
f (ϵ) = β(ϵ−µ) = 1 βϵ . (1.15)
e +1 ze +1

ˆ In the non-degenerate limit (T ≫ TF ), µ ≪ 0 and z ≪ 1, meaning f (ϵ) behaves

like a classical Maxwell-Boltzmann gas.

ˆ In the degenerate limit (T ≪ TF ), µ ≈ EF and z ≫ 1, meaning most states below

EF are occupied.

ˆ z = 1 when µ = 0 (this happens for a photon gas at equilibrium).

ˆ z > 1 for a highly degenerate Fermi gas, meaning many states are occupied.

ˆ z < 1 for a low-density Fermi gas, meaning the system behaves more like a classical
gas.
1.6. IDEAL FERMI GAS 9

1.6 Ideal Fermi gas

If i is the degeneracy or the statistical weight of the ith quantum state, then according to
FD Distribution law, the most probable distribution is given by
gi
ni =
eα+βϵi + 1
gi
ni = 1 βϵi
A e +1
µ 1
with α = −
, β= , and A = e−α
kT kT
The number of particle states lying between momentum space p and p+dp is the volume of a thi
p and p + dp, which in spherical polar coordinates is:

d3 p = p2 sin θ dp dθ dϕ
Integrating over the angular part:
Z π Z 2π
3 2
d p = p · dp sin θ dθ · dϕ = 4πp2 dp
0 0

The phase space volume is dx dy dz d3 p = V · d3 p.

In phase space (i.e., the 6D space of position and momentum), each independent quantum state oc
h3 .
Therefore, to count the number of available quantum states, we divide the classical phase space vo

V d3 p
g(p) dp =
h3
If gs = 2s + 1 is the spin degeneracy factor, then:

V d3 p
g(p) dp = gs ·
h3
The number of particles between momentum p and p + dp is:
g(p) dp
dn(p) dp = βp2
eα+ 2m + 1

4πV p2 dp 1
dn(p) dp = gs · · βp2
h3 α+
e 2m + 1
p2
Now, using ϵ = 2m ⇒ p dp = m dϵ:

4πV p · (p dp) 1
dn(ϵ) dϵ = gs · 3
· α+βϵ
h e +1
4πmV √ 1
dn(ϵ) dϵ = gs · 3
2mϵ dϵ · 1 βϵ
h Ae +1
√
4πmV 2m ϵ1/2
dn(ϵ) dϵ = gs · · 1 βϵ dϵ
h3 Ae +1
10 CHAPTER 1. FERMI-DIRAC STATISTICS

The total number of particles is:

Z ∞ √
4πmV 2m ∞ ϵ1/2
Z
n= dn = gs · 1 βϵ dϵ
0 h3 0 Ae +1
The total energy is:
∞
√
4πmV 2m ∞ ϵ3/2
Z Z
E= ϵ dn = gs · 1 βϵ dϵ
0 h3 0 Ae +1
Let x = f racϵkT , so that, The total number of particles rewritten as
Z ∞
2gs V 3/2 x1/2
n = · √ 3 (2πmkT ) 1 x
dϵ
πh 0 Ae + 1

gs V 3/2
n=· (2πmkT ) · f1 (α)
h3
Where Z ∞
2 x1/2
f1 (α) = √ 1 dx
π 0 A ex + 1
Similarly, The total energy is:
Z ∞
3 V 3/2
E= ϵ dn = gs 3 · (2πmKT ) KT f2 (α)
0 2 h
Where
∞
x3/2
Z
4
f2 (α) = √ 1 x dx
3 π 0 Ae +1
We can find the expressions for positive and negative values of α, but before that we have
to bring the concept of fermi energy into our picture.
As we know, in FD distribution, the quantity α is related to chemical potential µ or fermi
energy ϵF (T ) at temperature T as
µ ϵF (T )
α=− =−
kT kT
For a fermionic system at absolute Zero, all states with energies 0 < ϵ < ϵF (0) are com-
pletely filled and all states with energies ϵ > ϵF (0) are completely empty. The value of
ϵF (0) at T = 0K is determined by replacing the upper limit of integration with ϵF (0) in
the expression for total number of particles n:
√
4πmV 2m ϵF (0) 1/2
Z
n = gs · ϵ dϵ
h3 0
√
8πmV 2m
n = gs · ϵF (0)3/2
3h3
Which gives the energy of the highest occupied level at absolute zero as :
2/3
h2

3n
ϵF (0) = µ =
2m 4πV gs
Now that we have got the idea about fermi energy, we can proceed with our original dis-
cussion.
1.6. IDEAL FERMI GAS 11

1.6.1 Degeneracy of a Fermi Gas

The degree of degeneracy of a Fermi gas indicates how strongly quantum mechanical effects
dominate over classical behavior. It is typically measured using the ratio of thermal energy
to the Fermi energy:

kT
ϵF

kT
1.6.2 Interpretation of ϵF
ˆ kT ≫ ϵF : The gas behaves classically. It is in the non-degenerate regime.

ˆ kT ∼ ϵF : The gas is moderately degenerate, with partial quantum effects.

ˆ kT ≪ ϵF : The gas is strongly degenerate. Quantum statistics (Fermi-Dirac) are

essential.

1.6.3 Alternative Measure: Dimensionless Chemical Potential

Another widely used indicator of degeneracy is the dimensionless chemical potential:

µ
α=
kT
Here, µ is the chemical potential of the system at temperature T .

ˆ α ≫ 1: Strongly degenerate Fermi gas.

ˆ α ∼ 0: Weakly degenerate gas.

ˆ α ≪ 0: Classical limit (Maxwell-Boltzmann statistics valid).

1.6.4 Case 1: Slight Degeneracy, α > 0

At T > TF , the degeneracy is weak. In such a case, kT > ϵF (0), so α > 0, and A = e−α < 1.
The Fermi-Dirac distribution becomes:

1
−1
= Ae−x 1 + Ae−x = Ae−x 1 − Ae−x + A2 e−2x − A3 e−3x + . . .


1 x
Ae +1

Now,
∞ ∞
x1/2
Z Z
2 2
Ae−x 1 − Ae−x + A2 e−2x − . . .


f1 (α) = √ 1 x dx = √ dx
π 0 Ae + 1
π 0

A2 A3
f1 (α) = A − + − ...
23/2 33/2
Similarly,
A2 A3
f2 (α) = A − + − ...
25/2 35/2
12 CHAPTER 1. FERMI-DIRAC STATISTICS

Using these expressions, the total particle number n and energy E are:

A2 A3
 
gs V 3/2
n = 3 (2πmkT ) A − 3/2 + 3/2 − . . .
h 2 3
2
A3
 
3gs V 3/2 A
E= (2πmkT ) A − 5/2 + 5/2 − . . .
2h3 2 3
Taking the ratio:
A2 A3
E 3 A− 25/2
+ 35/2
− ... 3
= kT · A2 A3
= kT · η
n 2 A− + − ... 2
23/2 33/2
where the correction factor η represents deviation from ideal gas behavior. For A ≪ 1,
we can write:

A A2
η =1+ −+ ...
25/235/2
Thus, in the limit η → 1, a weakly degenerate Fermi gas behaves like an ideal gas.

1.6.5 Case 2: Strong Degeneracy, α ≪ 0

At T ≪ TF , the gas is strongly degenerate. In this case, kT ≪ ϵF (0), and α ≪ 0. The
Fermi-Dirac distribution becomes approximately a step function:
(
1 1 E<µ
f (E) = (E−µ)/kT ≈
e +1 0 E>µ
To account for finite temperature effects, we apply the Sommerfeld expansion. The
number density and total energy are:
Z ∞
gs V 4πp2
n= 3 dp
h 0 e(ϵ−µ)/kT + 1
Z ∞
gs V 4πp2
E= 3 ϵ · (ϵ−µ)/kT dp
h 0 e +1
p2
Changing variables to energy, with ϵ = 2m :
Z ∞ √
gs V ϵ
n= 4π(2m)3/2 dϵ
h3 0 e(ϵ−µ)/kT + 1
∞
ϵ3/2
Z
gs V
E= 4π(2m)3/2 dϵ
h3 0 e(ϵ−µ)/kT + 1
Applying Sommerfeld expansion:
Z ∞ Z µ
f (ϵ) π2
dϵ = f (ϵ)dϵ + (kT )2 f ′ (µ) + · · ·
0 e(ϵ−µ)/kT + 1 0 6
For number density:

π2
 
gs V 3/2 2 3/2 2 −1/2
n= 4π(2m) µ + (kT ) µ
h3 3 8
1.6. IDEAL FERMI GAS 13

For total energy:

5π 2
 
gs V 3/2 2 5/2 2 1/2
E = 3 4π(2m) µ + (kT ) µ
h 5 8

Now, define ϵF = µ(T = 0). At low temperatures:

" 2 #
π 2 kT

µ ≈ ϵF 1 −
12 ϵF
Substituting into the energy per particle:
" 2 #
5π 2 kT

E 3
= ϵF 1 +
n 5 12 ϵF

Final Result:
" 2 #
5π 2 kT

3
E = nϵF 1 +
5 12 ϵF
This result shows:

ˆ At T = 0, E = 53 nϵF , which is purely quantum mechanical.

ˆ The leading temperature correction is quadratic in T /TF , a hallmark of Fermi de-

generacy.

Hence, a strongly degenerate Fermi gas deviates significantly from classical ideal gas
behavior.