Fermi Gas Thermodynamics

In our last lesson, we explored the ground state of an ideal Fermi gas at absolute zero. We saw how the Pauli exclusion principle dictates that fermions fill all available energy states up to a maximum, the Fermi energy $E_{F}$ , creating a sharp step-function in the occupation number. This T=0 picture provides a powerful, if simplified, model for electrons in metals.

Now, we will move beyond absolute zero to understand how a Fermi gas behaves at any finite temperature $T$ . Our objective is to derive the integral expressions for the fundamental thermodynamic quantities: the total particle number (N), the internal energy (U), and the pressure (P). These general expressions form the bedrock for analyzing the properties of Fermi systems in more realistic, non-zero temperature scenarios. They are the essential toolkit we will need before we can tackle specific low-temperature phenomena like the electronic heat capacity of metals.

1. The Grand Canonical Ensemble: A Natural Framework

When dealing with a large number of identical quantum particles, enforcing the constraint of a fixed particle number $N$ within the canonical ensemble is notoriously cumbersome. The need to sum over states while respecting either symmetrization (bosons) or anti-symmetrization (fermions) under a fixed-N constraint leads to significant combinatorial challenges. As you appreciate the elegance of mathematical structure in physical theories, you'll see that transitioning to the grand canonical ensemble, where the particle number is allowed to fluctuate, dramatically simplifies the problem.

In the grand canonical picture, we fix the chemical potential $μ$ and temperature $T$ , and the fixed-N constraint is replaced by an average value $⟨ N ⟩ = N$ . This allows us to treat the occupation of each single-particle state as statistically independent, making the calculations tractable. The following video from MIT OpenCourseWare provides an excellent discussion on why the coordinate-space path integral formulation for identical particles becomes fraught with "statistical interactions" and how switching to the occupation number basis within the grand canonical ensemble resolves these issues elegantly.

2. General Integral Expressions for N, U, and P

Having established the grand canonical ensemble as our framework, we can now derive the macroscopic properties of the Fermi gas. The strategy is to convert the discrete sums over quantum states into continuous integrals by introducing the density of states, $g (E)$ . For a non-relativistic particle of mass $m$ and spin $s$ in a 3D volume $V$ , you will recall this is given by:

g (E) = g_{s} \frac{V}{4 π ^{2}} (\frac{2 m}{ℏ ^{2}})^{3/2} E^{1/2}

where $g_{s} = 2 s + 1$ is the spin degeneracy factor (for electrons, $s = 1/2$ , so $g_{s} = 2$ ).

The central quantity is the Fermi-Dirac distribution, which gives the mean occupation number of a state with energy $E$ :

⟨ n (E)⟩ = \frac{1}{e ^{β (E - μ)} + 1}

where $β = 1/ (k_{B} T)$ and $μ$ is the chemical potential.

David Tong's lecture notes offer a concise and clear derivation of the integral expressions for N, U, and P starting from these two ingredients.

Let's consolidate and analyze these results.

Particle Number (N): The total number of particles is the integral of the mean occupation number over all available states.

N (μ, T, V) = \int_{0}^{\infty} d E g (E) ⟨ n (E)⟩ = \int_{0}^{\infty} \frac{g ( E )}{e ^{β (E - μ)} + 1} d E

This equation implicitly defines the relationship between the chemical potential $μ$ and the particle number $N$ at a given temperature. In a typical experiment, $N$ is fixed, so one must, in principle, solve this integral equation to find how $μ$ depends on $T$ .

Internal Energy (U): Similarly, the total internal energy is the first moment of the distribution.

U (μ, T, V) = \int_{0}^{\infty} d E E g (E) ⟨ n (E)⟩ = \int_{0}^{\infty} \frac{E g ( E )}{e ^{β (E - μ)} + 1} d E

Pressure (P) and the Equation of State:
The pressure is derived from the grand potential, $Φ = - P V = - k_{B} T ln Ξ$ . For fermions, the grand partition function $Ξ$ gives:

P V = k_{B} T \int_{0}^{\infty} d E g (E) ln (1 + e^{- β (E - μ)})

As shown in Tong's notes and a common technique in this field, we can relate this to the internal energy $U$ via integration by parts. Since $g (E) \propto E^{1/2}$ , let's define a primitive function $G (E) = \int_{0}^{E} g (E^{'}) d E^{'} \propto E^{3/2}$ .
Then, integrating by parts:

P V = k_{B} T [G (E) ln (1 + e^{- β (E - μ)})]_{0}^{\infty} - k_{B} T \int_{0}^{\infty} G (E) \frac{- β ( - 1 ) e ^{- β (E - μ)}}{1 + e ^{- β (E - μ)}} d E

The boundary term vanishes at both ends. The remaining integral simplifies to:

P V = \int_{0}^{\infty} G (E) \frac{1}{e ^{β (E - μ)} + 1} d E

Since $G (E) = \int_{0}^{E} C E^{' 1/2} d E^{'} = \frac{2}{3} C E^{3/2} = \frac{2}{3} E \cdot (C E^{1/2}) = \frac{2}{3} E g (E)$ (this is a slight abuse of notation, but the proportionality is key), we arrive at a profound result:

P V = \frac{2}{3} \int_{0}^{\infty} \frac{E g ( E )}{e ^{β (E - μ)} + 1} d E = \frac{2}{3} U

This relationship, $P = \frac{2}{3} (U / V)$ , is identical to that of a classical ideal gas and holds for any non-relativistic ideal gas, regardless of quantum statistics, at any temperature. It stems directly from the energy-momentum relation $E \propto p^{2}$ in 3D.

For a walkthrough of these derivations in a lecture format, the following video provides a clear, step-by-step calculation.

3. A More Formal Notation: Fermi-Dirac Integrals

The integrals for $N$ , $U$ , and $P$ appear frequently in the study of Fermi gases. To handle them systematically, it is conventional to define a standardized family of functions called the Fermi-Dirac integrals. While the notation can vary, a common form is:

f_{ν} (z) = \frac{1}{Γ ( ν )} \int_{0}^{\infty} \frac{x ^{ν - 1}}{z ^{- 1} e ^{x} + 1} d x

where $z = e^{β μ}$ is the fugacity, $x = β E$ is the dimensionless energy, and $Γ (ν)$ is the Gamma function.

Using this definition, our expressions for $N$ and $U$ (and thus $P$ ) can be written more compactly. After substituting $E = x / β$ and the form of $g (E)$ into our integrals for $N$ and $U$ , and collecting constants, we find:

Particle Density: $\frac{N}{V} = g_{s} (\frac{m k _{B} T}{2 π ℏ ^{2}})^{3/2} f_{3/2} (z) = \frac{g _{s}}{λ ^{3}} f_{3/2} (z)$
Internal Energy: $U = \frac{3}{2} N k_{B} T \frac{f _{5/2} ( z )}{f _{3/2} ( z )}$
Pressure: $P V = N k_{B} T \frac{f _{5/2} ( z )}{f _{3/2} ( z )}$

This formalism, which is detailed in the UMD Physics notes, is extremely useful in advanced treatments and numerical solutions, as it packages the complex temperature and chemical potential dependence into these standard functions.

Ideal Quantum Gases

These notes introduce the formal definition of the Fermi functions and use them to express the thermodynamic quantities.

Please review the section "Thermodynamics of nearly degenerate Fermi gases" (p. 14-15). Focus on how the total number of particles N and the internal energy U are re-expressed using the family of functions f_n(z). Seeing this compact notation will be valuable as you encounter more advanced literature.

Conclusion

In this lesson, we have taken the crucial step of generalizing our description of the ideal Fermi gas from the simple T=0 ground state to a full thermodynamic description at arbitrary temperature. We leveraged the power of the grand canonical ensemble to derive the fundamental integral expressions that govern the system's properties.

Key Takeaways:

The grand canonical ensemble is the most convenient framework for studying ideal quantum gases, as it circumvents the difficult combinatorial constraints of fixing particle number.
The macroscopic properties of an ideal Fermi gas—particle number $N$ , internal energy $U$ , and pressure $P$ —are given by integrals over the single-particle density of states weighted by the Fermi-Dirac distribution.
The key integral expressions are: $N = \int_{0}^{\infty} \frac{g ( E )}{e ^{β (E - μ)} + 1} d E U = \int_{0}^{\infty} \frac{E g ( E )}{e ^{β (E - μ)} + 1} d E$
For any non-relativistic ideal gas (classical, Bose, or Fermi), the pressure and internal energy are related by $P V = \frac{2}{3} U$ . This is a direct consequence of the kinetic energy scaling with momentum squared in three dimensions.

These integrals cannot be solved analytically in a simple closed form. However, they are the exact starting point for powerful approximation methods. In our next lesson, we will develop one such method, the Sommerfeld expansion, which allows us to solve these integrals as an asymptotic series in the physically crucial limit of low temperatures ( $T ≪ T_{F}$ ). This will lead directly to one of the landmark results of condensed matter physics: the linear temperature dependence of the electronic heat capacity in metals.

Recommended resources

1. The Grand Canonical Ensemble: A Natural Framework

2. General Integral Expressions for N, U, and P

3. A More Formal Notation: Fermi-Dirac Integrals

Conclusion