A Brief Introduction to Bose and Fermi Gases

This post explains the physical picture behind Bose gases and Fermi gases. The mathematics is kept light so that it is accessible with little background.

A problem with the classical ideal gas

By the equipartition theorem, each particle of a classical monatomic ideal gas has mean energy \(\frac{3}{2}kT\). The heat capacity per particle is therefore \(C_V = \frac{3}{2}k\), or \(\frac{3}{2}R\) per mole, which is constant and non-zero at all temperatures.

This contradicts the third law of thermodynamics, which requires \(C_V \to 0\) as \(T \to 0\). At low temperatures the classical picture must break down, and quantum statistics takes over.

Fermi gases

Quantum particles come in two types. Bosons have integer spin, and fermions have half-integer spin (\(\tfrac12, \tfrac32, \ldots\)).

The crucial difference is that fermions obey the Pauli exclusion principle: no two identical fermions can occupy the same quantum state. At low temperatures, fermions therefore cannot all crowd into the lowest-energy states; they fill the states one by one up to the Fermi energy. A Fermi gas consequently has a much higher energy than a classical gas at the same low temperature. As \(T\) increases this effect becomes less important, and the energy approaches that of the classical gas, an example of the correspondence principle.

The heat capacity of a Fermi gas tends to \(0\) as \(T \to 0\), in agreement with the third law:

Heat capacity of an ideal Fermi gas. (Source)

Bose gases

Bosons are not restricted in this way. Below a critical temperature, a macroscopic fraction of them occupies the single lowest-energy state: this is Bose–Einstein condensation. As a result, the energy of a Bose gas at low temperatures is much lower than that of a classical gas. At high temperatures it again approaches the classical limit.

The heat capacity of a Bose gas also tends to \(0\) as \(T \to 0\):

Specific heat of an ideal Bose gas. (Source)

Comparing the energies

Energy of Bose, classical and Fermi gases. (Source)

Quantum effects explain where classical physics fails at low temperatures. Exclusion pushes the energy of a Fermi gas up, condensation pulls the energy of a Bose gas down, and both bring the heat capacity to zero as the third law requires.

Changes on 25 October 2026

What was wrong. Nothing in the physics. Some statements about the heat capacity did not say whether they were per particle or per mole.

What changed. These are clarified, with a little more on what happens as \(T \to 0\). Previously titled “An Easy and Brief Introduction to Bose Gas and Fermi Gas”.