Common Pitfalls

Confusing the microcanonical, canonical, and grand canonical ensembles. The microcanonical ensemble describes an isolated system with fixed $E, V, N$. The canonical ensemble describes a system in contact with a heat bath at fixed $T, V, N$. The grand canonical ensemble describes a system exchanging both energy and particles, at fixed $\mu, V, T$.

Forgetting the $1/N!$ for indistinguishable particles. Without this factor, entropy is not extensive and the Gibbs paradox arises. This is essential for all quantum statistical mechanics.

Applying the equipartition theorem to quantum systems. At temperatures below the characteristic energy spacing ($k_BT \ll \Delta E$), the relevant degrees of freedom are “frozen out” and do not contribute to $C_V$.

Assuming the classical limit always applies. Electrons in metals are degenerate ($T \ll T_F$) and must be treated with Fermi-Dirac …/4-statistics-and-probability/2*statistics. Helium-4 at low temperatures exhibits Bose-Einstein condensation and superfluidity. The classical limit $n\lambda*{\mathrm{th}^3 \ll 1}$ is violated in these cases.

Confusing $\mu = 0$ for bosons with $\mu$ for fermions. For bosons, $\mu \leq \varepsilon_0$ and $\mu \to 0$ at BEC. For fermions, $\mu \approx \varepsilon_F$ at low temperatures and can be much larger than $\varepsilon_0$.

Using mean field critical exponents in 2D. Mean field theory gives $\beta = 1/2$ everywhere, but the exact 2D Ising result is $\beta = 1/8$. Mean field theory is qualitatively wrong in low dimensions.