Quantum Statistics in Detail

For non-interacting quantum particles:

$\langle n_i \rangle = \frac{1}{e^{\beta(\epsilon_i - \mu)} \pm 1}$

Where $+$ is for fermions (Fermi—Dirac) and $-$ for bosons (Bose—Einstein).

Fermions (half-integer spin): Pauli exclusion limits $\langle n_i \rangle \leq 1$.

Bosons (integer spin): No restriction on occupation number; $\langle n_i \rangle$ can diverge when $\epsilon_i = \mu$.

For a 3D gas of $N$ non-interacting fermions in volume $V$:

$N = \sum_{\mathbf{k}} \frac{1}{e^{\beta(\hbar^2 k^2/2m - \mu)} + 1} \xrightarrow{\text{continuum} \frac{V}{(2\pi)^3}\int d^3k\, f(\epsilon_k)}$

The Fermi energy at $T = 0$:

$\epsilon_F = \frac{\hbar^2}{2m}(3\pi^2 n)^{2/3}$

Where $n = N/V$ is the number density. The Fermi temperature is $T_F = \epsilon_F/k_B$.

At low temperature ($T \ll T_F$), the Sommerfeld expansion gives:

$E = \frac{3}{5}N\epsilon_F\left[1 + \frac{5\pi^2}{12}\left(\frac{T}{T_F}\right)^2 + \cdots\right]$

$C_V = Nk_B\frac{\pi^2}{2}\frac{T}{T_F} + \cdots$

The linear specific heat is a hallmark of degenerate Fermi systems.

For bosons, the chemical potential must satisfy $\mu \leq \epsilon_0$ (ground state energy). When $\mu \to \epsilon_0$A macroscopic fraction of particles condenses into the ground state.

The critical temperature for BEC in 3D:

$T_c = \frac{2\pi\hbar^2}{mk_B}\left(\frac{n}{\zeta(3/2)}\right)^{2/3}$

Where $\zeta(3/2) \approx 2.612$.

Below $T_c$The condensate fraction is:

$\frac{N_0}{N} = 1 - \left(\frac{T}{T_c}\right)^{3/2}$

1. Confusing FD and BE limits: At high temperature ($T \gg T_F$ or $T \gg T_c$), both distributions reduce to the Maxwell—Boltzmann distribution. The quantum statistical corrections vanish when the interparticle spacing is much larger than the thermal de Broglie wavelength.

2. Chemical potential for bosons: For bosons, $\mu$ must always be less than the ground state energy. Setting $\mu > \epsilon_0$ gives negative occupation numbers, which is unphysical. At $T_c$, $\mu \to 0$ (for a free gas with $\epsilon_0 = 0$).

3. Continuum approximation validity: The $g(\epsilon) \propto \sqrt{\epsilon}$ density of states assumes a large volume. For small systems (nanoparticles, quantum dots), the discrete level structure becomes important and the integral approximation fails.

4. BEC requires dimensionality: In 1D and 2D, Bose—Einstein condensation does not occur in a uniform gas (Hohenberg’s theorem). Trapping potentials can, however, create quasi-condensates in lower dimensions.

• Metals and degenerate fermions: The electron specific heat $C_V = \gamma T$ in metals directly reflects the Fermi degeneracy. The Sommerfeld parameter $\gamma$ measures the density of states at the Fermi level.
• Ultracold atoms: BEC in dilute atomic gases (Rb, Na, Li) enables studies of superfluidity, quantised vortices, and matter-wave interferometry.
• Neutron stars: Degenerate neutron Fermi pressure supports neutron stars against gravitational collapse, with $\epsilon_F \sim 50$ MeV.
• White dwarfs: Electron degeneracy pressure balances gravity, with the Chandrasekhar mass limit arising from relativistic Fermi gas physics.

• Particle physics: Neutrinos in the early universe follow FD statistics. The neutrino background temperature ($T_\nu \approx 1.95$ K) is slightly lower than the CMB due to $e^+e^-$ annihilation heating the photon bath.
• Condensed matter: Heavy fermion materials have effective masses $m^* \gg m_e$, producing Fermi temperatures $T_F \sim 10$—$100$ K where quantum degeneracy meets accessible laboratory conditions.
• Cosmology: The Bose—Einstein condensate has been proposed as a dark matter candidate (fuzzy dark matter), with a de Broglie wavelength of kiloparsec scales smoothing small-scale structure.
• Quantum information: Degenerate Fermi gases in optical lattices simulate the Hubbard model, with the Pauli exclusion principle naturally enforcing the no-double-occupancy constraint at half-filling.