Quantum Statistics

Classical statistical mechanics treats particles as distinguishable: you can label them, track them, and tell them apart. Quantum mechanics changes everything. Identical particles are truly indistinguishable: not just hard to tell apart, but fundamentally identical in a way that has no classical analog. This indistinguishability leads to two radically different types of statistics, with consequences that range from the stability of matter to the glow of the cosmic microwave background.

Counting States: Why Classical Fails

Consider two particles, each able to occupy states $A$ or $B$. Classically, there are four configurations: $(1A, 2B)$, $(1B, 2A)$, $(1A, 2A)$, $(1B, 2B)$. The "both in $A$" configuration has probability $1/4$.

For indistinguishable particles, swapping labels changes nothing. There are only three distinct configurations: both in $A$, both in $B$, and one in each. For bosons, all three are equally likely, giving probability $1/3$ for "both in $A$", bosons are more likely to bunch together than classical particles.

For fermions, the Pauli exclusion principle forbids "both in $A$" and "both in $B$". Only the configuration "one in each" is allowed. Fermions are forced apart.

The Distribution Functions

Using the grand canonical ensemble, we can derive the average occupation number $\langle n_i \rangle$ for a single-particle state with energy $\epsilon_i$:

The only difference is $-1$ versus $+1$ in the denominator, yet this single sign change produces utterly different physics. In the Fermi-Dirac case, $\langle n_i \rangle$ can never exceed 1, the exclusion principle is automatically enforced. In the Bose-Einstein case, $\langle n_i \rangle$ can be arbitrarily large, allowing macroscopic occupation of a single state.

The Classical Limit

When $e^{\beta(\epsilon_i - \mu)} \gg 1$ (high temperature, low density), both quantum distributions reduce to the Maxwell-Boltzmann distribution. The criterion for quantum effects to be important is that the thermal de Broglie wavelength $\lambda_{\text{th}}$ be comparable to the inter-particle spacing:

$$\lambda_{\text{th}} = \sqrt{\frac{2\pi\hbar^2}{mk_BT}} \gtrsim \left(\frac{V}{N}\right)^{1/3}$$

For air at room temperature, $\lambda_{\text{th}} \sim 0.02$ nm while the spacing is $\sim 3$ nm, so classical statistics work perfectly. For electrons in a metal or helium-4 near absolute zero, quantum effects dominate completely.

Bose-Einstein Condensation

Something extraordinary happens to bosons at very low temperature. As $T$ decreases, the chemical potential $\mu$ rises toward the ground state energy. At a critical temperature $T_c$, a macroscopic fraction of particles suddenly collapses into the single lowest-energy state.

This is not merely a large occupation, it is a genuine phase transition. The condensate behaves as a single quantum object on a macroscopic scale. Liquid helium-4 below 2.17 K becomes a superfluid: it flows without viscosity, climbs walls, and exhibits quantized vortices. In 1995, BEC was achieved in dilute atomic gases (rubidium, sodium), earning Cornell, Wieman, and Ketterle the 2001 Nobel Prize.

The Fermi Gas

Fermions tell a completely different story. At zero temperature, the exclusion principle forces particles to stack up into progressively higher energy states, filling a "Fermi sea" up to the Fermi energy $E_F$:

$$E_F = \frac{\hbar^2}{2m}\left(\frac{3\pi^2 N}{V}\right)^{2/3}$$

At $T = 0$, all states below $E_F$ are filled and all states above are empty. The distribution function is a sharp step. At small but nonzero $T$, only particles within $\sim k_B T$ of $E_F$ can be thermally excited, because states deeper in the Fermi sea have no nearby empty states to jump to.

Electrons in Metals

Conduction electrons in a metal form a nearly free Fermi gas. The Fermi energy is typically $E_F \sim 5$ eV, corresponding to a Fermi temperature $T_F = E_F/k_B \sim 50{,}000$ K. Since room temperature ($\sim 300$ K) is far below $T_F$, the electron gas is deeply quantum, or "degenerate." This explains why metals have a much smaller heat capacity than the classical equipartition prediction of $\frac{3}{2}k_B$ per electron.

The electronic heat capacity is linear in $T$:

$$C_V^{\text{el}} = \frac{\pi^2}{2} N k_B \frac{T}{T_F}$$

At room temperature, this is only about 1% of the classical value. Most electrons are frozen deep in the Fermi sea, unable to absorb thermal energy.

White Dwarfs and Neutron Stars

Fermi pressure, the resistance of fermions to compression due to the exclusion principle, supports dead stars against gravitational collapse:

White dwarfs are supported by electron degeneracy pressure. The Chandrasekhar limit of $\sim 1.4 M_\odot$ is the maximum mass that electron degeneracy can support. Beyond this, gravity wins.

Neutron stars are supported by neutron degeneracy pressure. They are essentially giant atomic nuclei, with densities $\sim 10^{17}$ kg/m$^3$ and radii of only $\sim 10$ km.

Blackbody Radiation and Planck's Law

Photons are bosons with a crucial simplification: they can be created and destroyed freely, so their chemical potential is $\mu = 0$. The average number of photons in a mode with frequency $\nu$ is:

$$\langle n \rangle = \frac{1}{e^{h\nu/k_BT} - 1}$$

Combined with the density of electromagnetic modes, this gives:

The Stefan-Boltzmann law for total radiated power follows by integrating Planck's law:

$$P = \sigma A T^4, \quad \sigma = \frac{2\pi^5 k_B^4}{15 h^3 c^2}$$

Wien's displacement law gives the peak frequency: $\nu_{\text{max}} \propto T$. The cosmic microwave background is an almost perfect blackbody at $T = 2.725$ K, a relic of the early universe, and one of the most precisely measured spectra in all of physics.

Phonons and the Debye Model

Lattice vibrations in a solid can be quantized as phonons: bosons with $\mu = 0$, just like photons. Applying Bose-Einstein statistics to phonons, Debye derived the heat capacity of solids:

$$C_V = 9Nk_B \left(\frac{T}{\Theta_D}\right)^3 \int_0^{\Theta_D/T} \frac{x^4 e^x}{(e^x - 1)^2} dx$$

where $\Theta_D$ is the Debye temperature. At high $T$, this reproduces the classical Dulong-Petit law $C_V = 3Nk_B$. At low $T$, it gives $C_V \propto T^3$, in excellent agreement with experiment. The classical equipartition theorem could not explain the observed low-temperature behavior.

Key Insights

• Identical quantum particles are fundamentally indistinguishable, leading to Bose-Einstein or Fermi-Dirac statistics depending on spin
• Bosons have occupation $\langle n \rangle = 1/(e^{\beta(\epsilon - \mu)} - 1)$ and can pile into one state; fermions have $\langle n \rangle = 1/(e^{\beta(\epsilon - \mu)} + 1)$ and are limited to one per state
• Bose-Einstein condensation is a phase transition where a macroscopic number of bosons occupy the ground state
• Fermi degeneracy pressure supports white dwarfs and neutron stars against gravitational collapse
• Planck's blackbody law follows from Bose-Einstein statistics applied to photons
• The classical limit applies when the thermal de Broglie wavelength is much smaller than the inter-particle spacing
• Quantum statistics resolves failures of classical physics: the ultraviolet catastrophe, the low-temperature heat capacity of solids, and the heat capacity of metals

Looking Ahead

Quantum statistics governs the behavior of matter in extreme conditions, but some of the most dramatic phenomena occur at phase transitions, where the character of a system changes abruptly:

• Phase transitions: how systems change state, from water boiling to magnets demagnetizing
• Universality: why utterly different systems show the same critical behavior
• Spontaneous symmetry breaking: the mechanism behind the Higgs field, superconductivity, and ferromagnetism

These ideas connect statistical physics directly to the frontiers of particle physics and cosmology.