11.3 Maxwell-Boltzmann, Fermi-Dirac, and Bose-Einstein statistics

Classical vs Quantum Statistics

The three major distribution functions in statistical mechanics each describe how particles spread across energy levels. The key difference between them comes down to two questions: can you tell the particles apart, and can more than one particle sit in the same quantum state?

Distinguishability and Quantum Effects

Classical statistics (Maxwell-Boltzmann) treats particles as distinguishable, like labeled billiard balls. You can, in principle, track which particle is which. Quantum statistics, by contrast, treat particles as fundamentally indistinguishable. Two electrons in the same system have no hidden label that tells them apart.

This distinction matters because of the Gibbs paradox: if you use classical statistics to calculate the entropy of mixing two containers of the same gas, you get a nonzero entropy change even when the gases are identical. That's physically wrong. Quantum statistics resolve this by properly accounting for indistinguishability, dividing out the overcounting of identical configurations.

Fermions and Bosons

All quantum particles fall into one of two categories based on their spin:

• Fermions have half-integer spin ($1/2, 3/2, \ldots$) and obey the Pauli exclusion principle, meaning no two identical fermions can occupy the same quantum state. Examples: electrons, protons, neutrons.
• Bosons have integer spin ($0, 1, 2, \ldots$) and face no such restriction. Any number of identical bosons can pile into the same state. Examples: photons, gluons, the Higgs boson.

The underlying reason is wavefunction symmetry under particle exchange. Swapping two fermions introduces a minus sign (antisymmetric wavefunction), while swapping two bosons leaves the wavefunction unchanged (symmetric). These different symmetries lead directly to different distribution functions.

Distribution Functions for Quantum Systems

Derivation Using Ensemble Theory

Each distribution function is derived by maximizing the system's entropy subject to constraints on total particle number $N$ and total energy $E$. The method of Lagrange multipliers enforces these constraints, introducing the parameters $\beta = 1/k_BT$ (inverse temperature) and $\mu$ (chemical potential).

The three resulting distribution functions give the mean occupation number at energy $\epsilon$:

• Maxwell-Boltzmann:

$f_{MB}(\epsilon) = e^{-\beta(\epsilon - \mu)}$

Assumes distinguishable particles. Derived from the canonical ensemble. No restriction on occupation numbers, but the exponential decay means high-energy states are exponentially unlikely.

• Fermi-Dirac:

$f_{FD}(\epsilon) = \frac{1}{e^{\beta(\epsilon - \mu)} + 1}$

Derived from the grand canonical ensemble for indistinguishable fermions. The $+1$ in the denominator enforces the Pauli exclusion principle, capping the occupation at 1.

• Bose-Einstein:

$f_{BE}(\epsilon) = \frac{1}{e^{\beta(\epsilon - \mu)} - 1}$

Derived from the grand canonical ensemble for indistinguishable bosons. The $-1$ in the denominator allows occupation numbers to grow without bound, enabling macroscopic state occupation.

Comparing the Three Distributions

The three functions differ only in the denominator term ($0$, $+1$, or $-1$), but this small difference has large physical consequences:

• At high temperatures and low densities, $e^{\beta(\epsilon - \mu)} \gg 1$ for all occupied states, so the $\pm 1$ becomes negligible. Both quantum distributions converge to Maxwell-Boltzmann. This is the classical limit.
• At low temperatures, the distributions diverge sharply. The Fermi-Dirac function develops a sharp step at $\epsilon = \mu$ (the Fermi energy), while the Bose-Einstein function can diverge at the ground state.

Occupation Numbers and Thermodynamic Properties

Average Occupation Numbers

The distribution function evaluated at energy $\epsilon_i$ directly gives the average occupation number $\langle n_i \rangle = f(\epsilon_i)$ for that level.

• For fermions: $0 \leq \langle n_i \rangle \leq 1$. Each state is either empty or singly occupied on average.
• For bosons: $\langle n_i \rangle$ can be any non-negative number, including values much greater than 1.
• For Maxwell-Boltzmann particles: $\langle n_i \rangle$ can take any non-negative value, but in practice it's typically much less than 1 in the classical regime.

Calculating Thermodynamic Quantities

Once you have the occupation numbers, macroscopic thermodynamic quantities follow from summation over all energy levels (or integration over the density of states):

• Total particle number: $N = \sum_i \langle n_i \rangle$
• Total energy: $E = \sum_i \epsilon_i \langle n_i \rangle$
• Heat capacity, entropy, and other quantities are derived from the appropriate partition function $Z$, which sums over all microstates weighted by the relevant statistics.

For Fermi-Dirac and Bose-Einstein systems, the natural framework is the grand partition function $\mathcal{Z}$, since particle number fluctuates in the grand canonical ensemble. The grand potential $\Phi = -k_BT \ln \mathcal{Z}$ then gives access to pressure, entropy, and particle number through standard thermodynamic relations.

Pauli Exclusion Principle and Bose-Einstein Condensation

Consequences of the Pauli Exclusion Principle

The Pauli exclusion principle (no two identical fermions in the same quantum state) shapes the structure of matter at every scale.

• Atomic structure: Electrons fill orbitals in order of increasing energy rather than all collapsing to the ground state. This is why the periodic table exists.
• The Fermi sea: In a metal at $T = 0$, electrons fill all available states up to the Fermi energy $\epsilon_F$. States above $\epsilon_F$ are empty. At finite but low temperatures, only electrons within roughly $k_BT$ of $\epsilon_F$ can be thermally excited, which explains why the electronic heat capacity of metals is much smaller than classical predictions.
• Electrical and thermal conductivity: The Fermi sea structure determines which electrons can participate in conduction. Only those near the Fermi surface respond to applied fields.

Bose-Einstein Condensation

When a gas of bosons is cooled below a critical temperature $T_c$, a macroscopic fraction of the particles drops into the single lowest-energy quantum state. This is Bose-Einstein condensation (BEC).

The condensate is a coherent, macroscopic quantum state where the usual thermal fluctuations are suppressed. Key phenomena associated with BEC and bosonic condensation include:

• Superfluidity: Liquid helium-4 below about 2.17 K flows without viscosity, a direct consequence of bosonic condensation.
• Superconductivity: Electrons (fermions) can form Cooper pairs that behave as composite bosons, condensing into a state with zero electrical resistance.

Applications of BEC physics include atomic clocks (precision measurements using ultracold atoms), superconducting qubits for quantum computing, and fundamental tests of quantum mechanics at macroscopic scales.

Role in Determining Properties of Quantum Systems

The fermion/boson distinction propagates through all of physics. Fermionic statistics govern atomic shell structure, the stability of white dwarfs (electron degeneracy pressure), and the behavior of quarks and leptons. Bosonic statistics underlie laser operation (stimulated emission into an already-occupied photon mode), superfluidity, and the force-carrying gauge bosons of the Standard Model.

Understanding which statistics apply to a given system is the starting point for predicting its thermodynamic behavior, phase transitions, and response to external fields.