Statistical Distributions

Statistical ensembles and occupancy distribution functions for distinguishable and quantum particles.

Statistical distributions represent the cornerstone of statistical mechanics, providing the mathematical framework to predict how energy, velocities, and microscopic quantum states are distributed across vast ensembles of particles in thermal equilibrium. While macroscopic thermodynamics models bulk state variables such as pressure, volume, and temperature, distribution functions bridge microscopic quantum laws and observable laboratory measurements.

Depending on particle indistinguishability and the quantum spin constraints imposed by nature, physical systems are classified into three major statistical regimes:

1. Classical Maxwell-Boltzmann Statistics

The Maxwell-Boltzmann distribution governs systems composed of distinguishable particles, or quantum systems in the dilute limit where the thermal De Broglie wavelength is much smaller than the average interparticle separation (high-temperature and low-density regimes).

• Core Postulate: Every accessible microstate has equal a priori probability, yielding the canonical exponential Boltzmann factor: $f_{\text{MB}}(E) \propto e^{-\frac{E}{k_B T}}$
• Velocity Distribution: Derives the isotropic molecular speed profile in an ideal gas, characterizing the most probable speed, the mean speed, and the root-mean-square speed ($v_{\text{rms}}$).
• Equipartition Theorem: Each quadratic degree of freedom in the Hamiltonian contributes an average thermal energy of $\frac{1}{2} k_B T$, explaining classical heat capacities in monoatomic and diatomic gases.

2. Quantum Bose-Einstein Statistics

Particles with integer intrinsic spin ($s = 0, 1, 2, \dots$), known as bosons, are quantum mechanically indistinguishable and feature symmetric wavefunctions under particle exchange. They do not obey Pauli's exclusion principle, meaning any number of bosons can occupy the same quantum state.

• Mean Occupation Number: The expected number of particles in a state with energy $\epsilon$ is: $\langle n_{\text{BE}}(\epsilon) \rangle = \frac{1}{e^{\frac{\epsilon - \mu}{k_B T}} - 1}$ where $\mu \le 0$ denotes the chemical potential.
• Bose-Einstein Condensation (BEC): Below a critical transition temperature, a macroscopic fraction of particles condenses into the zero-momentum ground state ($\epsilon = 0$), giving rise to macroscopic quantum coherence, superfluidity, and vortex quantization.
• Photon and Phonon Radiation: Blackbody radiation and lattice vibrations correspond to boson gases with zero chemical potential ($\mu = 0$), recovering Planck's radiation law.

3. Quantum Fermi-Dirac Statistics

Particles with half-integer spin ($s = 1/2, 3/2, \dots$), such as electrons, protons, and neutrons, are fermions. Their total wavefunction is antisymmetric under particle exchange, and they strictly obey the Pauli exclusion principle: no two identical fermions can occupy the same quantum state simultaneously.

• Mean Occupation Number: The average occupation is strictly bounded between 0 and 1: $\langle n_{\text{FD}}(\epsilon) \rangle = \frac{1}{e^{\frac{\epsilon - \mu}{k_B T}} + 1}$
• Fermi Energy ($E_F$): At absolute zero ($T = 0\text{ K}$), fermions fill all available energy levels up to the Fermi energy, establishing the Fermi sea.
• Degeneracy Pressure and Electron Gas: Governs the electrical and thermal properties of metals, semiconductor physics, and prevents white dwarf and neutron stars from gravitational collapse.

Classical Correspondence Principle

When the system temperature is high or the particle density is low ($e^{(\epsilon - \mu)/k_B T} \gg 1$), both Bose-Einstein and Fermi-Dirac statistics smoothly converge to the classical Maxwell-Boltzmann limit, confirming the correspondence principle between quantum and classical mechanics.