Quantum statistics dive into how particles behave in groups. Fermi-Dirac and Bose-Einstein distributions explain the quirks of fermions and bosons, respectively. These concepts are key to understanding particle behavior in various systems.

This topic builds on earlier ideas in statistical thermodynamics. It shows how quantum mechanics affects large-scale systems, connecting microscopic particle properties to macroscopic observables we can measure in experiments.

Quantum Statistical Distributions

Fermi-Dirac Distribution

• Describes the statistical behavior of fermions (particles with half-integer spin, such as electrons, protons, and neutrons)
• Takes into account the Pauli exclusion principle, which states that no two identical fermions can occupy the same quantum state simultaneously
• Characterized by the Fermi-Dirac distribution function $f(E) = \frac{1}{e^{(E-\mu)/kT}+1}$, where $E$ is the energy of the state, $\mu$ is the chemical potential, $k$ is the Boltzmann constant, and $T$ is the temperature
• At absolute zero temperature $(T=0)$, the distribution becomes a step function, with all states below the Fermi energy $(\mu)$ being occupied and all states above it being empty
• As temperature increases, the distribution smoothly transitions from the step function to a more spread-out distribution, allowing some states above the Fermi energy to be occupied

Bose-Einstein Distribution and Maxwell-Boltzmann Distribution

• Bose-Einstein distribution describes the statistical behavior of bosons (particles with integer spin, such as photons and certain atoms)
• Unlike fermions, bosons can occupy the same quantum state simultaneously, leading to phenomena such as Bose-Einstein condensation (a state of matter where a large fraction of bosons occupy the lowest energy state)
• Characterized by the Bose-Einstein distribution function $f(E) = \frac{1}{e^{(E-\mu)/kT}-1}$, where the symbols have the same meaning as in the Fermi-Dirac distribution
• Maxwell-Boltzmann distribution is a classical approximation that describes the statistical behavior of particles in a system when quantum effects are negligible
• Applies to systems where the particle density is low and the temperature is high enough that the quantum nature of the particles can be ignored
• Characterized by the Maxwell-Boltzmann distribution function $f(E) = e^{-(E-\mu)/kT}$, which is a simplified version of the Fermi-Dirac and Bose-Einstein distributions

Occupation Number and Density of States

• Occupation number $n_i$ represents the average number of particles occupying a particular energy state $i$
• For fermions, the occupation number can only be 0 or 1 due to the Pauli exclusion principle
• For bosons, the occupation number can be any non-negative integer
• Density of states $g(E)$ is a function that describes the number of quantum states available per unit energy interval at a given energy level
• Depends on the dimensionality and the dispersion relation of the system (e.g., free electrons in a metal, photons in a cavity, or phonons in a solid)
• The total number of particles in the system can be calculated by integrating the product of the occupation number and the density of states over all energy levels: $N = \int n(E) g(E) dE$

Particle Classifications

Fermions and the Pauli Exclusion Principle

• Fermions are particles with half-integer spin $(1/2, 3/2, ...)$ that obey the Pauli exclusion principle
• Examples of fermions include electrons, protons, neutrons, quarks, and neutrinos
• The Pauli exclusion principle states that no two identical fermions can occupy the same quantum state simultaneously
• This principle is responsible for the stability of matter, as it prevents electrons from collapsing into the lowest energy state, which would result in the collapse of atoms
• The Pauli exclusion principle also explains the electronic structure of atoms and the periodic table of elements

Bosons and Quantum Degeneracy

• Bosons are particles with integer spin $(0, 1, 2, ...)$ that do not obey the Pauli exclusion principle
• Examples of bosons include photons, gluons, W and Z bosons, and the Higgs boson
• Bosons can occupy the same quantum state simultaneously, which leads to phenomena such as Bose-Einstein condensation and superfluidity
• Quantum degeneracy occurs when the average inter-particle distance becomes comparable to the thermal de Broglie wavelength of the particles
• In this regime, the quantum nature of the particles becomes important, and the system can no longer be described by classical statistical mechanics
• Fermi-Dirac statistics and Bose-Einstein statistics are used to describe the behavior of particles in the quantum degenerate regime, depending on whether the particles are fermions or bosons