Graduate
  1. Classical mechanics  1.1. Advanced Kinematics  1.1.1. Generalized coordinate systems  1.1.2. Parametric equations of motion  1.1.3. Non-inertial frames and fictitious forces  1.1.4. Covariant formulation of momentum  1.1.5. Action-angle variable  1.2. Lagrangian and Hamiltonian mechanics  1.2.1. Principle of minimum action  1.2.2. Euler–Lagrange equations  1.2.3. Noether's theorem and conservation laws  1.2.4. Normalized Speed  1.2.5. Canonical conversion  1.2.6. Poisson bracket and Hamilton–Jacobi theory  1.2.7. Hamiltonian chaos and integrability  1.3. Rigid body mobility  1.3.1. Euler's equations of motion  1.3.2. Principal moments of inertia  1.3.3. Gyroscopic motion and precession  1.3.4. Inertia tensor  1.3.5. Stability of rotational speed  1.4. Nonlinear dynamics and chaos  1.4.1. Phase space and stability analysis  1.4.2. Poincaré sections and bifurcation theory  1.4.3. Strange Attractors and Fractals  1.4.4. Lyapunov exponent  1.4.5. KAM theorem and quasi-periodic motion
  2. Electromagnetism  2.1. Advanced Electrodynamics  2.1.1. Green's function and potential theory  2.1.2. Multipole expansion  2.1.3. Image charging method  2.1.4. Boundary conditions and uniqueness theorem  2.2. Electromagnetic wave propagation  2.2.1. Plane waves in dielectrics and conductors  2.2.2. Waveguides and cavity resonators  2.2.3. Radiation pressure and optical tweezers  2.2.4. Polarization and Jones calculus  2.3. Relativistic electrodynamics  2.3.1. Covariant formulation of electrodynamics  2.3.2. Lienard–Wiechert potentials  2.3.3. Synchrotron radiation and bremsstrahlung  2.3.4. Electromagnetic field tensor and Lorentz transformations  2.4. Plasma physics  2.4.1. Magnetohydrodynamics  2.4.2. Debye shielding and plasma oscillations  2.4.3. Alfvén waves and tokamak confinement  2.4.4. Plasma instabilities and turbulence
  3. Statistical mechanics and thermodynamics  3.1. Advanced Thermodynamics  3.1.1. Legendre transforms and thermodynamic potentials  3.1.2. Fluctuation–dissipation theorem  3.1.3. Onsager interpersonal relations  3.1.4. Critical events and phase transitions  3.2. Quantum statistical mechanics  3.2.1. Density Matrix and Ensemble Theory  3.2.2. Bose–Einstein condensates  3.2.3. Quantum phase transition  3.2.4. Fermi–Dirac and Bose–Einstein statistics  3.2.5. Partition functions and grand canonical ensembles
  4. Quantum mechanics  4.1. Advanced wave mechanics  4.1.1. WKB approximation  4.1.2. Path integral formulation  4.1.3. Quantum harmonic oscillator and coherent states  4.2. Angular momentum and spin  4.2.1. Spherical Harmonics  4.2.2. Clebsch–Gordan coefficient  4.2.3. Wigner–Eckart theorem  4.2.4. Spin–orbit coupling  4.3. Quantum scattering theory  4.3.1. Born approximation  4.3.2. Partial wave analysis  4.3.3. Optical theorem  4.3.4. S-matrix theory  4.4. Quantum field theory  4.4.1. Second Quantization  4.4.2. Feynman diagrams and propagators  4.4.3. Renormalization theory  4.4.4. Gauge theory and Yang–Mills fields  4.4.5. Spontaneous symmetry breaking and the Higgs mechanism
  5. General relativity and cosmology  5.1. Tensor calculus and differential geometry  5.1.1. Einstein field equations  5.1.2. Schwarzschild and Kerr metrics  5.1.3. Geodesics and Christoffel symbols  5.2. Cosmology and the Universe  5.2.1. The FLRW metric and cosmic inflation  5.2.2. Dark energy and structure formation  5.2.3. Cosmic microwave background radiation
  6. Condensed matter physics  6.1. Band structure and transport theory  6.1.1. Block theorem and the Kronig–Penney model  6.1.2. Fermi surface topology  6.1.3. Quantum Hall Effect  6.2. Superconductivity  6.2.1. BCS theory and Cooper pairs  6.2.2. Meissner effect and flux quantization  6.2.3. The Josephson Effect and SQUIDs  6.3. Topological phases of matter  6.3.1. Topological Insulators  6.3.2. Majorana fermions in topological phases of matter
  7. Nuclear and Particle Physics  7.1. Quantum Chromodynamics (QCD)  7.1.1. Quark–gluon plasma  7.1.2. Asymptotic freedom  7.1.3. Confinement and hadronization  7.2. Beyond the Standard Model  7.2.1. Grand Unified Theories  7.2.2. Supersymmetry and extra dimensions  7.2.3. Neutrino oscillations

Graduate → Statistical mechanics and thermodynamics → Quantum statistical mechanics ↓


Fermi–Dirac and Bose–Einstein statistics


In the field of quantum statistical mechanics, it is important to understand how particles are distributed between different energy levels. Two important statistical distributions describing these systems are Fermi-Dirac and Bose-Einstein statistics. These arise due to different intrinsic properties of particles known as fermions and bosons. The quantum world is divided into these two categories, each of which has unique characteristics and implications for how particles behave in a system.

Quantum particles: fermions and bosons

Before going into the details of statistics, let us briefly understand the difference between fermions and bosons:

Fermions

  • Particles that obey the Pauli exclusion principle, which means that no two fermions can be in the same quantum state at the same time.
  • Examples include electrons, protons, and neutrons.
  • They have half-integer spins, such as 1/2, -1/2.

Bosons

  • Particles that do not obey the Pauli exclusion principle allow multiple bosons to exist in the same quantum state.
  • Examples include the photon and the Higgs boson.
  • They have integer spins, such as 0, 1, 2.

Fermi–Dirac statistics

Fermi-Dirac statistics describes the distribution of fermionic particles over different energy states. This statistical model is essential for understanding the behavior of electrons in metals and other systems where quantum effects are important.

The Fermi-Dirac distribution function, f(E), gives the probability that the energy level E is occupied by a fermion. It is given by the following formula:

f(E) = 1 / (exp((E - μ) / kT) + 1)

Where:

  • E is the energy level.
  • μ is the chemical potential (Fermi energy at absolute zero).
  • k is the Boltzmann constant.
  • T is the temperature in Kelvin.
fermi-diracEnergy (E)f(e)Fermi energy (μ)

Importance in metals

In metals, the behavior of conduction electrons can be understood using Fermi-Dirac statistics. At absolute zero, all energy levels up to the Fermi energy are filled, and those above are empty. As the temperature increases, electrons gain energy and move to higher energy states, but the efficiency of this occupation is governed by the distribution function.

Example: Electrons in a metal

Consider a metal with a Fermi energy of 5 eV at absolute zero. As we increase the temperature, say up to 300 K, the distribution of electron occupancy is given by the Fermi-Dirac equation. For energy levels below 5 eV, the probability is high, close to 1, while for those above 5 eV, it decreases according to the model defined earlier.

Bose–Einstein statistics

Bose-Einstein statistics describes the distribution of identical, indistinguishable bosons. Unlike fermions, bosons tend to aggregate in the same energy state, especially at low temperatures. This property gives rise to phenomena such as Bose-Einstein condensation, where a large fraction of bosons are in the lowest energy state.

The Bose–Einstein distribution function, n(E), is expressed by:

n(E) = 1 / (exp((E - μ) / kT) - 1)

where the terms correspond to the Fermi–Dirac distribution:

  • E represents energy levels.
  • μ is the chemical potential, which can be zero for bosons at low temperatures.
  • k represents the Boltzmann constant.
  • T is the absolute temperature.
bose einsteinEnergy (E)N(E)

Bose–Einstein condensation

A remarkable implication of Bose-Einstein statistics is the phenomenon of Bose-Einstein condensation (BEC). At very low temperatures, typically close to absolute zero, a macroscopic number of bosons occupy the lowest energy state. This leads to unique macroscopic quantum phenomena that are observable on a large scale, challenging classical physics intuition.

Example: helium-4

Helium-4 is a common example of Bose-Einstein statistics. When it is cooled to temperatures below 2.17 K, it undergoes a phase transition and goes into a superfluid state. In this state, the liquid flows without any resistance, demonstrating the fascinating effects of BEC.

Comparison and implications

Fermi-Dirac and Bose-Einstein statistics provide profound information about the microscopic world. Let's compare the two:

Propertyfermi-diracbose einstein
applies toFermions (e.g., electrons)Bosons (e.g., photons)
Exclusion principleobey Pauli's exclusion principledoes not follow orders
Behavior at low TEnergy levels fill up to the Fermi levelTendency to occupy similar states (BEC)

These data are fundamental to understanding many phenomena in physics, from the electronic properties of solids to the functioning of complex celestial bodies.

Final thoughts

Both Fermi-Dirac and Bose-Einstein statistics are key pillars in the study of quantum mechanics and thermodynamics, offering detailed explanations of particle behavior in a variety of systems. From the conductivity of metals to the unique properties of superfluids, these distributions shape our understanding of the physical universe.

In applied fields such as electronics, understanding Fermi-Dirac statistics is crucial for designing semiconductors. Conversely, Bose-Einstein statistics have implications in the development of technologies dependent on low-temperature physics, including quantum computing and advanced materials.


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Fermi–Dirac and Bose–Einstein statistics

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