Fermi–Dirac statistics

In quantum statistics, a branch of physics, Fermi–Dirac statistics describe a distribution of particles over energy states in systems consisting of many identical particles that obey the 'Pauli exclusion principle'. It is named after Enrico Fermi and Paul Dirac, each of whom discovered the method independently (although Fermi defined the statistics earlier than Dirac). n ¯ i = 1 e ( ϵ i − μ ) / k B T + 1 {displaystyle {ar {n}}_{i}={frac {1}{e^{(epsilon _{i}-mu )/k_{ m {B}}T}+1}}} Energy dependence. More gradual at higher T. n ¯ {displaystyle {ar {n}}} = 0.5 when ϵ {displaystyle epsilon ;} = μ {displaystyle mu ;} . Not shown is that μ {displaystyle mu } decreases for higher T. In quantum statistics, a branch of physics, Fermi–Dirac statistics describe a distribution of particles over energy states in systems consisting of many identical particles that obey the 'Pauli exclusion principle'. It is named after Enrico Fermi and Paul Dirac, each of whom discovered the method independently (although Fermi defined the statistics earlier than Dirac). Fermi–Dirac (F–D) statistics apply to identical particles with half-integer spin in a system with thermodynamic equilibrium. Additionally, the particles in this system are assumed to have negligible mutual interaction. That allows the many-particle system to be described in terms of single-particle energy states. The result is the F–D distribution of particles over these states which includes the condition that no two particles can occupy the same state; this has a considerable effect on the properties of the system. Since F–D statistics apply to particles with half-integer spin, these particles have come to be called fermions. It is most commonly applied to electrons, which are fermions with spin 1/2. Fermi–Dirac statistics are a part of the more general field of statistical mechanics and use the principles of quantum mechanics. The opposite of F–D statistics are the Bose–Einstein statistics, that apply to bosons (full integer spin or no spin, like the Higgs boson), particles that do not follow the Pauli exclusion principle, meaning that more than one boson can take up the same quantum configuration simultaneously. Before the introduction of Fermi–Dirac statistics in 1926, understanding some aspects of electron behavior was difficult due to seemingly contradictory phenomena. For example, the electronic heat capacity of a metal at room temperature seemed to come from 100 times fewer electrons than were in the electric current. It was also difficult to understand why those emission currents generated by applying high electric fields to metals at room temperature were almost independent of temperature. The difficulty encountered by the Drude model, the electronic theory of metals at that time, was due to considering that electrons were (according to classical statistics theory) all equivalent. In other words, it was believed that each electron contributed to the specific heat an amount on the order of the Boltzmann constant kB.This statistical problem remained unsolved until the discovery of F–D statistics. F–D statistics was first published in 1926 by Enrico Fermi and Paul Dirac. According to Max Born, Pascual Jordan developed in 1925 the same statistics which he called Pauli statistics, but it was not published in a timely manner. According to Dirac, it was first studied by Fermi, and Dirac called it Fermi statistics and the corresponding particles fermions. F–D statistics was applied in 1926 by Ralph Fowler to describe the collapse of a star to a white dwarf. In 1927 Arnold Sommerfeld applied it to electrons in metals and developed the free electron model, and in 1928 Fowler and Lothar Wolfgang Nordheim applied it to field electron emission from metals. Fermi–Dirac statistics continues to be an important part of physics. For a system of identical fermions with thermodynamic equilibrium, the average number of fermions in a single-particle state i is given by a logistic function, or sigmoid function: the Fermi–Dirac (F–D) distribution, where kB is Boltzmann's constant, T is the absolute temperature, εi is the energy of the single-particle state i, and μ is the total chemical potential.