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Fermi-Dirac statistics

Quantum statistical description of indistinguishable fermions obeying the Pauli exclusion principle; gives occupation probabilities of single-particle states and underlies electronic behavior in matter.

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Fermi-Dirac statistics is the quantum statistical framework that describes the equilibrium distribution of fermions — particles with half-integer spin that obey the Pauli exclusion principle. It gives the probability that an available single-particle state of energy E is occupied at a given temperature and chemical potential and is fundamental to understanding the electronic properties of solids, the behavior of degenerate gases, and many phenomena in atomic and astrophysical contexts. The distribution is named after Enrico Fermi and Paul Dirac.

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Definition and basic formula

The Fermi-Dirac occupation function is commonly written as f(E) = 1 / (exp((E - μ)/(k_B T)) + 1), where μ is the chemical potential, k_B is Boltzmann's constant and T the temperature. This function interpolates between a sharp step at zero temperature — all states with energy below the chemical potential are occupied and those above are empty — and a smooth occupation profile at finite temperature. Physically μ at low temperature is near the Fermi energy, which characterizes the highest occupied single-particle levels in a many-fermion system.

Underlying assumptions and principles

  • Pauli exclusion: No single quantum state can be occupied by more than one fermion (for a given set of quantum numbers). This restriction is central to the statistics; see the Pauli exclusion principle.
  • Identical particles and antisymmetry: Exchanging two identical fermions leaves the physical state unchanged up to a sign; the many-body wavefunction is antisymmetric under particle exchange. This antisymmetry directly leads to the occupancy constraints used in the statistical counting.
  • Thermal equilibrium: The formula assumes a system in thermal and chemical equilibrium described by a grand canonical ensemble, so the chemical potential controls the average particle number.

Important consequences

Because fermions cannot occupy the same quantum state in unlimited numbers, many macroscopic properties arise: atoms acquire shell structure, solids form electronic bands with characteristic filling, and dense fermion systems develop a degeneracy pressure that is important in compact astrophysical objects. In metals, the distribution of electrons close to the Fermi energy determines electrical and thermal transport; for electrons in materials see references on electrons and metals. The Fermi-Dirac distribution also approaches the Maxwell-Boltzmann form in the classical limit of low occupation probability, providing a continuity with classical statistical mechanics.

Low-temperature behavior and expansions

At temperatures small compared with characteristic energy scales, only states within a narrow range around the chemical potential are thermally excited. This leads to distinctive low-temperature dependencies of observable quantities: for example, the electronic contribution to the heat capacity of a metal is proportional to temperature at sufficiently low T, reflecting the linear increase in available excitations near the Fermi surface. Such expansions are widely used in condensed-matter theory and in calculations of response functions.

Density of states and practical calculations

To compute macroscopic quantities one combines the occupation function with the single-particle density of states appropriate to the system. Integrals of the form ∫ g(E) f(E) dE yield average particle number, energy, and other thermodynamic properties. Numerical and analytic methods handle different dimensionalities and band structures; these techniques underpin models of conductivity and optical response in solids and of population distributions in trapped atomic gases. Practical introductions discuss how to use the distribution to calculate average occupations and thermal averages and how the crossover to classical behavior occurs; for compact interpretations of occupation see occupation probability.

Applications

  • Condensed matter: band filling, electrical conductivity and thermal properties in conductors, semiconductors and semimetals; see materials on electrical conductivity.
  • Atomic physics: behavior of fermionic ultracold gases in traps and optical lattices, where Fermi statistics determines the filling of quantum states and collective behavior.
  • Astrophysics: electron degeneracy pressure in white dwarfs and related effects in compact objects, where the statistical pressure plays a central stabilizing role.

Historical and theoretical context

The statistical form appeared in the mid-1920s as quantum mechanics matured and was independently formulated by Fermi and Dirac. It linked microscopic quantum rules for indistinguishable particles to macroscopic thermodynamic properties and clarified why electrons in atoms and solids fill discrete shells and bands. For broader background about macroscopic or many-particle states see materials on macroscopic state and summaries about fermions.

Further notes

The Fermi-Dirac distribution is a basic tool in theoretical and computational work across physics and materials science. It provides the starting point for transport theory, many-body perturbation approaches and numerical simulations used to compare with experiments. Introductory expositions typically derive the function from combinatorial arguments and from the grand canonical ensemble and then apply it to concrete problems in metals, cold atoms and astrophysical contexts. Researchers and students consult textbooks and reviews for detailed derivations, low-temperature expansions and worked examples.

Description

General formula

In a system of temperature T\!\, the Fermi distribution is W(E), which measures the occupation probability:

W(E)={\frac {1}{\exp {\left({\frac {E-\mu }{k_{{\mathrm {B}}}T}}\right)}+1}}

with

If the energy E\,calculated from the lowest possible single-particle state, {\displaystyle E_{\rm {F}}\,}also called Fermi energy. The occupation probability Wfor a state with Fermi level energy {\displaystyle E=E_{\rm {F}}\!\,}is at all temperatures:

{\displaystyle W(E=E_{\rm {F}})={\frac {1}{e^{0}+1}}={\frac {1}{2}}\ .}

To calculate the particle density ⟨ \langle n(E)\rangle E\,prevailing at energy e.g. for electrons in a metal, the Fermi distribution must still be D(E)\!\,multiplied by the density of states

\langle n(E)\rangle =W(E)\cdot D(E)\ .

At absolute temperature zero

At the absolute temperature zero point T=0\,{\mathrm {K}} the Fermi gas as a whole is in its energetically lowest possible state, i.e. in the ground state of the many-particle system. Since (given a sufficiently large number of particles) not all particles can occupy the one-particle ground state according to the Pauli principle, there must be particles in excited one-particle states even at the absolute temperature zero T=0\,{\mathrm {K}}particles must be in excited one-particle states. This can be vividly described with the notion of a Fermi lake: each added fermion occupies the lowest possible energy state, which is not yet occupied by another fermion. The "filling level" is determined by the density of the occupiable states and the number of particles to be accommodated.

Accordingly, the Fermi distribution for temperature T=0\,\mathrm {K} a sharp jump at the Fermi energy E_{{\mathrm {F}}}=\mu \!\,, which is therefore also called the Fermi edge or Fermi limit (see figure).

  • All states with {\displaystyle E<E_{\rm {F}}}are occupied, since here {\displaystyle W(E)=1}, i.e., the probability of encountering one of the fermions in such a state is one.
  • None of the states with {\displaystyle E>E_{\rm {F}}} is occupied, since here holds: {\displaystyle W(E)=0}, i.e. the probability of encountering one of the fermions in such a state is zero.

The Fermi level at T=0\,{\mathrm {K}}is therefore determined by the number and energetic distribution of states and the number of fermions to be accommodated in those states. Only one energy difference appears in the formula. If one gives the magnitude of the Fermi energy alone, it is the energy difference of the highest occupied to the lowest possible one-particle state. For illustration or quick estimation of temperature-dependent effects, this quantity is often expressed as a temperature value - the Fermi temperature:

T_{{\mathrm {F}}}={\frac {E_{{\mathrm {F}}}}{k_{{\mathrm {B}}}}}\!\,.

At the Fermi temperature, the thermal energy k_{{\mathrm {B}}}T\!\,equal to the Fermi energy. This term has nothing to do with the real temperature of the fermions, it is only used to characterize energy ratios.

At finite temperatures

The Fermi distribution gives the occupation probability in the equilibrium state at temperature }{\displaystyle T>0\,\mathrm {K} } Starting from T=0\,{\mathrm {K}}, states above the Fermi energy {\displaystyle E_{\mathrm {F} }(T=0\,\mathrm {K} )}occupied by fermions. In exchange, an equal number of states below the Fermi energy remain empty and are called holes.

The sharp Fermi edge is symmetrically located around E_{\mathrm {F} }located in an interval of total width {\displaystyle \sim 4k_{\mathrm {B} }T}rounded off ("softened", see Fig.). States at lower energies are still nearly full ( {\displaystyle W\lessapprox 1}), while those at higher energies are very weak ( {\displaystyle 0<W\ll 1}).

Since the same number of particles is still to be distributed among the possible states with density of states D(E)\!\,, the Fermi energy can shift with temperature: If the density of states in the excited particle region is smaller than in the holes, the Fermi energy increases, in the opposite case it decreases.

In the temperature range {\displaystyle T\ll T_{\mathrm {F} }\,} the system is called a degenerate Fermi gas, because the occupation of the states is largely determined by the Pauli principle (exclusion principle). This leads to the fact that all states with {\displaystyle E<E_{\mathrm {F} }} have the same probability (of nearly one) of being occupied; this involves a large energy range compared to the softening interval.

For energies E of at least some k_{\mathrm {B} }T above E_{\mathrm {F} }, i.e., for {\displaystyle E-E_{\mathrm {F} }\gg k_{\mathrm {B} }T}, the Fermi distribution can be approximated by the classical Boltzmann distribution:

{\displaystyle W(E)\propto \exp {\left(-{\frac {E-E_{\mathrm {F} }}{k_{\mathrm {B} }T}}\right)}}.

At very high temperatures

"Very high temperatures" are those well above the Fermi temperature, i.e., {\displaystyle T\gg T_{\mathrm {F} }\Leftrightarrow k_{\mathrm {B} }T\gg E_{\mathrm {F} }}. Because this makes the softening interval very large, so that even for energies well above the Fermi energy the occupation probability is noticeably different from zero, particle number conservation leads to the Fermi energy being below the lowest occupiable level. The Fermi gas then behaves like a classical gas, it is not degenerate.

Fermi distribution for different temperatures, increasing rounding with increasing temperature (red line: T = 0 K)
Fermi distribution for different temperatures, increasing rounding with increasing temperature (red line: T = 0 K)

Fermi distribution for metals

For the conduction electrons in a metal, the Fermi energy {\displaystyle E_{\rm {F}}\!\,} is a few electron volts, corresponding to a Fermi temperature T_\mathrm{F} \!\, of a few 10,000 K. As a consequence, the thermal energy k_{\mathrm {B} }Tis much smaller than the typical width of the conduction band. This is a degenerate electron gas. Therefore, the contribution of electrons to the heat capacity is negligible even at room temperature and can be accounted for in perturbation theory. The temperature dependence of the Fermi energy is very small (meV range) and is often neglected.

Questions and answers

Q: What is Fermi-Dirac statistics?

A: Fermi-Dirac statistics is a branch of quantum statistics that is used to describe the macroscopic state of a system made of many similar particles.

Q: Who is Fermi-Dirac statistics named after?

A: Fermi-Dirac statistics is named after Enrico Fermi and Paul Dirac.

Q: What is an example of a system that can be described using Fermi-Dirac statistics?

A: One example of a system that can be described using Fermi-Dirac statistics is the state of electrons in metals and semimetals, in order to describe electrical conductivity.

Q: What assumptions are made in Fermi-Dirac statistics?

A: Fermi-Dirac statistics makes two assumptions: 1) none of the states of the particles can hold more than one particle (known as Pauli exclusion principle), and 2) exchanging a particle for another similar particle will not lead to a new state, but will give the same state (known as identical particles).

Q: What does the Fermi distribution tell us?

A: The Fermi distribution tells us with what probability a Fermi gas, at a given temperature and energy level, will have a particle in the given state.

Q: What is another name for the Pauli exclusion principle?

A: The Pauli exclusion principle is also known as the exclusion principle.

Q: What is a Fermi gas?

A: A Fermi gas is a group of fermions that are at a low enough temperature to exhibit quantum effects.

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