Fermi–Dirac statistics. In quantum statistics, a branch of physics, Fermi–Dirac statistics describes a distribution of particles in certain systems comprising many identical particles that obey the Pauli exclusion principle. It is named after Enrico Fermi and Paul Dirac, who each discovered it independently, although Enrico Fermi defined the statistics earlier than Paul Dirac.[1][2] History[edit] 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.[3] It was also difficult to understand why the emission currents, generated by applying high electric fields to metals at room temperature, were almost independent of temperature.
Fermi–Dirac distribution[edit] For a system of identical fermions, the average number of fermions in a single-particle state When . Bose–Einstein statistics. In quantum statistics, Bose–Einstein statistics (or more colloquially B–E statistics) is one of two possible ways in which a collection of non-interacting indistinguishable particles may occupy a set of available discrete energy states, at thermodynamic equilibrium. The aggregation of particles in the same state, which is a characteristic of particles obeying Bose–Einstein statistics, accounts for the cohesive streaming of laser light and the frictionless creeping of superfluid helium. The theory of this behaviour was developed (1924–25) by Satyendra Nath Bose, who recognized that a collection of identical and indistinguishable particles can be distributed in this way.
The idea was later adopted and extended by Albert Einstein in collaboration with Satyendra Nath Bose. Concept[edit] At low temperatures, bosons behave differently from fermions (which obey the Fermi–Dirac statistics) in that an unlimited number of them can "condense" into the same energy state. . , namely History[edit] ). Boltzmann distribution. Where is state energy (which varies from state to state), and (a constant of the distribution) is the product of Boltzmann's constant and thermodynamic temperature.
The ratio of a Boltzmann distribution computed for two states is known as the Boltzmann factor and characteristically only depends on the states' energy difference. The Boltzmann distribution is named after Ludwig Boltzmann who first formulated it in 1868 during his studies of the statistical mechanics of gases in thermal equilibrium. [citation needed] The distribution was later investigated extensively, in its modern generic form, by Josiah Willard Gibbs in 1902.[2]:Ch.IV In statistical mechanics[edit] The Boltzmann distribution appears in statistical mechanics when considering isolated (or nearly-isolated) systems of fixed composition that are in thermal equilibrium (equilibrium with respect to energy exchange).
Canonical ensemble (general case) Statistical frequencies of subsystems' states (in a non-interacting collection) Partition function (statistical mechanics) The canonical partition function is where the "inverse temperature", β, is conventionally defined as with kB denoting Boltzmann's constant.
The exponential factor exp(−βEs) is known as the Boltzmann factor. In systems with multiple quantum states s sharing the same Es, it is said that the energy levels of the system are degenerate. In the case of degenerate energy levels, we can write the partition function in terms of the contribution from energy levels (indexed by j ) as follows: where gj is the degeneracy factor, or number of quantum states s which have the same energy level defined by Ej = Es. The above treatment applies to quantum statistical mechanics, where a physical system inside a finite-sized box will typically have a discrete set of energy eigenstates, which we can use as the states s above.
Where pi indicate particle momenta xi indicate particle positions d3 is a shorthand notation serving as a reminder that the pi and xi are vectors in three-dimensional space, and and .