In condensed matter physics, a **Bose–Einstein condensate** (**BEC**) is a state of matter (also called the fifth state of matter) which is typically formed when a gas of bosons at low densities is cooled to temperatures very close to absolute zero (−273.15 °C or −459.67 °F). Under such conditions, a large fraction of bosons occupy the lowest quantum state, at which point microscopic quantum mechanical phenomena, particularly wavefunction interference, become apparent macroscopically. A BEC is formed by cooling a gas of extremely low density (about one-hundred-thousandth (1/100,000) the density of normal air) to ultra-low temperatures.

This state was first predicted, generally, in 1924–1925 by Albert Einstein following and crediting a pioneering paper by Satyendra Nath Bose on the new field now known as quantum statistics.

This transition to BEC occurs below a critical temperature, which for a uniform three-dimensional gas consisting of non-interacting particles with no apparent internal degrees of freedom is given by:{\displaystyle T_{\rm {c}}=\left({\frac {n}{\zeta (3/2)}}\right)^{2/3}{\frac {2\pi \hbar ^{2}}{mk_{\rm {B}}}}\approx 3.3125\ {\frac {\hbar ^{2}n^{2/3}}{mk_{\rm {B}}}}}

where:

is the critical temperature, | |

the particle density, | |

the mass per boson, | |

the reduced Planck constant, | |

the Boltzmann constant and | |

the Riemann zeta function; {\displaystyle \,\zeta (3/2)\approx 2.6124.} |

Interactions shift the value and the corrections can be calculated by mean-field theory. This formula is derived from finding the gas degeneracy in the Bose gas using Bose–Einstein statistics.

### Superfluidity of BEC and Landau criterion

The phenomena of superfluidity of a Bose gas and superconductivity of a strongly-correlated Fermi gas (a gas of Cooper pairs) are tightly connected to Bose–Einstein condensation. Under corresponding conditions, below the temperature of phase transition, these phenomena were observed in helium-4 and different classes of superconductors. In this sense, the superconductivity is often called the superfluidity of Fermi gas. In the simplest form, the origin of superfluidity can be seen from the weakly interacting bosons model.

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