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Velocity-distribution data of a gas of rubidium atoms, confirming the discovery of a new phase of matter, the Bose–Einstein condensate. Left: just before the appearance of a Bose–Einstein condensate. Center: just after the appearance of the condensate. Right: after further evaporation, leaving a sample of nearly pure condensate. A Bose–Einstein condensate (BEC) is a state of matter of a dilute gas of bosons cooled to temperatures very near absolute zero (0 K or −273.15 °C[1]). Under such conditions, a large fraction of the bosons occupy the lowest quantum state, at which point quantum effects become apparent on a macroscopic scale. These effects are called macroscopic quantum phenomena. This state of matter was first predicted by Satyendra Nath Bose and Albert Einstein in 1924–25. Bose first sent a paper to Einstein on the quantum statistics of light quanta (now called photons). Einstein was impressed, translated the paper himself from English to German and submitted it for Bose to the Zeitschrift für Physik, which published it (The Einstein manuscript, once believed to be lost, was found in a library at Leiden University in 2005.[2]). Einstein then extended Bose's ideas to material particles (or matter) in two other papers.[3] The result of the efforts of Bose and Einstein is the concept of a Bose gas, governed by Bose–Einstein statistics, which describes the statistical distribution of identical particles with integer spin, now known as bosons. Bosonic particles, which include the photon as well as atoms such as helium-4, are allowed to share quantum states with each other. Einstein demonstrated that cooling bosonic atoms to a very low temperature would cause them to fall (or "condense") into the lowest accessible quantum state, resulting in a new form of matter. In 1938 Fritz London proposed BEC as a mechanism for superfluidity in 4He and superconductivity.[4][5] In 1995 the first gaseous condensate was produced by Eric Cornell and Carl Wieman at the University of Colorado at Boulder NIST–JILA lab, using a gas of rubidium atoms cooled to 170 nanokelvin (nK) [6] (1.7×10−7 K). For their achievements Cornell, Wieman, and Wolfgang Ketterle at MIT received the 2001 Nobel Prize in Physics.[7] In November 2010 the first photon BEC was observed.[8] 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:

where:

is the critical temperature,

is the particle density,

is the mass per boson,

is the reduced Planck constant,

is the Boltzmann constant, and

is the Riemann zeta function; (sequence A078434 in OEIS)

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Einstein's argument

Consider a collection of N noninteracting particles, which can each be in one of two quantum states, and . If the two states are equal in energy, each different configuration is equally likely. If we can tell which particle is which, there are different configurations, since each particle can be in or independently. In almost all of the configurations, about half the particles are in and the other half in . The balance is a statistical effect: the number of configurations is largest when the particles are divided equally. If the particles are indistinguishable, however, there are only N+1 different configurations. If there are K particles in state , there are N − K particles in state . Whether any particular particle is in state or in state cannot be determined, so each value of K determines a unique quantum state for the whole system. If all these states are equally likely, there is no statistical spreading out; it is just as likely for all the particles to sit in as for the particles to be split half and half. Suppose now that the energy of state is slightly greater than the energy of state...