Bose-Einstein statistics, class of statistics that applies to elementary particles called bosons, which include the photon photon (fō`tŏn), the particle composing light and other forms of electromagnetic radiation , sometimes called light quantum.
..... Click the link for more information. , pion pion (pī`ŏn) or pi meson, lightest of the meson family of elementary particles .
..... Click the link for more information. , and the W and Z particles W and Z particles, elementary particles that mediate, or carry, the fundamental force associated with weak interactions . The discovery of the W and Z
..... Click the link for more information. . Bosons have integral values of the quantum mechanical property called spin and are "gregarious" in the sense that an unlimited number of bosons can be placed in the same state. All of the particles that mediate the fundamental forces of nature are bosons. See elementary particles elementary particles, the most basic physical constituents of the universe.

Basic Constituents of Matter

Molecules are built up from the atom , which is the basic unit of any chemical element .
..... Click the link for more information. ; Fermi-Dirac statistics Fermi-Dirac statistics, class of statistics that applies to particles called fermions. Fermions have half-integral values of the quantum mechanical property called spin and are "antisocial" in the sense that two fermions cannot exist in the same state.
..... Click the link for more information. ; statistical mechanics statistical mechanics, quantitative study of systems consisting of a large number of interacting elements, such as the atoms or molecules of a solid, liquid, or gas, or the individual quanta of light (see photon ) making up electromagnetic radiation.
..... Click the link for more information. .

Bose-Einstein statistics

One of two possible ways (the other is Fermi-Dirac statistics) in which a collection of indistinguishable particles may occupy a set of available discrete energy states. The gathering of particles in the same state, which is characteristic of particles that obey Bose-Einstein statistics, accounts for the cohesive streaming of laser light and the frictionless creeping of superfluid helium (see superfluidity). The theory of this behaviour was developed in 1924–25 by Satyendra Nath Bose (1894–1974) and Albert Einstein. Bose-Einstein statistics apply only to those particles, called bosons, which have integer values of spin and so do not obey the Pauli exclusion principle.

Bose-Einstein statistics [¦bōz ¦īn‚stīn stə′tis·tiks]

(statistical mechanics)

The statistical mechanics of a system of indistinguishable particles for which there is no restriction on the number of particles that may exist in the same state simultaneously. Also known as Einstein-Bose statistics.

The statistical description of quantum mechanical systems in which there is no restriction on the way in which particles can be distributed over the individual energy levels. This description applies when the system has a symmetric wave function. This in turn has to be the case when the particles described are of integer spin.

Suppose one describes a system by giving the number of particles ni in an energy state ε_i, where the n_i are called occupation numbers and the index i labels the various states. The energy level ε_i, is of finite width, being really a range of energies comprising, say, g_i, individual (nondegenerate) quantum levels. If any arrangement of particles over individual energy levels is allowed, one obtains for the probability of a specific distribution Eq. (1a). In Boltzmann statistics, this same probability would be written as Eq.

(1 a)

(1 b)

(1b). See Boltzmann statistics

The equilibrium state is defined as the most probable state of the system. To obtain it, one must maximize Eq. (1a) under the conditions given by Eqs. (2a)

(2 a)

(2 b)

and (2b), which express the fact that the total number of particles N and the total energy E are fixed. One finds for the most probable distribution that Eq. (3) holds.

(3)

Here, A and ß are parameters to be determined from Eqs. (2a) and (2b); actually, ß\,\, = 1/kT, where k is the Boltzmann constant and T is the absolute temperature.

An interesting and important result emerges when Eq. (3) is applied to a gas of photons, that is, a large number of photons in an enclosure. (Since photons have integer spin, this is legitimate.) The formula for the energy density (energy per unit volume) in a given frequency range is found to be the celebrated Planck radiation formula for blackbody radiation. Thus, black-body radiation must be considered as a photon gas, with the photons satisfying Bose-Einstein statistics. See Heat radiation

If an ideal Bose gas, consisting of a fixed number of material particles, is compressed beyond a certain point, some of the particles will condense in a zero state, where they do not contribute to the density or the pressure. If the volume is decreased, this curious condensation phenomenon results, yielding the zero state which has the paradoxical properties of not contributing to the pressure, volume, or density. There is now considerable evidence that many of the superfluid properties exhibited by liquid helium are in fact manifestations of an Einstein condensation. See Fermi-Dirac statistics, Quantum statistics, Statistical mechanics