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Bose–Einstein statistics

Statistical mechanics

Thermodynamics

Kinetic theory

VTE

From Wikipedia, the free encyclopedia

Inquantum 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 ofsuperfluid 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 byAlbert

Einstein in collaboration with Bose.

The Bose–Einstein statistics apply only to those particles not

limited to single occupancy of the same state—that is, particles that do not obey the Pauli exclusion

principle restrictions. Such particles have integer values of spin and are named bosons, after the statistics

that correctly describe their behaviour. There must also be no significant interaction between the particles.

Contents [hide]

1 Concept

2 History

3 Two derivations of the Bose–Einstein distribution

3.1 Derivation from the grand canonical ensemble

3.2 Derivation in the canonical approach

4 Interdisciplinary applications

5 See also

6 Notes

7 References

Concept [edit]

At low temperatures, bosons behave differently fromfermions (which obey the Fermi–Dirac statistics) in a

way that an unlimited number of them can "condense" into the same energy state. This apparently unusual

property also gives rise to the special state of matter –Bose Einstein Condensate. Fermi–Dirac and Bose–

Einstein statistics apply when quantum effects are important and the particles are "indistinguishable".

Quantum effects appear if the concentration of particles satisfies,

Particle Statistics [show ]

Thermodynamic Ensembles [show ]

Models [show ]

Potentials [show ]

Scientists [show ]

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where N is the number of particles and V is the volume and nq is the quantum concentration, for which the

interparticle distance is equal to the thermal de Broglie wavelength, so that the wavefunctions of the

particles are barely overlapping. Fermi–Dirac statistics apply to fermions (particles that obey the Pauli

exclusion principle), and Bose–Einstein statistics apply tobosons. As the quantum concentration depends

on temperature, most systems at high temperatures obey the classical (Maxwell–Boltzmann) limit unless

they have a very high density, as for a white dwarf. Both Fermi–Dirac and Bose–Einstein become Maxwell–

Boltzmann statistics at high temperature or at low concentration.

B–E statistics was introduced for photons in 1924 byBose and generalized to atoms by Einstein in 1924–25.

The expected number of particles in an energy state i for B–E statistics is

with εi > μ and where ni is the number of particles in state i, gi is the degeneracy of state i, εi is

the energyof the ith state, μ is the chemical potential, k is theBoltzmann constant, and T is

absolute temperature. For comparison, the average number of fermions with energy given by Fermi–

Dirac particle-energy distribution has a similar form,

B–E statistics reduces to the Rayleigh–Jeans Lawdistribution for , namely .

History [edit]

While presenting a lecture at the University of Dhakaon the theory of radiation and the ultraviolet

catastrophe, Satyendra Nath Bose intended to show his students that the contemporary theory was

inadequate, because it predicted results not in accordance with experimental results. During this lecture,

Bose committed an error in applying the theory, which unexpectedly gave a prediction that agreed with the

experiment. The error was a simple mistake—similar to arguing that flipping two fair coins will produce two

heads one-third of the time—that would appear obviously wrong to anyone with a basic understanding of

statistics (remarkably, this error resembled the famous blunder by d'Alembert known from his "Croix ou

Pile " Article) . However, the results it predicted agreed with experiment, and Bose realized it might not be

a mistake after all. He for the first time took the position that the Maxwell–Boltzmann distribution would not

be true for microscopic particles where fluctuations due to Heisenberg's uncertainty principle will be

significant. Thus he stressed the probability of finding particles in the phase space, each state having

volume h3, and discarding the distinct position and momentum of the particles.

Bose adapted this lecture into a short article called "Planck's Law and the Hypothesis of Light

Quanta"[1][2]and submitted it to the Philosophical Magazine. However, the referee's report was negative,

and the paper was rejected. Undaunted, he sent the manuscript to Albert Einstein requesting publication in

the Zeitschrift für Physik. Einstein immediately agreed, personally translated the article into German (Bose

had earlier translated Einstein's article on the theory of General Relativity from German to English), and

saw to it that it was published. Bose's theory achieved respect when Einstein sent his own paper in support

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of Bose's to Zeitschrift für Physik, asking that they be published together. This was done in 1924.

The reason Bose produced accurate results was that since photons are indistinguishable from each other,

one cannot treat any two photons having equal energy as being two distinct identifiable photons. By

analogy, if in an alternate universe coins were to behave like photons and other bosons, the probability of

producing two heads would indeed be one-third, and so is the probability of getting a head and a tail which

equals one-half for the conventional (classical, distinguishable) coins. Bose's "error" lead to what is now

called Bose–Einstein statistics.

Bose and Einstein extended the idea to atoms and this led to the prediction of the existence of phenomena

which became known as Bose–Einstein condensate, a dense collection of bosons (which are particles with

integer spin, named after Bose), which was demonstrated to exist by experiment in 1995.

Two derivations of the Bose–Einstein distribution [edit]

Derivation from the grand canonical ensemble [edit]

The Bose–Einstein distribution, which applies only to a quantum system of non-interacting bosons, is easily

derived from the grand canonical ensemble.[3] In this ensemble, the system is able to exchange energy and

exchange particles with a reservoir (temperature T and chemical potential µ fixed by the reservoir).

Due to the non-interacting quality, each available single-particle level (with energy level ϵ) forms a separate

thermodynamic system in contact with the reservoir. In other words, each single-particle level is a separate,

tiny grand canonical ensemble. With bosons there is no limit on the number of particles N in the level, but

due to indistinguishability each possible Ncorresponds to only one microstate (with energy Nϵ). The

resulting partition function for that single-particle level therefore forms a geometric series:

and the average particle number for that single-particle substate is given by

This result applies for each single-particle level and thus forms the Bose–Einstein distribution for the entire

state of the system.[3] [4]

The variance in particle number (due to thermal fluctuations) may also be derived:

This level of fluctuation is much larger than fordistinguishable particles, which would instead showPoisson

statistics ( ). This is because the probability distribution for the number of bosons in a

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given energy level is a geometric distribution, not a Poisson distribution.

Derivation in the canonical approach [edit]

It is also possible to derive approximate Bose–Einstein statistics in the canonical ensemble. These

derivations are lengthy and only yield the above results in the asymptotic limit of a large number of

particles. The reason is that the total number of bosons is fixed in the canonical ensemble. That contradicts

the implication in Bose–Einstein statistics that each energy level is filled independently from the others

(which would require the number of particles to be flexible).

Interdisciplinary applications [edit]

Viewed as a pure probability distribution, the Bose–Einstein distribution has found application in other

fields:

In recent years, Bose Einstein statistics have also been used as a method for term weighting

ininformation retrieval. The method is one of a collection of DFR ("Divergence From Randomness")

models,[6] the basic notion being that Bose Einstein statistics may be a useful indicator in cases where a

particular term and a particular document have a significant relationship that would not have occurred

purely by chance. Source code for implementing this model is available from the Terrier project at the

University of Glasgow.

Main article: Bose–Einstein condensation (network theory)

The evolution of many complex systems, including the World Wide Web, business, and citation

networks, is encoded in the dynamic web describing the interactions between the system's constituents.

Despite their irreversible and nonequilibrium nature these networks follow Bose statistics and can

undergo Bose–Einstein condensation. Addressing the dynamical properties of these nonequilibrium

systems within the framework of equilibrium quantum gases predicts that the "first-mover-advantage,"

"fit-get-rich(FGR)," and "winner-takes-all" phenomena observed in competitive systems are

thermodynamically distinct phases of the underlying evolving networks.[6]

See also [edit]

Bose–Einstein correlations

Higgs boson

Parastatistics

Planck's law of black body radiation

Superconductivity

Notes [edit]

1. ^ See p. 14, note 3, of the Ph.D. Thesis entitledBose–Einstein condensation: analysis of problems and

Derivation [show]

Notes [show]

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rigorous results, presented by Alessandro Michelangeli to the International School for Advanced Studies,

Mathematical Physics Sector, October 2007 for the degree of Ph.D.

See:http://digitallibrary.sissa.it/handle/1963/5272?show=full , and download

fromhttp://digitallibrary.sissa.it/handle/1963/5272

2. ^ To download the Bose paper, see:http://www.condmat.uni-oldenburg.de/TeachingSP/bose.ps

3. ̂a b Chapter 7 of Srivastava, R. K.; Ashok, J. (2005).Statistical Mechanics. New Delhi: PHI Learning Pvt.

Ltd. ISBN 9788120327825. edit

4. ^ The BE distribution can be derived also from thermal field theory.

5. ^ See McQuarrie in citations

6. ̂a b Amati, G.; C. J. Van Rijsbergen (2002). "Probabilistic models of information retrieval based on measuring

the divergence from randomness "ACM TOIS 20 (4):357–389.

References [edit]

Annett, James F. (2004). Superconductivity, Superfluids and Condensates. New York: Oxford University

Press. ISBN 0-19-850755-0.

Bose (1924). "Plancks Gesetz und Lichtquantenhypothese", Zeitschrift für Physik26:178–

181. doi:10.1007/BF01327326 (Einstein's translation into German of Bose's paper on Planck's law).

Carter, Ashley H. (2001). Classical and Statistical Thermodynamics. Upper Saddle River, New Jersey:

Prentice Hall. ISBN 0-13-779208-5.

Griffiths, David J. (2005). Introduction to Quantum Mechanics (2nd ed.). Upper Saddle River, New

Jersey: Pearson, Prentice Hall. ISBN 0-13-191175-9.

McQuarrie, Donald A. (2000). Statistical Mechanics(1st ed.). Sausalito, California 94965: University

Science Books. p. 55. ISBN 1-891389-15-7.