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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Bose–Einstein statistics | 4/4 | https://en.wikipedia.org/wiki/Bose–Einstein_statistics | reference | science, encyclopedia | 2026-05-05T13:41:28.703444+00:00 | kb-cron |
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{\displaystyle \sigma _{N}^{2}=k_{\text{B}}T\left({\frac {d\langle N\rangle }{d\mu }}\right)_{V,T}={\frac {\exp((\varepsilon -\mu )/k_{\text{B}}T)}{(\exp((\varepsilon -\mu )/k_{\text{B}}T)-1)^{2}}}=\langle N\rangle (1+\langle N\rangle ).}
As a result, for highly occupied states the standard deviation of the particle number of an energy level is very large, slightly larger than the particle number itself:
σ
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{\displaystyle \sigma _{N}\approx \langle N\rangle }
. This large uncertainty is due to the fact that the probability distribution for the number of bosons in a given energy level is a geometric distribution; somewhat counterintuitively, the most probable value for N is always 0. (In contrast, classical particles have instead a Poisson distribution in particle number for a given state, with a much smaller uncertainty of
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{\textstyle \sigma _{N,{\rm {classical}}}={\sqrt {\langle N\rangle }}}
, and with the most-probable N value being near
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{\displaystyle \langle N\rangle }
.)
=== Derivation in the canonical approach === 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. The Bose–Einstein distribution in this case can be derived as in most texts by maximization, but the mathematically best derivation is by the Darwin–Fowler method of mean values as emphasized by Dingle. See also Müller-Kirsten. The fluctuations of the ground state in the condensed region are however markedly different in the canonical and grand-canonical ensembles.
== Interdisciplinary applications ==
Viewed as a pure probability distribution, the Bose–Einstein distribution has found application in other fields:
In recent years, Bose–Einstein statistics has also been used as a method for term weighting in information retrieval. The method is one of a collection of DFR ("Divergence From Randomness") models, 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. 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.
== See also ==
== Notes ==
== References == Annett, James F. (2004). Superconductivity, Superfluids and Condensates. New York: Oxford University Press. ISBN 0-19-850755-0. Carter, Ashley H. (2001). Classical and Statistical Thermodynamics. Upper Saddle River, NJ: Prentice Hall. ISBN 0-13-779208-5. Griffiths, David J. (2005). Introduction to Quantum Mechanics (2nd ed.). Upper Saddle River, NJ: Pearson, Prentice Hall. ISBN 0-13-191175-9. McQuarrie, Donald A. (2000). Statistical Mechanics (1st ed.). Sausalito, CA: University Science Books. p. 55. ISBN 1-891389-15-7.