Editing Negative binomial distribution (section)

===Multiplicity observations (physics)===

The negative binomial distribution has been the most effective statistical model for a broad range of multiplicity observations in [[particle collision]] experiments, e.g., <math>p\bar p,\ hh,\ hA,\ AA,\ e^{+}e^-</math> <ref>{{Cite journal |last1=Grosse-Oetringhaus |first1=Jan Fiete |last2=Reygers |first2=Klaus |date=2010-08-01 |title=Charged-particle multiplicity in proton–proton collisions |url=https://iopscience.iop.org/article/10.1088/0954-3899/37/8/083001 |journal=Journal of Physics G: Nuclear and Particle Physics |volume=37 |issue=8 |pages=083001 |doi=10.1088/0954-3899/37/8/083001 |issn=0954-3899|arxiv=0912.0023 |s2cid=119233810 }}</ref><ref>{{Cite journal |last1=Rybczyński |first1=Maciej |last2=Wilk |first2=Grzegorz |last3=Włodarczyk |first3=Zbigniew |date=2019-05-31 |title=Intriguing properties of multiplicity distributions |journal=Physical Review D |language=en |volume=99 |issue=9 |page=094045 |doi=10.1103/PhysRevD.99.094045 |arxiv=1811.07197 |bibcode=2019PhRvD..99i4045R |issn=2470-0010|doi-access=free }}</ref><ref>{{Cite journal |last1=Tarnowsky |first1=Terence J. |last2=Westfall |first2=Gary D. |date=2013-07-09 |title=First study of the negative binomial distribution applied to higher moments of net-charge and net-proton multiplicity distributions |journal=Physics Letters B |volume=724 |issue=1 |pages=51–55 |doi=10.1016/j.physletb.2013.05.064 |arxiv=1210.8102 |bibcode=2013PhLB..724...51T |issn=0370-2693|doi-access=free }}</ref><ref>{{Cite journal |last1=Derrick |first1=M. |last2=Gan |first2=K. K. |last3=Kooijman |first3=P. |last4=Loos |first4=J. S. |last5=Musgrave |first5=B. |last6=Price |first6=L. E. |last7=Repond |first7=J. |last8=Schlereth |first8=J. |last9=Sugano |first9=K. |last10=Weiss |first10=J. M. |last11=Wood |first11=D. E. |last12=Baranko |first12=G. |last13=Blockus |first13=D. |last14=Brabson |first14=B. |last15=Brom |first15=J. M. |date=1986-12-01 |title=<nowiki>Study of quark fragmentation in ${e}^{+}$${e}^{\mathrm{\ensuremath{-}}}$ annihilation at 29 GeV: Charged-particle multiplicity and single-particle rapidity distributions</nowiki> |url=https://link.aps.org/doi/10.1103/PhysRevD.34.3304 |journal=Physical Review D |volume=34 |issue=11 |pages=3304–3320 |doi=10.1103/PhysRevD.34.3304|pmid=9957066 |hdl=1808/15222 |hdl-access=free }}</ref><ref>{{Cite journal |last=Zborovský |first=I. |date=2018-10-10 |title=Three-component multiplicity distribution, oscillation of combinants and properties of clans in pp collisions at the LHC |journal=The European Physical Journal C |language=en |volume=78 |issue=10 |pages=816 |doi=10.1140/epjc/s10052-018-6287-x |arxiv=1811.11230 |bibcode=2018EPJC...78..816Z |issn=1434-6052|doi-access=free }}</ref> (See <ref>{{Cite book |last1=Kittel |first1=Wolfram |title=Soft multihardon dynamics |last2=De Wolf |first2=Eddi A |publisher=World Scientific |year=2005}}</ref> for an overview), and is argued to be a [[scale-invariant]] property of matter,<ref>{{Cite journal |last=Schaeffer |first=R |date=1984 |title=Determination of the galaxy N-point correlation function |journal=Astronomy and Astrophysics |volume=134 |issue=2 |pages=L15|bibcode=1984A&A...134L..15S }}</ref><ref>{{Cite journal |last=Schaeffer |first=R |date=1985 |title=The probability generating function for galaxy clustering |journal=Astronomy and Astrophysics |volume=144 |issue=1 |pages=L1–L4|bibcode=1985A&A...144L...1S }}</ref> providing the best fit for astronomical observations, where it predicts the number of galaxies in a region of space.<ref>{{Cite journal |last1=Perez |first1=Lucia A. |last2=Malhotra |first2=Sangeeta |last3=Rhoads |first3=James E. |last4=Tilvi |first4=Vithal |date=2021-01-07 |title=Void Probability Function of Simulated Surveys of High-redshift Ly α Emitters |journal=The Astrophysical Journal |volume=906 |issue=1 |pages=58 |doi=10.3847/1538-4357/abc88b |arxiv=2011.03556 |bibcode=2021ApJ...906...58P |issn=1538-4357 |doi-access=free }}</ref><ref>{{Cite journal |last1=Hurtado-Gil |first1=Lluís |last2=Martínez |first2=Vicent J. |last3=Arnalte-Mur |first3=Pablo |last4=Pons-Bordería |first4=María-Jesús |last5=Pareja-Flores |first5=Cristóbal |last6=Paredes |first6=Silvestre |date=2017-05-01 |title=The best fit for the observed galaxy counts-in-cell distribution function |url=https://www.aanda.org/articles/aa/abs/2017/05/aa29097-16/aa29097-16.html |journal=Astronomy & Astrophysics |language=en |volume=601 |pages=A40 |doi=10.1051/0004-6361/201629097 |arxiv=1703.01087 |bibcode=2017A&A...601A..40H |issn=0004-6361|doi-access=free }}</ref><ref>{{Cite journal |last1=Elizalde |first1=E. |last2=Gaztanaga |first2=E. |date=January 1992 |title=Void probability as a function of the void's shape and scale-invariant models |journal=Monthly Notices of the Royal Astronomical Society |volume=254 |issue=2 |pages=247–256 |doi=10.1093/mnras/254.2.247 |issn=0035-8711|doi-access=free |hdl=2060/19910019799 |hdl-access=free }}</ref><ref>{{Cite journal |last1=Hameeda |first1=M |last2=Plastino |first2=Angelo |last3=Rocca |first3=M C |date=2021-03-01 |title=Generalized Poisson distributions for systems with two-particle interactions |journal=IOP SciNotes |volume=2 |issue=1 |pages=015003 |doi=10.1088/2633-1357/abec9f |bibcode=2021IOPSN...2a5003H |issn=2633-1357|doi-access=free |hdl=11336/181371 |hdl-access=free }}</ref> The phenomenological justification for the effectiveness of the negative binomial distribution in these contexts remained unknown for fifty years, since their first observation in 1973.<ref>{{Cite journal |last=Giovannini |first=A. |date=June 1973 |title="Thermal chaos" and "coherence" in multiplicity distributions at high energies |url=http://dx.doi.org/10.1007/bf02734689 |journal=Il Nuovo Cimento A |volume=15 |issue=3 |pages=543–551 |doi=10.1007/bf02734689 |bibcode=1973NCimA..15..543G |s2cid=118805136 |issn=0369-3546|url-access=subscription }}</ref> In 2023, a proof from [[first principle]]s was eventually demonstrated by Scott V. Tezlaf, where it was shown that the negative binomial distribution emerges from [[Spacetime symmetries|symmetries]] in the [[Dynamics (mechanics)|dynamical equations]] of a [[canonical ensemble]] of particles in [[Minkowski space]].<ref name=":1">{{Cite journal |last=Tezlaf |first=Scott V. |date=2023-09-29 |title=Significance of the negative binomial distribution in multiplicity phenomena |url=https://iopscience.iop.org/article/10.1088/1402-4896/acfead |journal=Physica Scripta |volume=98 |issue=11 |doi=10.1088/1402-4896/acfead |arxiv=2310.03776 |bibcode=2023PhyS...98k5310T |s2cid=263300385 |issn=0031-8949}}</ref> Roughly, given an expected number of trials <math>\langle n \rangle</math> and expected number of successes <math>\langle r \rangle</math>, where

: <math>\langle \mathcal{n} \rangle - \langle r \rangle = k,  \quad \quad  \langle p \rangle  = \frac{\langle r \rangle}{\langle \mathcal{n} \rangle}  \quad\quad \quad \implies \quad\quad \quad
\langle \mathcal{n} \rangle = \frac{k}{1-\langle p \rangle},  \quad \quad \langle {r} \rangle = \frac{k\langle p \rangle}{1 - \langle p \rangle},</math>

an [[Isomorphism|isomorphic]] set of equations can be identified with the parameters of a [[Special relativity|relativistic]] [[current density]] of a canonical ensemble of massive particles, via

: <math>c^2\langle \rho^2 \rangle  - \langle j^2 \rangle = c^2\rho_0^2, \quad \quad  \quad \langle \beta^2_v \rangle = \frac{\langle j^2 \rangle}{c^2\langle \rho^2 \rangle}  \quad \quad \implies \quad \quad
c^2\langle \rho^2 \rangle = \frac{c^2\rho_0^2}{1-\langle \beta^2_v \rangle},  \quad \quad  \quad \langle j^2 \rangle = \frac{c^2\rho_0^2  \langle \beta^2_v \rangle}{1-\langle \beta^2_v \rangle},</math>

where <math>\rho_0</math> is the rest [[density]], <math>\langle \rho ^2 \rangle</math> is the relativistic mean square density, <math>\langle j ^2 \rangle</math> is the relativistic mean square current density, and <math>\langle \beta^2_v \rangle=\langle v^2 \rangle /c^2</math>, where <math>\langle v ^2 \rangle</math> is the [[Maxwell–Boltzmann distribution|mean square speed]] of the particle ensemble and <math>c</math> is the [[speed of light]]—such that one can establish the following [[Bijection|bijective map]]:

: <math>c^2\rho_0^2 \mapsto k, \quad \quad \langle \beta^2_v \rangle \mapsto \langle p \rangle, \quad \quad c^2\langle\rho^2 \rangle  \mapsto \langle \mathcal{n} \rangle, \quad \quad \langle j^2 \rangle \mapsto \langle r \rangle.</math>

A rigorous alternative proof of the above correspondence has also been demonstrated through [[quantum mechanics]] via the Feynman [[Path integral formulation|path integral]].<ref name=":1" />