Thor Wiki - User contributions [en]

Jets

2021-05-10T08:11:32Z

Bleicher:

In [[high-energy physics]], '''jet quenching''' is a phenomenon that can occur in the [[collision]] of ultra-high-energy particles. In general, the collision of high-energy particles can produce [[jet (particle physics)|jet]]s of [[elementary particle]]s that emerge from these collisions. Collisions of [[ultra-relativistic]] [[heavy-ion]] particle beams create a hot and dense medium comparable to the conditions in the [[early universe]], and then these jets interact strongly with the medium, leading to a marked reduction of their energy. This energy reduction is called "jet quenching".

==Physics background==
In the context of high-energy [[hadron]] collisions, [[quark]]s and [[gluon]]s are collectively called [[parton (particle physics)|parton]]s. The jets emerging from the collisions originally consist of partons, which quickly combine to form hadrons, a process called [[hadronization]]. Only the resulting hadrons can be directly observed. The hot, dense medium produced in the collisions is also composed of partons; it is known as a [[quark–gluon plasma]] (QGP). In this realm, the laws of physics that apply are those of [[quantum chromodynamics]] (QCD).

High-energy nucleus-nucleus collisions make it possible to study the properties of the [[Quark–gluon plasma|QGP]] medium through the observed changes in the jet fragmentation functions as compared to the unquenched case. According to [[Quantum chromodynamics|QCD]], high-momentum [[parton (particle physics)|partons]] produced in the initial stage of a nucleus-nucleus collision will undergo multiple interactions inside the collision region prior to [[hadronization]]. In these interactions, the energy of the partons is reduced through collisional energy loss<ref>D. H. Perkins (2000). ''Introduction to High Energy Physics'', Cambridge University Press.</ref> and medium-induced gluon radiation,<ref>{{cite journal | last1=Gross | first1=David J. | last2=Wilczek | first2=Frank | title=Ultraviolet Behavior of Non-Abelian Gauge Theories | journal=Physical Review Letters | volume=30 | issue=26 | date=25 June 1973 | doi=10.1103/physrevlett.30.1343 |doi-access=free | pages=1343–1346| bibcode=1973PhRvL..30.1343G }}</ref> the latter being the dominant mechanism in a QGP. The effect of jet quenching in QGP is the main motivation for studying jets as well as high-momentum particle spectra and particle correlations in heavy-ion collisions. Accurate jet reconstruction will allow measurements of the jet fragmentation functions and consequently the degree of quenching and therefore provide insight on the properties of the hot dense QGP medium created in the collisions.

==Experimental evidence of jet quenching==
First evidence of parton energy loss has been observed at the [[Relativistic Heavy Ion Collider]] (RHIC) from the suppression of high-pt particles studying the nuclear modification factor<ref>{{cite journal | last=Adcox | first=K. |display-authors=etal |collaboration=PHENIX Collaboration | title=Suppression of Hadrons with Large Transverse Momentum in Central Au+Au Collisions at {{sqrt|sNN}} = 130 GeV | journal=Physical Review Letters | volume=88 | issue=2 | year=2002 | doi=10.1103/physrevlett.88.022301 | pmid=11801005 | page=022301| arxiv=nucl-ex/0109003 | s2cid=119347728 }}</ref><ref name=Adler2003>{{cite journal | last=Adler | first=C. |display-authors=etal |collaboration=STAR Collaboration | title=Disappearance of Back-To-Back High-pT Hadron Correlations in Central Au + Au Collisions at {{sqrt|sNN}} = 200 GeV | journal=Physical Review Letters | volume=90 | issue=8 | date=26 February 2003 | doi=10.1103/physrevlett.90.082302 | page=082302| pmid=12633419 | arxiv=nucl-ex/0210033 | s2cid=41635379 }}</ref> and the suppression of back-to-back correlations.<ref name=Adler2003 />

In ultra-relativistic heavy-ion collisions at center-of-mass energy of 2.76 and 5.02 TeV at the [[Large Hadron Collider]] (LHC), interactions between the high-momentum [[Parton (particle physics)|parton]] and the hot, dense medium produced in the collisions, are expected to lead to jet quenching. Indeed, in November 2010 [[CERN]] announced the first direct observation of jet quenching, based on experiments with ''heavy-ion collisions'', which involved [[ATLAS experiment|ATLAS]], [[Compact Muon Solenoid|CMS]] and [[A Large Ion Collider Experiment|ALICE]].<ref>{{cite press release |title=LHC experiments bring new insight into primordial universe |url=http://press.web.cern.ch/press-releases/2010/11/lhc-experiments-bring-new-insight-primordial-universe|publisher=[[CERN]] |date=November 26, 2010 |accessdate=December 2, 2010}}</ref><ref>{{cite journal | last=Aad | first=G. |display-authors=etal |collaboration=ATLAS Collaboration | title=Observation of a Centrality-Dependent Dijet Asymmetry in Lead-Lead Collisions at {{sqrt|sNN}} = 2.76 TeV with the ATLAS Detector at the LHC | journal=Physical Review Letters | volume=105 | issue=25 | date=13 December 2010 | doi=10.1103/physrevlett.105.252303 |doi-access=free | page=252303 | pmid=21231581 }}</ref><ref>{{cite journal | last=Chatrchyan | first=S. |display-authors=etal |collaboration=CMS Collaboration | title=Observation and studies of jet quenching in Pb-Pb collisions at {{sqrt|sNN}} = 2.76 TeV | journal=Physical Review C | volume=84 | issue=2 | date=12 August 2011 | doi=10.1103/physrevc.84.024906 |doi-access=free | page=024906 }}</ref><ref>{{cite web|url=http://home.web.cern.ch/about/physics/heavy-ions-and-quark-gluon-plasma|title=Heavy ions and quark-gluon plasma|author=CERN|date=18 July 2012}}</ref>

== See also ==
* [[Parity (physics)]]

== References ==
{{reflist|2}}

== External links ==
* [https://web.archive.org/web/20090330204507/http://www.suteka.za.net/?p=35 Jet Suppression in Heavy Ion Collisions]
* [http://web.mit.edu/physics/news/physicsatmit/physicsatmit_17_yen-jie_lee.pdf Jetting through the Quark Soup]
* [https://arxiv.org/abs/1705.01974 Review of Jet Quenching (2017)]
* [https://inspirehep.net/search?p=find+eprint+0902.2011 Review of Jet Quenching (2009)]
[[Category:Particle physics]]

Heavy quarks

2021-05-10T08:11:02Z

Bleicher:

{{Short description|Type of quark}}
{{Redirect|Charm (physics)||Charm (disambiguation)#Science and technology}}
{{Infobox Particle
|bgcolour =
|name = Charm quark
|image =[[FILE:Charm quark.svg|150px]]
|caption =
|num_types =
|composition = [[Elementary particle]]
|statistics = [[Fermionic]]
|group = [[Quark]]
|generation = Second
|interaction = [[Strong interaction|strong]], [[Weak interaction|weak]], [[electromagnetic force]], [[gravity]]
|particle =
|antiparticle = Charm antiquark ({{SubatomicParticle|Charm antiquark}})
|theorized = [[Sheldon Lee Glashow|Sheldon Glashow]], [[John Iliopoulos]], [[Luciano Maiani]] (1970)
|discovered = {{plainlist|
*[[Burton Richter]] ''et al.'' ([[SLAC]], 1974)
*[[Samuel C. C. Ting|Samuel Ting]] ''et al.'' ([[Brookhaven National Laboratory|BNL]], 1974)}}
|symbol = {{SubatomicParticle|Charm quark}}
|mass = {{val|1.275|+0.025|-0.035|ul=GeV/c2}}<ref name="PDG2018">
{{cite journal
|author=M. Tanabashi et al. (Particle Data Group)
|title=Review of Particle Physics
|year= 2018
|doi=10.1103/PhysRevD.98.030001
|volume=98
|issue=3
|pages=030001
|journal=Physical Review D
|url=http://pdglive.lbl.gov/DataBlock.action?node=Q004M
|doi-access=free
|bibcode=2018PhRvD..98c0001T
}}</ref>
|decay_time =
|decay_particle = [[Strange quark]] (~95%), [[down quark]] (~5%)<ref name="hyperphysics">
{{cite web
|quote=The c quark has about 5% probability of decaying into a d quark instead of an s quark.
|author=R. Nave
|url=http://hyperphysics.phy-astr.gsu.edu/hbase/particles/qrkdec.html
|title=Transformation of Quark Flavors by the Weak Interaction
|access-date=2010-12-06
}}</ref><ref name="PDG2010_CKM">
{{cite journal
|author=K. Nakamura ''et al.'' ([[Particle Data Group]])
|year=2010
|title=Review of Particles Physics: The CKM Quark-Mixing Matrix
|url=http://pdg.lbl.gov/2010/reviews/rpp2010-rev-ckm-matrix.pdf
|journal=[[Journal of Physics G]]
|volume=37 |issue=75021 |page=150
|doi=10.1088/0954-3899/37/7a/075021
|display-authors=etal
|bibcode=2010JPhG...37g5021N}}</ref>
|electric_charge = +{{sfrac|2|3}} [[Elementary charge|''e'']]
|color_charge = Yes
|spin = {{sfrac|1|2}}
|num_spin_states =
|weak_isospin = {{nowrap|[[Chirality (physics)|LH]]: +{{sfrac|1|2}}, [[Chirality (physics)|RH]]: 0}}
|weak_hypercharge= {{nowrap|[[Chirality (physics)|LH]]: +{{sfrac|1|3}}, [[Chirality (physics)|RH]]: +{{sfrac|4|3}}}}
}}

The '''charm quark''', '''charmed quark''' or '''c quark''' (from its symbol, c) is the third most massive of all [[quark]]s, a type of [[elementary particle]]. Charm quarks are found in [[hadron]]s, which are [[subatomic particle]]s made of quarks. Examples of hadrons containing charm quarks include the [[J/ψ]] meson ({{SubatomicParticle|J/Psi}}), [[D meson]]s ({{SubatomicParticle|D}}), [[charmed Sigma baryon]]s ({{SubatomicParticle|charmed Sigma}}), and other charmed particles.

It, along with the [[strange quark]], is part of the [[generation (particle physics)|second generation]] of matter, and has an [[electric charge]] of +{{sfrac|2|3}} [[elementary charge|''e'']] and a [[Quark#Mass|bare mass]] of {{val|1.275|+0.025|-0.035|ul=GeV/c2}}.<ref name="PDG2018"/> Like all [[quark]]s, the charm quark is an [[elementary particle|elementary]] [[fermion]] with [[spin (physics)|spin]] [[spin-1/2|{{sfrac|1|2}}]], and experiences all four [[fundamental interaction]]s: [[gravitation]], [[electromagnetism]], [[weak interaction]]s, and [[strong interaction]]s. The [[antiparticle]] of the charm quark is the '''charm antiquark''' (sometimes called ''anticharm quark'' or simply ''anticharm''), which differs from it only in that some of its properties have [[additive inverse|equal magnitude but opposite sign]].

The existence of a fourth quark had been speculated by a number of authors around 1964 (for instance by [[James Bjorken]] and [[Sheldon Lee Glashow|Sheldon Glashow]]<ref>
{{cite journal
|author=B.J. Bjorken, S.L. Glashow
|year=1964
|title=Elementary particles and SU(4)
|journal=[[Physics Letters]]
|volume=11 |pages=255–257
|doi=10.1016/0031-9163(64)90433-0
|bibcode = 1964PhL....11..255B
|issue=3 |last2=Glashow
}}</ref>), but its prediction is usually credited to [[Sheldon Lee Glashow|Sheldon Glashow]], [[John Iliopoulos]] and [[Luciano Maiani]] in 1970 (see [[GIM mechanism]]).<ref>
{{cite journal
|author=S.L. Glashow, J. Iliopoulos, L. Maiani
|year=1970
|title=Weak Interactions with Lepton–Hadron Symmetry
|journal=[[Physical Review D]]
|volume=2 |pages=1285–1292
|doi=10.1103/PhysRevD.2.1285
|bibcode = 1970PhRvD...2.1285G
|issue=7 |last2=Iliopoulos
|last3=Maiani
}}</ref> Glashow is quoted as saying, "We called our construct the 'charmed quark', for we were fascinated and pleased by the symmetry it brought to the subnuclear world."<ref>
{{cite book
|author=M. Riordan
|title=The Hunting of the Quark: A True Story of Modern Physics
|page=[https://archive.org/details/huntingofquarktr00mich/page/210 210]
|publisher=[[Simon & Schuster]]
|year=1987
|isbn=978-0-671-50466-3
|url=https://archive.org/details/huntingofquarktr00mich/page/210
}}</ref> The first charmed particle (a particle containing a charm quark) to be discovered was the [[J/ψ meson]]. It was discovered in 1974 by a team at the [[Stanford Linear Accelerator Center]] (SLAC), led by [[Burton Richter]],<ref>
{{cite journal
|author=J.-E. Augustin
|year=1974
|title=Discovery of a Narrow Resonance in ''e''+''e''− Annihilation
|journal=[[Physical Review Letters]]
|volume=33 |issue=23 |page=1406
|bibcode=1974PhRvL..33.1406A
|doi=10.1103/PhysRevLett.33.1406
|display-authors=etal|doi-access=free
}}</ref> and one at the [[Brookhaven National Laboratory]] (BNL), led by [[Samuel C. C. Ting|Samuel Ting]].<ref>
{{cite journal
|author=J.J. Aubert
|year=1974
|title=Experimental Observation of a Heavy Particle ''J''
|journal=[[Physical Review Letters]]
|volume=33 |issue=23 |page=1404
|bibcode=1974PhRvL..33.1404A
|doi=10.1103/PhysRevLett.33.1404
|display-authors=etal|doi-access=free
}}</ref>

The 1974 discovery of the {{SubatomicParticle|J/Psi}} (and thus the charm quark) ushered in a series of breakthroughs which are collectively known as the ''[[November Revolution (physics)|November Revolution]]''.

== Hadrons containing charm quarks ==
{{Main list|List of baryons|list of mesons}}

Some of the [[hadron]]s containing charm quarks include:
* [[D meson]]s contain a charm quark (or its [[antiparticle]]) and an [[up quark|up]] or [[down quark]].
* {{SubatomicParticle|Strange D}} mesons contain a charm quark and a [[strange quark]].
* There are many [[charmonium]] states, for example the {{SubatomicParticle|J/Psi}} particle. These consist of a charm quark and its antiparticle.
* [[Charmed baryon]]s have been observed, and are named in analogy with strange baryons (e.g. {{SubatomicParticle|Charmed Lambda+}}).

==See also==
*[[Quark model]]

== References ==
{{Reflist}}

==Further reading==
*{{cite web
|author=R. Nave
|title=Quarks
|url=http://hyperphysics.phy-astr.gsu.edu/hbase/Particles/quark.html
|work=[[HyperPhysics]]
|publisher=[[Georgia State University]], Department of Physics and Astronomy
|access-date=2008-06-29
}}
*{{cite book
|author=A. Pickering
|title=Constructing Quarks
|pages=114–125
|publisher=[[University of Chicago Press]]
|year=1984
|isbn=978-0-226-66799-7
}}

{{Particles}}
{{Authority control}}

[[Category:Quarks]]
[[Category:Elementary particles]]

Strangeness production

2021-05-10T08:10:15Z

Bleicher:

{{see also|Strangeness}}
'''Strangeness production''' in [[Relativistic heavy-ion collisions|relativistic heavy ion collisions]] is a signature and a [[diagnostic tool]] of [[quark–gluon plasma]] (QGP) formation and properties.<ref>{{Cite journal|last1=Margetis|first1=Spyridon|last2=Safarík|first2=Karel|last3=Villalobos Baillie|first3=Orlando|year=2000|title=Strangeness Production in Heavy-Ion Collisions|journal=Annual Review of Nuclear and Particle Science|language=en|volume=50|issue=1|pages=299–342|doi=10.1146/annurev.nucl.50.1.299|bibcode=2000ARNPS..50..299S|issn=0163-8998|doi-access=free}}</ref> Unlike [[up quark|up]] and [[down quark]]s, from which everyday matter is made, heavier quark flavors such as [[strangeness]] and [[Charm quark|charm]] typically approach chemical equilibrium in a dynamic evolution process. QGP (also known as [[QCD matter|quark matter]]) is an interacting localized assembly of [[quark]]s and [[gluon]]s at [[Thermodynamic equilibrium#Local and global equilibrium|thermal (kinetic)]] and not necessarily chemical (abundance) equilibrium. The word plasma signals that color charged particles (quarks and/or gluons) are able to move in the volume occupied by the plasma. The abundance of [[strange quark]]s is formed in [[pair production|pair-production]] processes in collisions between constituents of the plasma, creating the chemical abundance equilibrium. The dominant mechanism of production involves [[gluon]]s only present when matter has become a quark–gluon plasma. When quark–gluon plasma disassembles into [[hadron]]s in a breakup process, the high availability of strange [[quark|antiquarks]] helps to produce antimatter containing multiple strange quarks, which is otherwise rarely made. Similar considerations are at present made for the heavier [[charm quark|charm]] flavor, which is made at the beginning of the collision process in the first interactions and is only abundant in the high-energy environments of [[CERN]]'s [[Large Hadron Collider]].

==Quark–gluon plasma in the early universe and in the laboratory==
[[File:strange production 3.svg|thumb|280px|Collision between two highly-energetic [[atomic nucleus|nuclei]] create an extremely dense environment, in which quarks and gluons may interact as free particles for brief moments. The collisions happened at such extreme velocities that the nuclei are "pancaked" because of [[Lorentz contraction]].]]

Free quarks probably existed in the extreme conditions of the very early universe until about 30 [[microsecond]]s after the Big Bang,<ref name=":11">
{{cite book
|author=J. Letessier
|author2=J. Rafelski
|date=2002
|title=Hadrons and Quark–Gluon Plasma
|publisher=[[Cambridge University Press]]
|isbn=978-0-521-38536-7
}}</ref> in a very hot [[gas]] of free quarks, antiquarks and [[gluon]]s. This gas is called [[quark–gluon plasma]] (QGP), since the quark-interaction charge ([[color charge]]) is mobile and quarks and gluons move around. This is possible because at a high temperature the early universe is in a different [[vacuum decay|vacuum state]], in which normal matter cannot exist but quarks and gluons can; they are [[deconfinement|deconfined]] (able to exist independently as separate unbound particles). In order to recreate this [[deconfinement|deconfined]] [[Phase (matter)|phase of matter]] in the laboratory it is necessary to exceed a minimum temperature, or its equivalent, a minimum [[energy density]]. Scientists achieve this using [[Relativistic nuclear collisions|particle collisions]] at extremely high speeds, where the energy released in the collision can raise the subatomic particles' energies to an exceedingly high level, sufficient for them to briefly form a tiny amount of quark–gluon plasma that can be studied in laboratory experiments for little more than the time light needs to cross the QGP fireball, thus about 10−22 s. After this brief time the hot drop of quark plasma evaporates in a process called [[hadronization]]. This is so since practically all QGP components flow out at relativistic speed. In this way, it is possible to study conditions akin to those in the early Universe at the age of 10–40 microseconds.

[[QCD matter#Laboratory experiments|Discovery]] of this new QGP [[state of matter]] has been announced both at [[CERN]]<ref>{{Cite journal|last=Abbott|first=Alison|year=2000|title=CERN claims first experimental creation of quark–gluon plasma|url=http://www.nature.com/articles/35001196|journal=Nature|language=en|volume=403|issue=6770|pages=581|doi=10.1038/35001196|pmid=10688162|bibcode=2000Natur.403..581A|issn=0028-0836|doi-access=free}}</ref> and at [[Brookhaven National Laboratory]] (BNL).<ref>{{Cite journal|last1=Jacak|first1=Barbara|last2=Steinberg|first2=Peter|year=2010|title=Creating the perfect liquid in heavy-ion collisions|journal=Physics Today|language=en|volume=63|issue=5|pages=39–43|doi=10.1063/1.3431330|bibcode=2010PhT....63e..39J|issn=0031-9228}}</ref> Preparatory work, allowing for these discoveries, was carried out at the [[Joint Institute for Nuclear Research]] (JINR) and [[Lawrence Berkeley National Laboratory]] (LBNL) at the [[Bevalac]].<ref name=":8">{{Cite journal|last1=Gazdzicki|first1=Marek|last2=Gorenstein|first2=Mark|last3=Seyboth|first3=Peter|date=2020-04-05|title=Brief history of the search for critical structures in heavy-ion collisions|journal=Acta Physica Polonica B|volume=51|issue=5|page=1033|doi=10.5506/APhysPolB.51.1033|arxiv=2004.02255|s2cid=214802159}}</ref> New experimental facilities, [[Facility for Antiproton and Ion Research|FAIR]] at the [[GSI Helmholtz Centre for Heavy Ion Research]] (GSI) and NICA at JINR, are under construction. Strangeness as a signature of QGP was first explored in 1983.<ref>{{Cite journal|last1=Anikina|first1=M.|last2=Gaździcki|first2=M.|last3=Golokhvastov|first3=A.|last4=Goncharova|first4=L.|last5=Iovchev|first5=K.|last6=Khorozov|first6=S.|last7=Kuznetzova|first7=E.|last8=Lukstins|first8=J.|last9=Okonov|first9=E.|last10=Ostanievich|first10=T.|last11=Sidorin|first11=S.|year=1983|title=Λ Hyperons Produced in Central Nucleus-Nucleus Interactions at 4.5 GeV/ c Momentum per Incident Nucleon|journal=Physical Review Letters|language=en|volume=50|issue=25|pages=1971–1974|doi=10.1103/PhysRevLett.50.1971|bibcode=1983PhRvL..50.1971A|issn=0031-9007}}</ref> Comprehensive experimental evidence about its properties is being assembled. Recent work by the [[ALICE experiment|ALICE collaboration]]<ref name=":0">{{Cite journal|last=ALICE Collaboration|year=2017|title=Enhanced production of multi-strange hadrons in high-multiplicity proton–proton collisions|url=http://www.nature.com/articles/nphys4111|journal=Nature Physics|language=en|volume=13|issue=6|pages=535–539|doi=10.1038/nphys4111|arxiv=1606.07424|bibcode=2017NatPh..13..535A|issn=1745-2473|doi-access=free}}</ref> at CERN has opened a new path to study of QGP and strangeness production in very high energy pp collisions.

==Strangeness in quark–gluon plasma==
The diagnosis and the study of the properties of quark–gluon plasma can be undertaken using quarks not present in matter seen around us. The experimental and theoretical work relies on the idea of strangeness enhancement. This was the first observable of quark–gluon plasma proposed in 1980 by [[Johann Rafelski]] and [[Rolf Hagedorn]].<ref>
{{cite conference
|author=J. Rafelski
|author2=R. Hagedorn
|date=1981
|title=From Hadron Gas to Quark Matter II
|url=http://www.physics.arizona.edu/~rafelski/Books/80BielRH2.pdf
|pages=253–272
|book-title=Statistical mechanics of quarks and hadrons
|editor=H. Satz
|publisher=[[North-Holland Publishing Company|North-Holland]] and [[Elsevier]]
|id=CERN-TH-2969 (1980)
|isbn=0-444-86227-7
}}</ref> Unlike the up and down quarks, strange quarks are not brought into the reaction by the colliding nuclei. Therefore, any strange quarks or antiquarks observed in experiments have been "freshly" made from the kinetic energy of colliding nuclei, with gluons being the catalyst.<ref name=":4" /> Conveniently, the [[mass]] of strange quarks and antiquarks is equivalent to the temperature or energy at which protons, neutrons and other [[hadron]]s dissolve into quarks. This means that the abundance of strange quarks is sensitive to the conditions, structure and dynamics of the deconfined matter phase, and if their number is large it can be assumed that deconfinement conditions were reached. An even stronger signature of strangeness enhancement is the highly enhanced production of [[#Strangeness (and charm) hadronization|strange antibaryons]].<ref>{{Cite journal|last=Rafelski|first=Johann|year=2015|orig-year=1980|title=Extreme states of nuclear matter - 1980: From: "Workshop on Future Relativistic Heavy Ion Experiments" held 7-10 October 1980 at: GSI, Darmstadt, Germany|journal=The European Physical Journal A|language=en|volume=51|issue=9|pages=115|doi=10.1140/epja/i2015-15115-y|bibcode=2015EPJA...51..115R|issn=1434-6001|doi-access=free}}</ref><ref>{{Cite journal|last=Rafelski|first=Johann|year=2015|orig-year=1983|title=Strangeness and phase changes in hot hadronic matter - 1983: From: "Sixth High Energy Heavy Ion Study" held 28 June - 1 July 1983 at: LBNL, Berkeley, CA, USA|journal=The European Physical Journal A|language=en|volume=51|issue=9|pages=116|doi=10.1140/epja/i2015-15116-x|bibcode=2015EPJA...51..116R|issn=1434-6001|doi-access=free}}</ref> An early comprehensive review of strangeness as a signature of QGP was presented by Koch, Müller and Rafelski,<ref name=":3" /> which was recently updated.<ref name=":9" /> The abundance of produced strange anti-baryons, and in particular anti-omega <math>\bar{\Omega}(\bar{s}\bar{s}\bar{s})</math>, allowed to distinguish fully deconfined large QGP domain<ref>{{Cite journal|last1=Soff|first1=S.|last2=Bass|first2=S.A.|last3=Bleicher|first3=M.|last4=Bravina|first4=L.|last5=Gorenstein|first5=M.|last6=Zabrodin|first6=E.|last7=Stöcker|first7=H.|last8=Greiner|first8=W.|year=1999|title=Strangeness enhancement in heavy ion collisions – evidence for quark–gluon matter?|journal=Physics Letters B|language=en|volume=471|issue=1|pages=89–96|doi=10.1016/S0370-2693(99)01318-0|arxiv=nucl-th/9907026|bibcode=1999PhLB..471...89S|s2cid=16805966}}</ref> from transient collective quark models such as the color rope model proposed by Biró, [[Holger Bech Nielsen|Nielsen]] and Knoll.<ref>{{Cite journal|last1=Biro|first1=T.S.|last2=Nielsen|first2=H.B.|last3=Knoll|first3=J.|year=1984|title=Colour rope model for extreme relativistic heavy ion collisions|journal=Nuclear Physics B|language=en|volume=245|pages=449–468|doi=10.1016/0550-3213(84)90441-3|bibcode=1984NuPhB.245..449B}}</ref> The relative abundance of <math>\phi (s\bar{s})/\bar{\Xi}(\bar{q}\bar{s}\bar{s})</math> resolves<ref name=":10" /> questions raised by the canonical model of strangeness enhancement.<ref name=":5" />

== Equilibrium of strangeness in quark–gluon plasma ==

One cannot assume that under all conditions the yield of strange quarks is in thermal equilibrium. In general, the quark-flavor composition of the plasma varies during its ultra short lifetime as new flavors of quarks such as strangeness are cooked up inside. The up and down quarks from which normal matter is made are easily produced as quark-antiquark pairs in the hot fireball because they have small masses. On the other hand, the next lightest quark flavor—strange quarks—will reach its high quark–gluon plasma thermal abundance provided that there is enough time and that the temperature is high enough.<ref name=":9">{{Cite journal|last1=Koch|first1=Peter|last2=Müller|first2=Berndt|last3=Rafelski|first3=Johann|year=2017|title=From strangeness enhancement to quark–gluon plasma discovery|journal=International Journal of Modern Physics A|language=en|volume=32|issue=31|pages=1730024–272|doi=10.1142/S0217751X17300241|arxiv=1708.08115|bibcode=2017IJMPA..3230024K|s2cid=119421190|issn=0217-751X}}</ref> This work elaborated the kinetic theory of strangness production proposed by T. Biro and J. Zimanyi who demonstrated that strange quarks could not be produced fast enough alone by quark-antiquark reactions.<ref>{{Cite journal|last1=Biró|first1=T.S.|last2=Zimányi|first2=J.|year=1982|title=Quarkochemistry in relativistic heavy-ion collisions|journal=Physics Letters B|language=en|volume=113|issue=1|pages=6–10|doi=10.1016/0370-2693(82)90097-1|bibcode=1982PhLB..113....6B|url=http://real-eod.mtak.hu/7402/1/KFKIreports_81-069.pdf}}</ref> A new mechanism operational alone in QGP was proposed.

==Gluon fusion into strangeness==
[[File:strange production 8.gif|thumb|280px| Feynman diagrams for the lowest order in strong coupling constant <math>\alpha_{s}</math> strangeness production processes: gluon fusion, top, dominate the light quark based production.|alt=]]

Yield equilibration of strangeness yield in QGP is only possible due to a new process, gluon fusion, as shown by [[Johann Rafelski|Rafelski]] and [[Berndt Müller|Müller]].<ref name=":4">{{Cite journal|last1=Rafelski|first1=Johann|last2=Müller|first2=Berndt|year=1982|title=Strangeness Production in the Quark–Gluon Plasma|journal=Physical Review Letters|language=en|volume=48|issue=16|pages=1066–1069|doi=10.1103/PhysRevLett.48.1066|bibcode=1982PhRvL..48.1066R|issn=0031-9007}} {{erratum|doi=10.1103/PhysRevLett.56.2334|checked=yes}}</ref> The top section of the [[Feynman diagram]]s figure, shows the new gluon fusion processes: gluons are the wavy lines; strange quarks are the solid lines; time runs from left to right. The bottom section is the process where the heavier quark pair arises from the lighter pair of quarks shown as dashed lines. The gluon fusion process occurs almost ten times faster than the quark-based strangeness process, and allows achievement of the high thermal yield where the quark based process would fail to do so during the duration of the "micro-bang".<ref>{{Cite journal|last=Rafelski|first=Johann|year=1984|title=Strangeness production in the quark gluon plasma|journal=Nuclear Physics A|language=en|volume=418|pages=215–235|doi=10.1016/0375-9474(84)90551-7|bibcode=1984NuPhA.418..215R|url=http://cds.cern.ch/record/147788}}</ref>

The ratio of newly produced <math>\bar{s}s</math> pairs with the normalized light quark pairs <math>\bar{u}u+\bar{d}d/2</math>—the Wroblewski ratio<ref>{{Cite journal|last=Wroblewski|first=A.|year=1985|title=On the strange quark suppression factor in high-energy collisions|url=https://www.actaphys.uj.edu.pl/R/16/4/379/pdf|journal=Acta Phys. Polon. B|volume=16|pages=379–392}}</ref>—is considered a measure of efficacy of strangeness production. This ratio more than doubles in heavy ion collisions,<ref>{{Citation|last1=Becattini|first1=Francesco|title=The QCD Confinement Transition: Hadron Formation|date=2010|url=http://materials.springer.com/lb/docs/sm_lbs_978-3-642-01539-7_8|work=Relativistic Heavy Ion Physics|volume=23|pages=208–239|editor-last=Stock|editor-first=R.|publisher=Springer Berlin Heidelberg|arxiv=0907.1031|doi=10.1007/978-3-642-01539-7_8|isbn=978-3-642-01538-0|quote=Fig. 10|access-date=2020-04-20|last2=Fries|first2=Rainer J.|bibcode=2010LanB...23..208B|s2cid=14306761}}</ref> providing a model independent confirmation of a new mechanism of strangeness production operating in collisions that are producing QGP.

Regarding [[Flavour (particle physics)|charm and bottom flavour]]:<ref name=":7">{{Cite journal|last1=Dong|first1=Xin|last2=Lee|first2=Yen-Jie|last3=Rapp|first3=Ralf|year=2019|title=Open Heavy-Flavor Production in Heavy-Ion Collisions|journal=Annual Review of Nuclear and Particle Science|language=en|volume=69|issue=1|pages=417–445|doi=10.1146/annurev-nucl-101918-023806|arxiv=1903.07709|bibcode=2019ARNPS..69..417D|s2cid=119328093|issn=0163-8998}}</ref><ref>{{Citation|last1=Kluberg|first1=Louis|title=Color deconfinement and charmonium production in nuclear collisions|url=http://materials.springer.com/lb/docs/sm_lbs_978-3-642-01539-7_13|work=Relativistic Heavy Ion Physics|volume=23|pages=373–423|year=2010|editor-last=Stock|editor-first=R.|publisher=Springer Berlin Heidelberg|arxiv=0901.3831|doi=10.1007/978-3-642-01539-7_13|isbn=978-3-642-01538-0|access-date=2020-04-20|last2=Satz|first2=Helmut|bibcode=2010LanB...23..373K|s2cid=13953895}}</ref> the gluon collisions here are occurring within the thermal matter phase and thus are different from the high energy processes that can ensue in the early stages of the collisions when the nuclei crash into each other. The heavier, charm and bottom quarks are produced there dominantly. The study in relativistic nuclear (heavy ion) collisions of charmed and soon also bottom hadronic particle production—beside strangeness—will provide complementary and important confirmation of the mechanisms of formation, evolution and hadronization of quark–gluon plasma in the laboratory.<ref name=":0" />

==Strangeness (and charm) hadronization==
[[File:Strange production 5.gif|thumb|280px|Illustration of the two step process of strange antibaryon production, a key signature of QGP: strangeness is produced inside the fireball and later on in an independent process at hadronization several (anti)strange quarks form (anti)baryons. The production of triple strange <math>\Omega</math> and <math>\bar{\Omega}</math> is the strongest signature to date of QGP formation.|alt=]]
These newly cooked strange quarks find their way into a multitude of different final particles that emerge as the hot quark–gluon plasma fireball breaks up, see the scheme of different processes in figure. Given the ready supply of antiquarks in the "fireball", one also finds a multitude of antimatter particles containing more than one strange quark. On the other hand, in a system involving a cascade of nucleon-nucleon collisions, multi-strange antimatter are produced less frequently considering that several relatively improbable events must occur in the same collision process. For this reason one expects that the yield of multi-strange antimatter particles produced in the presence of quark matter is enhanced compared to conventional series of reactions.<ref>{{Cite thesis |type=PhD|last=Petran|first=Michal|year=2013|title=Strangeness and charm in quark–gluon hadronization|publisher=University of Arizona|arxiv=1311.6154}}</ref><ref name=":1">
{{cite journal|author=R. Stock|author2=NA35 Collaboration|author2-link=NA35 experiment|date=1991|title=Strangeness enhancement in central S + S collisions at 200 GeV/nucleon|journal=[[Nuclear Physics A]]|volume=525|pages=221–226|bibcode=1991NuPhA.525..221S|doi=10.1016/0375-9474(91)90328-4}}</ref> Strange quarks also bind with the heavier charm and bottom quarks which also like to bind with each other. Thus, in the presence of a large number of these quarks, quite unusually abundant exotic particles can be produced; some of which have never been observed before. This should be the case in the forthcoming exploration at the new [[Large Hadron Collider]] at CERN of the particles that have charm and strange quarks, and even bottom quarks, as components.<ref>{{Cite journal|last1=Kuznetsova|first1=I.|last2=Rafelski|first2=J.|year=2007|title=Heavy flavor hadrons in statistical hadronization of strangeness-rich QGP|journal=The European Physical Journal C|language=en|volume=51|issue=1|pages=113–133|doi=10.1140/epjc/s10052-007-0268-9|arxiv=hep-ph/0607203|bibcode=2007EPJC...51..113K|s2cid=18266326|issn=1434-6044}}</ref>

==Strange hadron decay and observation==
[[File:WA97TransverseLXSlop.png|thumb|280px|Universality of transverse mass spectra of strange baryons and antibaryons as measured by CERN-WA97 collaboration.<ref name=":2">{{Cite journal|last=The WA97 Collaboration|year=2000|title=Transverse mass spectra of strange and multi–strange particles in Pb–Pb collisions at 158 A GeV/c|journal=The European Physical Journal C|language=en|volume=14|issue=4|pages=633–641|doi=10.1007/s100520000386|bibcode=2000EPJC...14..633W|s2cid=195312472|issn=1434-6044|url=http://cds.cern.ch/record/429846}}</ref> Collisions at 158 A GeV. These results demonstrate that all these particles are produced in explosively hadronizing fireball (of QGP) and do not undergo further interaction once produced. This key result shows therefore formation a new state of matter announced at CERN in February 2000.]]
Strange quarks are naturally [[radioactive decay|radioactive]] and decay by [[weak interaction]]s into lighter quarks on a timescale that is extremely long compared with the nuclear-collision times. This makes it relatively easy to detect [[strange particle]]s through the tracks left by their decay products. Consider as an example the decay of a negatively charged <math>\Xi</math> [[Xi baryon|baryon]] (green in figure, dss), into a negative [[pion]] ({{Subatomic particle|Up antiquark}}d) and a neutral [[Lambda baryon|<math>\Lambda</math>]] (uds) [[Lambda baryon|baryon]]. Subsequently, the <math>\Lambda</math> decays into a proton and another negative pion. In general this is the signature of the decay of a <math>\Xi</math>. Although the negative <math>\Omega</math> (sss) [[Omega baryon|baryon]] has a similar final state decay topology, it can be clearly distinguished from the <math>\Xi</math> because its decay products are different.

Measurement of abundant formation of <math>\Xi</math> (uss/dss), <math>\Omega</math> (sss) and especially their antiparticles is an important cornerstone of the claim that quark–gluon plasma has been formed.<ref name=":2" /> This abundant formation is often presented in comparison with the scaled expectation from normal proton-proton collisions; however, such a comparison is not a necessary step in view of the large absolute yields which defy conventional model expectations.<ref name=":3">
{{cite journal
|author=P. Koch
|author2=B. Müller
|author3=J. Rafelski
|title=Strangeness in relativistic heavy ion collisions
|journal=[[Physics Reports]]
|volume=142 |pages=167
|doi=10.1016/0370-1573(86)90096-7
|date=1986
|bibcode = 1986PhR...142..167K
|issue=4 |citeseerx=10.1.1.462.8703
}}</ref> The overall yield of strangeness is also larger than expected if the new form of matter has been achieved. However, considering that the light quarks are also produced in gluon fusion processes, one expects increased production of all [[hadron]]s. The study of the relative yields of strange and non strange particles provides information about the competition of these processes and thus the reaction mechanism of particle production.

==Systematics of strange matter and antimatter creation==
[[File:14AliceSEnhance1.png|thumb|280px|Enhancement of antibaryon yield increases with number of newly made quarks (s, anti-s, anti-q) and the size of the colliding system represented by the number of nucleons "damaged=wounded" in the collision of relativistic heavy ions. SPS, RHIC, and ALICE results shown as function of participating nucleons scaled—this represents residual enhancement after removal of scaling with number of participant.|alt=]]
The work of Koch, Muller, Rafelski<ref name=":3" /> predicts that in a quark–gluon plasma hadronization process the enhancement for each particle species increases with the strangeness content of the particle. The enhancements for particles carrying one, two and three strange or antistrange quarks were measured and this effect was demonstrated by the CERN [[WA97 experiment]]<ref>
{{cite journal
|author=E. Andersen
|author2=WA97 Collaboration
|author2-link=WA97 experiment
|date=1999
|title=Strangeness enhancement at mid-rapidity in Pb–Pb collisions at 158 A GeV/c
|journal=[[Physics Letters B]]
|volume=449 |pages=401
|doi=10.1016/S0370-2693(99)00140-9
|bibcode = 1999PhLB..449..401W
|issue=3–4 |url=http://cds.cern.ch/record/380839
}}</ref> in time for the CERN announcement in 2000<ref>{{Cite web|url=https://home.cern/news/press-release/cern/new-state-matter-created-cern|title=New State of Matter created at CERN|date=10 February 2000|website=CERN|language=en|access-date=2020-04-24}}</ref> of a possible quark–gluon plasma formation in its experiments.<ref>{{Cite arxiv|last1=Heinz|first1=Ulrich|last2=Jacob|first2=Maurice|date=2000-02-16|title=Evidence for a New State of Matter: An Assessment of the Results from the CERN Lead Beam Programme|eprint=nucl-th/0002042}}</ref> These results were elaborated by the successor collaboration [[NA57 experiment|NA57]]<ref>
{{cite journal
|author=F. Antinori
|author2=NA57 Collaboration
|author2-link=NA57 experiment
|date=2006
|title=Enhancement of hyperon production at central rapidity in 158 ''A'' GeV/''c'' Pb+Pb collisions
|journal=[[Journal of Physics G]]
|volume= 32|pages=427–442
|doi=10.1088/0954-3899/32/4/003
|arxiv=nucl-ex/0601021
|bibcode = 2006JPhG...32..427N
|issue=4 |s2cid=119102482
}}</ref> as shown in the enhancement of antibaryon figure. The gradual rise of the enhancement as a function of the variable representing the amount of nuclear matter participating in the collisions, and thus as a function of the geometric centrality of nuclear collision strongly favors the quark–gluon plasma source over normal matter reactions.

A similar enhancement was obtained by the [[STAR detector|STAR]] experiment at the [[Relativistic Heavy Ion Collider|RHIC]].<ref>
{{cite journal
|author=A.R. Timmins
|author2=STAR Collaboration
|author2-link=STAR experiment
|year=2009
|title=Overview of strangeness production at the STAR experiment
|journal=[[Journal of Physics G]]
|volume= 36|pages=064006
|arxiv=0812.4080
|doi=10.1088/0954-3899/36/6/064006
|bibcode = 2009JPhG...36f4006T
|issue=6 |s2cid=12853074
}}</ref> Here results obtained when two colliding systems at 100 A GeV in each beam are considered: in red the heavier Gold-Gold collisions and in blue the smaller Copper-Copper collisions. The energy at RHIC is 11 times greater in the CM frame of reference compared to the earlier CERN work. The important result is that enhancement observed by STAR also increases with the number of participating nucleons. We further note that for the most peripheral events at the smallest number of participants, copper and gold systems show, at the same number of participants, the same enhancement as expected.

Another remarkable feature of these results, comparing CERN and STAR, is that the enhancement is of similar magnitude for the vastly different collision energies available in the reaction. This near energy independence of the enhancement also agrees with the quark–gluon plasma approach regarding the mechanism of production of these particles and confirms that a quark–gluon plasma is created over a wide range of collision energies, very probably once a minimal energy threshold is exceeded.

==ALICE: Resolution of remaining questions about strangeness as signature of quark–gluon plasma==
[[File:AliceXiPhiRev.png|thumb|280px| LHC-ALICE results for <math>(\bar{\Xi}+\Xi/\phi)</math> obtained in three different collision systems at highest available energy as a function of charged hadron multiplicity produced.<ref name=":6">{{Cite journal|last=Rafelski|first=Johann|year=2020|title=Discovery of Quark–Gluon Plasma: Strangeness Diaries|journal=The European Physical Journal Special Topics|language=en|volume=229|issue=1|pages=1–140|doi=10.1140/epjst/e2019-900263-x|arxiv=1911.00831|bibcode=2020EPJST.229....1R|s2cid=207869782|issn=1951-6355}}</ref><ref>{{Cite journal|last=Tripathy|first=Sushanta|year=2019|title=Energy dependence of ϕ(1020) production at mid-rapidity in pp collisions with ALICE at the LHC|journal=Nuclear Physics A|language=en|volume=982|pages=180–182|doi=10.1016/j.nuclphysa.2018.09.078|arxiv=1807.11186|bibcode=2019NuPhA.982..180T|s2cid=119223653}}</ref><ref>{{cite arxiv|last=Tripathy|first=Sushanta|date=2019-07-01|title=An insight into strangeness with $\phi$(1020) production in small to large collision systems with ALICE at the LHC|class=hep-ex|eprint=1907.00842}}</ref>|alt=]]
[[File:AliceEventMultiplicityPrelim159143.png|thumb|280px|Ratio to pion of integrated yields for <math>p, K^0_s, \Lambda, \phi, \Xi</math> and <math>\Omega</math>. The evolution with multiplicity at mid-rapidity, <math>\operatorname{d}\!N_{ch}/\operatorname{d}\!\eta{{|}}_{<0.5}</math>, is reported for several systems and energies, including pp at <math>\sqrt{s}=7</math> TeV, p-Pb at <math>\sqrt{s_{\operatorname{N}\!\operatorname{N}\!}}=5.02</math> TeV, and also the ALICE preliminary results for pp at <math>\sqrt{s}=13</math> TeV, Xe-Xe at <math>\sqrt{s_{\operatorname{N}\!\operatorname{N}\!}}=5.44</math> TeV and Pb-Pb at <math>\sqrt{s_{\operatorname{N}\!\operatorname{N}\!}}=5.02</math> TeV are included for comparison. Error bars show the statistical uncertainty, whereas the empty boxes show the total systematic uncertainty.<ref>{{Cite journal|last=Albuquerque|first=D.S.D.|year=2019|title=Hadronic resonances, strange and multi-strange particle production in Xe-Xe and Pb-Pb collisions with ALICE at the LHC|journal=Nuclear Physics A|language=en|volume=982|pages=823–826|doi=10.1016/j.nuclphysa.2018.08.033|arxiv=1807.08727|bibcode=2019NuPhA.982..823A|s2cid=119404602}}</ref>|alt=]]
The very high precision of (strange) particle spectra and large transverse momentum coverage reported by the [[ALICE experiment|ALICE]] Collaboration at the [[Large Hadron Collider]] (LHC) allows in-depth exploration of lingering challenges, which always accompany new physics, and here in particular the questions surrounding strangeness signature. Among the most discussed challenges has been the question if the abundance of particles produced is enhanced or if the comparison base line is suppressed. Suppression is expected when a otherwise absent quantum number, such as strangeness, is rarely produced. This situation was recognized by [[Rolf Hagedorn|Hagedorn]] in his early analysis of particle production<ref>{{Cite journal|last=Hagedorn|first=Rolf|year=1968|title=Statistical thermodynamics of strong interactions at high energies - III : heavy pair(quark) production rates|url=http://cds.cern.ch/record/936485|journal=Supplemento al Nuovo Cimento|volume=6|pages=311–354}}</ref> and solved by [[Johann Rafelski|Rafelski]] and Danos.<ref>{{Cite journal|last1=Rafelski|first1=Johann|last2=Danos|first2=Michael|year=1980|title=The importance of the reaction volume in hadronic collisions|journal=Physics Letters B|language=en|volume=97|issue=2|pages=279–282|doi=10.1016/0370-2693(80)90601-2|bibcode=1980PhLB...97..279R}}</ref> In that work it was shown that even if just a few new pairs of strange particles were produced the effect disappears. However, the matter was revived by Hamieh et al.<ref name=":5">{{Cite journal|last1=Hamieh|first1=Salah|last2=Redlich|first2=Krzysztof|last3=Tounsi|first3=Ahmed|year=2000|title=Canonical description of strangeness enhancement from p–A to Pb–Pb collisions|journal=Physics Letters B|language=en|volume=486|issue=1–2|pages=61–66|doi=10.1016/S0370-2693(00)00762-0|arxiv=hep-ph/0006024|bibcode=2000PhLB..486...61H|s2cid=8566125}}</ref> who argued that is possible that small sub-volumes in QGP are of relevance. This argument can be resolved by exploring specific sensitive experimental signatures for example the ratio of double strange particles of different type, such yield of <math>ssq</math> (<math>\Xi</math>) compared to <math>\bar{s}s</math>(<math>\phi</math>). The [[ALICE experiment]] obtained this ratio for several collision systems in a wide range of [[hadronization]] volumes as described by the total produced particle multiplicy. The results show that this ratio assumes the expected value for a large range volumes (two orders of magnitude). At small particle volume or multiplicity, the curve shows the expected reduction: The <math>ssq</math> (<math>\Xi</math>) must be smaller compared to <math>\bar{s}s</math>(<math>\phi</math>) as the number of produced strange pairs decreases and thus it easier to make <math>\bar{s}s</math>(<math>\phi</math>) compared to <math>ssq</math> (<math>\Xi</math>) that requires two pairs minimum to be made. However, we also see an increase at very high volume—this is an effect at the level of one-two standard deviations. Similar results were already recognized before by Petran et al. .<ref name=":10">{{Cite journal|last1=Petráň|first1=Michal|last2=Rafelski|first2=Johann|year=2010|title=Multistrange particle production and the statistical hadronization model|journal=Physical Review C|language=en|volume=82|issue=1|pages=011901|doi=10.1103/PhysRevC.82.011901|arxiv=0912.1689|bibcode=2010PhRvC..82a1901P|s2cid=119179477|issn=0556-2813}}</ref>

Another highly praised [[ALICE experiment|ALICE]] result<ref name=":0" /> is the observation of same strangeness enhancement, not only on AA (nucleus-nucleus) but also in pA (proton-nucleus) and pp (proton-proton) collisions when the particle production yields are presented as a function of the multiplicity, which, as noted, corresponds to the available [[hadronization]] volume. ALICE results display a smooth volume dependence of total yield of all studied particles as function of volume, there is no additional "canonical" suppression.<ref name=":5" /> This is so since the yield of strange pairs in QGP is sufficiently high and tracks well the expected abundance increase as the volume and lifespan of QGP increases. This increase is incompatible with the hypothesis that for all reaction volumes QGP is always in chemical (yield) equilibrium of strangeness. Instead, this confirms the theoretical kinetic model proposed by Rafelski and [[Berndt Müller|Müller]].<ref name=":4" /> The production of QGP in pp collisions was not expected by all, but should not be a surprise. The [[onset of deconfinement]] is naturally a function of both energy and collision system size. The fact that at extreme LHC energies we cross this boundary also in experiments with the smallest elementary collision systems, such as pp, confirms the unexpected strength of the processes leading to QGP formation. Onset of deconfinement in pp and other "small" system collisions remains an active research topic.

Beyond strangeness the great advantage offered by LHC energy range is the abundant production of [[Flavour (particle physics)|charm and bottom flavor]].<ref name=":7" /> When QGP is formed, these quarks are embedded in a high density of strangeness present. This should lead to copious production of exotic heavy particles, for example {{Subatomic particle|Strange D}}. Other heavy flavor particles, some which have not even been discovered at this time, are also likely to appear.<ref>
{{cite journal|author=I. Kuznetsova|author2=J. Rafelski|year=2007|title=Heavy Flavor Hadrons in Statistical Hadronization of Strangeness-rich QGP|journal=[[European Physical Journal C]]|volume=51|issue=1|pages=113–133|arxiv=hep-ph/0607203|bibcode=2007EPJC...51..113K|doi=10.1140/epjc/s10052-007-0268-9|s2cid=18266326}}</ref><ref>
{{cite journal|author=N. Armesto|display-authors=etal|year=2008|title=Heavy-ion collisions at the LHC—Last call for predictions|journal=[[Journal of Physics G]]|volume=35|issue=5|pages=054001|arxiv=0711.0974|doi=10.1088/0954-3899/35/5/054001|s2cid=118529585}}</ref>

== S-S and S-W collisions at SPS-CERN with projectile energy 200 GeV per nucleon on fixed target ==
[[File:Strangeness production 4.gif|thumb|280px|Illustration of self-analyzing strange hadron decay: a double strange <math>\Xi^{-}</math> decays producing a <math>\pi^{-}</math> and invisible <math>\Lambda</math> which decays making a characteristic V-signature (<math>\pi^{-}</math>and p). This figure is created from actual picture taken at the NA35 CERN experiment. More details at page 28 in Letessier and Rafelski.<ref name=":11" />|alt=]]
[[File:strange production 1.gif|thumb|280px|Quantitative comparison of <math>\bar{\Lambda}</math> yield created in S-S with that in up-scaled p-p (squares) collision as a function of rapidity. Collisions at 200 A GeV.<ref>{{Cite book|last=Foka|first=P.|title=Study of strangness production in central nucleus-nucleus collisions at 200 GeV/nucleon by developing a new analysis method for the NA35 streamer chamber pictures|publisher=University of Geneva|year=1994|series=Thesis number 2723|location=Geneva|quote=The figure is a re-work of the original figure appearing on the top of page 271.}}</ref>]]

Looking back to the beginning of the CERN heavy ion program one sees de facto announcements of quark–gluon plasma discoveries. The CERN-[[NA35 experiment|NA35]]<ref name=":1" /> and [[WA85 experiment|CERN-WA85]]<ref>{{Cite journal|last1=Abatzis|first1=S.|last2=Barnes|first2=R.P.|last3=Benayoun|first3=M.|last4=Beusch|first4=W.|last5=Bloodworth|first5=I.J.|last6=Bravar|first6=A.|last7=Carney|first7=J.N.|last8=Dufey|first8=J.P.|last9=Evans|first9=D.|last10=Fini|first10=R.|last11=French|first11=B.R.|year=1991|title=Λ and anti-Λ production in 32S+W and p+W interactions at 200 A GeV/c|journal=Nuclear Physics A|language=en|volume=525|pages=445–448|doi=10.1016/0375-9474(91)90361-9|bibcode=1991NuPhA.525..445A}}</ref> experimental collaborations announced <math>\bar{\Lambda}</math> formation in heavy ion reactions in May 1990 at the Quark Matter Conference, [[Menton]], [[France]]. The data indicates a significant enhancement of the production of this antimatter particle comprising one antistrange quark as well as antiup and antidown quarks. All three constituents of the <math>\bar{\Lambda}</math> particle are newly produced in the reaction. The WA85 results were in agreement with theoretical predictions.<ref name=":3" /> In the published report, WA85 interpreted their results as QGP.<ref>{{Cite journal|last1=Abatzis|first1=S.|last2=Antinori|first2=F.|last3=Barnes|first3=R.P.|last4=Benayoun|first4=M.|last5=Beusch|first5=W.|last6=Bloodworth|first6=I.J.|last7=Bravar|first7=A.|last8=Carney|first8=J.N.|last9=de la Cruz|first9=B.|last10=Di Bari|first10=D.|last11=Dufey|first11=J.P.|year=1991|title=production in sulphur-tungsten interactions at 200 GeV/c per nucleon|journal=Physics Letters B|language=en|volume=270|issue=1|pages=123–127|doi=10.1016/0370-2693(91)91548-A|url=http://cds.cern.ch/record/222848}}</ref> NA35 had large systematic errors in its data, which were improved in the following years. Moreover, the collaboration needed to evaluate the pp-background.These results are presented as function of the variable called [[rapidity]] which characterizes the speed of the source. The peak of emission indicates that the additionally formed antimatter particles do not originate from the colliding nuclei themselves, but from a source that moves at a speed corresponding to one-half of the rapidity of the incident nucleus that is a common center of momentum frame of reference source formed when both nuclei collide, that is, the hot quark–gluon plasma fireball.

==Horn in <math>K \rightarrow \pi </math> ratio and the onset of deconfinement==
{{See also|Onset of deconfinement}}[[File:NA61-NA49_horn.png|thumb|280px|The ratio of mean multiplicities of positively charged [[kaon]]s and [[pion]]s as a function of collision energy in collisions of two [[lead]] [[atomic nucleus|nuclei]] and [[proton]]–proton interactions.]]One of most interesting questions is if there is a threshold in reaction energy and/or volume size which needs to be exceeded in order to form a domain in which quarks can move freely.<ref>{{cite journal|last1=Gazdzicki|first1=Marek|last2=Gorenstein|first2=Mark|last3=Seyboth|first3=Peter|year=2020|title=Brief history of the search for critical structures in heavy-ion collisions|journal=Acta Physica Polonica B|volume=51|issue=5|page=1033|doi=10.5506/APhysPolB.51.1033|arxiv=2004.02255|s2cid=214802159}}</ref> It is natural to expect that if such a threshold exists the particle yields/ratios we have shown above should indicate that.<ref>{{Cite journal|last=Becattini|first=F.|year=2012|title=Strangeness and onset of deconfinement|journal=Physics of Atomic Nuclei|language=en|volume=75|issue=5|pages=646–649|doi=10.1134/S106377881205002X|bibcode=2012PAN....75..646B|s2cid=120504052|issn=1063-7788}}</ref> One of the most accessible signatures would be the relative [[Kaon]] yield ratio.<ref>
{{cite journal
|author=N.K. Glendenning
|author2=J. Rafelski
|date=1985
|title=Kaons and quark–gluon plasma
|journal=[[Physical Review C]]
|volume=31 |pages=823–827
|doi=10.1103/PhysRevC.31.823
|bibcode = 1985PhRvC..31..823G
|issue=3 |pmid=9952591
|url=http://www.escholarship.org/uc/item/2bc9q3gh
}}</ref> A possible structure has been predicted,<ref>
{{cite journal
|author=M. Gazdzicki
|author2=M.I. Gorenstein
|date=1999
|title=On the Early Stage of Nucleus--Nucleus Collisions
|url=http://th-www.if.uj.edu.pl/acta/vol30/abs/v30p2705.htm
|journal=[[Acta Physica Polonica B]]
|volume=30 |issue=9
|pages=2705
|arxiv=hep-ph/9803462
|bibcode = 1999AcPPB..30.2705G
}}</ref> and indeed, an unexpected structure is seen in the ratio of particles comprising the positive kaon K (comprising anti s-quarks and up-quark) and positive [[pion]] particles, seen in the figure (solid symbols). The rise and fall (square symbols) of the ratio has been reported by the CERN [[NA49 experiment|NA49]].<ref>
{{cite journal
|author=M. Gazdzicki
|author2=NA49 Collaboration
|author2-link=NA49 experiment
|date=2004
|title=Report from NA49
|journal=[[Journal of Physics G]]
|volume=30 |pages=S701–S708
|arxiv=nucl-ex/0403023
|doi=10.1088/0954-3899/30/8/008
|bibcode = 2004JPhG...30S.701G
|issue=8 |s2cid=119197566
}}</ref><ref>
{{cite journal
|author=C. Alt
|author2=NA49 Collaboration
|author2-link=NA49 experiment
|year=2008
|title=Pion and kaon production in central Pb+Pb collisions at 20A and 30A GeV: Evidence for the onset of deconfinement
|journal=[[Physical Review C]]
|volume=77 |pages=024903
|doi=10.1103/PhysRevC.77.024903
|bibcode = 2008PhRvC..77b4903A
|issue=2 |arxiv = 0710.0118 |url=http://cds.cern.ch/record/1060039
}}</ref> The reason the negative kaon particles do not show this "horn" feature is that the s-quarks prefer to hadronize bound in the Lambda particle, where the counterpart structure is observed. Data point from [[STAR detector|BNL-RHIC-STAR]] (red stars) in figure agree with the CERN data.

In view of these results the objective of ongoing [[NA61/SHINE]] experiment at CERN [[Super Proton Synchrotron|SPS]] and the proposed low energy run at BNL [[RHIC]] where in particular the [[STAR detector]] can search for the onset of production of quark–gluon plasma as a function of energy in the domain where the horn maximum is seen, in order to improve the understanding of these results, and to record the behavior of other related quark–gluon plasma observables.

==Outlook==

The strangeness production and its diagnostic potential as a signature of quark–gluon plasma has been discussed for nearly 30 years. The theoretical work in this field today focuses on the interpretation of the overall particle production data and the derivation of the resulting properties of the bulk of quark–gluon plasma at the time of breakup.<ref name=":6" /> The global description of all produced particles can be attempted based on the picture of hadronizing hot drop of quark–gluon plasma or, alternatively, on the picture of confined and equilibrated hadron matter. In both cases one describes the data within the statistical thermal production model, but considerable differences in detail differentiate the nature of the source of these particles. The experimental groups working in the field also like to develop their own data analysis models and the outside observer sees many different analysis results. There are as many as 10–15 different particles species that follow the pattern predicted for the QGP as function of reaction energy, reaction centrality, and strangeness content. At yet higher LHC energies saturation of strangeness yield and binding to heavy flavor open new experimental opportunities.

== Conferences and meetings ==
Scientists studying strangeness as signature of quark gluon plasma present and discuss their results at specialized meetings. Well established is the series International Conference on Strangeness in Quark Matter, first organized in [[Tucson, Arizona|Tucson]], [[Arizona]], in 1995.<ref>{{Cite book|url=http://scitation.aip.org/content/aip/proceeding/aipcp/340|title=Strangeness in hadronic matter : S'95, Tucson, AZ January 1995|date=1995|publisher=AIP Press|others=Rafelski, Johann|isbn=1-56396-489-9|location=New York|oclc=32993061}}</ref><ref>{{Cite web|title=History – Strangeness in Quark Matter 2019|url=https://sqm2019.ba.infn.it/index.php/previous-editions/|language=en-US|access-date=2020-05-01}}</ref> The latest edition, 10–15 June 2019, of the conference was held in Bari, Italy, attracting about 300 participants.<ref>{{Cite web|title=Strangeness in Quark Matter 2019|url=https://sqm2019.ba.infn.it/|language=en-US|access-date=2020-05-05}}</ref><ref>{{Cite web|title=Quark-matter mysteries on the run in Bari|url=https://cerncourier.com/a/quark-matter-mysteries-on-the-run-in-bari/|date=2019-09-11|website=CERN Courier|language=en-GB|access-date=2020-05-05}}</ref> A more general venue is the Quark Matter conference, which last time took place from 4–9 November 2019 in [[Wuhan]], [[China]], attracting 800 participants.<ref>{{Cite web|title=Quark Matter 2019 - the XXVIIIth International Conference on Ultra-relativistic Nucleus-Nucleus Collisions|url=https://indico.cern.ch/event/792436/|website=Indico|access-date=2020-05-01}}</ref><ref>{{Cite web|title=LHC and RHIC heavy ions dovetail in Wuhan|url=https://cerncourier.com/a/lhc-and-rhic-heavy-ions-dovetail-in-wuhan/|date=2020-03-14|website=CERN Courier|language=en-GB|access-date=2020-05-05}}</ref>

== Further reading ==

* Brief history of the search for critical structures in heavy-ion collisions, Marek Gazdzicki, Mark Gorenstein, Peter Seyboth, 2020.<ref name=":8" />
* Discovery of quark–gluon plasma: strangeness diaries, Johann Rafelski, 2020.<ref name=":6" />
* Four heavy-ion experiments at the CERN-SPS: A trip down memory lane, Emanuele Quercigh, 2012.<ref>{{Cite journal|last=Quercigh|first=E.|year=2012|title=Four heavy-ion experiments at the CERN-SPS: A trip down memory lane|url=http://www.actaphys.uj.edu.pl/vol43/abs/v43p0771|journal=Acta Physica Polonica B|volume=43|issue=4|pages=771|doi=10.5506/APhysPolB.43.771}}</ref>
* On the history of multi-particle production in high energy collisions, Marek Gazdzicki, 2012.<ref>{{Cite journal|last=Gazdzicki|first=M.|date=2012|title=On the history of multi-particle production in high energy collisions|url=http://www.actaphys.uj.edu.pl/vol43/abs/v43p0791|journal=Acta Physica Polonica B|language=en|volume=43|issue=4|pages=791|doi=10.5506/APhysPolB.43.791|arxiv=1201.0485|bibcode=2012arXiv1201.0485G|s2cid=118418649|issn=0587-4254}}</ref>
* Strangeness and the quark–gluon plasma: thirty years of discovery, Berndt Müller, 2012.<ref>{{Cite journal|last=Müller|first=B.|year=2012|title=Strangeness and the quark–gluon plasma: thirty years of discovery|url=http://www.actaphys.uj.edu.pl/vol43/abs/v43p0761|journal=Acta Physica Polonica B|volume=43|issue=4|pages=761|doi=10.5506/APhysPolB.43.761|arxiv=1112.5382|s2cid=119280137}}</ref>

==See also==
* [[Quark–gluon plasma]]
* [[Quark matter]]
* [[Hadronization]]
* [[Strangelet]]
*[[Strange particle]]

==References==
{{reflist}}

{{physics-footer}}
{{Phase of matter}}

{{|}}
{{DEFAULTSORT:Strangeness and quark-gluon plasma}}
[[Category:Quark matter]]
[[Category:Strange quark|Production]]
[[Category:Exotic matter]]
[[Category:Nuclear physics]]
[[Category:Particle physics]]
[[Category:Phases of matter]]
[[Category:Plasma physics]]
[[Category:Quantum chromodynamics]]

Femtoscopy

2021-05-10T08:09:32Z

Bleicher:

{{Use dmy dates|date=June 2016}}
In [[physics]], the '''Hanbury Brown and Twiss''' ('''HBT''') '''effect''' is any of a variety of [[correlation]] and anti-correlation effects in the [[intensity (physics)|intensities]] received by two detectors from a beam of particles. HBT effects can generally be attributed to the [[wave–particle duality]] of the beam, and the results of a given experiment depend on whether the beam is composed of [[fermion]]s or [[boson]]s. Devices which use the effect are commonly called [[intensity interferometer]]s and were originally used in [[astronomy]], although they are also heavily used in the field of [[quantum optics]].

== History ==
In 1954, [[Robert Hanbury Brown]] and [[Richard Q. Twiss]] introduced the [[intensity interferometer]] concept to [[radio astronomy]] for measuring the tiny angular size of stars, suggesting that it might work with visible light as well.<ref name="BrownTwiss2010">{{cite journal|last1=Brown|first1=R. Hanbury|last2=Twiss|first2=R.Q.|title=A new type of interferometer for use in radio astronomy|journal=[[Philosophical Magazine]]|volume=45|issue=366|year=1954|pages=663–682|issn=1941-5982|doi=10.1080/14786440708520475}}</ref> Soon after they successfully tested that suggestion: in 1956 they published an in-lab experimental mockup using blue light from a [[mercury-vapor lamp]],<ref name="BrownTwiss1956">{{cite journal|last1=Brown|first1=R. Hanbury|last2=Twiss|first2=R. Q.|title=Correlation between Photons in two Coherent Beams of Light|journal=Nature|volume=177|issue=4497|year=1956|pages=27–29|issn=0028-0836|doi=10.1038/177027a0}}</ref> and later in the same year, they applied this technique to measuring the size of [[Sirius]].<ref>{{Cite journal |doi = 10.1038/1781046a0|title = A Test Of A New Type Of Stellar Interferometer On Sirius |journal = Nature|volume = 178|pages = 1046-1048|year = 1956|last1 = Hanbury Brown|first1 = R.|last2 = Twiss|first2 = Dr R.Q.|url = http://www.cmp.caltech.edu/refael/league/hanbury.pdf|bibcode = 1956Natur.178.1046H}}</ref> In the latter experiment, two [[photomultiplier tube]]s, separated by a few meters, were aimed at the star using crude telescopes, and a correlation was observed between the two fluctuating intensities. Just as in the radio studies, the correlation dropped away as they increased the separation (though over meters, instead of kilometers), and they used this information to determine the apparent [[angular size]] of Sirius.

[[File:Correlation-interferometer.svg|frame|150px|right|An example of an intensity interferometer that would observe no correlation if the light source is a coherent laser beam, and positive correlation if the light source is a filtered one-mode thermal radiation. The theoretical explanation of the difference between the correlations of photon pairs in thermal and in laser beams was first given by [[Roy J. Glauber]], who was awarded the 2005 [[Nobel Prize in Physics]] "for his contribution to the quantum theory of [[Coherence (physics)|optical coherence]]".]]

This result was met with much skepticism in the physics community. The radio astronomy result was justified by [[Maxwell's equations]], but there were concerns that the effect should break down at optical wavelengths, since the light would be quantised into a relatively small number of [[photon]]s that induce discrete [[photoelectron]]s in the detectors. Many [[physicists]] worried that the correlation was inconsistent with the laws of thermodynamics. Some even claimed that the effect violated the [[uncertainty principle]]. Hanbury Brown and Twiss resolved the dispute in a neat series of articles (see [[#References|References]] below) that demonstrated, first, that wave transmission in quantum optics had exactly the same mathematical form as Maxwell's equations, albeit with an additional noise term due to quantisation at the detector, and second, that according to Maxwell's equations, intensity interferometry should work. Others, such as [[Edward Mills Purcell]] immediately supported the technique, pointing out that the clumping of bosons was simply a manifestation of an effect already known in [[statistical mechanics]]. After a number of experiments, the whole physics community agreed that the observed effect was real.

The original experiment used the fact that two bosons tend to arrive at two separate detectors at the same time. Morgan and Mandel used a thermal photon source to create a dim beam of photons and observed the tendency of the photons to arrive at the same time on a single detector. Both of these effects used the wave nature of light to create a correlation in arrival time – if a single photon beam is split into two beams, then the particle nature of light requires that each photon is only observed at a single detector, and so an anti-correlation was observed in 1977 by [[H. Jeff Kimble]].<ref>{{Cite journal |doi = 10.1103/PhysRevLett.39.691|title = Photon Antibunching in Resonance Fluorescence|journal = Physical Review Letters|volume = 39|issue = 11|pages = 691–695|year = 1977|last1 = Kimble|first1 = H. J.|last2 = Dagenais|first2 = M.|last3 = Mandel|first3 = L.|url = https://authors.library.caltech.edu/6051/1/KIMprl77.pdf|bibcode = 1977PhRvL..39..691K}}</ref> Finally, bosons have a tendency to clump together, giving rise to [[Bose–Einstein correlations]], while fermions due to the [[Pauli exclusion principle]], tend to spread apart, leading to Fermi–Dirac (anti)correlations. Bose–Einstein correlations have been observed between pions, kaons and photons, and Fermi–Dirac (anti)correlations between protons, neutrons and electrons. For a general introduction in this field, see the textbook on Bose–Einstein correlations by [[Richard M. Weiner]]<ref>Richard M. Weiner, Introduction to Bose–Einstein Correlations and Subatomic Interferometry, John Wiley, 2000.</ref> A difference in repulsion of [[Bose–Einstein condensate]] in the "trap-and-free fall" analogy of the HBT effect<ref>[https://arxiv.org/abs/cond-mat/0612278 Comparison of the Hanbury Brown-Twiss effect for bosons and fermions].</ref> affects comparison.

Also, in the field of [[particle physics]], [[Gerson Goldhaber|Goldhaber]] et al. performed an experiment in 1959 in [[University of California, Berkeley|Berkeley]] and found an unexpected angular correlation among identical [[pion]]s, discovering the [[rho meson|ρ0 resonance]], by means of <math>\rho^0 \to \pi^-\pi^+</math> decay.<ref>
{{cite journal
|author1=G. Goldhaber
|author2=W. B. Fowler
|author3=S. Goldhaber
|author4=T. F. Hoang
|author5=T. E. Kalogeropoulos
|author6=W. M. Powell
|year=1959
|title=Pion-pion correlations in antiproton annihilation events
|journal=Phys. Rev. Lett.
|volume=3 |issue=4 |page=181
|doi=10.1103/PhysRevLett.3.181
|bibcode=1959PhRvL...3..181G|url=http://www.escholarship.org/uc/item/7nw6p1br
}}</ref> From then on, the HBT technique started to be used by the [[High-energy nuclear physics|heavy-ion community]] to determine the space–time dimensions of the particle emission source for heavy-ion collisions. For recent developments in this field, see for example the review article by Lisa.<ref>M. Lisa, et al., ''Annu. Rev. Nucl. Part. Sci.'' '''55''', p. 357 (2005), [https://arxiv.org/abs/nucl-ex/0505014 ArXiv 0505014].</ref>

== Wave mechanics ==
The HBT effect can, in fact, be predicted solely by treating the incident [[electromagnetic radiation]] as a classical [[wave]]. Suppose we have a monochromatic wave with frequency <math>\omega</math> on two detectors, with an amplitude <math>E(t)</math> that varies on timescales slower than the wave period <math>2\pi/\omega</math>. (Such a wave might be produced from a very distant [[point source]] with a fluctuating intensity.)

Since the detectors are separated, say the second detector gets the signal delayed by a time <math>\tau</math>, or equivalently, a [[Phase (waves)|phase]] <math>\phi = \omega\tau</math>; that is,

:<math> E_1(t) = E(t) \sin(\omega t),</math>

:<math> E_2(t) = E(t - \tau) \sin(\omega t - \phi).</math>

The intensity recorded by each detector is the square of the wave amplitude, averaged over a timescale that is long compared to the wave period <math>2\pi/\omega</math> but short compared to the fluctuations in <math>E(t)</math>:

:<math>
\begin{align}
i_1(t) &= \overline{E_1(t)^2} = \overline{E(t)^2 \sin^2(\omega t)} = \tfrac{1}{2} E(t)^2, \\
i_2(t) &= \overline{E_2(t)^2} = \overline{E(t - \tau)^2 \sin^2(\omega t - \phi)} = \tfrac{1}{2} E(t - \tau)^2,
\end{align}
</math>
where the overline indicates this time averaging. For wave frequencies above a few [[Terahertz radiation|terahertz]] (wave periods less than a [[picosecond]]), such a time averaging is unavoidable, since detectors such as [[photodiode]]s and [[photomultiplier tube]]s cannot produce photocurrents that vary on such short timescales.

The correlation function <math>\langle i_1 i_2 \rangle(\tau)</math> of these time-averaged intensities can then be computed:
:<math>
\begin{align}
\langle i_1 i_2 \rangle(\tau) &= \lim_{T \to \infty} \frac{1}{T} \int\limits_0^T i_1(t) i_2(t)\, \mathrm{d}t \\
&= \lim_{T \to \infty} \frac{1}{T} \int\limits_0^T \tfrac{1}{4} E(t)^2 E(t-\tau)^2 \, \mathrm{d}t.
\end{align}
</math>

Most modern schemes actually measure the correlation in intensity fluctuations at the two detectors, but it is not too difficult to see that if the intensities are correlated, then the fluctuations <math>\Delta i = i - \langle i\rangle</math>, where <math>\langle i\rangle</math> is the average intensity, ought to be correlated, since
:<math>\begin{align}
\langle\Delta i_1\Delta i_2\rangle &= \big\langle(i_1 - \langle i_1\rangle)(i_2 - \langle i_2\rangle)\big\rangle = \langle i_1 i_2\rangle - \big\langle i_1\langle i_2\rangle\big\rangle - \big\langle i_2\langle i_1\rangle\big\rangle + \langle i_1\rangle \langle i_2\rangle \\
&=\langle i_1 i_2\rangle -\langle i_1\rangle \langle i_2\rangle.
\end{align}</math>

In the particular case that <math>E(t)</math> consists mainly of a steady field <math>E_0</math> with a small sinusoidally varying component <math>\delta E \sin(\Omega t)</math>, the time-averaged intensities are
:<math>
\begin{align}
i_1(t) &= \tfrac{1}{2} E_0^2 + E_0\,\delta E \sin(\Omega t) + \mathcal{O}(\delta E^2), \\
i_2(t) &= \tfrac{1}{2} E_0^2 + E_0\,\delta E \sin(\Omega t-\Phi) + \mathcal{O}(\delta E^2),
\end{align}
</math>
with <math>\Phi = \Omega \tau</math>, and <math>\mathcal{O}(\delta E^2)</math> indicates terms proportional to <math>(\delta E)^2</math>, which are small and may be ignored.

The correlation function of these two intensities is then
:<math>
\begin{align}
\langle \Delta i_1 \Delta i_2 \rangle(\tau) &= \lim_{T \to \infty} \frac{(E_0\delta E)^2}{T} \int\limits_0^T \sin(\Omega t) \sin(\Omega t - \Phi) \, \mathrm{d}t \\
&= \tfrac{1}{2} (E_0 \delta E)^2 \cos(\Omega\tau),
\end{align}
</math>
showing a sinusoidal dependence on the delay <math>\tau</math> between the two detectors.

== Quantum interpretation ==
[[File:Photon bunching.svg|thumb|400px|Photon detections as a function of time for a) antibunching (e.g. light emitted from a single atom), b) random (e.g. a coherent state, laser beam), and c) bunching (chaotic light). τc is the coherence time (the time scale of photon or intensity fluctuations).]]
The above discussion makes it clear that the Hanbury Brown and Twiss (or photon bunching) effect can be entirely described by classical optics. The quantum description of the effect is less intuitive: if one supposes that a thermal or chaotic light source such as a star randomly emits photons, then it is not obvious how the photons "know" that they should arrive at a detector in a correlated (bunched) way. A simple argument suggested by [[Ugo Fano]] [Fano, 1961] captures the essence of the quantum explanation. Consider two points <math>a</math> and <math>b</math> in a source that emit photons detected by two detectors <math>A</math> and <math>B</math> as in the diagram. A joint detection takes place when the photon emitted by <math>a</math> is detected by <math>A</math> and the photon emitted by <math>b</math> is detected by <math>B</math> (red arrows) ''or'' when <math>a</math>'s photon is detected by <math>B</math> and <math>b</math>'s by <math>A</math> (green arrows). The quantum mechanical probability amplitudes for these two possibilities are denoted by
<math>\langle A|a \rangle \langle B|b \rangle</math> and
<math>\langle B|a \rangle \langle A|b \rangle</math> respectively. If the photons are indistinguishable, the two amplitudes interfere constructively to give a joint detection probability greater than that for two independent events. The sum over all possible pairs <math>a, b</math> in the source washes out the interference unless the distance <math>AB</math> is sufficiently small.

[[File:Two-photon Amplitude.svg|thumb|right|Two source points ''a'' and ''b'' emit photons detected by detectors ''A'' and ''B''. The two colors represent two different ways to detect two photons.]]

Fano's explanation nicely illustrates the necessity of considering two-particle amplitudes, which are not as intuitive as the more familiar single-particle amplitudes used to interpret most interference effects. This may help to explain why some physicists in the 1950s had difficulty accepting the Hanbury Brown and Twiss result. But the quantum approach is more than just a fancy way to reproduce the classical result: if the photons are replaced by identical fermions such as electrons, the antisymmetry of wave functions under exchange of particles renders the interference destructive, leading to zero joint detection probability for small detector separations. This effect is referred to as antibunching of fermions [Henny, 1999]. The above treatment also explains [[photon antibunching]] [Kimble, 1977]: if the source consists of a single atom, which can only emit one photon at a time, simultaneous detection in two closely spaced detectors is clearly impossible. Antibunching, whether of bosons or of fermions, has no classical wave analog.

From the point of view of the field of quantum optics, the HBT effect was important to lead physicists (among them [[Roy J. Glauber]] and [[Leonard Mandel]]) to apply quantum electrodynamics to new situations, many of which had never been experimentally studied, and in which classical and quantum predictions differ.

== See also ==
*[[Bose–Einstein correlations]]
*[[Degree of coherence]]
*[[Timeline of electromagnetism and classical optics]]

== References ==
<references/>
Note that Hanbury Brown is not hyphenated.

* {{cite journal |author1=E. Brannen |author2=H. Ferguson | title=The question of correlation between photons in coherent light beams | journal=Nature | year=1956 | volume=178 | pages=481–482 | doi=10.1038/178481a0 | issue=4531|bibcode = 1956Natur.178..481B }} – paper which (incorrectly) disputed the existence of the Hanbury Brown and Twiss effect
* {{cite journal |author1=R. Hanbury Brown |author2=R. Q. Twiss | title=A Test of a New Type of Stellar Interferometer on Sirius | journal=Nature | year=1956 | volume=178 | pages=1046–1048 | doi=10.1038/1781046a0 | issue=4541|bibcode = 1956Natur.178.1046H }} – experimental demonstration of the effect
* {{cite journal | author=E. Purcell | title=The Question of Correlation Between Photons in Coherent Light Rays | journal=Nature | year=1956 | volume=178 | pages=1449–1450 | doi=10.1038/1781449a0 | issue=4548|bibcode = 1956Natur.178.1449P }}
* {{cite journal |author1=R. Hanbury Brown |author2=R. Q. Twiss | title=Interferometry of the intensity fluctuations in light. I. Basic theory: the correlation between photons in coherent beams of radiation | journal=Proceedings of the Royal Society A | year=1957 | volume=242 | pages=300–324 | doi=10.1098/rspa.1957.0177 | issue=1230|bibcode = 1957RSPSA.242..300B }} [https://web.archive.org/web/20050124033547/http://www.strw.leidenuniv.nl/~tubbs/classic_papers/hanbury_brown_et_twiss_1957.pdf download as PDF]
* {{cite journal |author1=R. Hanbury Brown |author2=R. Q. Twiss | title=Interferometry of the intensity fluctuations in light. II. An experimental test of the theory for partially coherent light | journal=Proceedings of the Royal Society A | year=1958 | volume=243 | pages=291–319 | doi=10.1098/rspa.1958.0001 | issue=1234|bibcode = 1958RSPSA.243..291B }} [https://web.archive.org/web/20050124033547/http://www.strw.leidenuniv.nl/~tubbs/classic_papers/hanbury_brown_et_twiss_1958a.pdf download as PDF]
*{{cite journal |last=Fano |first=U. |title=Quantum theory of interference effects in the mixing of light from phase independent sources |journal=American Journal of Physics |volume=29 |year=1961 |pages=539–545 |doi=10.1119/1.1937827 |issue=8 |bibcode = 1961AmJPh..29..539F }}
* {{cite journal |author1=B. L. Morgan |author2=L. Mandel | title=Measurement of Photon Bunching in a Thermal Light Beam | journal=Phys. Rev. Lett. | year=1966 | volume=16 | pages=1012–1014 | doi=10.1103/PhysRevLett.16.1012 | issue=22 | bibcode=1966PhRvL..16.1012M|citeseerx=10.1.1.713.7239 }}
*{{Cite journal |last1=Kimble |first1=H. J. |last2=Dagenais|first2=M. |last3=Mandel|first3=L.|title=Photon antibunching in resonance fluorescence |journal=Physical Review Letters |volume=39 |year=1977 |pages=691–695|doi=10.1103/PhysRevLett.39.691 |issue=11 |bibcode=1977PhRvL..39..691K|url=https://authors.library.caltech.edu/6051/1/KIMprl77.pdf }}
*{{Cite journal |last1=Dayan |first1=B. |last2=Parkins |first2=A. S.|last3=Aoki |first3=T.|last4=Ostby |first4=E. P.|last5=Vahala |first5=K. J. |last6=Kimble |first6=H. J.|title=A Photon Turnstile Dynamically Regulated by One Atom |journal=Science |volume=319 |issue=5866 |year=2008 |pages=1062–1065|doi=10.1126/Science.1152261|bibcode = 2008Sci...319.1062D |pmid=18292335|url=https://authors.library.caltech.edu/35067/2/Dayan.SOM.pdf }} – the cavity-QED equivalent for Kimble & Mandel's free-space demonstration of photon antibunching in resonance fluorescence
* {{cite journal |author1=P. Grangier |author2=G. Roger |author3=A. Aspect | title=Experimental Evidence for a Photon Anticorrelation Effect on a Beam Splitter: A New Light on Single-Photon Interferences | journal=Europhysics Letters | year=1986 | volume=1 | pages=173–179 | doi=10.1209/0295-5075/1/4/004 | issue=4|bibcode = 1986EL......1..173G |citeseerx=10.1.1.178.4356 }}
* {{cite journal | author=M. Henny| title=The Fermionic Hanbury Brown and Twiss Experiment | journal=Science | year=1999 | volume=284 | pages=296–298 | doi=10.1126/science.284.5412.296 | pmid=10195890 | issue=5412|bibcode = 1999Sci...284..296H |display-authors=etal| url=https://epub.uni-regensburg.de/3370/1/ScienceHBT.pdf }}
* {{cite book | author=R Hanbury Brown| title=BOFFIN : A Personal Story of the Early Days of Radar, Radio Astronomy and Quantum Optics | publisher=Adam Hilger | year=1991 | isbn=978-0-7503-0130-5}}
* {{cite book | author=Mark P. Silverman | title=More Than One Mystery: Explorations in Quantum Interference | url=https://archive.org/details/morethanonemyste0000silv | url-access=registration | publisher=Springer | year=1995 | isbn=978-0-387-94376-3}}
* {{cite book | author=R Hanbury Brown | title=The intensity interferometer; its application to astronomy | publisher=Wiley | year=1974 |id=ASIN B000LZQD3C | isbn=978-0-470-10797-3}}
* {{cite journal |author1=Y. Bromberg |author2=Y. Lahini |author3=E. Small |author4=Y. Silberberg | title=Hanbury Brown and Twiss Interferometry with Interacting Photons | journal=Nature Photonics| year=2010 | volume=4 | pages=721–726 | doi=10.1038/nphoton.2010.195 | issue=10|bibcode = 2010NaPho...4..721B }}

== External links ==
* http://adsabs.harvard.edu//full/seri/JApA./0015//0000015.000.html
* http://physicsweb.org/articles/world/15/10/6/1
* https://web.archive.org/web/20070609114114/http://www.du.edu/~jcalvert/astro/starsiz.htm
* http://www.2physics.com/2010/11/hanbury-brown-and-twiss-interferometry.html
*[https://www.becker-hickl.com/applications/antibunching-experiments/ Hanbury-Brown-Twiss Experiment] (Becker & Hickl GmbH, web page)

[[Category:Quantum optics]]

Transport models

2021-05-10T08:08:57Z

Bleicher:

{{short description|Equation of statistical mechanics}}
{{other uses|Boltzmann's entropy formula|Stefan–Boltzmann law|Maxwell–Boltzmann distribution}}
{{redirect|BTE}}

[[File:StairsOfReduction.svg|thumb|The place of the Boltzmann kinetic equation on the stairs of model reduction from microscopic dynamics to macroscopic continuum dynamics (illustration to the content of the book<ref>
{{cite book |last1=Gorban |first1= Alexander N.|last2= Karlin |first2= Ilya V. |date=2005 |title= Invariant Manifolds for Physical and Chemical Kinetics|url= https://www.academia.edu/17378865| location= Berlin, Heidelberg |publisher= Springer|series= Lecture Notes in Physics (LNP, vol. 660)| isbn= 978-3-540-22684-0|doi= 10.1007/b98103}} [https://archive.org/details/gorban-karlin-lnp-2005 Alt URL]</ref>)]]

The '''Boltzmann equation''' or '''Boltzmann transport equation''' ('''BTE''') describes the statistical behaviour of a [[thermodynamic system]] not in a state of [[Thermodynamic equilibrium|equilibrium]], devised by [[Ludwig Boltzmann]] in 1872.<ref name="Encyclopaediaof">Encyclopaedia of Physics (2nd Edition), [[Rita G. Lerner|R. G. Lerner]], G. L. Trigg, VHC publishers, 1991, ISBN (Verlagsgesellschaft) 3-527-26954-1, ISBN (VHC Inc.) 0-89573-752-3.</ref>
The classic example of such a system is a [[fluid]] with [[temperature gradient]]s in space causing heat to flow from hotter regions to colder ones, by the random but biased transport of the [[particle]]s making up that fluid. In the modern literature the term Boltzmann equation is often used in a more general sense, referring to any kinetic equation that describes the change of a macroscopic quantity in a thermodynamic system, such as energy, charge or particle number.

The equation arises not by analyzing the individual [[position vector|position]]s and [[momentum|momenta]] of each particle in the fluid but rather by considering a probability distribution for the position and momentum of a typical particle—that is, the [[probability]] that the particle occupies a given [[infinitesimal|very small]] region of space (mathematically the [[volume element]] <math>\mathrm{d}^3 \bf{r}</math>) centered at the position <math>\bf{r}</math>, and has momentum nearly equal to a given momentum vector <math> \bf{p}</math> (thus occupying a very small region of [[momentum space]] <math>\mathrm{d}^3 \bf{p}</math>), at an instant of time.

The Boltzmann equation can be used to determine how physical quantities change, such as [[heat]] energy and [[momentum]], when a fluid is in transport. One may also derive other properties characteristic to fluids such as [[viscosity]], [[thermal conductivity]], and [[electrical conductivity]] (by treating the charge carriers in a material as a gas).<ref name="Encyclopaediaof" /> See also [[convection–diffusion equation]].

The equation is a [[Nonlinear system|nonlinear]] [[integro-differential equation]], and the unknown function in the equation is a probability density function in six-dimensional space of a particle position and momentum. The problem of existence and uniqueness of solutions is still not fully resolved, but some recent results are quite promising.<ref>{{cite journal | last1=DiPerna | first1= R. J. |last2 = Lions | first2 = P.-L. | title= On the Cauchy problem for Boltzmann equations: global existence and weak stability | journal= Ann. of Math. |series= 2 | volume=130 | pages= 321–366 | year=1989 | doi=10.2307/1971423 | issue=2| jstor= 1971423 }}
</ref><ref>{{cite journal |author1=Philip T. Gressman |author-link1=Philip Gressman |author2=Robert M. Strain |name-list-style=amp|year=2010 |title= Global classical solutions of the Boltzmann equation with long-range interactions |journal= Proceedings of the National Academy of Sciences |volume=107 |pages= 5744–5749 | doi = 10.1073/pnas.1001185107 |bibcode = 2010PNAS..107.5744G |arxiv = 1002.3639 |issue= 13 |pmid=20231489 |pmc=2851887}}</ref>

==Overview==

===The phase space and density function===
The set of all possible positions '''r''' and momenta '''p''' is called the [[phase space]] of the system; in other words a set of three [[coordinates]] for each position coordinate ''x, y, z'', and three more for each momentum component ''px, py, pz''. The entire space is 6-[[dimension]]al: a point in this space is ('''r''', '''p''') = (''x, y, z, px, py, pz''), and each coordinate is [[Parametric equation|parameterized]] by time ''t''. The small volume ("differential [[volume element]]") is written
:<math> \text{d}^3\mathbf{r}\,\text{d}^3\mathbf{p} = \text{d}x\,\text{d}y\,\text{d}z\,\text{d}p_x\,\text{d}p_y\,\text{d}p_z. </math>

Since the probability of ''N'' molecules which ''all'' have '''r''' and '''p''' within <math> \mathrm{d}^3\bf{r}</math> <math> \mathrm{d}^3\bf{p}</math> is in question, at the heart of the equation is a quantity ''f'' which gives this probability per unit phase-space volume, or probability per unit length cubed per unit momentum cubed, at an instant of time ''t''. This is a [[probability density function]]: ''f''('''r''', '''p''', ''t''), defined so that,

:<math>\text{d}N = f (\mathbf{r},\mathbf{p},t)\,\text{d}^3\mathbf{r}\,\text{d}^3\mathbf{p}</math>

is the number of molecules which ''all'' have positions lying within a volume element <math> d^3\bf{r}</math> about '''r''' and momenta lying within a [[momentum space]] element <math> \mathrm{d}^3\bf{p}</math> about '''p''', at time ''t''.<ref>{{Cite book |last=Huang |first=Kerson |year=1987 |title=Statistical Mechanics |url=https://archive.org/details/statisticalmecha00huan_475 |url-access=limited |location=New York |publisher=Wiley |isbn=978-0-471-81518-1 |page=[https://archive.org/details/statisticalmecha00huan_475/page/n65 53] |edition=Second }}</ref> [[Integration (calculus)|Integrating]] over a region of position space and momentum space gives the total number of particles which have positions and momenta in that region:

: <math>
\begin{align}
N & = \int\limits_\mathrm{momenta} \text{d}^3\mathbf{p} \int\limits_\mathrm{positions} \text{d}^3\mathbf{r}\,f (\mathbf{r},\mathbf{p},t) \\[5pt]
& = \iiint\limits_\mathrm{momenta} \quad \iiint\limits_\mathrm{positions} f(x,y,z,p_x,p_y,p_z,t) \, \text{d}x \, \text{d}y \, \text{d}z \, \text{d}p_x \, \text{d}p_y \, \text{d}p_z
\end{align}
</math>

which is a [[multiple integral|6-fold integral]]. While ''f'' is associated with a number of particles, the phase space is for one-particle (not all of them, which is usually the case with [[deterministic]] [[many body problem|many-body]] systems), since only one '''r''' and '''p''' is in question. It is not part of the analysis to use '''r'''1, '''p'''1 for particle 1, '''r'''2, '''p'''2 for particle 2, etc. up to '''r'''''N'', '''p'''''N'' for particle ''N''.

It is assumed the particles in the system are identical (so each has an identical [[mass]] ''m''). For a mixture of more than one [[chemical species]], one distribution is needed for each, see below.

===Principal statement===

The general equation can then be written as<ref name="McGrawHill">McGraw Hill Encyclopaedia of Physics (2nd Edition), C. B. Parker, 1994, {{ISBN|0-07-051400-3}}.</ref>

:<math>
\frac{df}{dt} =
\left(\frac{\partial f}{\partial t}\right)_\text{force} +
\left(\frac{\partial f}{\partial t}\right)_\text{diff} +
\left(\frac{\partial f}{\partial t}\right)_\text{coll},
</math>

where the "force" term corresponds to the forces exerted on the particles by an external influence (not by the particles themselves), the "diff" term represents the [[diffusion]] of particles, and "coll" is the [[collision]] term – accounting for the forces acting between particles in collisions. Expressions for each term on the right side are provided below.<ref name="McGrawHill" />

Note that some authors use the particle velocity '''v''' instead of momentum '''p'''; they are related in the definition of momentum by '''p''' = ''m'''''v'''.

==The force and diffusion terms==

Consider particles described by ''f'', each experiencing an ''external'' force '''F''' not due to other particles (see the collision term for the latter treatment).

Suppose at time ''t'' some number of particles all have position '''r''' within element <math> d^3\bf{r}</math> and momentum '''p''' within <math> d^3\bf{p}</math>. If a force '''F''' instantly acts on each particle, then at time ''t'' + Δ''t'' their position will be '''r''' + Δ'''r''' = '''r''' + '''p'''Δ''t''/''m'' and momentum '''p''' + Δ'''p''' = '''p''' + '''F'''Δ''t''. Then, in the absence of collisions, ''f'' must satisfy

:<math>
f \left (\mathbf{r}+\frac{\mathbf{p}}{m} \, \Delta t,\mathbf{p}+\mathbf{F} \, \Delta t, t+\Delta t \right )\,d^3\mathbf{r}\,d^3\mathbf{p} = f(\mathbf{r}, \mathbf{p},t) \, d^3\mathbf{r} \, d^3\mathbf{p}
</math>

Note that we have used the fact that the phase space volume element <math> d^3\bf{r}</math> <math> d^3\bf{p}</math> is constant, which can be shown using [[Hamilton's equations]] (see the discussion under [[Liouville's theorem (Hamiltonian)|Liouville's theorem]]). However, since collisions do occur, the particle density in the phase-space volume <math> d^3\bf{r}</math> '<math> d^3\bf{p}</math> changes, so

{{NumBlk|:|
<math>\begin{align}
dN_\mathrm{coll} & = \left(\frac{\partial f}{\partial t} \right)_\mathrm{coll}\Delta td^3\mathbf{r} d^3\mathbf{p} \\[5pt]
& = f \left (\mathbf{r}+\frac{\mathbf{p}}{m}\Delta t,\mathbf{p} + \mathbf{F}\Delta t, t+\Delta t \right)d^3\mathbf{r}d^3\mathbf{p} - f(\mathbf{r}, \mathbf{p}, t) \, d^3\mathbf{r} \, d^3\mathbf{p} \\[5pt]
& = \Delta f \, d^3\mathbf{r} \, d^3\mathbf{p}
\end{align}</math>
|{{EquationRef|1}}}}

where Δ''f'' is the ''total'' change in ''f''. Dividing ({{EquationNote|1}}) by <math> d^3\bf{r}</math> <math> d^3\bf{p}</math> Δ''t'' and taking the limits Δ''t'' → 0 and Δ''f'' → 0, we have

{{NumBlk|:|
<math>\frac{d f}{d t} = \left(\frac{\partial f}{\partial t} \right)_\mathrm{coll}</math>
|{{EquationRef|2}}}}

The total [[differential of a function|differential]] of ''f'' is:

{{NumBlk|:|
<math>\begin{align}
d f & = \frac{\partial f}{\partial t} \, dt
+\left(\frac{\partial f}{\partial x} \, dx
+\frac{\partial f}{\partial y} \, dy
+\frac{\partial f}{\partial z} \, dz
\right)
+\left(\frac{\partial f}{\partial p_x} \, dp_x
+\frac{\partial f}{\partial p_y} \, dp_y
+\frac{\partial f}{\partial p_z} \, dp_z
\right)\\[5pt]
& = \frac{\partial f}{\partial t}dt +\nabla f \cdot d\mathbf{r} + \frac{\partial f}{\partial \mathbf{p}}\cdot d\mathbf{p} \\[5pt]
& = \frac{\partial f}{\partial t}dt +\nabla f \cdot \frac{\mathbf{p}}{m}dt + \frac{\partial f}{\partial \mathbf{p}}\cdot \mathbf{F} \, dt
\end{align}</math>
|{{EquationRef|3}}}}

where ∇ is the [[gradient]] operator, '''·''' is the [[dot product]],

:<math>
\frac{\partial f}{\partial \mathbf{p}} = \mathbf{\hat{e}}_x\frac{\partial f}{\partial p_x} + \mathbf{\hat{e}}_y\frac{\partial f}{\partial p_y} + \mathbf{\hat{e}}_z \frac{\partial f}{\partial p_z}= \nabla_\mathbf{p}f
</math>

is a shorthand for the momentum analogue of ∇, and '''ê'''''x'', '''ê'''''y'', '''ê'''''z'' are [[cartesian coordinates|Cartesian]] [[unit vector]]s.

===Final statement===

Dividing ({{EquationNote|3}}) by ''dt'' and substituting into ({{EquationNote|2}}) gives:

:<math>\frac{\partial f}{\partial t} + \frac{\mathbf{p}}{m}\cdot\nabla f + \mathbf{F} \cdot \frac{\partial f}{\partial \mathbf{p}} = \left(\frac{\partial f}{\partial t} \right)_\mathrm{coll}</math>

In this context, '''F'''('''r''', ''t'') is the [[Force field (chemistry)|force field]] acting on the particles in the fluid, and ''m'' is the [[mass]] of the particles. The term on the right hand side is added to describe the effect of collisions between particles; if it is zero then the particles do not collide. The collisionless Boltzmann equation, where individual collisions are replaced with long-range aggregated interactions, e.g. [[Coulomb interaction]]s, is often called the [[Vlasov equation]].

This equation is more useful than the principal one above, yet still incomplete, since ''f'' cannot be solved unless the collision term in ''f'' is known. This term cannot be found as easily or generally as the others – it is a statistical term representing the particle collisions, and requires knowledge of the statistics the particles obey, like the [[Maxwell–Boltzmann distribution|Maxwell–Boltzmann]], [[Fermi–Dirac distribution|Fermi–Dirac]] or [[Bose–Einstein distribution|Bose–Einstein]] distributions.

==The collision term (Stosszahlansatz) and molecular chaos==

=== Two-body collision term ===
A key insight applied by [[Ludwig Boltzmann|Boltzmann]] was to determine the collision term resulting solely from two-body collisions between particles that are assumed to be uncorrelated prior to the collision. This assumption was referred to by Boltzmann as the "{{lang|de|Stosszahlansatz}}" and is also known as the "[[molecular chaos]] assumption". Under this assumption the collision term can be written as a momentum-space integral over the product of one-particle distribution functions:<ref name="Encyclopaediaof" />
:<math>
\left(\frac{\partial f}{\partial t}\right)_\text{coll} =
\iint gI(g, \Omega)[f(\mathbf{r},\mathbf{p'}_A, t) f(\mathbf{r},\mathbf{p'}_B,t) - f(\mathbf{r},\mathbf{p}_A,t) f(\mathbf{r},\mathbf{p}_B,t)] \,d\Omega \,d^3\mathbf{p}_A \,d^3\mathbf{p}_B,
</math>
where '''p'''''A'' and '''p'''''B'' are the momenta of any two particles (labeled as ''A'' and ''B'' for convenience) before a collision, '''p′'''''A'' and '''p′'''''B'' are the momenta after the collision,
:<math>
g = |\mathbf{p}_B - \mathbf{p}_A| = |\mathbf{p'}_B - \mathbf{p'}_A|
</math>
is the magnitude of the relative momenta (see [[relative velocity]] for more on this concept), and ''I''(''g'', Ω) is the [[differential cross section]] of the collision, in which the relative momenta of the colliding particles turns through an angle θ into the element of the [[solid angle]] ''d''Ω, due to the collision.

=== Simplifications to the collision term ===
Since much of the challenge in solving the Boltzmann equation originates with the complex collision term, attempts have been made to "model" and simplify the collision term. The best known model equation is due to Bhatnagar, Gross and Krook.<ref>{{Cite journal|title = A Model for Collision Processes in Gases. I. Small Amplitude Processes in Charged and Neutral One-Component Systems|journal = Physical Review|date = 1954-05-01|pages = 511–525|volume = 94|issue = 3|doi = 10.1103/PhysRev.94.511|first1 = P. L.|last1 = Bhatnagar|first2 = E. P.|last2 = Gross|first3 = M.|last3 = Krook|bibcode = 1954PhRv...94..511B }}</ref> The assumption in the BGK approximation is that the effect of molecular collisions is to force a non-equilibrium distribution function at a point in physical space back to a Maxwellian equilibrium distribution function and that the rate at which this occurs is proportional to the molecular collision frequency. The Boltzmann equation is therefore modified to the BGK form:

:<math>\frac{\partial f}{\partial t} + \frac{\mathbf{p}}{m}\cdot\nabla f + \mathbf{F} \cdot \frac{\partial f}{\partial \mathbf{p}} = \nu (f_0 - f),</math>

where <math>\nu</math> is the molecular collision frequency, and <math>f_0</math> is the local Maxwellian distribution function given the gas temperature at this point in space.

==General equation (for a mixture)==

For a mixture of chemical species labelled by indices ''i'' = 1, 2, 3, ..., ''n'' the equation for species ''i'' is<ref name="Encyclopaediaof" />

:<math>\frac{\partial f_i}{\partial t} + \frac{\mathbf{p}_i}{m_i} \cdot \nabla f_i + \mathbf{F} \cdot \frac{\partial f_i}{\partial \mathbf{p}_i} = \left(\frac{\partial f_i}{\partial t} \right)_\text{coll},</math>

where ''fi'' = ''fi''('''r''', '''p'''''i'', ''t''), and the collision term is

:<math>
\left(\frac{\partial f_i}{\partial t} \right)_{\mathrm{coll}} = \sum_{j=1}^n \iint g_{ij} I_{ij}(g_{ij}, \Omega)[f'_i f'_j - f_if_j] \,d\Omega\,d^3\mathbf{p'},
</math>

where ''f′'' = ''f′''('''p′'''''i'', ''t''), the magnitude of the relative momenta is

:<math>g_{ij} = |\mathbf{p}_i - \mathbf{p}_j| = |\mathbf{p'}_i - \mathbf{p'}_j|,</math>

and ''Iij'' is the differential cross-section, as before, between particles ''i'' and ''j''. The integration is over the momentum components in the integrand (which are labelled ''i'' and ''j''). The sum of integrals describes the entry and exit of particles of species ''i'' in or out of the phase-space element.

==Applications and extensions==

===Conservation equations===

The Boltzmann equation can be used to derive the fluid dynamic conservation laws for mass, charge, momentum, and energy.<ref name="dG1984">{{cite book |last1=de Groot |first1=S. R.|last2=Mazur|first2=P.|title=Non-Equilibrium Thermodynamics |year=1984 |publisher=Dover Publications Inc. |location=New York |isbn=978-0-486-64741-8}}</ref>{{rp|p 163}} For a fluid consisting of only one kind of particle, the number density ''n'' is given by
:<math>n = \int f \,d^3p.</math>

The average value of any function ''A'' is
: <math>\langle A \rangle = \frac 1 n \int A f \,d^3p.</math>

Since the conservation equations involve tensors, the Einstein summation convention will be used where repeated indices in a product indicate summation over those indices. Thus <math>\mathbf{x} \mapsto x_i</math> and <math>\mathbf{p} \mapsto p_i = m w_i</math>, where <math>w_i</math> is the particle velocity vector. Define <math>A(p_i)</math> as some function of momentum <math>p_i</math> only, which is conserved in a collision. Assume also that the force <math>F_i</math> is a function of position only, and that ''f'' is zero for <math>p_i \to \pm\infty</math>. Multiplying the Boltzmann equation by ''A'' and integrating over momentum yields four terms, which, using integration by parts, can be expressed as

:<math>\int A \frac{\partial f}{\partial t} \,d^3p = \frac{\partial }{\partial t} (n \langle A \rangle),</math>

:<math>\int \frac{p_j A}{m}\frac{\partial f}{\partial x_j} \,d^3p = \frac{1}{m}\frac{\partial}{\partial x_j}(n\langle A p_j \rangle),</math>

:<math>\int A F_j \frac{\partial f}{\partial p_j} \,d^3p = -nF_j\left\langle \frac{\partial A}{\partial p_j}\right\rangle,</math>

:<math>\int A \left(\frac{\partial f}{\partial t}\right)_\text{coll} \,d^3p = 0,</math>

where the last term is zero, since ''A'' is conserved in a collision. Letting <math>A = m</math>, the mass of the particle, the integrated Boltzmann equation becomes the conservation of mass equation:<ref name="dG1984" />{{rp|pp 12,168}}

:<math>\frac{\partial}{\partial t}\rho + \frac{\partial}{\partial x_j}(\rho V_j) = 0,</math>

where <math>\rho = mn</math> is the mass density, and <math>V_i = \langle w_i\rangle</math> is the average fluid velocity.

Letting <math>A = p_i</math>, the momentum of the particle, the integrated Boltzmann equation becomes the conservation of momentum equation:<ref name="dG1984" />{{rp|pp 15,169}}

:<math>\frac{\partial}{\partial t}(\rho V_i) + \frac{\partial}{\partial x_j}(\rho V_i V_j+P_{ij}) - nF_i = 0,</math>

where <math>P_{ij} = \rho \langle (w_i-V_i) (w_j-V_j) \rangle</math> is the pressure tensor (the [[viscous stress tensor]] plus the hydrostatic [[pressure]]).

Letting <math>A =\frac{p_i p_i}{2m}</math>, the kinetic energy of the particle, the integrated Boltzmann equation becomes the conservation of energy equation:<ref name="dG1984" />{{rp|pp 19,169}}

:<math>\frac{\partial}{\partial t}(u + \tfrac{1}{2}\rho V_i V_i) + \frac{\partial}{\partial x_j} (uV_j + \tfrac{1}{2}\rho V_i V_i V_j + J_{qj} + P_{ij}V_i) - nF_iV_i = 0,</math>

where <math>u = \tfrac{1}{2} \rho \langle (w_i-V_i) (w_i-V_i) \rangle</math> is the kinetic thermal energy density, and <math>J_{qi} = \tfrac{1}{2} \rho \langle(w_i - V_i)(w_k - V_k)(w_k - V_k)\rangle</math> is the heat flux vector.

===Hamiltonian mechanics===

In [[Hamiltonian mechanics]], the Boltzmann equation is often written more generally as
:<math>\hat{\mathbf{L}}[f]=\mathbf{C}[f], \, </math>
where '''L''' is the [[Liouville operator]] (there is an inconsistent definition between the Liouville operator as defined here and the one in the article linked) describing the evolution of a phase space volume and '''C''' is the collision operator. The non-relativistic form of '''L''' is

:<math>\hat{\mathbf{L}}_\mathrm{NR} = \frac{\partial}{\partial t} + \frac{\mathbf{p}}{m} \cdot \nabla + \mathbf{F}\cdot\frac{\partial}{\partial \mathbf{p}}\,.</math>

=== Quantum theory and violation of particle number conservation ===

It is possible to write down relativistic [[quantum Boltzmann equation]]s for [[Quantum field theory|relativistic]] quantum systems in which the number of particles is not conserved in collisions. This has several applications in [[physical cosmology]],<ref name=KolbTurner>{{cite book|author1=Edward Kolb |author2=Michael Turner |name-list-style=amp|title=The Early Universe|year=1990|publisher=Westview Press|isbn=9780201626742}}</ref> including the formation of the light elements in [[Big Bang nucleosynthesis]], the production of [[dark matter]] and [[baryogenesis]]. It is not a priori clear that the state of a quantum system can be characterized by a classical phase space density ''f''. However, for a wide class of applications a well-defined generalization of ''f'' exists which is the solution of an effective Boltzmann equation that can be derived from first principles of [[quantum field theory]].<ref name=BEfromQFT>{{cite journal|author1=M. Drewes |author2=C. Weniger |author3=S. Mendizabal |journal=Phys. Lett. B|date=8 January 2013|volume=718|issue=3|pages=1119–1124|doi=10.1016/j.physletb.2012.11.046|arxiv = 1202.1301 |bibcode = 2013PhLB..718.1119D |title=The Boltzmann equation from quantum field theory|s2cid=119253828 }}</ref>

===General relativity and astronomy===

The Boltzmann equation is of use in galactic dynamics. A galaxy, under certain assumptions, may be approximated as a continuous fluid; its mass distribution is then represented by ''f''; in galaxies, physical collisions between the stars are very rare, and the effect of ''gravitational collisions'' can be neglected for times far longer than the [[age of the universe]].

Its generalization in [[general relativity]].<ref>Ehlers J (1971) General Relativity and Cosmology (Varenna), R K Sachs (Academic Press NY);Thorne K S (1980) Rev. Mod. Phys., 52, 299;
Ellis G F R, Treciokas R, Matravers D R, (1983) Ann. Phys., 150, 487}</ref> is

:<math>\hat{\mathbf{L}}_\mathrm{GR}=p^\alpha\frac{\partial}{\partial x^\alpha} - \Gamma^\alpha{}_{\beta\gamma}p^\beta p^\gamma\frac{\partial}{\partial p^\alpha},</math>

where Γαβγ is the [[Christoffel symbol]] of the second kind (this assumes there are no external forces, so that particles move along geodesics in the absence of collisions), with the important subtlety that the density is a function in mixed contravariant-covariant (''xi, pi'') phase space as opposed to fully contravariant (''xi, pi'') phase space.<ref>{{cite journal
| last = Debbasch
| first = Fabrice
|author2=Willem van Leeuwen
| title = General relativistic Boltzmann equation I: Covariant treatment
| journal = Physica A
| volume = 388
| issue = 7
| pages = 1079–1104
| year = 2009|bibcode = 2009PhyA..388.1079D |doi = 10.1016/j.physa.2008.12.023 }}</ref><ref>{{cite journal
| last = Debbasch
| first = Fabrice
|author2=Willem van Leeuwen
| title = General relativistic Boltzmann equation II: Manifestly covariant treatment
| journal = Physica A
| volume = 388
| issue = 9
| pages = 1818–34
| year = 2009|bibcode = 2009PhyA..388.1818D |doi = 10.1016/j.physa.2009.01.009 }}</ref>

In [[physical cosmology]] the fully covariant approach has been used to study the cosmic microwave background radiation.<ref>Maartens R, Gebbie T, Ellis GFR (1999). "Cosmic microwave background anisotropies: Nonlinear dynamics". Phys. Rev. D. 59 (8): 083506</ref> More generically the study of processes in the [[early universe]] often attempt to take into account the effects of [[quantum mechanics]] and [[general relativity]].<ref name=KolbTurner /> In the very dense medium formed by the primordial plasma after the [[Big Bang]], particles are continuously created and annihilated. In such an environment [[quantum coherence]] and the spatial extension of the [[wavefunction]] can affect the dynamics, making it questionable whether the classical phase space distribution ''f'' that appears in the Boltzmann equation is suitable to describe the system. In many cases it is, however, possible to derive an effective Boltzmann equation for a generalized distribution function from first principles of [[quantum field theory]].<ref name=BEfromQFT /> This includes the formation of the light elements in [[Big Bang nucleosynthesis]], the production of [[dark matter]] and [[baryogenesis]].

== Solving the equation ==
Exact solutions to the Boltzmann equations have been proven to exist in some cases;<ref>{{cite journal|doi=10.1090/S0894-0347-2011-00697-8 |arxiv=1011.5441 |title=Global Classical Solutions of the Boltzmann Equation without Angular Cut-off |authors=Philip T. Gressman, Robert M. Strain |journal=Journal of the American Mathematical Society |volume=24 |issue=3 |year=2011 |page=771|s2cid=115167686 }}</ref> this analytical approach provides insight, but is not generally usable in practical problems.

Instead, [[Numerical methods in fluid mechanics|numerical methods]] (including [[finite elements]] and [[lattice Boltzmann methods]]) are generally used to find approximate solutions to the various forms of the Boltzmann equation. Example applications range from [[Hypersonic speed|hypersonic aerodynamics]] in rarefied gas flows<ref>{{Cite journal|title = A discontinuous finite element solution of the Boltzmann kinetic equation in collisionless and BGK forms for macroscopic gas flows|url = https://cronfa.swan.ac.uk/Record/cronfa6256|journal = Applied Mathematical Modelling|date = 2011-03-01|pages = 996–1015|volume = 35|issue = 3|doi = 10.1016/j.apm.2010.07.027|first1 = Ben|last1 = Evans|first2 = Ken|last2 = Morgan|first3 = Oubay|last3 = Hassan}}</ref><ref>{{Cite journal|last1=Evans|first1=B.|last2=Walton|first2=S.P.|date=December 2017|title=Aerodynamic optimisation of a hypersonic reentry vehicle based on solution of the Boltzmann–BGK equation and evolutionary optimisation|journal=Applied Mathematical Modelling|volume=52|pages=215–240|doi=10.1016/j.apm.2017.07.024|issn=0307-904X|url=https://cronfa.swan.ac.uk/Record/cronfa34688}}</ref> to plasma flows.<ref>{{Cite journal|title = Numerical Solution of the Boltzmann Equation I: Spectrally Accurate Approximation of the Collision Operator|journal = SIAM Journal on Numerical Analysis|date = 2000-01-01|issn = 0036-1429|pages = 1217–1245|volume = 37|issue = 4|doi = 10.1137/S0036142998343300|first1 = L.|last1 = Pareschi|first2 = G.|last2 = Russo|citeseerx = 10.1.1.46.2853}}</ref> An application of the Boltzmann equation in electrodynamics is the calculation of the electrical conductivity - the result is in leading order identical with the semiclassical result.<ref>H.J.W. Müller-Kirsten, Basics of Statistical Mechanics, Chapter 13, 2nd ed., World Scientific (2013), {{ISBN|978-981-4449-53-3}}.</ref>

Close to [[Non-equilibrium thermodynamics#Local thermodynamic equilibrium|local equilibrium]], solution of the Boltzmann equation can be represented by an [[asymptotic expansion]] in powers of [[Knudsen number]] (the [[Chapman–Enskog theory|Chapman-Enskog]] expansion<ref>Sydney Chapman; Thomas George Cowling [https://books.google.com/books?id=JcjHpiJPKeIC&hl=en&source=gbs_navlinks_s The mathematical theory of non-uniform gases: an account of the kinetic theory of viscosity, thermal conduction, and diffusion in gases], Cambridge University Press, 1970. {{ISBN|0-521-40844-X}}</ref>). The first two terms of this expansion give the [[Euler equations (fluid dynamics)|Euler equations]] and the [[Navier-Stokes equations]]. The higher terms have singularities. The problem of developing mathematically the limiting processes, which lead from the atomistic view (represented by Boltzmann's equation) to the laws of motion of continua, is an important part of [[Hilbert's sixth problem]].<ref>{{cite journal | doi = 10.1098/rsta/376/2118 | volume=376 | year=2018 | journal=Philosophical Transactions of the Royal Society A | title = Theme issue 'Hilbert's sixth problem' | issue=2118 | doi-access=free }}</ref>

==See also==
{{Div col|colwidth=20em}}
* [[Vlasov equation]]

* [[H-theorem]]
* [[Fokker–Planck equation]]
* [[Williams spray equation|Williams-Boltzmann equation]]
* [[Navier–Stokes equations]]
* [[Vlasov equation#The Vlasov–Poisson equation|Vlasov–Poisson equation]]
* [[Lattice Boltzmann methods]]
{{Div col end}}

==Notes==
{{reflist|40em}}

==References==
* {{cite book
| last1= Harris
| first1= Stewart
| title= An introduction to the theory of the Boltzmann equation | publisher=Dover Books|pages=221 | year= 1971 | isbn=978-0-486-43831-3|url=https://books.google.com/books?id=KfYK1lyq3VYC}}. Very inexpensive introduction to the modern framework (starting from a formal deduction from Liouville and the Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy (BBGKY) in which the Boltzmann equation is placed). Most statistical mechanics textbooks like Huang still treat the topic using Boltzmann's original arguments. To derive the equation, these books use a heuristic explanation that does not bring out the range of validity and the characteristic assumptions that distinguish Boltzmann's from other transport equations like [[Fokker–Planck equation|Fokker–Planck]] or [[Landau equation]]s.

* {{cite journal | last1= Arkeryd | first1= Leif | title= On the Boltzmann equation part I: Existence | journal= Arch. Rational Mech. Anal. | volume= 45 | issue= 1 | pages= 1–16 | year= 1972 | doi = 10.1007/BF00253392 | bibcode = 1972ArRMA..45....1A | s2cid= 117877311 }}
* {{cite journal
| last1= Arkeryd
| first1= Leif
|author1-link= Leif Arkeryd
| title= On the Boltzmann equation part II: The full initial value problem | journal= Arch. Rational Mech. Anal. | volume= 45 | issue= 1
| pages= 17–34 | year= 1972 | doi = 10.1007/BF00253393 | bibcode = 1972ArRMA..45...17A | s2cid= 119481100
}}
* {{cite journal | last1= Arkeryd | first1= Leif | title= On the Boltzmann equation part I: Existence | journal= Arch. Rational Mech. Anal. | volume= 45 | issue= 1 | pages= 1–16 | year= 1972 | doi = 10.1007/BF00253392 | bibcode = 1972ArRMA..45....1A | s2cid= 117877311 }}
* {{cite journal | last1=DiPerna | first1= R. J. |last2 = Lions | first2 = P.-L. | title= On the Cauchy problem for Boltzmann equations: global existence and weak stability | journal= Ann. of Math. |series= 2 | volume=130 | issue= 2 | pages= 321–366 | year=1989 | doi=10.2307/1971423| jstor= 1971423 }}

==External links==
* [http://homepage.univie.ac.at/franz.vesely/sp_english/sp/node7.html The Boltzmann Transport Equation by Franz Vesely]
* [https://web.archive.org/web/20151123214334/http://www.upenn.edu/pennnews/news/university-pennsylvania-mathematicians-solve-140-year-old-boltzmann-equation-gaseous-behaviors Boltzmann gaseous behaviors solved]

{{Statistical mechanics topics}}

[[Category:Partial differential equations]]
[[Category:Statistical mechanics]]
[[Category:Transport phenomena]]
[[Category:Equations of physics]]
[[Category:Ludwig Boltzmann|Equation]]
[[Category:1872 in science]]
[[Category:1872 in Germany]]

Hydrodynamics

2021-05-10T08:08:20Z

Bleicher:

{{Short description|Study of the magnetic properties of electrically conducting fluids}}
{{For|the academic journal|Magnetohydrodynamics (journal)}}
{{more citations needed|date=April 2011}}
[[File:The sun is an MHD system that is not well understood- 2013-04-9 14-29.jpg|thumbnail|The sun is an MHD system that is not well understood.]]

'''Magnetohydrodynamics''' ('''MHD'''; '''also magneto-fluid dynamics''' or '''hydromagnetics''') is the study of the magnetic properties and behaviour of [[electrical conduction|electrically conducting]] [[fluid]]s. Examples of such magnetofluids include [[Plasma (physics)|plasmas]], [[liquid metal]]s, [[Brine|salt water]], and [[electrolyte]]s. The word "magnetohydrodynamics" is derived from ''magneto-'' meaning [[magnetic field]], ''hydro-'' meaning [[water]], and ''dynamics'' meaning movement. The field of MHD was initiated by [[Hannes Alfvén]],<ref>{{cite journal | last1 = Alfvén | first1 = H | year = 1942 | title = Existence of electromagnetic-hydrodynamic waves | bibcode=1942Natur.150..405A | journal = Nature | volume = 150 | issue = 3805| pages = 405–406 | doi=10.1038/150405d0| s2cid = 4072220 }}</ref> for which he received the [[Nobel Prize]] in Physics in 1970.

The fundamental concept behind MHD is that magnetic fields can [[Electromagnetic induction|induce]] currents in a moving conductive fluid, which in turn polarizes the fluid and reciprocally changes the magnetic field itself. The set of equations that describe MHD are a combination of the [[Navier–Stokes equations]] of [[fluid dynamics]] and [[Maxwell’s equations]] of [[electromagnetism|electromagnetism]]. These [[differential equation]]s must be solved [[simultaneous equation|simultaneously]], either analytically or [[Numerical analysis|numerically]].

== History ==
[[File:M Faraday Th Phillips oil 1842.jpg|thumb|180px|[[Michael Faraday]]]]
The first recorded use of the word ''magnetohydrodynamics'' is by [[Hannes Alfvén]] in 1942:
{{quote|At last some remarks are made about the transfer of momentum from the Sun to the planets, which is fundamental to the theory (§11). The importance of the Magnetohydrodynamic waves in this respect are pointed out.<ref>{{cite journal |last=Alfvén |first=H. |year=1942 |title=On the cosmogony of the solar system III |journal=Stockholms Observatoriums Annaler |volume=14 |pages=9.1–9.29 |bibcode=1942StoAn..14....9A}}</ref>}}

The ebbing salty water flowing past London's [[Waterloo Bridge]] interacts with the [[Earth's magnetic field]] to produce a potential difference between the two river-banks. [[Michael Faraday]] called this effect "magneto-electric induction" and tried this experiment in 1832 but the current was too small to measure with the equipment at the time,<ref>[http://www.phy6.org/earthmag/NSTA1C.htm Dynamos in Nature] by David P. Stern</ref> and the river bed contributed to short-circuit the signal. However, by a similar process the voltage induced by the tide in the English Channel was measured in 1851.<ref>McKetta, J. "[https://books.google.com/books?id=U8Bd7TOcma4C&pg=PA127 Encyclopedia of Chemical Processing and Design: Volume 66]" (1999) {{Dead link|date=December 2016}}</ref>

Faraday carefully omitted the term of hydrodynamics in this work.

== Ideal and resistive MHD ==
[[File:T3e troy.jpg|right|frame|MHD Simulation of the Solar Wind]]
The simplest form of MHD, Ideal MHD, assumes that the fluid has so little [[resistivity]] that it can be treated as a [[perfect conductor]]. This is the limit of infinite [[magnetic Reynolds number]]. In ideal MHD, [[Lenz's law]] dictates that the fluid is in a sense ''tied'' to the magnetic field lines. To explain, in ideal MHD a small rope-like volume of fluid surrounding a field line will continue to lie along a magnetic field line,
even as it is twisted and distorted by fluid flows in the system. This is sometimes referred to as the magnetic field lines being "frozen" in the fluid.<ref>Eric Priest and Terry Forbes, "Magnetic Reconnection: MHD Theory and Applications", Cambridge University Press, First Edition, 2000, pp 25.</ref>
The connection between magnetic field lines and fluid in ideal MHD fixes the [[topology]] of the magnetic field in the fluid—for example, if a set of magnetic field lines are tied into a knot, then they will remain so as long as the fluid/plasma has negligible resistivity. This difficulty in reconnecting magnetic field lines makes it possible to store energy by moving the fluid or the source of the magnetic field. The energy can then become available if the conditions for ideal MHD break down, allowing [[magnetic reconnection]] that releases the stored energy from the magnetic field.

=== Ideal MHD equations ===
[[File:Magnetohydrodynamic Flow Simulation.jpg|thumb|Magnetohydrodynamic flow simulation showing magnetic flux density]]
The ideal MHD equations consist of the [[continuity equation]], the [[Cauchy momentum equation]], [[Ampere's Law]] neglecting displacement current, and a [[Conservation of energy|temperature evolution equation]]. As with any fluid description to a kinetic system, a closure approximation must be applied to highest moment of the particle distribution equation. This is often accomplished with approximations to the heat flux through a condition of [[Adiabatic process|adiabaticity]] or [[Isothermal process|isothermality]].

The main quantities which characterize the electrically conducting fluid are the bulk plasma [[velocity field]] {{math|'''v'''}}, the [[current density]] {{math|'''J'''}}, the [[mass density]] {{mvar|ρ}}, and the plasma [[pressure]] {{mvar|p}}. The flowing electric charge in the plasma is the source of a [[magnetic field]] {{math|'''B'''}} and [[electric field]] {{math|'''E'''}}. All quantities generally vary with time {{mvar|t}}. [[Vector operator]] notation will be used, in particular {{math|∇}} is [[gradient]], {{math|∇ ⋅}} is [[divergence]], and {{math|∇ ×}} is [[curl (mathematics)|curl]].

The mass [[continuity equation]] is
:<math> \frac{\partial \rho}{\partial t} + \nabla \cdot \left(\rho\mathbf{v}\right)=0.</math>
The [[Cauchy momentum equation]] is
:<math> \rho\left(\frac{\partial }{\partial t} + \mathbf{v}\cdot\nabla \right)\mathbf{v} = \mathbf{J}\times\mathbf{B} - \nabla p.</math>
The [[Lorentz force]] term {{math|'''J''' × '''B'''}} can be expanded using [[Ampère's law]] and the [[vector calculus identity]]

:<math>\tfrac12\nabla(\mathbf{B}\cdot \mathbf{B})=(\mathbf{B}\cdot\nabla)\mathbf{B}+\mathbf{B}\times(\nabla\times \mathbf{B})</math>

to give
:<math>\mathbf{J}\times\mathbf{B} = \frac{\left(\mathbf{B}\cdot\nabla\right)\mathbf{B}}{\mu_0} - \nabla\left(\frac{B^2}{2\mu_0}\right),</math>
where the first term on the right hand side is the [[magnetic tension force]] and the second term is the [[magnetic pressure]] force.

The ideal [[Ohm's law#Magnetic effects|Ohm's law]] for a plasma is given by
:<math>\mathbf{E} + \mathbf{v}\times\mathbf{B} = 0. </math>
Faraday's law is
:<math>\frac{\partial \mathbf{B}}{\partial t} = - \nabla \times \mathbf{E}. </math>
The low-frequency Ampère's law neglects displacement current and is given by
:<math>\mu_0 \mathbf{J} = \nabla\times\mathbf{B}.</math>
The magnetic divergence constraint is
:<math>\nabla\cdot\mathbf{B} = 0.</math>
The energy equation is given by
:<math>\frac{\mathrm{d}}{\mathrm{d}t} \left(\frac{p}{\rho^\gamma}\right) = 0,</math>
where {{math|''γ'' {{=}} {{sfrac|5|3}}}} is the ratio of [[specific heat]]s for an [[adiabatic]] [[equation of state]]. This energy equation is only applicable in the absence of shocks or heat conduction as it assumes that the entropy of a fluid element does not change.

=== Applicability of ideal MHD to plasmas ===

Ideal MHD is only strictly applicable when:

# The plasma is strongly collisional, so that the time scale of collisions is shorter than the other characteristic times in the system, and the particle distributions are therefore close to [[Maxwell–Boltzmann distribution|Maxwellian]].
# The resistivity due to these collisions is small. In particular, the typical magnetic diffusion times over any scale length present in the system must be longer than any time scale of interest.
# Interest in length scales much longer than the ion [[Plasma parameters#Lengths|skin depth]] and [[Larmor radius]] perpendicular to the field, long enough along the field to ignore [[Landau damping]], and time scales much longer than the ion gyration time (system is smooth and slowly evolving).

=== Importance of resistivity ===

In an imperfectly conducting fluid the magnetic field can generally move through the fluid following a [[Diffusion equation|diffusion law]] with the resistivity of the plasma serving as a [[diffusion constant]]. This means that solutions to the ideal MHD equations are only applicable for a limited time for a region of a given size before diffusion becomes too important to ignore. One can estimate the diffusion time across a [[solar active region]] (from collisional resistivity) to be hundreds to thousands of years, much longer than the actual lifetime of a sunspot—so it would seem reasonable to ignore the resistivity. By contrast, a meter-sized volume of seawater has a magnetic diffusion time measured in milliseconds.

Even in physical systems – which are large and conductive enough that simple estimates of the [[Lundquist number]] suggest that the resistivity can be ignored – resistivity may still be important: many [[Instability|instabilities]] exist that can increase the effective resistivity of the plasma by factors of more than 109. The enhanced resistivity is usually the result of the formation of small scale structure like current sheets or fine scale magnetic [[Magnetohydrodynamic turbulence|turbulence]], introducing small spatial scales into the system over which ideal MHD is broken and magnetic diffusion can occur quickly. When this happens, magnetic reconnection may occur in the plasma to release stored magnetic energy as waves, bulk mechanical acceleration of material, [[particle acceleration]], and heat.

Magnetic reconnection in highly conductive systems is important because it concentrates energy in time and space, so that gentle forces applied to a plasma for long periods of time can cause violent explosions and bursts of radiation.

When the fluid cannot be considered as completely conductive, but the other conditions for ideal MHD are satisfied, it is possible to use an extended model called resistive MHD. This includes an extra term in Ohm's Law which models the collisional resistivity. Generally MHD computer simulations are at least somewhat resistive because their computational grid introduces a [[numerical resistivity]].

=== Importance of kinetic effects ===

Another limitation of MHD (and fluid theories in general) is that they depend on the assumption that the plasma is strongly collisional (this is the first criterion listed above), so that the time scale of collisions is shorter than the other characteristic times in the system, and the particle distributions are [[Maxwell–Boltzmann distribution|Maxwellian]]. This is usually not the case in fusion, space and astrophysical plasmas. When this is not the case, or the interest is in smaller spatial scales, it may be necessary to use a kinetic model which properly accounts for the non-Maxwellian shape of the distribution function. However, because MHD is relatively simple and captures many of the important properties of plasma dynamics it is often qualitatively accurate and is therefore often the first model tried.

Effects which are essentially kinetic and not captured by fluid models include [[Double layer (plasma)|double layers]], [[Landau damping]], a wide range of instabilities, chemical separation in space plasmas and electron runaway. In the case of ultra-high intensity laser interactions, the incredibly short timescales of energy deposition mean that hydrodynamic codes fail to capture the essential physics.

== Structures in MHD systems ==
{{Further|Magnetosphere particle motion}}
[[File:Currents.jpg|right|thumb|250px|Schematic view of the different current systems which shape the Earth's magnetosphere]]

In many MHD systems most of the electric current is compressed into thin nearly-two-dimensional ribbons termed [[current sheet]]s. These can divide the fluid into magnetic domains, inside of which the currents are relatively weak. Current sheets in
the solar corona are thought to be between a few meters and a few kilometers in thickness, which is quite thin compared to the magnetic domains (which are thousands to hundreds of thousands of kilometers across). Another example is in the Earth's [[magnetosphere]], where current sheets separate topologically distinct domains, isolating most of the Earth's [[ionosphere]] from the [[solar wind]].

== Waves ==
{{See also|Waves in plasmas}}
{{Main|Magnetosonic wave}}
The wave modes derived using MHD plasma theory are called '''magnetohydrodynamic waves''' or '''MHD waves'''. In general there are three MHD wave modes:
* Pure (or oblique) Alfvén wave
* Slow MHD wave
* Fast MHD wave
{{multiple image
| width=250
| direction=vertical
| align=right
| header=Phase velocity plotted with respect to {{mvar|θ}}
| image1=MHD wave mode 1.svg
| alt1=<math>v_A>v_s</math>
| caption1={{math|''vA'' > ''vs''}}
| image2=MHD wave mode 2.svg
| alt2=<math>v_A<v_s</math>
| caption2={{math|''vA'' < ''vs''}}
}}
All these waves have constant phase velocities for all frequencies, and hence there is no dispersion. At the limits when the angle between the wave propagation vector {{math|'''k'''}} and magnetic field {{math|'''B'''}} is either 0° (180°) or 90°, the wave modes are called:<ref>[http://www.oulu.fi/~spaceweb/textbook/mhdwaves.html MHD waves [Oulu]] {{webarchive|url=https://web.archive.org/web/20070810223700/http://www.oulu.fi/~spaceweb/textbook/mhdwaves.html |date=2007-08-10 }}</ref>
{|class="wikitable" style="margin: 1em auto 1em auto"
!Name||Type||Propagation||Phase velocity||Association||Medium||Other names
|-
| Sound wave || longitudinal || {{math|'''k''' ∥ '''B'''}} || adiabatic sound velocity || none ||compressible, nonconducting fluid ||
|-
| [[Alfvén wave]] || transverse || {{math|'''k''' ∥ '''B'''}} || Alfvén velocity || {{math|'''B'''}} || ||shear Alfvén wave, the slow Alfvén wave, torsional Alfvén wave
|-
| [[Magnetosonic wave]] || longitudinal || {{math|'''k''' ⟂ '''B'''}} || || {{math|'''B'''}}, {{math|'''E'''}} || ||compressional Alfvén wave, fast Alfvén wave, magnetoacoustic wave
|}
The phase velocity depends on the angle between the wave vector {{math|'''k'''}} and the magnetic field {{math|'''B'''}}. An MHD wave propagating at an arbitrary angle {{mvar|θ}} with respect to the time independent or bulk field {{math|'''B'''0}} will satisfy the dispersion relation
:<math>\frac{\omega}{k} = v_A \cos\theta</math>
where
:<math>v_A = \frac{B_0}{\sqrt{\mu_0\rho}}</math>
is the Alfvén speed. This branch corresponds to the shear Alfvén mode. Additionally the dispersion equation gives
:<math>\frac{\omega}{k} = \left( \tfrac12\left(v_A^2+v_s^2\right) \pm \tfrac12\sqrt{\left(v_A^2+v_s^2\right)^2 - 4v_s^2v_A^2\cos^2\theta}\right)^\frac12</math>
where
:<math>v_s = \sqrt{\frac{\gamma p}{\rho}}</math>
is the ideal gas speed of sound. The plus branch corresponds to the fast-MHD wave mode and the minus branch corresponds to the slow-MHD wave mode.

The MHD oscillations will be damped if the fluid is not perfectly conducting but has a finite conductivity, or if viscous effects are present.

MHD waves and oscillations are a popular tool for the remote diagnostics of laboratory and astrophysical plasmas, for example, the [[solar corona|corona]] of the Sun ([[Coronal seismology]]).

== Extensions ==
; Resistive
: Resistive MHD describes magnetized fluids with finite electron diffusivity ({{math|''η'' ≠ 0}}). This diffusivity leads to a breaking in the magnetic topology; magnetic field lines can 'reconnect' when they collide. Usually this term is small and reconnections can be handled by thinking of them as not dissimilar to [[Shocks and discontinuities (magnetohydrodynamics)|shocks]]; this process has been shown to be important in the Earth-Solar magnetic interactions.
; Extended
: Extended MHD describes a class of phenomena in plasmas that are higher order than resistive MHD, but which can adequately be treated with a single fluid description. These include the effects of Hall physics, electron pressure gradients, finite Larmor Radii in the particle gyromotion, and electron inertia.
; Two-fluid
: Two-fluid MHD describes plasmas that include a non-negligible Hall [[electric field]]. As a result, the electron and ion momenta must be treated separately. This description is more closely tied to Maxwell's equations as an evolution equation for the electric field exists.
; Hall
: In 1960, M. J. Lighthill criticized the applicability of ideal or resistive MHD theory for plasmas.<ref>M. J. Lighthill, "Studies on MHD waves and other anisotropic wave motion," ''Phil. Trans. Roy. Soc.'', London, vol. 252A, pp. 397–430, 1960.</ref> It concerned the neglect of the "Hall current term", a frequent simplification made in magnetic fusion theory. Hall-magnetohydrodynamics (HMHD) takes into account this electric field description of magnetohydrodynamics. The most important difference is that in the absence of field line breaking, the magnetic field is tied to the electrons and not to the bulk fluid.<ref>{{cite journal | last1 = Witalis | first1 = E.A. | year = 1986 | title = Hall Magnetohydrodynamics and Its Applications to Laboratory and Cosmic Plasma | journal = IEEE Transactions on Plasma Science | volume = PS-14 | issue = 6| pages = 842–848 | bibcode=1986ITPS...14..842W|doi = 10.1109/TPS.1986.4316632 | s2cid = 31433317 }}</ref>
; Electron MHD
: Electron Magnetohydrodynamics (EMHD) describes small scales plasmas when electron motion is much faster than the ion one. The main effects are changes in conservation laws, additional resistivity, importance of electron inertia. Many effects of Electron MHD are similar to effects of the Two fluid MHD and the Hall MHD. EMHD is especially important for [[z-pinch]], [[magnetic reconnection]], [[ion thrusters]], [[neutron stars]], and plasma switches.
; Collisionless
: MHD is also often used for collisionless plasmas. In that case the MHD equations are derived from the [[Vlasov equation]].<ref name="space">W. Baumjohann and R. A. Treumann, ''Basic Space Plasma Physics'', Imperial College Press, 1997</ref>
; Reduced
: By using a [[Multiple-scale analysis|multiscale analysis]] the (resistive) MHD equations can be reduced to a set of four closed scalar equations. This allows for, amongst other things, more efficient numerical calculations.<ref name=paper:hegna>{{cite web|last1=Kruger|first1=S.E.|last2=Hegna|first2=C.C.|last3=Callen|first3=J.D.|title=Reduced MHD equations for low aspect ratio plasmas|url=http://epsppd.epfl.ch/Praha/98icpp/g096pr.pdf|archive-url=https://web.archive.org/web/20150925113607/http://epsppd.epfl.ch/Praha/98icpp/g096pr.pdf|url-status=dead|archive-date=25 September 2015|publisher=University of Wisconsin|access-date=27 April 2015}}</ref>

== Applications ==

=== Geophysics ===
Beneath the Earth's mantle lies the core, which is made up of two parts: the solid inner core and liquid outer core. Both have significant quantities of iron. The liquid outer core moves in the presence of the magnetic field and eddies are set up into the same due to the Coriolis effect. These eddies develop a magnetic field which boosts Earth's original magnetic field—a process which is self-sustaining and is called the geomagnetic dynamo.<ref name = "pbs">[https://www.pbs.org/wgbh/nova/magnetic/reve-drives.html NOVA | Magnetic Storm | What Drives Earth's Magnetic Field? | PBS]</ref>

[[File:NASA 54559main comparison1 strip.gif|thumb|center|350px|Reversals of [[Earth's magnetic field]]]]

Based on the MHD equations, Glatzmaier and Paul Roberts have made a supercomputer model of the Earth's interior. After running the simulations for thousands of years in virtual time, the changes in Earth's magnetic field can be studied. The simulation results are in good agreement with the observations as the simulations have correctly predicted that the Earth's magnetic field flips every few hundred thousand years. During the flips, the magnetic field does not vanish altogether—it just gets more complex.
<ref name = "glatz">[https://science.nasa.gov/science-news/science-at-nasa/2003/29dec_magneticfield/ Earth's Inconstant Magnetic Field – NASA Science]</ref>

====Earthquakes====
Some monitoring stations have reported that [[earthquakes]] are sometimes preceded by a spike in [[ultra low frequency]] (ULF) activity. A remarkable example of this occurred before the [[1989 Loma Prieta earthquake]] in [[California]],<ref>{{cite journal |first1=Antony C. |last1=Fraser-Smith |first2=A. |last2=Bernardi |first3=P. R. |last3=McGill |first4=M. E. |last4=Ladd |first5=R. A. |last5=Helliwell |first6=O. G. |last6=Villard Jr. |date=August 1990 |title=Low-Frequency Magnetic Field Measurements Near the Epicenter of the Ms 7.1 Loma Prieta Earthquake |journal=[[Geophysical Research Letters]] |volume=17 |issue=9 |pages=1465–1468 |issn=0094-8276 |oclc=1795290 |access-date=December 18, 2010 |url=http://ee.stanford.edu/~acfs/LomaPrietaPaper.pdf |doi=10.1029/GL017i009p01465 |bibcode=1990GeoRL..17.1465F}}</ref> although a subsequent study indicates that this was little more than a sensor malfunction.<ref>{{Cite journal | last1 = Thomas | first1 = J. N. | last2 = Love | first2 = J. J. | last3 = Johnston | first3 = M. J. S. | title = On the reported magnetic precursor of the 1989 Loma Prieta earthquake | doi = 10.1016/j.pepi.2008.11.014 | journal = Physics of the Earth and Planetary Interiors | volume = 173 | issue = 3–4 | pages = 207–215 |date=April 2009 | bibcode=2009PEPI..173..207T}}</ref> On December 9, 2010, geoscientists announced that the [[Demeter (satellite)|DEMETER]] satellite observed a dramatic increase in ULF radio waves over [[Haiti]] in the month before the magnitude 7.0 Mw [[2010 Haiti earthquake|2010 earthquake]].<ref>{{cite web|url=http://www.technologyreview.com/blog/arxiv/26114/|title=Spacecraft Saw ULF Radio Emissions over Haiti before January Quake|author=KentuckyFC |date=December 9, 2010|website=Physics arXiv Blog|publisher=[[TechnologyReview.com]]|location=[[Cambridge, Massachusetts]]|access-date=December 18, 2010}} {{Cite journal|last1=Athanasiou|first1=M|last2=Anagnostopoulos|first2=G|last3=Iliopoulos|first3=A|last4=Pavlos|first4=G|last5=David|first5=K|year=2010|title=Enhanced ULF radiation observed by DEMETER two months around the strong 2010 Haiti earthquake|journal=Natural Hazards and Earth System Sciences|volume=11|issue=4|pages=1091|arxiv=1012.1533|doi=10.5194/nhess-11-1091-2011|bibcode=2011NHESS..11.1091A|s2cid=53456663}}</ref> Researchers are attempting to learn more about this correlation to find out whether this method can be used as part of an early warning system for earthquakes.

=== Astrophysics ===
MHD applies to [[astrophysics]], including stars, the [[interplanetary medium]] (space between the planets), and possibly within the [[interstellar medium]] (space between the stars) and [[Relativistic jet|jets]].<ref>{{cite journal|last1=Kennel|first1=C.F. |last2=Arons|first2=J.|last3=Blandford|first3=R.|last4=Coroniti|first4=F.|last5=Israel |first5=M.|last6=Lanzerotti|first6=L.|last7=Lightman|first7=A.|title=Perspectives on Space & Astrophysical Plasma Physics|url=https://authors.library.caltech.edu/96134/2/1985IAUS__107__537K.pdf|date=1985|journal=Unstable Current Systems & Plasma Instabilities in Astrophysics|volume=107 |doi=10.1007/978-94-009-6520-1_63|pages=537–552 |access-date=2019-07-22|isbn=978-90-277-1887-7 |bibcode=1985IAUS..107..537K }}</ref> Most astrophysical systems are not in local thermal equilibrium, and therefore require an additional kinematic treatment to describe all the phenomena within the system (see [[Astrophysical plasma]]).{{Citation needed|date=December 2015}}

[[Sunspot]]s are caused by the Sun's magnetic fields, as [[Joseph Larmor]] theorized in 1919. The [[solar wind]] is also governed by MHD. The differential [[solar rotation]] may be the long-term effect of magnetic drag at the poles of the Sun, an MHD phenomenon due to the [[Parker spiral]] shape assumed by the extended magnetic field of the Sun.

Previously, theories describing the formation of the Sun and planets could not explain how the Sun has 99.87% of the mass, yet only 0.54% of the [[angular momentum]] in the [[solar system]]. In a [[closed system]] such as the cloud of gas and dust from which the Sun was formed, mass and angular momentum are both [[Conservation law|conserved]]. That conservation would imply that as the mass concentrated in the center of the cloud to form the Sun, it would spin faster, much like a skater pulling their arms in. The high speed of rotation predicted by early theories would have flung the proto-Sun apart before it could have formed. However, magnetohydrodynamic effects transfer the Sun's angular momentum into the outer solar system, slowing its rotation.

Breakdown of ideal MHD (in the form of magnetic reconnection) is known to be the likely cause of [[solar flare]]s.{{Citation needed|date=December 2015}} The magnetic field in a solar [[active region]] over a sunspot can store energy that is released suddenly as a burst of motion, [[X-ray]]s, and [[radiation]] when the main current sheet collapses, reconnecting the field.{{Citation needed|date=December 2015}}

===Sensors===

Magnetohydrodynamic sensors are used for precision measurements of [[Angular velocity|angular velocities]] in [[inertial navigation system]]s such as in [[aerospace engineering]]. Accuracy improves with the size of the sensor. The sensor is capable of surviving in harsh environments.<ref>{{cite web |url=http://read.pudn.com/downloads165/ebook/756655/Strapdown%20Inertial%20Navigation%20Technology/13587_04b.pdf |title=Archived copy |access-date=2014-08-19 |url-status=dead |archive-url=https://web.archive.org/web/20140820045250/http://read.pudn.com/downloads165/ebook/756655/Strapdown%20Inertial%20Navigation%20Technology/13587_04b.pdf |archive-date=2014-08-20 }} D.Titterton, J.Weston, Strapdown Inertial Navigation Technology, chapter 4.3.2</ref>

=== Engineering ===

MHD is related to engineering problems such as [[fusion power|plasma confinement]], liquid-metal cooling of [[nuclear reactor]]s, and [[Electromagnetism|electromagnetic]] casting (among others).

A [[magnetohydrodynamic drive]] or MHD propulsor is a method for propelling seagoing vessels using only electric and magnetic fields with no moving parts, using magnetohydrodynamics. The working principle involves electrification of the propellant (gas or water) which can then be directed by a magnetic field, pushing the vehicle in the opposite direction. Although some working prototypes exist, MHD drives remain impractical.

The first prototype of this kind of propulsion was built and tested in 1965 by Steward Way, a professor of mechanical engineering at the [[University of California, Santa Barbara]]. Way, on leave from his job at [[Westinghouse Electric (1886)|Westinghouse Electric]], assigned his senior-year undergraduate students to develop a submarine with this new propulsion system.<ref>{{cite journal |title=Run Silent, Run Electromagnetic |date=1966-09-23 |journal=[[Time (magazine)|Time]] |url=http://www.time.com/time/magazine/article/0,9171,842848-1,00.html}}</ref> In the early 1990s, a foundation in Japan (Ship & Ocean Foundation (Minato-ku, Tokyo)) built an experimental boat, the ''[[Yamato 1|Yamato-1]]'', which used a [[magnetohydrodynamic drive]] incorporating a [[superconductor]] cooled by [[helium|liquid helium]], and could travel at 15 km/h.<ref name = "yamato">Setsuo Takezawa et al. (March 1995) ''Operation of the Thruster for Superconducting Electromagnetohydrodynamic Propu1sion Ship YAMATO 1''</ref>

[[MHD generator|MHD power generation]] fueled by potassium-seeded coal combustion gas showed potential for more efficient energy conversion (the absence of solid moving parts allows operation at higher temperatures), but failed due to cost-prohibitive technical difficulties.<ref>''[http://navier.stanford.edu/PIG/PIGdefault.html Partially Ionized Gases] {{webarchive|url=https://web.archive.org/web/20080905113821/http://navier.stanford.edu/PIG/PIGdefault.html |date=2008-09-05 }}'', M. Mitchner and Charles H. Kruger, Jr., Mechanical Engineering Department, [[Stanford University]]. See Ch. 9 "Magnetohydrodynamic (MHD) Power Generation", pp. 214–230.</ref> One major engineering problem was the failure of the wall of the primary-coal combustion chamber due to abrasion.

In [[microfluidics]], MHD is studied as a fluid pump for producing a continuous, nonpulsating flow in a complex microchannel design.<ref name=Nguyen>{{cite book | author=Nguyen, N.T. |author2=Wereley, S. | title=Fundamentals and Applications of Microfluidics | date=2006 | publisher =[[Artech House]] }}</ref>

MHD can be implemented in the [[continuous casting]] process of metals to suppress instabilities and control the flow.<ref>{{cite conference |last=Fujisaki |first=Keisuke |date=Oct 2000 |title=In-mold electromagnetic stirring in continuous casting |doi=10.1109/IAS.2000.883188 |conference=Industry Applications Conference |publisher=IEEE |volume=4 |pages= 2591–2598 }}</ref><ref>{{cite journal |last1=Kenjeres |first1=S. |last2=Hanjalic |first2=K. |date=2000 |title=On the implementation of effects of Lorentz force in turbulence closure models |journal=International Journal of Heat and Fluid Flow |volume=21 |issue=3 |pages=329–337 |doi=10.1016/S0142-727X(00)00017-5 }}</ref>

Industrial MHD problems can be modeled using the open-source software EOF-Library.<ref>{{Cite journal|last1=Vencels|first1=Juris|last2=Råback|first2=Peter|last3=Geža|first3=Vadims|date=2019-01-01|title=EOF-Library: Open-source Elmer FEM and OpenFOAM coupler for electromagnetics and fluid dynamics|journal=SoftwareX|volume=9|pages=68–72|doi=10.1016/j.softx.2019.01.007|issn=2352-7110|bibcode=2019SoftX...9...68V|doi-access=free}}</ref> Two simulation examples are 3D MHD with a free surface for [[Magnetic levitation|electromagnetic levitation]] melting,<ref>{{Cite journal|last1=Vencels|first1=Juris|last2=Jakovics|first2=Andris|last3=Geza|first3=Vadims|date=2017|title=Simulation of 3D MHD with free surface using Open-Source EOF-Library: levitating liquid metal in an alternating electromagnetic field|journal=Magnetohydrodynamics|volume=53|issue=4|pages=643–652|doi=10.22364/mhd.53.4.5|issn=0024-998X}}</ref> and liquid metal stirring by rotating permanent magnets.<ref>{{Cite journal|last1=Dzelme|first1=V.|last2=Jakovics|first2=A.|last3=Vencels|first3=J.|last4=Köppen|first4=D.|last5=Baake|first5=E.|date=2018|title=Numerical and experimental study of liquid metal stirring by rotating permanent magnets|url=http://stacks.iop.org/1757-899X/424/i=1/a=012047|journal=IOP Conference Series: Materials Science and Engineering|language=en|volume=424|issue=1|pages=012047|doi=10.1088/1757-899X/424/1/012047|issn=1757-899X|bibcode=2018MS&E..424a2047D|doi-access=free}}</ref>

===Magnetic drug targeting===
An important task in cancer research is developing more precise methods for delivery of medicine to affected areas. One method involves the binding of medicine to biologically compatible magnetic particles (such as ferrofluids), which are guided to the target via careful placement of permanent magnets on the external body. Magnetohydrodynamic equations and finite element analysis are used to study the interaction between the magnetic fluid particles in the bloodstream and the external magnetic field.<ref>{{Cite journal|last1=Nacev|first1=A.|last2=Beni|first2=C.|last3=Bruno|first3=O.|last4=Shapiro|first4=B.|date=2011-03-01|title=The Behaviors of Ferro-Magnetic Nano-Particles In and Around Blood Vessels under Applied Magnetic Fields|journal=Journal of Magnetism and Magnetic Materials|volume=323|issue=6|pages=651–668|doi=10.1016/j.jmmm.2010.09.008|issn=0304-8853|pmc=3029028|pmid=21278859|bibcode=2011JMMM..323..651N}}</ref>

== See also ==
{{div col|colwidth=30em}}
* [[Computational magnetohydrodynamics]]
* [[Electrohydrodynamics]]
* [[Electromagnetic pump]]
* [[Ferrofluid]]
* [[List of plasma physics articles]]
* [[Lorentz force velocimetry|Lorentz force velocity meter]]
* [[Magnetic flow meter]]
* [[Magnetohydrodynamic generator]]
* [[Magnetohydrodynamic turbulence]]
* [[Molten salt]]
* [[Plasma stability]]
* [[Shocks and discontinuities (magnetohydrodynamics)]]
{{div col end}}

== Notes ==
{{Reflist|30em}}

== References ==
{{ref begin}}
* Bansal, J. L. (1994) ''Magnetofluiddynamics of Viscous Fluids'' Jaipur Publishing House, Jaipur, India, {{OCLC|70267818}}
* {{cite journal | last1 = Barbu | first1 = V. | display-authors = etal | year = 2003 | title = Exact controllability magneto-hydrodynamic equations | journal = Communications on Pure and Applied Mathematics | volume = 56 | issue = 6| pages = 732–783 | doi=10.1002/cpa.10072}}
* Biskamp, Dieter. ''Nonlinear Magnetohydrodynamics''. Cambridge, England: Cambridge University Press, 1993. 378 p. {{ISBN|0-521-59918-0}}
* Calvert, James B. (20 October 2002) [https://web.archive.org/web/20070625105802/http://www.du.edu/~jcalvert/phys/mhd.htm "Magnetohydrodynamics: The dynamics of conducting fluids in an electromagnetic field"] (self published by an Associate Professor Emeritus of Engineering, University of Denver, U.S.A.)
* Davidson, Peter Alan (May 2001) ''An Introduction to Magnetohydrodynamics'' Cambridge University Press, Cambridge, England, {{ISBN|0-521-79487-0}}
* Faraday, M. (1832). "Experimental Researches in Electricity." First Series, ''Philosophical Transactions of the Royal Society,'' pp. 125–162.
* Ferraro, Vincenzo Consolato Antonio and Plumpton, Charles. ''An Introduction to Magneto-Fluid Mechanics'', 2nd ed.
* Galtier, Sebastien. [http://www.cambridge.org/fr/academic/subjects/physics/plasma-physics-and-fusion-physics/introduction-modern-magnetohydrodynamics?format=HB#FKTJat1pcSURPkL4.97 "Introduction to Modern Magnetohydrodynamics"], Cambridge University Press, Cambridge, England, 2016. 288 p. {{ISBN|9781107158658}}
*{{cite journal | last1 = Havarneanu | first1 = T. | last2 = Popa | first2 = C. | last3 = Sritharan | first3 = S. S. | year = 2006 | title = Exact Internal Controllability for Magneto-Hydrodynamic Equations in Multi-connected Domains | journal = Advances in Differential Equations | volume = 11 | issue = 8| pages = 893–929 }}
* Haverkort, J.W. (2009) ''Magnetohydrodynamics'' short introduction for fluid dynamicists, [http://jwhaverkort.net23.net/documents/MHD.pdf Magnetohydrodynamics]
* Hughes, William F. and Young, Frederick J. (1966) ''The Electromagnetodynamics of Fluids'' John Wiley, New York, {{OCLC|440919050}}
* {{cite journal | last1 = Hurricane | first1 = O. A. | last2 = Fong | first2 = B. H. | last3 = Cowley | first3 = S. C. | year = 1997 | title = Nonlinear magnetohydrodynamic detonation: Part I | doi = 10.1063/1.872252 | journal = Physics of Plasmas | volume = 4 | issue = 10| pages = 3565–3580 |bibcode = 1997PhPl....4.3565H }}
* {{cite journal | last1 = Jordan | first1 = R. | year = 1995 | title = A statistical equilibrium model of coherent structures in magnetohydrodynamics | url = http://www.iop.org/EJ/abstract/-search=5216637.3/0951-7715/8/4/007 | archive-url = https://archive.today/20130113022823/http://www.iop.org/EJ/abstract/-search=5216637.3/0951-7715/8/4/007 | url-status = dead | archive-date = 2013-01-13 | journal = Nonlinearity | volume = 8 | issue = 4 | pages = 585–613 | doi = 10.1088/0951-7715/8/4/007 | bibcode = 1995Nonli...8..585J }}
* {{cite journal | last1 = Kerrebrock | first1 = J. L. | year = 1965 | title = Magnetohydrodynamic Generators with Nonequilibrium Ionization | journal = AIAA Journal | volume = 3 | issue = 4| pages = 591–601 | doi = 10.2514/3.2934 |bibcode = 1965AIAAJ...3..591. }}
* Kulikovskiy, Andreĭ G. and Lyubimov, Grigoriĭ A. (1965)''Magnetohydrodynamics''. Addison-Wesley, Reading, Massachusetts, {{OCLC|498979430}}
* Lorrain, Paul ; Lorrain, François and Houle, Stéphane (2006) ''Magneto-fluid dynamics: fundamentals and case studies of natural phenomena'' Springer, New York, {{ISBN|0-387-33542-0}}
* Pai, Shih-I (1962) ''Magnetogasdynamics and Plasma Dynamics'' Springer-Verlag, Vienna, {{ISBN|0-387-80608-3}}
* {{cite journal | last1 = Popa | first1 = C. | last2 = Sritharan | first2 = S. S. | year = 2003 | title = Fluid-magnetic splitting methods for magneto-hydrodynamics | journal = Mathematical Methods and Models in Applied Sciences | volume = 13 | issue = 6| pages = 893–917 | doi=10.1142/s0218202503002763}}
* Roberts, Paul H. (1967) ''An Introduction to Magnetohydrodynamics'' Longmans Green, London, {{OCLC|489632043}}
* Rosa, Richard J. (1987) ''Magnetohydrodynamic Energy Conversion'' (2nd edition) Hemisphere Publishing, Washington, D.C., {{ISBN|0-89116-690-4}}
* Sritharan, S. S. and Sundar, P. (1999) "The stochastic magneto-hydrodynamic system" ''Infinite Dimensional Analysis, Quantum Probability and Related Topics'' (e-journal) 2(2): pp. 241–265.
* Stern, David P. [http://pwg.gsfc.nasa.gov/earthmag/sunspots.htm "The Sun's Magnetic Cycle"] ''In'' Stern, David P. ''The Great Magnet, the Earth'' [[NASA|United States National Aeronautics and Space Administration]]
* Sutton, George W., and Sherman, Arthur (1965) ''Engineering Magnetohydrodynamics'', McGraw-Hill Book Company, New York, {{OCLC|537669}}
* {{Cite journal|arxiv=hep-th/9503005|last1=Rahimitabar|first1=M. R|title=Turbulent Two Dimensional Magnetohydrodynamics and Conformal Field Theory|journal=Annals of Physics|volume=246|issue=2|pages=446–458|last2=Rouhani|first2=S|year=1996|doi=10.1006/aphy.1996.0033|bibcode=1996AnPhy.246..446R|s2cid=21720348}}
* Van Wie, D. M. (2005) [https://web.archive.org/web/20110722013255/http://ftp.rta.nato.int/public//PubFullText/RTO/EN/RTO-EN-AVT-116///EN-AVT-116-15.pdf Future Technologies – Application of Plasma Devices for Vehicle Systems], The Johns Hopkins University, Applied Physics Laboratory – Laurel, Maryland, USA – [[NATO]] Document
* {{cite journal | last1 = West | first1 = Jonathan | display-authors = etal | year = 2002 | title = Application of magnetohydrodynamic actuation to continuous flow chemistry | journal = Lab on a Chip | volume = 2 | issue = 4| pages = 224–230 | doi=10.1039/b206756k| pmid = 15100815 }}
* [http://www.bookrags.com/research/magnetohydrodynamics-mee-02/ "Magnetohydrodynamics"] ''In'' Zumerchik, John (editor) (2001) ''Macmillan Encyclopedia of Energy'' Macmillan Reference USA, New York, {{ISBN|0-02-865895-7}}
{{ref end}}

{{fusion power}}
{{Authority control}}

[[Category:Magnetohydrodynamics| ]]
[[Category:Plasma physics]]

Parton distribution functions

2021-05-10T08:06:40Z

Bleicher:

{{short description|Model in particle physics}}
In [[particle physics]], the '''parton model''' is a model of [[hadron]]s, such as [[proton]]s and [[neutron]]s, proposed by [[Richard Feynman]]. It is useful for interpreting the cascades of radiation (a '''parton shower''') produced from [[QCD]] processes and interactions in high-energy particle collisions.

==Model==
[[File:Parton scattering.PNG|thumb|250px| The scattering particle only sees the valence partons. At higher energies, the scattering particles also detects the sea partons.]]
Parton showers are simulated extensively in Monte Carlo [[event generator]]s, in order to calibrate and interpret (and thus understand) processes in collider experiments.<ref>Davison E. Soper, [https://docs.google.com/viewer?a=v&q=cache:JrKIMbwEl4YJ:www.phys.psu.edu/~cteq/schools/summer09/talks/Soper1.pdf+&hl=en&gl=uk&pid=bl&srcid=ADGEESiyBbiaJiSgY045z3NIZYG5aDzrh5SpSCRS0cxdxkDawW8zsXyhDZqLN2ReCKPfHC2Eqna6jlf-cWAtG5e9LkAxHfcn9tx-pcuPxL2fukN_mRnwk2JeF43jTM1WxZjWVqHgZTXn&sig=AHIEtbQo8ZGoenn0I44T3p4HzIMDkDrsIg The physics of parton showers]. Accessed 17 Nov 2013.</ref> As such, the name is also used to refer to algorithms that approximate or simulate the process.

===Motivation===
The parton model was proposed by [[Richard Feynman]] in 1969 as a way to analyze high-energy hadron collisions.<ref name="feynman">
{{cite conference
|last=Feynman |first=R. P.
|year=1969
|title=The Behavior of Hadron Collisions at Extreme Energies
|book-title=High Energy Collisions: Third International Conference at Stony Brook, N.Y.
|pages=237–249
|publisher=[[Gordon & Breach]]
|isbn=978-0-677-13950-0
}}</ref> Any hadron (for example, a [[proton]]) can be considered as a composition of a number of point-like constituents, termed "partons". The parton model was immediately applied to [[electron]]-[[proton]] [[deep inelastic scattering]] by [[James Bjorken|Bjorken]] and [[Emmanuel Anthony Paschos|Paschos]].<ref name="bjorken">
{{cite journal
|title=Inelastic Electron-Proton and γ-Proton Scattering and the Structure of the Nucleon
|year=1969
|last1=Bjorken |first1=J.
|last2=Paschos |first2=E.
|journal=[[Physical Review]]
|volume=185 |issue=5 |pages=1975–1982
|bibcode=1969PhRv..185.1975B
|doi=10.1103/PhysRev.185.1975
}}</ref>

===Component particles===

A hadron is composed of a number of point-like constituents, termed "partons". Later, with the experimental observation of [[Bjorken scaling]], the validation of the [[quark model]], and the confirmation of [[asymptotic freedom]] in [[quantum chromodynamics]], partons were matched to [[quark]]s and [[gluon]]s. The parton model remains a justifiable approximation at high energies, and others have extended the theory over the years.{{who|date=August 2018}}

Just as accelerated electric charges emit QED radiation (photons), the accelerated coloured partons will emit QCD radiation in the form of gluons. Unlike the uncharged photons, the gluons themselves carry colour charges and can therefore emit further radiation, leading to parton showers.<ref>[[Bryan Webber]] (2011). [http://www.scholarpedia.org/article/Parton_shower_Monte_Carlo_event_generators Parton shower Monte Carlo event generators.] Scholarpedia, 6(12):10662., revision #128236.
* {{webarchive|url=https://web.archive.org/web/20130402235841/http://www.scholarpedia.org/article/Parton_shower_Monte_Carlo_event_generators |date=2013-04-02 }}</ref><ref>*[http://indico.cern.ch/getFile.py/access?contribId=0&resId=0&materialId=slides&confId=49675 Parton Shower Monte Carlo Event Generators.] Mike Seymour, MC4LHC EU Networks’ Training Event
May 4th – 8th 2009.</ref><ref>*[http://www.stfc.ac.uk/PPD/resources/pdf/Krauss_08_Pheno_5.pdf Phenomenology at collider experiments. Part 5: MC generators] {{webarchive|url=https://web.archive.org/web/20120703153443/http://www.stfc.ac.uk/PPD/resources/pdf/Krauss_08_Pheno_5.pdf |date=2012-07-03 }}, Frank Krauss. HEP Summer School 31.8.-12.9.2008, RAL.</ref>

===Reference frame===

{{see also|DGLAP}}

The [[hadron]] is defined in a [[frame of reference|reference frame]] where it has infinite momentum—a valid approximation at high energies. Thus, parton motion is slowed by [[time dilation]], and the hadron charge distribution is [[Length contraction|Lorentz-contracted]], so incoming particles will be scattered "instantaneously and incoherently".{{cn|date=August 2018}}

Partons are defined with respect to a physical scale (as probed by the inverse of the momentum transfer).{{clarify|date=August 2018}} For instance, a quark parton at one length scale can turn out to be a superposition of a quark parton state with a quark parton and a gluon parton state together with other states with more partons at a smaller length scale. Similarly, a gluon parton at one scale can resolve into a superposition of a gluon parton state, a gluon parton and quark-antiquark partons state and other multiparton states. Because of this, the number of partons in a hadron actually goes up with momentum transfer.<ref>{{cite journal | author = G. Altarelli and G. Parisi | title = Asymptotic Freedom in Parton Language | journal = Nuclear Physics | volume = B126 | issue = 2 | pages = 298–318 | year = 1977 | doi= 10.1016/0550-3213(77)90384-4| bibcode = 1977NuPhB.126..298A }}</ref> At low energies (i.e. large length scales), a baryon contains three valence partons (quarks) and a meson contains two valence partons (a quark and an antiquark parton). At higher energies, however, observations show ''sea partons'' (nonvalence partons) in addition to valence partons.<ref name="DrellYan">
{{cite journal
|last1=Drell |first1=S.D.
|last2=Yan |first2=T.-M.
|s2cid=16827178
|year=1970
|title=Massive Lepton-Pair Production in Hadron-Hadron Collisions at High Energies
|journal=[[Physical Review Letters]]
|volume=25 |issue=5 |pages=316–320
|bibcode=1970PhRvL..25..316D
|doi=10.1103/PhysRevLett.25.316
}}
::And ''erratum'' in {{cite journal
|last1=Drell |first1=S. D.
|last2=Yan |first2=T.-M.
|year=1970
|title=none
|journal=[[Physical Review Letters]]
|volume=25 |issue=13 |page=902
|bibcode=1970PhRvL..25..902D
|doi=10.1103/PhysRevLett.25.902.2
|osti=1444835
}}</ref>

==History==

The parton model was proposed by [[Richard Feynman]] in 1969, used originally for analysis of high-energy collisions.<ref name="feynman" />
It was applied to [[electron]]/[[proton]] [[deep inelastic scattering]] by [[James Bjorken|Bjorken]] and Paschos.<ref name="bjorken" />
Later, with the experimental observation of [[Bjorken scaling]], the validation of the [[quark model]], and the confirmation of [[asymptotic freedom]] in [[quantum chromodynamics]], partons were matched to quarks and gluons. The parton model remains a justifiable approximation at high energies, and others have extended the theory over the years{{how|date=December 2016}}.

It was recognized{{when|date=December 2016}} that partons describe the same objects now more commonly referred to as [[quark]]s and [[gluon]]s. A more detailed presentation of the properties and physical theories pertaining indirectly to partons can be found under [[quark]]s.

== Parton distribution functions ==

[[File:CTEQ6 parton distribution functions.png|thumb|250px|The ''CTEQ6'' parton distribution functions in the {{overline|MS}} renormalization scheme and ''Q'' = 2 GeV for gluons (red), up (green), down (blue), and strange (violet) quarks. Plotted is the product of longitudinal momentum fraction ''x'' and the distribution functions ''f'' versus ''x''.]]

A parton distribution function (PDF) within so called ''collinear factorization'' is defined as the [[probability amplitude|probability density]] for finding a particle with a certain longitudinal momentum fraction ''x'' at resolution scale ''Q''2. Because of the inherent [[non-perturbative]] nature of partons which cannot be observed as free particles, parton densities cannot be calculated using perturbative QCD. Within QCD one can, however, study variation of parton density with resolution scale provided by external probe. Such a scale is for instance provided by a [[virtual particle|virtual photon]] with virtuality ''Q''2 or by a [[Jet (particle physics)|jet]]. The scale can be calculated from the energy and the momentum of the virtual photon or jet; the larger the momentum and energy, the smaller the resolution scale—this is a consequence of Heisenberg's [[uncertainty principle]]. The variation of parton density with resolution scale has been found to agree well with experiment;<ref>'''PDG''': Aschenauer, Thorne, and Yoshida, (2019). "Structure Functions", [http://pdg.lbl.gov/2019/reviews/rpp2019-rev-structure-functions.pdf online]. </ref> this is an important test of QCD.

Parton distribution functions are obtained by fitting observables to experimental data; they cannot be calculated using perturbative QCD. Recently, it has been found that they can be calculated directly in [[lattice QCD]] using large-momentum effective field theory.<ref>{{Cite journal|last=Ji|first=Xiangdong|date=2013-06-26|title=Parton Physics on a Euclidean Lattice|journal=Physical Review Letters|volume=110|issue=26|pages=262002|doi=10.1103/PhysRevLett.110.262002|pmid=23848864|arxiv=1305.1539|bibcode=2013PhRvL.110z2002J|s2cid=27248761}}</ref><ref>{{Cite journal|last=Ji|first=Xiangdong|date=2014-05-07|title=Parton physics from large-momentum effective field theory|journal=Science China Physics, Mechanics & Astronomy|language=en|volume=57|issue=7|pages=1407–1412|doi=10.1007/s11433-014-5492-3|issn=1674-7348|arxiv=1404.6680|bibcode=2014SCPMA..57.1407J|s2cid=119208297}}</ref>

Experimentally determined parton distribution functions are available from various groups worldwide. The major unpolarized data sets are:
* [http://mail.ihep.ru/~alekhin/pdfs.html ''ABM''] by S. Alekhin, J. Bluemlein, S. Moch
*''[http://www.cteq.org/#PDFs CTEQ]'', from the CTEQ Collaboration
* [http://doom.physik.tu-dortmund.de/pdfserver/index.html ''GRV/GJR''], from M. Glück, P. Jimenez-Delgado, E. Reya, and A. Vogt
* [https://www.desy.de/h1zeus/combined_results/index.php?do=proton_structure ''HERA''] PDFs, by H1 and ZEUS collaborations from the Deutsches Elektronen-Synchrotron center (DESY) in Germany
* [http://www.hep.ucl.ac.uk/mmht/ MSHT/MRST/MSTW/MMHT], from [[Alan Martin (physicist)|A. D. Martin]], R. G. Roberts, W. J. Stirling, R. S. Thorne, and collaborators
* [http://nnpdf.hepforge.org/ ''NNPDF''], from the NNPDF Collaboration

The [http://lhapdf.hepforge.org/ ''LHAPDF''] <ref>
{{cite arxiv
|last1=Whalley |first1=M. R.
|last2=Bourilkov |first2=D
|last3=Group |first3=R. C.
|year=2005
|title=The Les Houches accord PDFs (LHAPDF) and LHAGLUE
|eprint=hep-ph/0508110
}}</ref> library provides a unified and easy-to-use [[Fortran]]/[[C++]] interface to all major PDF sets.

''Generalized parton distributions'' (GPDs) are a more recent approach to better understand [[hadron]] structure by representing the parton distributions as functions of more variables, such as the transverse momentum and [[Spin (physics)|spin]] of the parton.<ref>{{cite journal | journal = Phys. Rev. D| volume = 29 | issue = 3 | pages= 567–569 | year= 1984 | title= Spin structure of the nucleon | doi= 10.1103/PhysRevD.29.567 | author1 = DJE Callaway | author2 = SD Ellis | s2cid = 15798912 | bibcode = 1984PhRvD..29..567C}}</ref> They can be used to study the spin structure of the proton, in particular, the Ji sum rule relates the integral of GPDs to angular momentum carried by quarks and gluons.<ref>{{Cite journal|last=Ji|first=Xiangdong|date=1997-01-27|title=Gauge-Invariant Decomposition of Nucleon Spin|journal=Physical Review Letters|volume=78|issue=4|pages=610–613|doi=10.1103/PhysRevLett.78.610|arxiv=hep-ph/9603249|bibcode=1997PhRvL..78..610J|s2cid=15573151}}</ref> Early names included "non-forward", "non-diagonal" or "skewed" parton distributions. They are accessed through a new class of exclusive processes for which all particles are detected in the final state, such as the deeply virtual Compton scattering.<ref>{{Cite journal|last=Ji|first=Xiangdong|date=1997-06-01|title=Deeply virtual Compton scattering|journal=Physical Review D|volume=55|issue=11|pages=7114–7125|doi=10.1103/PhysRevD.55.7114|bibcode=1997PhRvD..55.7114J|arxiv=hep-ph/9609381|s2cid=1975588}}</ref> Ordinary parton distribution functions are recovered by setting to zero (forward limit) the extra variables in the generalized parton distributions. Other rules show that the [[electric form factor]], the [[magnetic form factor]], or even the form factors associated to the energy-momentum tensor are also included in the GPDs. A full 3-dimensional image of partons inside hadrons can also be obtained from GPDs.<ref>
{{cite journal
|last1=Belitsky |first1=A. V.
|last2=Radyushkin |first2=A. V.
|year=2005
|title=Unraveling hadron structure with generalized parton distributions
|journal=[[Physics Reports]]
|volume=418 |issue=1–6
|pages=1–387
|arxiv=hep-ph/0504030
|bibcode=2005PhR...418....1B
|doi=10.1016/j.physrep.2005.06.002
|s2cid=119469719
}}</ref>

==Simulation==
Parton showers simulations are of use in [[computational particle physics]] either in [[automatic calculation of particle interaction or decay]] or [[event generators]], and are particularly important in LHC phenomenology, where they are usually explored using Monte Carlo simulation. The scale at which partons are given to hadronization is fixed by the Shower Monte Carlo program. Common choices of Shower Monte Carlo are [[PYTHIA]] and HERWIG.<ref>Johan Alwall, [https://web.archive.org/web/20140114023830/http://phys.cts.ntu.edu.tw/workshop/2012/1010525LHC/PDF/LEC2.pdf Complete simulation of collider events], pg 33. NTU MadGraph school, May 25–27, 2012.</ref><ref>M Moretti.[https://docs.google.com/viewer?a=v&q=cache:LVQ_MpmPgdsJ:www1b.physik.rwth-aachen.de/~kolleg/fileadmin/data/gk/de/veranstaltungen/moretti.pdf+&hl=en&gl=uk&pid=bl&srcid=ADGEESiJijmZ58A4afChs_OhQWmJn5dPP-MGV0RRl9gv33DIq0yxt7E7QkOO37FtiTnIUnzNMZd9D_xyjlFTnlYmyZwFm907Qwm6L7khQFRG5MUu_gLH5xQl4nNk_YSoLXPqdZPP-1pP&sig=AHIEtbR6MMStdsy0OMF7ZihtTJBHTJs0SA Understunding events at the LHC: Parton Showers and Matrix Element tools for physics simulation at the hadronic colliders], p. 19. 28/11/2006.</ref>

==See also==
* [[Hadronization]]
* [[Jet (particle physics)]]
* [[Particle shower]]

== References ==
{{reflist|25em}}

This article contains material from Scholarpedia.

==Further reading==
*{{cite journal
|last1=Glück |first1=M.
|last2=Reya |first2=E.
|last3=Vogt |first3=A.
|year=1998
|journal=[[European Physical Journal C]]
|title=Dynamical Parton Distributions Revisited
|volume=5 |issue=3 |pages=461–470
|arxiv=hep-ph/9806404
|bibcode=1998EPJC....5..461G
|doi=10.1007/s100529800978
|s2cid=119842774
}}
*{{cite web
|last=Hoodbhoy | first =P. A.
|year=2006
|title=Generalized Parton Distributions
|url=http://www.ncp.edu.pk/docs/12th_rgdocs/Pervez-Hoodbhoy.pdf
|publisher=[[National Center for Physics]] and [[Quaid-e-Azam University]]
|access-date=2011-04-06
}}
*{{cite journal
|last=Ji
|first=X.
|year=2004
|title=Generalized Parton Distributions
|journal=[[Annual Review of Nuclear and Particle Science]]
|volume=54
|pages=413–450
|bibcode=2004ARNPS..54..413J
|doi=10.1146/annurev.nucl.54.070103.181302 | doi-access=free
|arxiv=hep-ph/9807358}}
*{{cite journal
|last1=Kretzer |first1=S.
|last2=Lai |first2=H.
|last3=Olness |first3=F.
|last4=Tung |first4=W.
|year=2004
|title=CTEQ6 Parton Distributions with Heavy Quark Mass Effects
|journal=[[Physical Review D]]
|volume=69 |issue=11 |page=114005
|arxiv=hep-ph/0307022
|bibcode=2004PhRvD..69k4005K
|doi=10.1103/PhysRevD.69.114005
|s2cid=119379329
}}
*{{cite journal
|last1=Martin |first1=A. D.
|last2=Roberts |first2=R. G.
|last3=Stirling |first3=W. J.
|last4=Thorne |first4=R. S.
|year=2005
|title=Parton distributions incorporating QED contributions
|journal=[[European Physical Journal C]]
|volume=39 |issue=2 |pages=155–161
|arxiv=hep-ph/0411040
|bibcode=2005EPJC...39..155M
|doi=10.1140/epjc/s2004-02088-7
|s2cid=14743824
}}

==External links==
{{toomanylinks|date=July 2020}}
* [http://durpdg.dur.ac.uk/hepdata/pdf.html Parton distribution functions] – from [http://durpdg.dur.ac.uk/hepdata/pdf.html HEPDATA: The Durham HEP Databases]
* [http://hep.pa.msu.edu/cteq/public/cteq6.html CTEQ6 parton distribution functions]
* [https://docs.google.com/viewer?a=v&q=cache:AWqGr4WO6twJ:www.ippp.dur.ac.uk/~mschoenherr/talks/20121121_CERN.pdf+&hl=en&gl=uk&pid=bl&srcid=ADGEESjB8DagVzodcwxJVD0XE_5CrTkt4kZPk_zwnt7lOsxDMy9sPoSczl6zt1r9zxQwso5-piTWESlw_dOxHYM5QTLBZAD7zFJzxUzu1dfSe0aDSC8Ld0hq4tU2TqVW2SwxdJ4kKox6&sig=AHIEtbQq3w5dzV3Dy7JWNE5xDhThkCKd9Q Electroweak radiation in parton showers]
* http://home.fnal.gov/~mrenna/lutp0613man2/node22.html
* [https://docs.google.com/viewer?a=v&q=cache:xIc1VR3m1i8J:www.hep.phy.cam.ac.uk/theory/webber/MCnet/MClecture2.pdf+&hl=en&gl=uk&pid=bl&srcid=ADGEESj8yjHM5i6G2jXizVSX2umPLLdGX3BtYZQJeImPc3WqmqfSFjbJUIcvvGKdyOvZNbV7b0hoCY3G9tkvRKNR3ggPy0l9B49ABofLpk1Y75jN_Dbats1kitAHLl2WYwlShv5hGm0z&sig=AHIEtbQj3NfqnF0OYou0K0Ba7nnAFy-Dtw Event Generator Physics] (http://www.hep.phy.cam.ac.uk/theory/webber/MCnet/MClecture2.pdf)
* [https://docs.google.com/viewer?a=v&q=cache:yVb7erYPCX8J:home.fnal.gov/~skands/slides/uwm06.ppt+&hl=en&gl=uk&pid=bl&srcid=ADGEESjSLpMkR3NcPgXZlCU4R9dOBlKIklZFuLTmnL6ciJOfaCVDuH65nHA9jb0jaOrZwbm5uuxCImkFT7c6kkXiykSkrGBx7xm-KSG2Mf2bWbD3OV34KZyBYW3rDZb0kfkM9b0zZu-m&sig=AHIEtbRYp2mMptgxQRkv01GVSwCo-1920A Frontiers of Chromodynamics]
* http://www.phys.ethz.ch/~pheno/QCDcourse/
* http://www.physnet.uni-hamburg.de/services/fachinfo/___Volltexte/Sebastian___Schmidt/Sebastian___Schmidt.pdf{{dead link|date=May 2017 |bot=InternetArchiveBot |fix-attempted=yes }}
* http://hep.ps.uci.edu/~wclhc07/LOShowers.pdf
* http://www.kceta.kit.edu/grk1694/img/2013_10_01_Hangst.pdf
* https://books.google.com/books?id=EAAHQ9XsEyQC&pg=PA47&lpg=PA47&dq=parton+shower#v=onepage&q=parton%20shower&f=false page 47
* [https://www.google.com/?gws_rd=cr&ei=5Ge8Us_1D5SZ0QXY9YAI#q=parton+shower Google search for 'parton shower']
* http://www.nikhef.nl/pub/theory/masters-theses/bart_verouden.pdf
* http://www.pa.msu.edu/~huston/tev4lhc/skands.pdf
* http://d-nb.info/1008230227/34
* http://www.rhul.ac.uk/physics/documents/pdf/events/particlephysicsseminars/10-11/ppseminprichardson250511.pdf
* http://dspace.mit.edu/handle/1721.1/62649
* http://www2.ph.ed.ac.uk/particle/Theory/seminars_12/andersen.pdf
* http://rchep.pku.edu.cn/filespath/files/20131202113805.pdf{{dead link|date=May 2017 |bot=InternetArchiveBot |fix-attempted=yes }}
* http://particle-theory.group.shef.ac.uk/si2008/participant_talks/Latunde-Dada.pdf
* http://www.science.uva.nl/onderwijs/thesis/centraal/files/f935119543.pdf
* http://www.science.uva.nl/onderwijs/thesis/centraal/files/f1299590863.pdf

{{Richard Feynman}}

{{DEFAULTSORT:Parton (Particle Physics)}}
[[Category:Particle physics]]
[[Category:Quantum chromodynamics]]
[[Category:Richard Feynman]]

Transport coefficients

2021-05-10T08:04:46Z

Bleicher:

A '''transport coefficient''' <math>\gamma</math> measures how rapidly a perturbed system returns to equilibrium.

The transport coefficients occur in [[transport phenomenon]] with transport laws
<math> {\mathbf{J} {_k}} \, = \, \gamma_k \, \mathbf{X} {_k}</math>
where:
: <math> {\mathbf{J}{_k}} </math> is a flux of the property <math> k </math>
: the transport coefficient <math> \gamma _k </math> of this property <math> k </math>
: <math> {\mathbf{X}{_k}} </math>, the gradient force which acts on the property <math> k </math>.

Transport coefficients can be expressed via a [[Green–Kubo relations|Green–Kubo relation]]:

:<math>\gamma= \int_0^\infty \langle \dot{A}(t) \dot{A}(0) \rangle \, dt,</math>

where <math>A</math> is an observable occurring in a perturbed Hamiltonian, <math>\langle \cdot \rangle</math> is an ensemble average and the dot above the ''A'' denotes the time derivative.<ref>Water in Biology, Chemistry, and Physics: Experimental Overviews and Computational Methodologies, G. Wilse Robinson, {{ISBN|9789810224516}}, p. 80, [https://books.google.com/books?id=FHEh1C4GuZcC&lpg=PA80&dq=ficks%20law%20%22transport%20coefficient%22&pg=PA80#v=onepage&q=ficks%20law%20%22transport%20coefficient%22&f=false Google Books]</ref>
For times <math>t</math> that are greater than the correlation time of the fluctuations of the observable the transport coefficient obeys a generalized [[Einstein relation (kinetic theory)|Einstein relation]]:

:<math>2t\gamma=\langle |A(t)-A(0)|^2 \rangle.</math>

In general a transport coefficient is a tensor.

Transport coefficients, like [https://en.wikipedia.org/wiki/Viscosity shear viscosity] and [https://en.wikipedia.org/wiki/Volume_viscosity bulk viscosity], are important for the description of fireball expansion in heavy-ion collisions.

== Examples ==
* [[Diffusion constant]], relates the flux of particles with the negative gradient of the concentration (see [[Fick's laws of diffusion]])
* [[Thermal conductivity]] (see [[Fourier's law]])
* [[Mass transfer coefficient|Mass transport coefficient]]
* [[Shear viscosity]] <math>\eta = \frac{1}{k_BT V} \int_0^\infty dt \, \langle \sigma_{xy}(0) \sigma_{xy} (t) \rangle</math>, where <math>\sigma</math> is the [[viscous stress tensor]] (see [[Newtonian fluid]])
* [[Electrical conductivity]]

== See also ==
*[[Linear response theory]]
*[[Onsager reciprocal relations]]

== References ==
<references />

[[Category:Thermodynamics]]
[[Category:Statistical mechanics]]

Statistical model

2021-05-10T08:03:51Z

Bleicher:

{{short description|Temperature at which the partition function of a statistical-mechanical system diverges}}
The '''Hagedorn temperature,''' ''T''H, is the temperature in [[theoretical physics]] where [[hadron]]ic matter (i.e. ordinary matter) is no longer stable, and must either "evaporate" or convert into [[quark matter]]; as such, it can be thought of as the "boiling point" of hadronic matter. The Hagedorn temperature exists because the amount of energy available is high enough that matter particle ([[quark]]–[[antiquark]]) pairs can be spontaneously pulled from vacuum. Thus, naively considered, a system at Hagedorn temperature can accommodate as much energy as one can put in, because the formed quarks provide new degrees of freedom, and thus the Hagedorn temperature would be an impassable [[absolute hot]]. However, if this phase is viewed as quarks instead, it becomes apparent that the matter has transformed into [[quark matter]], which can be further heated.

The Hagedorn temperature, ''T''H, is about {{val|150|u=MeV}} or about {{val|1.7|e=12|u=K}},<ref>{{Citation|last=Gaździcki|first=Marek|title=Hagedorn's Hadron Mass Spectrum and the Onset of Deconfinement|date=2016|work=Melting Hadrons, Boiling Quarks – From Hagedorn Temperature to Ultra-Relativistic Heavy-Ion Collisions at CERN|pages=87–92|editor-last=Rafelski|editor-first=Johann|publisher=Springer International Publishing|language=en|doi=10.1007/978-3-319-17545-4_11|isbn=978-3-319-17544-7|last2=Gorenstein|first2=Mark I.|editor-link=Johann Rafelski|doi-access=free}}</ref> the same as the mass–energy of the lightest hadrons, the [[pion]].<ref name="iop">{{cite web |last=Cartlidge |first=Edwin |title=Quarks break free at two trillion degrees |url=http://physicsworld.com/cws/article/news/2011/jun/23/quarks-break-free-at-two-trillion-degrees |work=[[Physics World]] |access-date=27 January 2014 |date=23 June 2011 }}</ref> Matter at Hagedorn temperature or above will spew out fireballs of new particles, which can again produce new fireballs, and the ejected particles can then be detected by particle detectors. This [[QCD matter|quark matter]] has been detected in heavy-ion collisions at [[Super Proton Synchrotron|SPS]] and [[Large Hadron Collider|LHC]] in [[CERN]] (France and Switzerland) and at [[Relativistic Heavy Ion Collider|RHIC]] in [[Brookhaven National Laboratory]] (USA).

In string theory, a separate Hagedorn temperature can be defined for strings rather than hadrons. This temperature is extremely high (1030 K) and thus of mainly theoretical interest.<ref name="NPB">{{cite journal |last=Atick |first=Joseph J. |last2=Witten |first2=Edward |year=1988 |title=The Hagedorn transition and the number of degrees of freedom of string theory |journal=[[Nuclear Physics B]] |volume=310 |issue=2 |pages=291 |bibcode=1988NuPhB.310..291A |doi=10.1016/0550-3213(88)90151-4 }}</ref>

== History ==
The Hagedorn temperature was discovered by German physicist [[Rolf Hagedorn]] in the 1960s while working at CERN. His work on the [[statistical bootstrap model]] of hadron production showed that because increases in energy in a system will cause new particles to be produced, an increase of collision energy will increase the entropy of the system rather than the temperature, and "the temperature becomes stuck at a limiting value".<ref name="tale" /><ref>{{Cite book|title=Melting Hadrons, Boiling Quarks – From Hagedorn Temperature to Ultra-Relativistic Heavy-Ion Collisions at CERN|date=2016|publisher=Springer International Publishing|isbn=978-3-319-17544-7|editor-last=Rafelski|editor-first=Johann|location=Cham|language=en|doi=10.1007/978-3-319-17545-4}}</ref>

== Technical explanation ==
Hagedorn temperature is the temperature above which the [[partition sum]] diverges in a system with exponential growth in the density of states.<ref name="tale">{{cite web |last=Ericson |first=Torleif |last2=Rafelski |first2=Johann |date=4 September 2003 |title=The tale of the Hagedorn temperature |url=http://cerncourier.com/cws/article/cern/28919 |work=[[CERN Courier]] |access-date=2016-12-09 }}</ref><ref>{{cite web |last=Tyson |first=Peter |date=December 2007 |title=Absolute Hot: Is There an Opposite to Absolute Zero? |url=https://www.pbs.org/wgbh/nova/zero/hot.html |work=[[Nova (American TV program)|NOVA]] |access-date=2008-12-21 }}</ref>

:<math>\lim_{T\rightarrow T_H^-} \operatorname{Tr}\left[e^{-\beta H}\right] = \infty</math>

Because of the divergence, people may come to the incorrect conclusion that it is impossible to have temperatures above the Hagedorn temperature, which would make it the [[absolute hot]] temperature, because it would require an infinite amount of [[energy]]. In equations:

:<math>\lim_{T\rightarrow T_H^-}E = \lim_{T\rightarrow T_H^-}\frac{\operatorname{Tr}\left[H e^{-\beta H}\right]}{\operatorname{Tr}\left[e^{-\beta H}\right]} = \infty</math>

This line of reasoning was well known to be false even to Hagedorn. The partition function for creation of hydrogen–antihydrogen pairs diverges even more rapidly, because it gets a finite contribution from energy levels that accumulate at the ionization energy. The states that cause the divergence are spatially big, since the electrons are very far from the protons. The divergence indicates that at a low temperature hydrogen–antihydrogen will not be produced, rather proton/antiproton and electron/antielectron. The Hagedorn temperature is only a maximum temperature in the physically unrealistic case of exponentially many species with energy E and finite size.

The concept of exponential growth in the number of states was originally proposed in the context of [[condensed matter physics]]. It was incorporated into high-energy physics in the early 1970s by [[Steven Frautschi]] and Hagedorn. In hadronic physics, the Hagedorn temperature is the deconfinement temperature.

== In string theory ==
In [[string theory]], it indicates a phase transition: the transition at which very long strings are copiously produced. It is controlled by the size of the string tension, which is smaller than the [[Planck energy|Planck scale]] by the some power of the coupling constant. By adjusting the tension to be small compared to the Planck scale, the Hagedorn transition can be much less than the [[Planck temperature]]. Traditional [[Grand Unified Theory|grand unified]] string models place this in the magnitude of {{val|e=30|u=[[kelvin]]}}, two orders of magnitude smaller than the Planck temperature. Such temperatures have not been reached in any experiment and are far beyond the reach of current, or even foreseeable technology.

==See also==
* [[Heat]]
* [[Absolute hot]]
* [[Thermodynamic temperature]]
* [[Non-extensive self-consistent thermodynamical theory]]

==References==
{{reflist}}

[[Category:Particle physics]]
[[Category:Nuclear physics]]
[[Category:Statistical mechanics]]
[[Category:String theory]]
[[Category:Quantum chromodynamics]]
[[Category:Quark matter]]

Phase diagram

2021-05-10T08:02:11Z

Bleicher:

{{short description|Theorized phases of matter whose degrees of freedom include quarks and gluons}}
'''Quark matter''' or '''QCD matter''' ([[quantum chromodynamics|quantum chromodynamic]]) refers to any of a number of hypothetical [[phase (matter)|phases]] of matter whose [[degrees of freedom (physics and chemistry)|degrees of freedom]] include [[quark]]s and [[gluon]]s, of which the prominent example is [[Quark–gluon plasma|quark-gluon plasma]].<ref>{{Cite book|last1=Letessier|first1=Jean|url=https://www.cambridge.org/core/product/identifier/9780511534997/type/book|title=Hadrons and Quark–Gluon Plasma|last2=Rafelski|first2=Johann|publisher=Cambridge University Press|year=2002|isbn=978-0-521-38536-7|edition=1|doi=10.1017/cbo9780511534997}}</ref> Several series of conferences in 2019, 2020, and 2021 are devoted to this topic.<ref>{{Cite web|title=Quark Matter 2021: The 29th International Conference on Ultrarelativistic Nucleus-Nucleus Collisions|url=https://indico.cern.ch/event/895086/|access-date=2020-06-26|website=Indico}}</ref><ref>{{Cite web|title=CPOD2020 - International Conference on Critical Point and Onset of Deconfinement|url=https://indico.cern.ch/event/851194/|access-date=2020-06-26|website=Indico}}</ref><ref>{{Cite web|title=Strangeness in Quark Matter 2019|url=https://indico.cern.ch/event/755366/|access-date=2020-06-26|website=Indico}}</ref>

Quarks are liberated into quark matter at extremely high temperatures and/or densities, and some of them are still only theoretical as they require conditions so extreme that they can not be produced in any laboratory, especially not at equilibrium conditions. Under these extreme conditions, the familiar structure of [[matter]], where the basic constituents are [[atomic nucleus|nuclei]] (consisting of [[nucleon]]s which are bound states of quarks) and electrons, is disrupted. In quark matter it is more appropriate to treat the quarks themselves as the basic degrees of freedom.

In the [[standard model]] of particle physics, the [[strong interaction|strong force]] is described by the theory of [[quantum chromodynamics|QCD]]. At ordinary temperatures or densities this force just [[color confinement|confines]] the quarks into composite particles ([[hadrons]]) of size around 10−15 m = 1 [[femtometer]] = 1 fm (corresponding to the QCD energy scale ΛQCD ≈ 200 [[MeV]]) and its effects are not noticeable at longer distances.

However, when the temperature reaches the QCD energy scale ([[Electronvolt|T]] of order 1012 [[kelvin]]s) or the density rises to the point where the average inter-quark separation is less than 1 fm (quark [[chemical potential]] μ around 400 MeV), the hadrons are melted into their constituent quarks, and the strong interaction becomes the dominant feature of the physics. Such phases are called quark matter or QCD matter.

The strength of the color force makes the properties of quark matter unlike gas or plasma, instead leading to a state of matter more reminiscent of a liquid. At high densities, quark matter is a [[Fermi liquid theory|Fermi liquid]], but is predicted to exhibit [[color superconductivity]] at high densities and temperatures below 1012 K.

{{unsolved|physics|QCD in the non-[[Perturbation theory (quantum mechanics)|perturbative]] regime: '''quark matter'''. The equations of QCD predict that a [[quark–gluon plasma|sea of quarks and gluons]] should be formed at high temperature and density. What are the properties of this [[phase of matter]]?}}

==Occurrence==

===Natural occurrence===

* According to the [[Big Bang]] theory, in the early universe at high temperatures when the universe was only a few tens of microseconds old, the phase of matter took the form of a hot phase of quark matter called the [[quark–gluon plasma]] (QGP).<ref>See ''"Hadrons and quark-gluon plasma"'' for example.</ref>
* [[Compact star]]s ([[neutron star]]s). A neutron star is much cooler than 1012 K, but gravitational collapse has compressed it to such high densities, that it is reasonable to surmise that quark matter may exist in the core.<ref>Shapiro and Teukolsky: ''Black Holes, White Dwarfs and Neutron Stars: The Physics of Compact Objects'', Wiley 2008</ref> Compact stars composed mostly or entirely of quark matter are called [[quark star]]s or [[Quark star#Strange stars|strange star]]s.
* QCD matter may exist within the [[Hypernova|collapsar]] of a [[gamma-ray burst]], where temperatures as high as 6.7 x 1013 K may be generated.

At this time no star with properties expected of these objects has been observed, although some evidence has been provided for quark matter in the cores of large neutron stars.<ref name=":0">{{Cite journal|last1=Annala|first1=Eemeli|last2=Gorda|first2=Tyler|last3=Kurkela|first3=Aleksi|last4=Nättilä|first4=Joonas|last5=Vuorinen|first5=Aleksi|date=2020-06-01|title=Evidence for quark-matter cores in massive neutron stars|url=https://www.nature.com/articles/s41567-020-0914-9|journal=Nature Physics|language=en|volume=16|issue=9|pages=907–910|doi=10.1038/s41567-020-0914-9|issn=1745-2481|doi-access=free}}</ref>
* [[Strangelet]]s. These are theoretically postulated (but as yet unobserved) lumps of [[strange matter]] comprising nearly equal amounts of up, down and strange quarks. Strangelets are supposed to be present in the galactic flux of high energy particles and should therefore theoretically be detectable in [[cosmic rays]] here on Earth, but no strangelet has been detected with certainty.<ref>{{cite book|chapter-url=http://inspirehep.net/record/1483757/references |title=Proceedings: A production scenario of Galactic strangelets and an estimation of their possible flux in solar neighborhood |volume=ICRC2015 |pages=504 |last=Biswas |first=Sayan |display-authors=etal |publisher=inSpire |year=2016 |access-date=11 October 2016|doi=10.22323/1.236.0504 |chapter=A production scenario of Galactic strangelets and an estimation of their possible flux in solar neighborhood}}</ref><ref>{{cite journal|title=Strangelet propagation and cosmic ray flux|author=Madsen, Jes|date=18 November 2004|journal=Phys. Rev. D|volume=71|issue=1|pages=014026|doi=10.1103/PhysRevD.71.014026|arxiv=astro-ph/0411538|bibcode=2005PhRvD..71a4026M|s2cid=119485839}}</ref>
* [[Cosmic ray]] impacts. Cosmic rays comprise a lot of different particles, including highly accelerated atomic nuclei, particularly that of [[iron]].

Laboratory experiments suggests that the inevitable interaction with heavy [[noble gas]] nuclei in the [[upper atmosphere]] would lead to quark–gluon plasma formation.
* Quark matter with [[baryon number]] over about 300 may be more stable than nuclear matter. This form of baryonic matter could possibly form a [[continent of stability]].<ref>{{cite journal |last1=Holdom |first1=Bob |last2=Ren |first2=Jing |last3=Zhang |first3=Chen |title=Quark Matter May Not Be Strange |journal=Physical Review Letters |date=31 May 2018 |volume=120 |issue=22 |pages=222001 |doi=10.1103/PhysRevLett.120.222001|pmid=29906186 |bibcode=2018PhRvL.120v2001H |arxiv=1707.06610 |s2cid=49216916 }}</ref>

===Laboratory experiments===
[[File:Alicelead3.jpg|thumb|Particle debris trajectories from one of the first lead-ion collisions with the LHC, as recorded by the [[ALICE: A Large Ion Collider Experiment|ALICE]] detector. The extremely brief appearance of quark matter in the point of collision is inferred from the statistics of the trajectories.]]
Even though quark-gluon plasma can only occur under quite extreme conditions of temperature and/or pressure, it is being actively studied at [[particle collider]]s, such as the Large Hadron Collider [[Large Hadron Collider|LHC]] at [[CERN]] and the Relativistic Heavy Ion Collider [[Relativistic Heavy Ion Collider|RHIC]] at [[Brookhaven National Laboratory]].

In these collisions, the plasma only occurs for a very short time before it spontaneously disintegrates. The plasma's physical characteristics are studied by detecting the debris emanating from the collision region with large particle detectors <ref>{{cite web|url=http://home.cern/about/experiments/alice|title=ALICE |publisher=CERN |access-date= 16 December 2015}}</ref><ref>See ''"Hunting the quark gluoan plasma"'' as an example of the research at RHIC.</ref>

[[Heavy-ion collisions]] at very high energies can produce small short-lived regions of space whose energy density is comparable to that of the [[Chronology of the universe#The quark epoch|20-micro-second-old universe]]. This has been achieved by colliding heavy nuclei such as [[lead]] nuclei at high speeds, and a first time claim of formation of [[quark–gluon plasma]] came from the [[Super Proton Synchrotron|SPS]] accelerator at [[CERN]] in February 2000.<ref>{{cite arXiv|eprint=nucl-th/0002042|last1=Heinz|first1=Ulrich|title=Evidence for a New State of Matter: An Assessment of the Results from the CERN Lead Beam Programme|last2=Jacob|first2=Maurice|year=2000}}</ref>

This work has been continued at more powerful accelerators, such as RHIC in the US, and as of 2010 at the European LHC at CERN located in the border area of Switzerland and France. There is good evidence that the quark–gluon plasma has also been produced at RHIC.<ref>{{cite arXiv|eprint=nucl-th/0508062|last1=Heinz|first1=Ulrich|title=Quark Matter 2005 – Theoretical Summary|last2=Jacob|first2=Maurice|year=2005}}</ref>

==Thermodynamics==

The context for understanding the thermodynamics of quark matter is the [[standard model]] of particle physics, which contains six different [[flavor (particle physics)|flavors]] of quarks, as well as [[lepton]]s like [[electron]]s and [[neutrino]]s. These interact via the [[strong interaction]], [[electromagnetism]], and also the [[weak interaction]] which allows one flavor of quark to turn into another. Electromagnetic interactions occur between particles that carry electrical charge; strong interactions occur between particles that carry [[color charge]].

The correct thermodynamic treatment of quark matter depends on the physical context. For large quantities that exist for long periods of time (the "thermodynamic limit"), we must take into account the fact that the only conserved charges in the standard model are quark number (equivalent to [[baryon]] number), electric charge, the eight color charges, and lepton number. Each of these can have an associated chemical potential. However, large volumes of matter must be electrically and color-neutral, which determines the electric and color charge chemical potentials. This leaves a three-dimensional [[phase space]], parameterized by quark chemical potential, lepton chemical potential, and temperature.

In compact stars quark matter would occupy cubic kilometers and exist for millions of years, so the thermodynamic limit is appropriate. However, the neutrinos escape, violating lepton number, so the phase space for quark matter in compact stars only has two dimensions, temperature (''T'') and quark number chemical potential μ. A [[strangelet]] is not in the thermodynamic limit of large volume, so it is like an exotic nucleus: it may carry electric charge.

A heavy-ion collision is in neither the thermodynamic limit of large volumes nor long times. Putting aside questions of whether it is sufficiently equilibrated for thermodynamics to be applicable, there is certainly not enough time for weak interactions to occur, so flavor is conserved, and there are independent chemical potentials for all six quark flavors. The initial conditions (the [[impact parameter]] of the collision, the number of up and down quarks in the colliding nuclei, and the fact that they contain no quarks of other flavors) determine the chemical potentials. (Reference for this section:,<ref name='RMP'/><ref name='Rischke'/>).

==Phase diagram==
[[File:QCD phase diagram.png|thumb|300px|right|Conjectured form of the phase diagram of QCD matter, with temperature on the vertical axis and quark [[chemical potential]] on the horizontal axis, both in mega-[[electron volt]]s.<ref name='RMP'/>]]

The [[phase diagram]] of quark matter is not well known, either experimentally or theoretically. A commonly conjectured form of the
phase diagram is shown in the figure to the right.<ref name='RMP'>{{cite journal|author1=Alford, Mark G.|author2=Schmitt, Andreas|author3=Rajagopal, Krishna|author4=Schäfer, Thomas|title=Color superconductivity in dense quark matter|arxiv=0709.4635 |journal=Reviews of Modern Physics |volume=80|issue=4 |pages=1455–1515 |year=2008|doi=10.1103/RevModPhys.80.1455|bibcode=2008RvMP...80.1455A|s2cid=14117263}}</ref> It is applicable to matter in a compact star, where the only relevant thermodynamic potentials are quark [[chemical potential#Fundamental particle chemical potential|chemical potential]] μ and [[temperature]] T.

For guidance it also shows the typical values of μ and ''T'' in heavy-ion collisions and in the early universe. For readers who are not familiar with the concept of a chemical potential, it is helpful to think of μ as a measure of the imbalance between quarks and antiquarks in the system. Higher μ means a stronger bias favoring quarks over antiquarks. At low temperatures there are no antiquarks, and then higher μ generally means a higher density of quarks.

Ordinary atomic matter as we know it is really a mixed phase, droplets of nuclear matter (nuclei) surrounded by vacuum, which exists at the low-temperature phase boundary between vacuum and nuclear matter, at μ = 310 MeV and ''T'' close to zero. If we increase the quark density (i.e. increase μ) keeping the temperature low, we move into a phase of more and more compressed nuclear matter. Following this path corresponds to burrowing more and more deeply into a [[neutron star]].

Eventually, at an unknown critical value of μ, there is a transition to quark matter. At ultra-high densities we expect to find the [[Color-flavor locking|color-flavor-locked]] (CFL) phase of [[color superconductivity|color-superconducting]] quark matter. At intermediate densities we expect some other phases (labelled "non-CFL quark liquid" in the figure) whose nature is presently unknown,.<ref name='RMP'/><ref name='Rischke'/> They might be other forms of color-superconducting quark matter, or something different.

Now, imagine starting at the bottom left corner of the phase diagram, in the vacuum where μ = ''T'' = 0. If we heat up the system without introducing any preference for quarks over antiquarks, this corresponds to moving vertically upwards along the ''T'' axis. At first, quarks are still confined and we create a gas of hadrons ([[pion]]s, mostly). Then around ''T'' = 150 MeV there is a crossover to the quark gluon plasma: thermal fluctuations break up the pions, and we find a gas of quarks, antiquarks, and gluons, as well as lighter particles such as photons, electrons, positrons, etc. Following this path corresponds to travelling far back in time (so to say), to the state of the universe shortly after the big bang (where there was a very tiny preference for quarks over antiquarks).

The line that rises up from the nuclear/quark matter transition and then bends back towards the ''T'' axis, with its end marked by a star, is the conjectured boundary between confined and unconfined phases. Until recently it was also believed to be a boundary between phases where chiral symmetry is broken (low temperature and density) and phases where it is unbroken (high temperature and density). It is now known that the CFL phase exhibits chiral symmetry breaking, and other quark matter phases may also break chiral symmetry, so it is not clear whether this is really a chiral transition line. The line ends at the "chiral [[critical point (thermodynamics)|critical point]]", marked by a star in this figure, which is a special temperature and density at which striking physical phenomena, analogous to [[critical opalescence]], are expected. (Reference for this section:,<ref name='RMP'/><ref name='Rischke'/><ref name='TS'/>).

For a complete description of phase diagram it is required that one must have complete understanding of dense, strongly interacting hadronic matter and strongly interacting quark matter from some underlying theory e.g. quantum chromodynamics (QCD). However, because such a description requires the proper understanding of QCD in its non-perturbative regime, which is still far from being completely understood, any theoretical advance remains very challenging.

==Theoretical challenges: calculation techniques==

The phase structure of quark matter remains mostly conjectural because it is difficult to perform calculations predicting the properties of quark matter. The reason is that QCD, the theory describing the dominant interaction between quarks, is strongly coupled at the densities and temperatures of greatest physical interest, and hence it is very hard to obtain any predictions from it. Here are brief descriptions of some of the standard approaches.

===Lattice gauge theory===
The only first-principles calculational tool currently available is [[lattice QCD]], i.e. brute-force computer calculations. Because of a technical obstacle known as the fermion [[sign problem]], this method can only be used at low density and high temperature (μ < ''T''), and it predicts that the crossover to the quark–gluon plasma will occur around ''T'' = 150 MeV <ref>{{cite journal | author=P. Petreczky | title= Lattice QCD at non-zero temperature| journal=J. Phys. G | volume=39 | date=2012 | pages= 093002 | doi=10.1088/0954-3899/39/9/093002 | arxiv=1203.5320 | issue=9 |bibcode = 2012JPhG...39i3002P | s2cid= 119193093}}</ref> However, it cannot be used to investigate the interesting color-superconducting phase structure at high density and low temperature.<ref>{{cite journal |author1=Christian Schmidt |journal=PoS LAT2006 |volume=2006 |issue=21 |page=021 |title=Lattice QCD at Finite Density |year=2006 |arxiv=hep-lat/0610116|bibcode = 2006slft.confE..21S }}</ref>

===Weak coupling theory===

Because QCD is [[asymptotic freedom|asymptotically free]] it becomes weakly coupled at unrealistically high densities, and diagrammatic
methods can be used.<ref name='Rischke'>{{cite journal |doi=10.1016/j.ppnp.2003.09.002 |last1=Rischke |first1=D |title=The quark–gluon plasma in equilibrium |journal=Progress in Particle and Nuclear Physics |volume=52 |issue=1 |pages=197–296 |year=2004|arxiv = nucl-th/0305030 |bibcode = 2004PrPNP..52..197R |citeseerx=10.1.1.265.4175 |s2cid=119081533 }}</ref> Such methods show that the CFL phase occurs at very high density. At high temperatures, however, diagrammatic methods are still not under full control.

===Models===

To obtain a rough idea of what phases might occur, one can use a model that has some of the same properties as QCD, but is easier to manipulate. Many physicists use [[Nambu-Jona-Lasinio model]]s, which contain no gluons, and replace the strong interaction with a [[four-fermion interaction]]. Mean-field methods are commonly used to analyse the phases. Another approach is the [[bag model]], in which the effects of confinement are simulated by an additive energy density that penalizes unconfined quark matter.

===Effective theories===

Many physicists simply give up on a microscopic approach, and make informed guesses of the expected phases (perhaps based on NJL model results). For each phase, they then write down an effective theory for the low-energy excitations, in terms of a small number of parameters, and use it to make predictions that could allow those parameters to be fixed by experimental observations.<ref name='TS'>
{{cite conference
|author=T. Schäfer
|date=2004
|title=Quark matter
|editor=A. B. Santra
|book-title=Mesons and Quarks
|conference=14th National Nuclear Physics Summer School
|publisher=Alpha Science International
|arxiv=hep-ph/0304281
|isbn=978-81-7319-589-1
|bibcode=2003hep.ph....4281S
}}</ref>

===Other approaches===
{{main|AdS/QCD}}
There are other methods that are sometimes used to shed light on QCD, but for various reasons have not yet yielded useful results in studying quark matter.

====1/N expansion====
Treat the number of colors ''N'', which is actually 3, as a large number, and expand in powers of 1/''N''. It turns out that at high density the higher-order corrections are large, and the expansion gives misleading results.<ref name='RMP'/>

====Supersymmetry====
Adding scalar quarks (squarks) and fermionic gluons (gluinos) to the theory makes it more tractable, but the thermodynamics of quark matter depends crucially on the fact that only fermions can carry quark number, and on the number of degrees of freedom in general.

==Experimental challenges==

Experimentally, it is hard to map the phase diagram of quark matter because it has been rather difficult to learn how to tune to high enough temperatures and density in the laboratory experiment using collisions of relativistic heavy ions as experimental tools. However, these collisions ultimately will provide information about the crossover from [[Hadron|hadronic matter]] to QGP. It has been suggested that the observations of compact stars may also constrain the information about the high-density low-temperature region. Models of the cooling, spin-down, and precession of these stars offer information about the relevant properties of their interior. As observations become more precise, physicists hope to learn more.<ref name='RMP'/>

One of the natural subjects for future research is the search for the exact location of the chiral critical point. Some ambitious lattice QCD calculations may have found evidence for it, and future calculations will clarify the situation. Heavy-ion collisions might be able to measure its position experimentally, but this will require scanning across a range of values of μ and T.<ref>{{cite journal|doi=10.1016/S0375-9474(99)85017-9|last1=Rajagopal|first1=K|title=Mapping the QCD phase diagram|arxiv=hep-ph/9908360 |journal=[[Nuclear Physics A]] |volume=661 |issue=1–4|date=1999 |pages=150–161|bibcode = 1999NuPhA.661..150R |s2cid=15893165}}</ref>

== Evidence ==
In 2020, evidence was provided that the cores of neutron stars with mass ~2[[Solar mass|M⊙]] were likely composed of quark matter.<ref name=":0" /><ref>{{Cite web|title=A new type of matter discovered inside neutron stars|url=https://www.sciencedaily.com/releases/2020/06/200601120036.htm|website=ScienceDaily|language=en|access-date=2020-06-01}}</ref> Their result was based on neutron-star [[Spaghettification|tidal deformability]] during a [[neutron star merger]] as measured by [[Gravitational-wave observatory|gravitational-wave observatories]], leading to an estimate of star radius, combined with calculations of the equation of state relating the pressure and energy density of the star's core. The evidence was strongly suggestive but did not conclusively prove the existence of quark matter.

==See also==
* {{annotated link|Color–flavor locking}}
* {{annotated link|Lattice QCD}}
* {{annotated link|Quantum chromodynamics}}
* {{annotated link|Quark–gluon plasma}}
* {{annotated link|Quark star}}
* {{annotated link|SU(2) color superconductivity}}
* {{annotated link|Strange matter}}
* {{annotated link|Strangeness and quark-gluon plasma}}
* {{annotated link|1/N expansion}}

== Sources and further reading ==
* Aronson, S. and Ludlam, T.: [https://searchworks.stanford.edu/view/11327941 ''"Hunting the quark gluon plasma"''], U.S. Dept. of Energy (2005)
* Letessier, Jean: ''[https://searchworks.stanford.edu/view/4807502 Hadrons and quark-gluon plasma]'', Cambridge monographs on particle physics, nuclear physics, and cosmology (Vol. 18), Cambridge University Press (2002)
* {{cite journal |author=S. Hands |year=2001 |title=The phase diagram of QCD |arxiv=physics/0105022 |journal=Contemporary Physics |volume=42 |issue=4 |pages=209–225 |doi=10.1080/00107510110063843 |bibcode=2001ConPh..42..209H|s2cid=16835076 }}
* {{cite journal |author=K. Rajagopal |date=2001 |title=Free the quarks |url=http://www.slac.stanford.edu/pubs/beamline/31/2/31-2-rajagopal.pdf |journal=[[Beam Line]] |volume=32 |issue=2 |pages=9–15}}

== References ==
{{Reflist|30em}}

== External links ==
*[http://qgp.phy.duke.edu/ Virtual Journal on QCD Matter]
*[http://www.bnl.gov/rhic/news2/news.asp?a=1075&t=pr RHIC finds Exotic Antimatter]

{{states of matter}}

{{DEFAULTSORT:Qcd Matter}}
[[Category:Particle physics]]
[[Category:Quark matter| ]]
[[Category:Quantum chromodynamics]]
[[Category:Phases of matter]]
[[Category:Plasma physics]]
[[Category:Unsolved problems in physics]]

Equation of State

2021-05-10T08:00:25Z

Bleicher:

{{about||the use of this in cosmology|Equation of state (cosmology)| the use of this concept in optimal control theory|Optimal control#General method}}
{{short description|An equation describing the state of matter under a given set of physical conditions}}

In [[physics]] and [[thermodynamics]], an '''equation of state''' is a [[thermodynamic equations|thermodynamic equation]] relating [[state variable]]s which describe the state of matter under a given set of physical conditions, such as [[pressure]], [[Volume (thermodynamics)|volume]], [[temperature]] ('''PVT'''), or [[internal energy]].<ref name="Perrot" >{{cite book | author=Perrot, Pierre | title=A to Z of Thermodynamics | publisher=Oxford University Press | year=1998 | isbn=978-0-19-856552-9}}</ref> Equations of state are useful in describing the properties of [[fluid]]s, mixtures of fluids, [[solid]]s, and the interior of [[star]]s.
{{Thermodynamics|cTopic=[[Thermodynamic system|Systems]]}}
{{toclimit|limit=3}}

==Overview==
At present, there is no single equation of state that accurately predicts the properties of all substances under all conditions. An example of an equation of state correlates densities of gases and liquids to temperatures and pressures, known as the [[ideal gas law]], which is roughly accurate for weakly polar gases at low pressures and moderate temperatures. This equation becomes increasingly inaccurate at higher pressures and lower temperatures, and fails to predict condensation from a gas to a liquid.

Another common use is in modeling the interior of stars, including [[neutron star]]s, dense matter ([[quark–gluon plasma]]s) and radiation fields. A related concept is the [[perfect fluid]] [[equation of state (cosmology)|equation of state used in cosmology]].

Equations of state can also describe solids, including the transition of solids from one crystalline state to another.

In a practical context, equations of state are instrumental for PVT calculations in [[process engineering]] problems, such as petroleum gas/liquid equilibrium calculations. A successful PVT model based on a fitted equation of state can be helpful to determine the state of the flow regime, the parameters for handling the [[reservoir fluids]], and pipe sizing.

{{Anchor|lasers2016-01-29}}Measurements of equation-of-state parameters, especially at high pressures, can be made using lasers.<ref>{{cite journal|last1=Solem|first1=J. C.|last2=Veeser|first2=L.|year=1977|title=Exploratory laser-driven shock wave studies|journal=Los Alamos Scientific Laboratory Report LA-6997|volume=79|pages=14376|url=http://www.osti.gov/scitech/servlets/purl/5313279.pdf|bibcode=1977STIN...7914376S}}</ref><ref>{{cite journal|last1=Veeser|first1=L. R.|last2=Solem|first2=J. C.|year=1978|title=Studies of Laser-driven shock waves in aluminum|journal=Physical Review Letters|volume=40|issue=21|pages=1391|bibcode = 1978PhRvL..40.1391V |doi = 10.1103/PhysRevLett.40.1391 }}</ref><ref>{{cite journal|last1=Veeser|first1=L.|last2=Solem|first2=J. C.|last3=Lieber|first3=A.|year=1979|title=Impedance-match experiments using laser-driven shock waves|journal=Applied Physics Letters|volume=35|issue=10|pages=761|bibcode = 1979ApPhL..35..761V |doi = 10.1063/1.90961 }}</ref>

== Historical ==

=== Boyle's law (1662) ===
[[Boyle's Law]] was perhaps the first expression of an equation of state.{{citation needed|date=July 2016}} In 1662, the Irish physicist and chemist [[Robert Boyle]] performed a series of experiments employing a J-shaped glass tube, which was sealed on one end. [[Mercury (element)|Mercury]] was added to the tube, trapping a fixed quantity of air in the short, sealed end of the tube. Then the volume of gas was measured as additional mercury was added to the tube. The pressure of the gas could be determined by the difference between the mercury level in the short end of the tube and that in the long, open end. Through these experiments, Boyle noted that the gas volume varied inversely with the pressure. In mathematical form, this can be stated as:

:<math> pV = \mathrm{constant}.\,\!</math> 

The above relationship has also been attributed to [[Edme Mariotte]] and is sometimes referred to as Mariotte's law. However, Mariotte's work was not published until 1676.

=== Charles's law or Law of Charles and Gay-Lussac (1787) ===
In 1787 the French physicist [[Jacques Charles]] found that oxygen, nitrogen, hydrogen, carbon dioxide, and air expand to roughly the same extent over the same 80-kelvin interval. Later, in 1802, [[Joseph Louis Gay-Lussac]] published results of similar experiments, indicating a linear relationship between volume and temperature ([[Charles's Law]]):

:<math>\frac{V_1}{T_1} = \frac{V_2}{T_2}.</math>

=== Dalton's law of partial pressures (1801) ===
[[Dalton's Law]] of partial pressure states that the pressure of a mixture of gases is equal to the sum of the pressures of all of the constituent gases alone.

Mathematically, this can be represented for ''n'' species as:

::<math>
p_\text{total} = p_1 + p_2 + \cdots + p_n = \sum_{i=1}^n p_i.
</math>

=== The ideal gas law (1834) ===
In 1834, [[Émile Clapeyron]] combined Boyle's Law and Charles' law into the first statement of the ''[[ideal gas law]]''. Initially, the law was formulated as ''pVm'' = ''R''(''TC'' + 267) (with temperature expressed in degrees Celsius), where ''R'' is the [[gas constant]]. However, later work revealed that the number should actually be closer to 273.2, and then the Celsius scale was defined with 0{{nbsp}}°C = 273.15{{nbsp}}K, giving:

:<math>pV_m = R \left(T_C + 273.15\ {}^\circ\text{C}\right).</math>

=== Van der Waals equation of state (1873) ===
In 1873, [[J. D. van der Waals]] introduced the first [[van der Waals equation|equation of state]] derived by the assumption of a finite volume occupied by the constituent molecules.<ref name="van der Waals" >{{cite book |author1=van der Waals |author2=J. D. | title=On the Continuity of the Gaseous and Liquid States (doctoral dissertation) | publisher=Universiteit Leiden | year=1873}}</ref> His new formula revolutionized the study of equations of state, and was most famously continued via the [[Redlich–Kwong equation of state]]<ref name=":1">{{Cite journal|last1=Redlich|first1=Otto.|last2=Kwong|first2=J. N. S.|date=1949-02-01|title=On the Thermodynamics of Solutions. V. An Equation of State. Fugacities of Gaseous Solutions.|journal=Chemical Reviews|volume=44|issue=1|pages=233–244|doi=10.1021/cr60137a013|pmid=18125401|issn=0009-2665}}</ref> and the [[#Soave modification of Redlich-Kwong|Soave modification of Redlich-Kwong]].<ref name="Soave modification of Redlich-Kwong">{{cite journal |last1=Soave |first1=Giorgio |title=Equilibrium constants from a modified Redlich-Kwong equation of state |journal=Chemical Engineering Science |date=1972 |volume=27 |issue=6 |pages=1197–1203 |doi=10.1016/0009-2509(72)80096-4 }}</ref>

== General form of an equation of state ==
For a given amount of substance contained in a system, the temperature, volume, and pressure are not independent quantities; they are connected by a relationship of the general form

:<math>f(p, V, T) = 0</math>

An equation used to model this relationship is called an equation of state. In the following sections major equations of state are described, and the variables used here are defined as follows. Any consistent set of units may be used, although [[International System of Units|SI]] units are preferred. [[Thermodynamic temperature|Absolute temperature]] refers to use of the [[Kelvin]] (K) or [[Rankine scale|Rankine]] (°R) temperature scales, with zero being absolute zero.

:<math>\ p</math>, pressure (absolute)
:<math>\ V</math>, volume
:<math>\ n</math>, number of moles of a substance
:<math>\ V_m</math>, <math>\frac{V}{n}</math>, '''[[molar volume]]''', the volume of 1 mole of gas or liquid
:<math>\ T</math>, [[thermodynamic temperature|absolute temperature]]
:<math>\ R</math>, [[gas constant|ideal gas constant]] ≈ 8.3144621{{nbsp}}J/mol·K
:<math>\ p_c</math>, pressure at the critical point
:<math>\ V_c</math>, molar volume at the critical point
:<math>\ T_c</math>, absolute temperature at the critical point

== Classical ideal gas law ==
The classical [[ideal gas law]] may be written

:<math>pV = nRT.</math>

In the form shown above, the equation of state is thus

:<math>f(p, V, T) = pV - nRT = 0.</math>

If the [[Perfect gas|calorically perfect]] gas approximation is used, then the ideal gas law may also be expressed as follows

:<math>p = \rho(\gamma - 1)e</math>

where <math>\rho</math> is the density, <math>\gamma = C_p/C_v</math> is the adiabatic index ([[heat capacity ratio|ratio of specific heats]]), <math>e = C_v T</math> is the internal energy per unit mass (the "specific internal energy"), <math>C_v</math> is the specific heat at constant volume, and <math>C_p</math> is the specific heat at constant pressure.

== Quantum ideal gas law ==
Since for atomic and molecular gases, the classical ideal gas law is well suited in most cases, let us describe the equation of state for elementary particles with mass <math>m</math> and spin <math>s</math> that takes into account of quantum effects. In the following, the upper sign will always correspond to [[Fermi-Dirac statistics]] and the lower sign to [[Bose–Einstein statistics]]. The equation of state of such gases with <math>N</math> particles occupying a volume <math>V</math> with temperature <math>T</math> and pressure <math>p</math> is given by<ref>Landau, L. D., Lifshitz, E. M. (1980). Statistical physics: Part I (Vol. 5). page 162-166.</ref>

:<math>p= \frac{(2s+1)\sqrt{2m^3k_B^5T^5}}{3\pi^2\hbar^3}\int_0^\infty\frac{z^{3/2}\,\mathrm{d}z}{e^{z-\mu/(k_BT)}\pm 1}</math>

where <math>k_B</math> is the [[Boltzmann constant]] and <math>\mu(T,N/V)</math> the [[chemical potential]] is given by the following implicit function

:<math>\frac{N}{V}=\frac{(2s+1)(mk_BT)^{3/2}}{\sqrt 2\pi^2\hbar^3}\int_0^\infty\frac{z^{1/2}\,\mathrm{d}z}{e^{z-\mu/(k_BT)}\pm 1}.</math>

In the limiting case where <math>e^{\mu/(k_BT)}\ll 1</math>, this equation of state will reduce to that of the classical ideal gas. It can be shown that the above equation of state in the limit <math>e^{\mu/(k_BT)}\ll 1</math> reduces to

:<math>pV = Nk_BT\left[1\pm\frac{\pi^{3/2}}{2(2s+1)} \frac{N\hbar^3}{V(mk_BT)^{3/2}}+\cdots\right]</math>

With a fixed number density <math>N/V</math>, decreasing the temperature causes in [[Fermi gas]], a increase in the value for pressure from its classical value implying an effective repulsion between particles (this is an apparent repulsion due to quantum exchange effects not because of actual interactions between particles since in ideal gas, interactional forces are neglected) and in [[Bose gas]], a decrease in pressure from its classical value implying an effective attraction.

== Cubic equations of state ==
Cubic equations of state are called such because they can be rewritten as a [[cubic function]] of <math>V_m</math>.

=== Van der Waals equation of state ===
The [[Van der Waals equation]] of state may be written:

:<math>\left(p + \frac{a}{V_m^2}\right)\left(V_m - b\right) = RT</math>

where <math>V_m</math> is [[molar volume]]. The substance-specific constants <math>a</math> and <math>b</math> can be calculated from the [[critical properties]] <math>p_c</math>, <math>T_c</math>, and <math>V_c</math> (noting that <math>V_c</math> is the molar volume at the critical point) as:
:<math>a = 3 p_c V_c^2</math>
:<math>b = \frac{V_c}{3}.</math>

Also written as
:<math>a = \frac{27(R T_c)^2}{64p_c}</math>
:<math>b = \frac{R T_c}{8p_c}.</math>

Proposed in 1873, the van der Waals equation of state was one of the first to perform markedly better than the ideal gas law. In this landmark equation <math>a</math> is called the attraction parameter and <math>b</math> the repulsion parameter or the effective molecular volume. While the equation is definitely superior to the ideal gas law and does predict the formation of a liquid phase, the agreement with experimental data is limited for conditions where the liquid forms. While the van der Waals equation is commonly referenced in text-books and papers for historical reasons, it is now obsolete. Other modern equations of only slightly greater complexity are much more accurate.

The van der Waals equation may be considered as the ideal gas law, "improved" due to two independent reasons:
# Molecules are thought as particles with volume, not material points. Thus <math>V_m</math> cannot be too little, less than some constant. So we get (<math>V_m - b</math>) instead of <math>V_m</math>.
# While ideal gas molecules do not interact, we consider molecules attracting others within a distance of several molecules' radii. It makes no effect inside the material, but surface molecules are attracted into the material from the surface. We see this as diminishing of pressure on the outer shell (which is used in the ideal gas law), so we write (<math>p +</math> something) instead of <math>p</math>. To evaluate this ‘something’, let's examine an additional force acting on an element of gas surface. While the force acting on each surface molecule is ~<math>\rho</math>, the force acting on the whole element is ~<math>\rho^2</math>~<math>\frac{1}{V_m^2}</math>.

With the reduced state variables, i.e. <math>V_r=V_m/V_c</math>, <math>P_r=P/P_c</math> and <math>T_r=T/T_c</math>, the reduced form of the Van der Waals equation can be formulated:

:<math>\left(P_r + \frac{3}{V_r^2}\right)\left(3V_r - 1\right) = 8T_r</math>

The benefit of this form is that for given <math>T_r</math> and <math>P_r</math>, the reduced volume of the liquid and gas can be calculated directly using [[Cubic formula#Cardano's method|Cardano's method]] for the reduced cubic form:

:<math>V_r^3 - \left(\frac{1}{3} + \frac{8T_r}{3P_r}\right)V_r^2 + \frac{3V_r}{P_r} - \frac{1}{P_r} = 0</math>

For <math>P_r<1</math> and <math>T_r<1</math>, the system is in a state of vapor–liquid equilibrium. The reduced cubic equation of state yields in that case 3 solutions. The largest and the lowest solution are the gas and liquid reduced volume.

=== Redlich-Kwong equation of state<ref name=":1" />===
:<math>\begin{align}
p &= \frac{R\,T}{V_m - b} - \frac{a}{\sqrt{T}\,V_m\left(V_m + b\right)} \\[3pt]
a &\approx 0.42748\frac{R^2\,T_c^\frac{5}{2}}{p_c} \\[3pt]
b &\approx 0.08664\frac{R\,T_c}{p_c}
\end{align}</math>

Introduced in 1949, the [[Redlich-Kwong equation of state]] was a considerable improvement over other equations of the time. It is still of interest primarily due to its relatively simple form. While superior to the van der Waals equation of state, it performs poorly with respect to the liquid phase and thus cannot be used for accurately calculating [[vapor–liquid equilibria]]. However, it can be used in conjunction with separate liquid-phase correlations for this purpose.

The Redlich-Kwong equation is adequate for calculation of gas phase properties when the ratio of the pressure to the [[critical properties|critical pressure]] (reduced pressure) is less than about one-half of the ratio of the temperature to the [[critical properties|critical temperature]] (reduced temperature):

:<math>\frac{p}{p_c} < \frac{T}{2T_c}.</math>

=== Soave modification of Redlich-Kwong<ref name="Soave modification of Redlich-Kwong" />===
:<math>p = \frac{R\,T}{V_m-b} - \frac{a \alpha}{V_m\left(V_m+b\right)}</math>

:<math>a = \frac{0.42747\,R^2 T_c^2}{P_c}</math>
:<math>b = \frac{0.08664\,R T_c}{P_c}</math>
:<math>\alpha = \left(1 + \left(0.48508 + 1.55171\,\omega - 0.15613\,\omega^2\right) \left(1-T_r^{0.5}\right)\right)^2</math>

:<math>T_r = \frac{T}{T_c}</math>

Where ''ω'' is the [[acentric factor]] for the species.

This formulation for <math>\alpha</math> is due to Graboski and Daubert. The original formulation from Soave is:

:<math>\alpha = \left(1 + \left(0.480 + 1.574\,\omega - 0.176\,\omega^2\right) \left(1-T_r^{0.5}\right)\right)^2</math>

for hydrogen:

:<math>\alpha = 1.202 \exp\left(-0.30288\,T_r\right).</math>

We can also write it in the polynomial form, with:

:<math>A = \frac{a \alpha P}{R^2 T^2}</math>
:<math>B = \frac{bP}{RT}</math>
then we have:
:<math>0 = Z^3-Z^2+Z\left(A-B-B^2\right) - AB</math>
where <math>R</math> is the [[universal gas constant]] and ''Z''=''PV''/(''RT'') is the [[compressibility factor]].

In 1972 G. Soave<ref>{{Cite journal | doi=10.1016/0009-2509(72)80096-4|title = Equilibrium constants from a modified Redlich-Kwong equation of state| journal=Chemical Engineering Science| volume=27| issue=6| pages=1197–1203|year = 1972|last1 = Soave|first1 = Giorgio}}</ref> replaced the 1/{{radic|''T''}} term of the Redlich-Kwong equation with a function ''α''(''T'',''ω'') involving the temperature and the [[acentric factor]] (the resulting equation is also known as the Soave-Redlich-Kwong equation of state; SRK EOS). The ''α'' function was devised to fit the vapor pressure data of hydrocarbons and the equation does fairly well for these materials.

Note especially that this replacement changes the definition of ''a'' slightly, as the <math>T_c</math> is now to the second power.

=== Volume translation of Peneloux et al. (1982) ===
The SRK EOS may be written as
:<math>p = \frac{R\,T}{V_{m,\text{SRK}} - b} - \frac{a}{V_{m,\text{SRK}} \left(V_{m,\text{SRK}} + b\right)}</math>

where
:<math>\begin{align}
a &= a_c\, \alpha \\
a_c &\approx 0.42747\frac{R^2\,T_c^2}{P_c} \\
b &\approx 0.08664\frac{R\,T_c}{P_c}
\end{align}</math>

where <math>\alpha</math> and other parts of the SRK EOS is defined in the SRK EOS section.

A downside of the SRK EOS, and other cubic EOS, is that the liquid molar volume is significantly less accurate than the gas molar volume. Peneloux et alios (1982)<ref name="Peneloux1982">{{cite journal|last1=Peneloux|first1=A.|last2=Rauzy|first2=E.|last3=Freze|first3=R.|year=1982|title= A Consistent Correction for Redlich-Kwong-Soave Volumes|journal= Fluid Phase Equilibria|volume=8|issue=1982|pages=7–23|doi=10.1016/0378-3812(82)80002-2}}</ref> proposed a simple correction for this by introducing a volume translation

:<math>V_{m,\text{SRK}} = V_m + c</math>

where <math>c</math> is an additional fluid component parameter that translates the molar volume slightly. On the liquid branch of the EOS, a small change in molar volume corresponds to a large change in pressure. On the gas branch of the EOS, a small change in molar volume corresponds to a much smaller change in pressure than for the liquid branch. Thus, the perturbation of the molar gas volume is small. Unfortunately, there are two versions that occur in science and industry.

In the first version only <math>V_{m,\text{SRK}}</math> is translated,<ref name="Soave1990">{{cite journal|last1=Soave|first1=G.|last2=Fermeglia|first2=M.|year=1990|title= On the Application of Cubic Equation of State to Synthetic High-Pressure VLE Measurements|journal= Fluid Phase Equilibria|volume=60|issue=1990|pages=261–271|doi=10.1016/0378-3812(90)85056-G}}</ref>
<ref name="Zeberg2001">{{Cite book|last1=Zéberg-Mikkelsen|first1=C.K.|year=2001|title=Viscosity study of hydrocarbon fluids at reservoir conditions - modeling and measurements|journal=Ph.D. Thesis at the Technical University of Denmark. Department of Chemical Engineering|volume=June|issue=2001|pages=1–271|isbn=9788790142742}}</ref> and the EOS becomes

:<math>p = \frac{R\,T}{V_m + c - b} - \frac{a}{\left(V_m + c\right) \left(V_m + c + b\right)}</math>

In the second version both <math>V_{m,\text{SRK}}</math> and <math>b_\text{SRK}</math> are translated, or the translation of <math>V_{m,\text{SRK}}</math> is followed by a renaming of the composite parameter {{nowrap|b − c}}.<ref name="Pedersen1989">{{Cite book|last1=Pedersen|first1=K. S.|last2=Fredenslund|first2=Aa.|last3=Thomassen|first3=P.|year=1989|title=Properties of Oils and Natural Gases|journal=Book Published by Gulf Publishing Company, Houston|volume=1989|issue=1989|pages=1–252|isbn=9780872015883}}</ref> This gives

:<math>\begin{align}
b_\text{SRK} &= b + c \quad \text{or} \quad b - c \curvearrowright b \\
p &= \frac{R\,T}{V_m - b} - \frac{a}{\left(V_m + c\right) \left(V_m + 2c + b\right)}
\end{align}</math>

The c-parameter of a fluid mixture is calculated by
:<math>c = \sum_{i=1}^n z_i c_i</math>

The c-parameter of the individual fluid components in a petroleum gas and oil can be estimated by the correlation

:<math>c_i \approx 0.40768\ \frac{RT_{ci}}{P_{ci}} \left(0.29441 - Z_{\text{RA},i}\right) </math>

where the Rackett compressibility factor <math>Z_{\text{RA},i}</math> can be estimated by

:<math>Z_{\text{RA},i} \approx 0.29056 - 0.08775\ \omega_i</math>

A nice feature with the volume translation method of Peneloux et al. (1982) is that it does not affect the vapor-liquid equilibrium calculations.<ref name="Knudsen1992">{{cite journal|last1=Knudsen|first1=K.|year=1992|title=Phase Equilibria and Transport of Multiphase Systems|journal=Ph.D. Thesis at the Technical University of Denmark. Department of Chemical Engineering|issue=1992}}</ref> This method of volume translation can also be applied to other cubic EOSs if the c-parameter correlation is adjusted to match the selected EOS.

=== Peng–Robinson equation of state ===

:<math>\begin{align}
p &= \frac{R\,T}{V_m - b} - \frac{a\,\alpha}{V_m^2 + 2bV_m - b^2} \\[3pt]
a &\approx 0.45724 \frac{R^2\,T_c^2}{p_c} \\[3pt]
b &\approx 0.07780 \frac{R\,T_c}{p_c} \\[3pt]
\alpha &= \left(1 + \kappa \left(1 - T_r^\frac{1}{2}\right)\right)^2 \\[3pt]
\kappa &\approx 0.37464 + 1.54226\,\omega - 0.26992\,\omega^2 \\[3pt]
T_r &= \frac{T}{T_c}
\end{align}</math>

In polynomial form:
:<math>A = \frac{\alpha a p}{R^2\,T^2}</math>
:<math>B = \frac{bp}{RT}</math>
:<math>Z^3 - (1 - B)Z^2 + \left(A - 2B - 3B^2\right)Z - \left(AB - B^2 - B^3\right) = 0</math>

where <math>\omega</math> is the [[acentric factor]] of the species, <math>R</math> is the [[universal gas constant]] and <math>Z = PV/nRT</math> is [[compressibility factor]].

The Peng–Robinson equation of state (PR EOS) was developed in 1976 at The [[University of Alberta]] by [[Ding Yu Peng|Ding-Yu Peng]] and Donald Robinson in order to satisfy the following goals:<ref>{{cite journal | title = A New Two-Constant Equation of State | journal = Industrial and Engineering Chemistry: Fundamentals | volume = 15 | year = 1976 | pages = 59–64 |author1=Peng, D. Y. |author2=Robinson, D. B. | doi = 10.1021/i160057a011}}</ref>
# The parameters should be expressible in terms of the [[critical properties]] and the [[acentric factor]].
# The model should provide reasonable accuracy near the critical point, particularly for calculations of the [[compressibility factor]] and liquid density.
# The mixing rules should not employ more than a single binary interaction parameter, which should be independent of temperature, pressure, and composition.
# The equation should be applicable to all calculations of all fluid properties in natural gas processes.

For the most part the Peng–Robinson equation exhibits performance similar to the Soave equation, although it is generally superior in predicting the liquid densities of many materials, especially nonpolar ones.<ref>{{cite journal | title = Essentials of Reservoir Engineering | volume = 1 | year = 2007 | pages = 151 | author = Pierre Donnez }}</ref> The [[departure function]]s of the Peng–Robinson equation are given on a separate article.

The analytic values of its characteristic constants are:

: <math>Z_c = \frac{1}{32} \left( 11 - 2\sqrt{7} \sinh\left(\frac{1}{3} \operatorname{arsinh}\left(\frac{13}{7 \sqrt{7}}\right)\right) \right) \approx 0.307401</math>
: <math>b' = \frac{b}{V_{m,c}} = \frac{1}{3} \left( \sqrt{8} \sinh\left(\frac{1}{3} \operatorname{arsinh}\left(\sqrt{8}\right) \right) - 1 \right) \approx 0.253077 \approx \frac{0.07780}{Z_c} </math>
: <math> \frac{P_c V_{m,c}^2}{a\,b'} = \frac{3}{8} \left( 1 + \cosh\left(\frac{1}{3} \operatorname{arcosh}(3) \right) \right) \approx 0.816619 \approx \frac{Z_c^2}{0.45724 \, b'} </math>

===Peng–Robinson-Stryjek-Vera equations of state===

====PRSV1====
A modification to the attraction term in the Peng–Robinson equation of state published by Stryjek and Vera in 1986 (PRSV) significantly improved the model's accuracy by introducing an adjustable pure component parameter and by modifying the polynomial fit of the [[acentric factor]].<ref name="PRSV1">{{cite journal | title = PRSV: An improved Peng–Robinson equation of state for pure compounds and mixtures | journal = The Canadian Journal of Chemical Engineering | volume = 64 | issue = 2 | year = 1986 | pages = 323–333 |author1=Stryjek, R. |author2=Vera, J. H. | doi = 10.1002/cjce.5450640224}}</ref>

The modification is:
:<math>\begin{align}
\kappa &= \kappa_0 + \kappa_1 \left(1 + T_r^\frac{1}{2}\right) \left(0.7 - T_r\right) \\
\kappa_0 &= 0.378893+1.4897153\,\omega - 0.17131848\,\omega^2 + 0.0196554\,\omega^3
\end{align}</math>

where <math>\kappa_1</math> is an adjustable pure component parameter. Stryjek and Vera published pure component parameters for many compounds of industrial interest in their original journal article. At reduced temperatures above 0.7, they recommend to set <math>\kappa_1 = 0 </math> and simply use <math>\kappa = \kappa_0 </math>. For alcohols and water the value of <math> \kappa_1 </math> may be used up to the critical temperature and set to zero at higher temperatures.<ref name="PRSV1" />

====PRSV2====
A subsequent modification published in 1986 (PRSV2) further improved the model's accuracy by introducing two additional pure component parameters to the previous attraction term modification.<ref name="PRSV2">{{cite journal | title = PRSV2: A cubic equation of state for accurate vapor—liquid equilibria calculations | journal = The Canadian Journal of Chemical Engineering | volume = 64 | issue = 5 | year = 1986 | pages = 820–826 |author1=Stryjek, R. |author2=Vera, J. H. | doi = 10.1002/cjce.5450640516}}</ref>

The modification is:
:<math>\begin{align}
\kappa &= \kappa_0 + \left[\kappa_1 + \kappa_2\left(\kappa_3 - T_r\right)\left(1 - T_r^\frac{1}{2}\right)\right]\left(1 + T_r^\frac{1}{2}\right) \left(0.7 - T_r\right) \\
\kappa_0 &= 0.378893 + 1.4897153\,\omega - 0.17131848\,\omega^2 + 0.0196554\,\omega^3
\end{align}</math>

where <math>\kappa_1</math>, <math>\kappa_2</math>, and <math>\kappa_3</math> are adjustable pure component parameters.

PRSV2 is particularly advantageous for [[vapor–liquid equilibrium|VLE]] calculations. While PRSV1 does offer an advantage over the Peng–Robinson model for describing thermodynamic behavior, it is still not accurate enough, in general, for phase equilibrium calculations.<ref name="PRSV1" /> The highly non-linear behavior of phase-equilibrium calculation methods tends to amplify what would otherwise be acceptably small errors. It is therefore recommended that PRSV2 be used for equilibrium calculations when applying these models to a design. However, once the equilibrium state has been determined, the phase specific thermodynamic values at equilibrium may be determined by one of several simpler models with a reasonable degree of accuracy.<ref name="PRSV2" />

One thing to note is that in the PRSV equation, the parameter fit is done in a particular temperature range which is usually below the critical temperature. Above the critical temperature, the PRSV alpha function tends to diverge and become arbitrarily large instead of tending towards 0. Because of this, alternate equations for alpha should be employed above the critical point. This is especially important for systems containing hydrogen which is often found at temperatures far above its critical point. Several alternate formulations have been proposed. Some well known ones are by Twu et al or by Mathias and Copeman.

=== Peng-Robinson-Babalola equation of state (PRB) ===
Babalola <ref>{{Cite web|title=(PDF) A comparative analysis of the performance of various equations of state in thermodynamic property prediction of reservoir fluid systems|url=https://www.researchgate.net/publication/297878197_A_comparative_analysis_of_the_performance_of_various_equations_of_state_in_thermodynamic_property_prediction_of_reservoir_fluid_systems|access-date=2021-01-08|website=ResearchGate|language=en}}</ref> modified the Peng–Robinson Equation of state as:

<math>P =\left ( \frac{RT}{v-b} \right ) -\left [ \frac{(a_1P+a_2)\alpha}{v(v+b)+b(v-b)} \right ]</math>

The attractive force parameter ‘a’, which was considered to be a constant with respect to pressure in Peng–Robinson EOS. The modification, in which parameter ‘a’ was treated as a variable with respect to pressure for multicomponent multi-phase high density reservoir systems was to improve accuracy in the prediction of properties of complex reservoir fluids for PVT modeling. The variation was represented with a linear equation where a1 and a2 represent the slope and the intercept respectively of the straight line obtained when values of parameter ‘a’ are plotted against pressure.

This modification increases the accuracy of Peng–Robinson equation of state for heavier fluids particularly at pressure ranges (>30MPa) and eliminates the need for tuning the original Peng-Robinson equation of state. Values for a

=== Elliott, Suresh, Donohue equation of state ===
The Elliott, Suresh, and Donohue (ESD) equation of state was proposed in 1990.<ref name="ESD" >{{cite journal |author1=J. Richard, Jr. Elliott |author2=S. Jayaraman Suresh |author3=Marc D. Donohue | year=1990 | title=A Simple Equation of State for Nonspherical and Associating Molecules | journal=Ind. Eng. Chem. Res. |volume=29 | pages=1476–1485 | doi=10.1021/ie00103a057 | issue=7}}</ref> The equation seeks to correct a shortcoming in the Peng–Robinson EOS in that there was an inaccuracy in the van der Waals repulsive term. The EOS accounts for the effect of the shape of a non-polar molecule and can be extended to polymers with the addition of an extra term (not shown). The EOS itself was developed through modeling computer simulations and should capture the essential physics of the size, shape, and hydrogen bonding.

:<math>\frac{p V_m}{RT}=Z=1 + Z^{\rm{rep}} + Z^{\rm{att}}</math>

where:

:<math>Z^{\rm{rep}} = \frac{4 c \eta}{1-1.9 \eta}</math>

:<math>Z^{\rm{att}} = -\frac{z_m q \eta Y}{1+ k_1 \eta Y}</math>

and

:<math>c</math> is a "shape factor", with <math>c=1</math> for spherical molecules
:For non-spherical molecules, the following relation is suggested

:<math>c=1+3.535\omega+0.533\omega^2</math> where <math>\omega</math> is the [[acentric factor]].

:The reduced number density <math>\eta</math> is defined as <math>\eta=\frac{v^* n}{V}</math>
where
:<math>v^*</math> is the characteristic size parameter
:<math>n</math> is the number of molecules
:<math>V</math> is the volume of the container
The characteristic size parameter is related to the shape parameter <math>c</math> through
:<math>v^*=\frac{kT_c}{P_c}\Phi</math>
where

:<math>\Phi=\frac{0.0312+0.087(c-1)+0.008(c-1)^2}{1.000+2.455(c-1)+0.732(c-1)^2}</math> and <math>k</math> is [[Boltzmann's constant]].

Noting the relationships between Boltzmann's constant and the [[Universal gas constant]], and observing that the number of molecules can be expressed in terms of [[Avogadro's number]] and the [[molar mass]], the reduced number density <math>\eta</math> can be expressed in terms of the molar volume as

:<math>\eta=\frac{R T_c}{P_c}\Phi\frac{1}{V_m}.</math>

The shape parameter <math>q</math> appearing in the Attraction term and the term <math>Y</math> are given by
:<math>q=1+k_3(c-1)</math> (and is hence also equal to 1 for spherical molecules).

:<math>Y=\exp\left(\frac{\epsilon}{kT}\right) - k_2</math>

where <math>\epsilon</math> is the depth of the square-well potential and is given by

:<math>\frac{\epsilon}{k} =\frac{1.000+0.945(c-1)+0.134(c-1)^2}{1.023+2.225(c-1)+0.478(c-1)^2}</math>

:<math>z_m</math>, <math>k_1</math>, <math>k_2</math> and <math>k_3</math> are constants in the equation of state:
:<math>z_m = 9.49</math> for spherical molecules (c=1)
:<math>k_1 = 1.7745</math> for spherical molecules (c=1)
:<math>k_2 = 1.0617</math> for spherical molecules (c=1)
:<math>k_3 = 1.90476.</math>

The model can be extended to associating components and mixtures of nonassociating components. Details are in the paper by J.R. Elliott, Jr. ''et al.'' (1990).<ref name="ESD"/>

=== Cubic-Plus-Association ===
The Cubic-Plus-Association (CPA) equation of state combines the Soave-Redlich-Kwong equation with an association term from Wertheim theory.<ref name=":0">{{cite journal|last1=Kontogeorgis|first1=Georgios M.|last2=Michelsen|first2=Michael L.|last3=Folas|first3=Georgios K.|last4=Derawi|first4=Samer|last5=von Solms|first5=Nicolas|last6=Stenby|first6=Erling H.|title=Ten Years with the CPA (Cubic-Plus-Association) Equation of State. Part 1. Pure Compounds and Self-Associating Systems|journal=Industrial and Engineering Chemistry Research|date=2006|volume=45|issue=14|pages=4855–4868|doi=10.1021/ie051305v}}</ref> The development of the equation began in 1995 as a research project that was funded by Shell, and in 1996 an article was published which presented the CPA equation of state.<ref name=":0" /><ref>{{cite journal|last1=Kontogeorgis|first1=Georgios M.|last2=Voutsas|first2=Epaminondas C.|last3=Yakoumis|first3=Iakovos V.|last4=Tassios|first4=Dimitrios P.|title=An Equation of State for Associating Fluids|journal=Industrial & Engineering Chemistry Research|date=1996|volume=35|issue=11|pages=4310–4318|doi=10.1021/ie9600203}}</ref>

:<math>P = \frac{RT}{(V - b)} - \frac{a}{V (V + b)} + \frac{RT}{V} \rho \sum_{A} \left[ \frac{1}{X^A} - \frac{1}{2} \right] \frac{\partial X^A}{\partial \rho}</math>

In the association term <math>X^A</math> is the mole fraction of molecules not bonded at site A.

== Non-cubic equations of state ==

=== Dieterici equation of state ===
: <math>p(V - b) = RTe^{-\frac{a}{RTV}}</math>

where ''a'' is associated with the interaction between molecules and ''b'' takes into account the finite size of the molecules, similar to the Van der Waals equation.

The reduced coordinates are:
: <math>T_c = \frac{a}{4Rb},\ p_c = \frac{a}{4b^2 e^2},\ V_c = 2b.</math>

== Virial equations of state ==

=== Virial equation of state ===
{{main|Virial expansion}}
:<math>\frac{pV_m}{RT} = A + \frac{B}{V_m} + \frac{C}{V_m^2} + \frac{D}{V_m^3} + \cdots</math>

Although usually not the most convenient equation of state, the virial equation is important because it can be derived directly from [[statistical mechanics]]. This equation is also called the [[Heike Kamerlingh Onnes|Kamerlingh Onnes]] equation. If appropriate assumptions are made about the mathematical form of intermolecular forces, theoretical expressions can be developed for each of the [[virial coefficient|coefficients]]. ''A'' is the first virial coefficient, which has a constant value of 1 and makes the statement that when volume is large, all fluids behave like ideal gases. The second virial coefficient ''B'' corresponds to interactions between pairs of molecules, ''C'' to triplets, and so on. Accuracy can be increased indefinitely by considering higher order terms. The coefficients ''B'', ''C'', ''D'', etc. are functions of temperature only.

One of the most accurate equations of state is that from Benedict-Webb-Rubin-Starling<ref>{{Cite book |title=Fluid Properties for Light Petroleum Systems |last=Starling |first=Kenneth E. |publisher=Gulf Publishing Company |year=1973 |page=270}}</ref> shown next. It was very close to a virial equation of state. If the exponential term in it is expanded to two Taylor terms, a virial equation can be derived:
:<math>p=\rho RT + \left(B_0 RT-A_0 - \frac{C_0}{T^2} + \frac{D_0}{T^3} - \frac{E_0}{T^4}\right) \rho^2 + \left(bRT-a-\frac{d}{T} + \frac{c}{T^2}\right) \rho^3 + \alpha\left(a+\frac{d}{T}\right) \rho^6 </math>

Note that in this virial equation, the fourth and fifth virial terms are zero. The second virial coefficient is monotonically decreasing as temperature is lowered. The third virial coefficient is monotonically increasing as temperature is lowered.

=== The BWR equation of state ===
{{main|Benedict–Webb–Rubin equation}}
:<math>
p = \rho RT +
\left(B_0 RT - A_0 - \frac{C_0}{T^2} + \frac{D_0}{T^3} - \frac{E_0}{T^4}\right) \rho^2 +
\left(bRT - a - \frac{d}{T}\right) \rho^3 +
\alpha\left(a + \frac{d}{T}\right) \rho^6 +
\frac{c\rho^3}{T^2}\left(1 + \gamma\rho^2\right)\exp\left(-\gamma\rho^2\right)
</math>

where
:''p'' is pressure
:''ρ'' is molar density

Values of the various parameters for 15 substances can be found in {{cite book |author=K.E. Starling |year=1973 |title=Fluid Properties for Light Petroleum Systems |publisher=[[Gulf Publishing Company]] }}

=== Lee-Kesler equation of state ===
The Lee-Kesler equation of state is based on the corresponding states principle, and is a modification of the BWR equation of state.<ref>{{Cite journal|last1=Lee|first1=Byung Ik|last2=Kesler|first2=Michael G.|date=1975|title=A generalized thermodynamic correlation based on three-parameter corresponding states|journal=AIChE Journal|language=fr|volume=21|issue=3|pages=510–527|doi=10.1002/aic.690210313|issn=1547-5905}}</ref>

:<math>
P = \frac{RT}{V} \left( 1 + \frac{B}{V_r} + \frac{C}{V_r^2} + \frac{D}{V_r^5} + \frac{c_4}{T_r^3 V_r^2} \left( \beta + \frac{\gamma}{V_r^2} \right) \exp \left( \frac{-\gamma}{V_r^2} \right) \right)
</math>

== SAFT equations of state ==
[[Statistical associating fluid theory]] (SAFT) equations of state predict the effect of molecular size and shape and hydrogen bonding on fluid properties and phase behavior. The SAFT equation of state was developed using [[Statistical mechanics|statistical mechanical]] methods (in particular [[perturbation theory]]) to describe the interactions between molecules in a system.<ref name="Chapman1988">{{cite journal|last1=Chapman|first1=Walter G.|title=Theory and Simulation of Associating Liquid Mixtures|journal=Doctoral Dissertation, Cornell University|date=1988|language=en }}</ref><ref name="ChapmanGubbins1988">{{cite journal|last1=Chapman|first1=Walter G.|last2=Jackson|first2=G.|last3=Gubbins|first3=K.E.|title=Phase equilibria of associating fluids: Chain molecules with multiple bonding sites|journal=Molecular Physics|date=11 July 1988|volume=65|pages=1057–1079|doi=10.1080/00268978800101601 |language=en }}</ref><ref name="ChapmanGubbins1989">{{cite journal|last1=Chapman|first1=Walter G.|last2=Gubbins|first2=K.E.|last3=Jackson|first3=G.|last4=Radosz|first4=M.|title=SAFT: Equation-of-state solution model for associating fluids|journal=Fluid Phase Equilibria|date=1 December 1989|volume=52|pages=31–38|doi=10.1016/0378-3812(89)80308-5|language=en|issn=0378-3812}}</ref> The idea of a SAFT equation of state was first proposed by Chapman et al. in 1988 and 1989.<ref name="Chapman1988" /><ref name="ChapmanGubbins1988" /><ref name="ChapmanGubbins1989" /> Many different versions of the SAFT equation of state have been proposed, but all use the same chain and association terms derived by Chapman.<ref name="Chapman1988" /><ref name="ChapmanGubbins1990">{{cite journal|last1=Chapman|first1=Walter G.|last2=Gubbins|first2=K.E.|last3=Jackson|first3=G.|last4=Radosz|first4=M.|title=New Reference Equation of State for Associating Liquids|journal=Ind. Eng. Chem. Res.|date=1 August 1990|volume=29|issue=8|pages=1709–1721|doi=10.1021/ie00104a021|language=en }}</ref><ref>{{Cite journal | doi=10.1063/1.473101|title = Statistical associating fluid theory for chain molecules with attractive potentials of variable range| journal=The Journal of Chemical Physics| volume=106| issue=10| pages=4168–4186|year = 1997|last1 = Gil-Villegas|first1 = Alejandro| last2=Galindo| first2=Amparo| last3=Whitehead| first3=Paul J.| last4=Mills| first4=Stuart J.| last5=Jackson| first5=George| last6=Burgess| first6=Andrew N.|bibcode = 1997JChPh.106.4168G}}</ref> SAFT equations of state represent molecules as chains of typically spherical particles that interact with one another through short range repulsion, long range attraction, and hydrogen bonding between specific sites.<ref name="ChapmanGubbins1989" /> One popular version of the SAFT equation of state includes the effect of chain length on the shielding of the dispersion interactions between molecules ([[PC-SAFT]]).<ref>{{Cite journal | doi=10.1021/ie0003887|title = Perturbed-Chain SAFT: An Equation of State Based on a Perturbation Theory for Chain Molecules| journal=Industrial & Engineering Chemistry Research| volume=40| issue=4| pages=1244–1260|year = 2001|last1 = Gross|first1 = Joachim| last2=Sadowski| first2=Gabriele}}</ref> In general, SAFT equations give more accurate results than traditional cubic equations of state, especially for systems containing liquids or solids.<ref>{{Cite journal | doi=10.1021/ie010954d|title = Application of the Perturbed-Chain SAFT Equation of State to Associating Systems| journal=Industrial & Engineering Chemistry Research| volume=41| issue=22| pages=5510–5515|year = 2002|last1 = Gross|first1 = Joachim| last2=Sadowski| first2=Gabriele}}</ref><ref>{{Cite journal | doi=10.1016/j.fluid.2014.08.035|title = A modified continuous flow apparatus for gas solubility measurements at high pressure and temperature with camera system| journal=Fluid Phase Equilibria| volume=382| pages=150–157|year = 2014|last1 = Saajanlehto|first1 = Meri| last2=Uusi-Kyyny| first2=Petri| last3=Alopaeus| first3=Ville}}</ref>

== Multiparameter equations of state ==

=== Helmholtz Function form ===
Multiparameter equations of state (MEOS) can be used to represent pure fluids with high accuracy, in both the liquid and gaseous states. MEOS's represent the Helmholtz function of the fluid as the sum of ideal gas and residual terms. Both terms are explicit in reduced temperature and reduced density - thus:

: <math>\frac{a(T, \rho)}{RT} =
\frac{a^o(T, \rho) + a^r(T, \rho)}{RT} =
\alpha^o(\tau, \delta) + \alpha^r(\tau, \delta)
</math>

where:
: <math>\tau = \frac{T_r}{T}, \delta = \frac{\rho}{\rho_r}</math>

The reduced density and temperature are typically, though not always, the critical values for the pure fluid.

Other thermodynamic functions can be derived from the MEOS by using appropriate derivatives of the Helmholtz function; hence, because integration of the MEOS is not required, there are few restrictions as to the functional form of the ideal or residual terms.<ref name=":2">{{Cite journal|last=Span|first=R.|last2=Wagner|first2=W.|date=2003|title=Equations of State for Technical Applications. I. Simultaneously Optimized Functional Forms for Nonpolar and Polar Fluids|url=http://link.springer.com/10.1023/A:1022390430888|journal=International Journal of Thermophysics|volume=24|issue=1|pages=1–39|doi=10.1023/A:1022390430888}}</ref><ref name=":3">{{Cite journal|last=Span|first=Roland|last2=Lemmon|first2=Eric W.|last3=Jacobsen|first3=Richard T|last4=Wagner|first4=Wolfgang|last5=Yokozeki|first5=Akimichi|date=2000-11-XX|title=A Reference Equation of State for the Thermodynamic Properties of Nitrogen for Temperatures from 63.151 to 1000 K and Pressures to 2200 MPa|url=http://aip.scitation.org/doi/10.1063/1.1349047|journal=Journal of Physical and Chemical Reference Data|language=en|volume=29|issue=6|pages=1361–1433|doi=10.1063/1.1349047|issn=0047-2689}}</ref> Typical MEOS use upwards of 50 fluid specific parameters, but are able to represent the fluid's properties with high accuracy. MEOS are available currently for about 50 of the most common industrial fluids including refrigerants. The IAPWS95 reference equation of state for water is also an MEOS.<ref name=":4">{{Cite journal|last=Wagner|first=W.|last2=Pruß|first2=A.|date=2002-06-XX|title=The IAPWS Formulation 1995 for the Thermodynamic Properties of Ordinary Water Substance for General and Scientific Use|url=http://aip.scitation.org/doi/10.1063/1.1461829|journal=Journal of Physical and Chemical Reference Data|language=en|volume=31|issue=2|pages=387–535|doi=10.1063/1.1461829|issn=0047-2689}}</ref> Mixture models for MEOS exist, as well.

One example of such an equation of state is the form proposed by Span and Wagner.<ref name=":2" />

:<math>
a^{res} = \sum_{i=1}^8 \sum_{j=-8}^{12} n_{i,j} \delta^i \tau^{j/8} + \sum_{i=1}^5 \sum_{j=-8}^{24} n_{i,j} \delta^i \tau^{j/8} \exp \left( -\delta \right) + \sum_{i=1}^5 \sum_{j=16}^{56} n_{i,j} \delta^i \tau^{j/8} \exp \left( -\delta^2 \right) + \sum_{i=2}^4 \sum_{j=24}^{38} n_{i,j} \delta^i \tau^{j/2} \exp \left( -\delta^3 \right)
</math>

This is a somewhat simpler form that is intended to be used more in technical applications.<ref name=":2" /> Reference equations of state require a higher accuracy and use a more complicated form with more terms.<ref name=":4" /><ref name=":3" />

== Other equations of state of interest ==

===Stiffened equation of state===

When considering water under very high pressures, in situations such as [[Underwater explosion|underwater nuclear explosions]], [[Extracorporeal shock wave lithotripsy|sonic shock lithotripsy]], and [[sonoluminescence]], the stiffened equation of state<ref>{{Cite journal|last=Le Métayer|first=O|last2=Massoni|first2=J|last3=Saurel|first3=R|date=2004-03-01|title=Élaboration des lois d'état d'un liquide et de sa vapeur pour les modèles d'écoulements diphasiques|url=https://www.sciencedirect.com/science/article/pii/S1290072903001443|journal=International Journal of Thermal Sciences|language=fr|volume=43|issue=3|pages=265–276|doi=10.1016/j.ijthermalsci.2003.09.002|issn=1290-0729}}</ref> is often used:

:<math>p = \rho(\gamma - 1)e - \gamma p^0 \,</math>

where <math>e</math> is the internal energy per unit mass, <math>\gamma</math> is an empirically determined constant typically taken to be about 6.1, and <math>p^0</math> is another constant, representing the molecular attraction between water molecules. The magnitude of the correction is about 2 gigapascals (20,000 atmospheres).

The equation is stated in this form because the speed of sound in water is given by <math>c^2 = \gamma\left(p + p^0\right)/\rho</math>.

Thus water behaves as though it is an ideal gas that is ''already'' under about 20,000 atmospheres (2 GPa) pressure, and explains why water is commonly assumed to be incompressible: when the external pressure changes from 1 atmosphere to 2 atmospheres (100 kPa to 200 kPa), the water behaves as an ideal gas would when changing from 20,001 to 20,002 atmospheres (2000.1 MPa to 2000.2 MPa).

This equation mispredicts the [[specific heat capacity]] of water but few simple alternatives are available for severely nonisentropic processes such as strong shocks.

===Ultrarelativistic equation of state===

An [[ultrarelativistic fluid]] has equation of state

:<math>p = \rho_m c_s^2</math>

where <math>p</math> is the pressure, <math>\rho_m</math> is the mass density, and <math>c_s</math> is the [[speed of sound]].

===Ideal Bose equation of state===

The equation of state for an ideal [[Bose gas]] is

:<math>pV_m =
RT~\frac{\text{Li}_{\alpha+1}(z)}{\zeta(\alpha)}
\left(\frac{T}{T_c}\right)^\alpha
</math>

where α is an exponent specific to the system (e.g. in the absence of a potential field, α = 3/2), ''z'' is exp(''μ''/''kT'') where ''μ'' is the [[chemical potential]], Li is the [[polylogarithm]], ζ is the [[Riemann zeta function]], and ''T''''c'' is the critical temperature at which a [[Bose–Einstein condensate]] begins to form.

===Jones–Wilkins–Lee equation of state for explosives (JWL equation)===
The '''equation of state from Jones–Wilkins–Lee''' is used to describe the detonation products of explosives.

:<math>p = A \left( 1 - \frac{\omega}{R_1 V} \right) \exp(-R_1 V) + B \left( 1 - \frac{\omega}{R_2 V} \right) \exp\left(-R_2 V\right) + \frac{\omega e_0}{V}</math>

The ratio <math> V = \rho_e / \rho </math> is defined by using <math> \rho_e </math> = density of the explosive (solid part) and <math> \rho </math> = density of the detonation products. The parameters <math> A </math>, <math> B </math>, <math> R_1 </math>, <math> R_2 </math> and <math> \omega </math> are given by several references.<ref name="Dobratz" >{{Cite journal |author1=B. M. Dobratz |author2=P. C. Crawford | year=1985 | title=LLNL Explosives Handbook: Properties of Chemical Explosives and Explosive Simulants|journal=Ucrl-52997 |url=https://ci.nii.ac.jp/naid/10012469501/|access-date = 31 August 2018}}</ref> In addition, the initial density (solid part) <math> \rho_0 </math>, speed of detonation <math> V_D </math>, Chapman–Jouguet pressure <math> P_{CJ} </math> and the chemical energy of the explosive <math> e_0 </math> are given in such references. These parameters are obtained by fitting the JWL-EOS to experimental results. Typical parameters for some explosives are listed in the table below.
{| class="wikitable centre"
! scope=col | Material
! scope=col | <math>\rho_0\,</math> (g/cm3)
! scope=col | <math>v_D\,</math> (m/s)
! scope=col | <math>p_{CJ}\,</math> (GPa)
! scope=col | <math>A\,</math> (GPa)
! scope=col | <math>B\,</math> (GPa)
! scope=col | <math>R_1\,</math>
! scope=col | <math>R_2\,</math>
! scope=col | <math>\omega\,</math>
! scope=col | <math>e_0\,</math> (GPa)
|-
|[[Trinitrotoluene|TNT]]
| 1.630
| 6930
| 21.0
| 373.8
| 3.747
| 4.15
| 0.90
| 0.35
| 6.00
|-
|[[Composition B]]
| 1.717
| 7980
| 29.5
| 524.2
| 7.678
| 4.20
| 1.10
| 0.35
| 8.50
|-
| [[Polymer-bonded explosive|PBX 9501]]<ref name=Wilkins>{{Citation|last=Wilkins|first=Mark L.|title=Computer Simulation of Dynamic Phenomena| publisher=Springer|year=1999|page=80|url=https://books.google.com/books?id=b3npCAAAQBAJ&pg=PA1|access-date = 31 August 2018 |isbn=9783662038857}}</ref>
| 1.844
|
| 36.3
| 852.4
| 18.02
| 4.55
| 1.3
| 0.38
| 10.2
|}

== Equations of state for solids and liquids ==
Common abbreviations: <math>
\eta = \left(\frac{V}{V_0}\right)^\frac{1}{3}~,~~ K_0^\prime = \frac{dK_0}{dp}
</math>
* [[Tait equation]] for water and other liquids. Several equations are referred to as the '''Tait equation'''.
* [[Murnaghan equation of state]]
:<math>p(V) = \frac{K_0}{K_0'} \left[\eta^{-3K_0'} - 1\right]</math>

* [[Birch–Murnaghan equation of state]]
:<math>
p(V) = \frac{3K_0}{2}
\left(\frac{1 - \eta^2}{\eta^7}\right)
\left\{1 + \frac{3}{4}\left(K_0' - 4\right)
\left(\frac{1 - \eta^2}{\eta^2}\right)\right\}
</math>
* '''Stacey-Brennan-Irvine equation of state'''<ref name="StaceyBrennan1981">{{Cite journal |url= https://espace.library.uq.edu.au/view/UQ:399907 |title= Finite strain theories and comparisons with seismological data |last1= Stacey |first1=F.D. |journal= Surveys in Geophysics |volume=4 |issue=3 |pages=189–232 |doi= 10.1007/BF01449185 |access-date= 31 August 2018 |last2= Brennan |first2= B. J. |last3= Irvine |first3= R.D. |year= 1981 |bibcode= 1981GeoSu...4..189S|s2cid= 129899060 }}</ref> (falsely often refer to Rose-Vinet equation of state)
:<math>
p(V) = 3K_0\left(\frac{1 - \eta}{\eta^2}\right)\exp\left[\frac{3}{2}\left(K_0' - 1\right)(1 - \eta)\right]
</math>
* '''Modified Rydberg equation of state'''<ref>{{Cite book|title = "Equations of states and scaling rules for molecular solids under strong compression" in "Molecular systems under high pressure" ed. R. Pucci and G. Piccino |last= Holzapfel |first= W.B. |publisher= Elsevier |year= 1991 |location= North-Holland |pages= 61–68}}</ref><ref>{{Cite journal |title= Equations of state for solids under strong compression |last= Holzapfel |first= W.B. |date=1991 |journal= High Press. Res. |volume= 7 |pages= 290–293 |doi= 10.1080/08957959108245571|orig-year= 1991}}</ref><ref name="Holzapfel1996">{{Cite journal |title= Physics of solids under strong compression |last= Holzapfel |first= Wi.B. |journal= Rep. Prog. Phys. |volume=59 |pages=29–90 |doi= 10.1088/0034-4885/59/1/002 |issue=1 |year= 1996 |bibcode= 1996RPPh...59...29H |issn= 0034-4885}}</ref> (more reasonable form for strong compression)
:<math>
p(V) = 3K_0\left(\frac{1 - \eta}{\eta^5}\right)\exp\left[\frac{3}{2}\left(K_0' - 3\right)(1 - \eta)\right]
</math>
* '''Adapted Polynomial equation of state'''<ref name="Holzapfel1998">{{Cite journal |title= Equation of state for solids under strong compression |last= Holzapfel |first= W.B. |journal= High Press. Res. |volume=16 |issue=2 |pages=81–126 |issn=0895-7959 |doi= 10.1080/08957959808200283 |year= 1998 |bibcode= 1998HPR....16...81H}}</ref> (second order form = AP2, adapted for extreme compression)
:<math>
p(V) = 3K_0\left(\frac{1 - \eta}{\eta^5}\right)\exp\left[c_0(1 - \eta)\right]\left\{1 + c_2\eta(1 - \eta)\right\}
</math>
: with
:<math>
c_0 = -\ln\left(\frac{3K_0}{p_\text{FG0}}\right)~,~~p_\text{FG0}
= a_0\left(\frac{Z}{V_0}\right)^\frac{5}{3} ~,~~ c_2
= \frac{3}{2}\left(K_0' - 3\right) - c_0
</math>
:where <math>a_0</math> = 0.02337 GPa.nm5. The total number of electrons <math>Z</math> in the initial volume <math>V_0</math> determines the [[Fermi gas]] pressure <math>p_\text{FG0}</math>, which provides for the correct behavior at extreme compression. So far there are no known "simple" solids that require higher order terms.
* '''Adapted polynomial equation of state'''<ref name="Holzapfel1998"/> (third order form = AP3)
:<math>
p(V) = 3K_0\left(\frac{1 - \eta}{\eta^5}\right)\exp\left[c_0(1 - \eta)\right]\left\{1 + c_2\eta(1 - \eta) + c_3\eta(1 - \eta)^2\right\}
</math>
* [[Johnson–Holmquist damage model|Johnson–Holmquist equation of state]]
:<math>
p(V) = \begin{cases}
k_1~\xi + k_2~\xi^2 + k_3~\xi^3 + \Delta p & \qquad \text{Compression} \\
k_1~\xi & \qquad \text{Tension}
\end{cases}
~;~~ \xi := \cfrac{V_0}{V}-1
</math>
* [[Mie–Grüneisen equation of state]] (for a more detailed discussion see ref.<ref>{{cite book |last1=Holzapfel |first1=Wilfried B. |editor1-last=Katrusiak |editor1-first=A. |editor2-last=McMillan |editor2-first=P. |title=High-Pressure Crystallography |date=2004 |publisher=Kluver Academic |location=Dordrecht, Netherlands |doi= 10.1007/978-1-4020-2102-2_14 |isbn= 978-1-4020-1954-8 |pages=217–236 |chapter-url=https://physik.uni-paderborn.de/fileadmin/physik/Alumni/Holzapfel_LOP/ldv-238.pdf |access-date=31 August 2018 |language=en |chapter=Equations of state and thermophysical properties of solids under pressure|volume=140 |series=NATO Science Series}}</ref>)
:<math>
p(V) - p_0 = \frac{\Gamma}{V}\left(e - e_0\right)
</math>
* [[Anton-Schmidt equation of state]]
:<math>
p(V) = - \beta \left(\frac{V}{V_0}\right)^n \ln\left(\frac{V}{V_0}\right)
</math>
:where <math>\beta = K_0 </math> is the bulk modulus at equilibrium volume <math> V_0 </math> and <math> n = -\frac{K'_0}{2} </math> typically about −2 is often related to the [[Grüneisen parameter]] by <math> n = -\frac{1}{6} - \gamma_G </math>

== See also ==
* [[Gas laws]]
* [[Departure function]]
* [[Table of thermodynamic equations]]
* [[Real gas]]
* [[Cluster Expansion]]

==References==
{{reflist}}

== External links==
*Elliott & Lira, (1999). ''Introductory Chemical Engineering Thermodynamics'', Prentice Hall.
{{Topics in continuum mechanics}}
{{States of matter}}
{{Statistical mechanics topics}}

[[Category:Equations of physics]]
[[Category:Engineering thermodynamics]]
[[Category:Mechanical engineering]]
[[Category:Fluid mechanics]]
[[Category:Equations of state| ]]
[[Category:Thermodynamic models]]

Effective theories

2021-05-10T07:58:44Z

Bleicher:

In science, an '''effective theory''' is a [[scientific theory]] which proposes to describe a certain set of [[Experiment|observations]], but explicitly without the claim or implication that the mechanism employed in the theory has a direct counterpart in the actual causes of the observed phenomena to which the theory is fitted. That means, the theory proposes to model a certain ''effect'', without proposing to adequately model any of the ''causes'' which contribute to the effect.

For example, [[effective field theory]] is a set of tools used to describe physical theories when there is a hierarchy of scales. Effective field theories in physics can include [[quantum field theories]] in which the fields are treated as fundamental, and effective theories describing phenomena in [[solid-state physics]]. For instance, the [[BCS theory]] of [[superconduction]] treats vibrations of the solid-state lattice as a "[[field (physics)|field]]" (i.e. without claiming that there is "[[reality|really]]" a field), with its own field quanta, called [[phonon]]s. Such "effective particles" derived from effective fields are also known as [[quasiparticle]]s.

In a certain sense, [[quantum field theory]], and any other currently known physical theory, could be described as "effective", as in being the "low energy limit" of an as-yet unknown "[[Theory of Everything]]".<ref>c.f. {{cite book |first=Ion-Olimpiu |last=Stamatescu |first2=Erhard |last2=Seiler |title=Approaches to Fundamental Physics: An Assessment of Current Theoretical Ideas |series=Lecture Notes in Physics |volume=vol. 721 |publisher=Springer |year=2007 |isbn=978-3-540-71115-5 |page=47 }}</ref>

There are many effective theories which capture the symmetries of the quantum chromodynamics and are easier to handle. Examples are
* [https://en.wikipedia.org/wiki/Nambu–Jona-Lasinio_model Nambu Jona Lasinio model]
* [[Polyakov Nambu Jona-Lasinio model]]
* [https://en.wikipedia.org/wiki/Chiral_perturbation_theory Chiral perturbation theory]
and others...

==See also==
{{Div col|colwidth=20em}}
*[[Effective mass (solid-state physics)]]
*[[Emergence]]
*[[Empirism]]
*[[Epistemology]]
*[[Heuristics]]
*[[Scientific method]]
*[[Turing test]]
{{Div col end}}

==References==
{{reflist}}

[[Category:Scientific theories]]

Lattice QCD

2021-05-10T07:55:21Z

Bleicher:

{{Quantum field theory}}

In [[physics]], '''lattice gauge theory''' is the study of [[Gauge theory|gauge theories]] on a spacetime that has been [[Discretization|discretized]] into a [[lattice (group)|lattice]].

Gauge theories are important in [[particle physics]], and include the prevailing theories of [[elementary particle]]s: [[quantum electrodynamics]], [[quantum chromodynamics]] (QCD) and particle physics' [[Standard Model]]. [[Non-perturbative]] gauge theory calculations in continuous spacetime formally involve evaluating an infinite-dimensional [[Path integral formulation|path integral]], which is computationally intractable. By working on a discrete [[spacetime]], the [[Functional integration|path integral]] becomes finite-dimensional, and can be evaluated by [[stochastic simulation]] techniques such as the [[Monte Carlo method]]. When the size of the lattice is taken infinitely large and its sites infinitesimally close to each other, the continuum gauge theory is recovered.<ref name="wilson">{{cite journal | authorlink=Kenneth G. Wilson | first=K. | last= Wilson | journal=[[Physical Review D]]| volume=10 | issue=8 | page=2445 | title=Confinement of quarks | year= 1974 | doi=10.1103/PhysRevD.10.2445|bibcode = 1974PhRvD..10.2445W }}</ref>

==Basics==
In lattice gauge theory, the spacetime is [[Wick rotated]] into [[Euclidean space]] and discretized into a lattice with sites separated by distance <math>a</math> and connected by links. In the most commonly considered cases, such as [[lattice QCD]], [[fermion]] fields are defined at lattice sites (which leads to [[fermion doubling]]), while the [[Gauge boson|gauge fields]] are defined on the links. That is, an element ''U'' of the [[compact group|compact]] [[Lie group]] ''G'' (not [[Lie algebra|algebra]]) is assigned to each link. Hence, to simulate QCD with Lie group [[Special unitary group|SU(3)]], a 3×3 [[unitary matrix]] is defined on each link. The link is assigned an orientation, with the [[inverse element]] corresponding to the same link with the opposite orientation. And each node is given a value in ℂ3 (a color 3-vector, the space on which the [[fundamental representation]] of SU(3) acts), a [[bispinor]] (Dirac 4-spinor), an ''nf'' vector, and a [[Grassmann number|Grassmann variable]].

Thus, the composition of links' SU(3) elements along a path (i.e. the ordered multiplication of their matrices) approximates a [[path-ordered exponential]] (geometric integral), from which [[Wilson loop]] values can be calculated for closed paths.

==Yang–Mills action==
The [[Yang–Mills theory|Yang–Mills]] action is written on the lattice using [[Wilson loop]]s (named after [[Kenneth G. Wilson]]), so that the limit <math>a \to 0</math> formally reproduces the original continuum action.<ref name="wilson" /> Given a [[faithful representation|faithful]] [[irreducible representation]] ρ of ''G'', the lattice Yang-Mills action is the sum over all lattice sites of the (real component of the) [[trace (matrix)|trace]] over the ''n'' links ''e''1, ..., ''e''n in the Wilson loop,

:<math>S=\sum_F -\Re\{\chi^{(\rho)}(U(e_1)\cdots U(e_n))\}.</math>

Here, χ is the [[character (mathematics)|character]]. If ρ is a [[real representation|real]] (or [[pseudoreal representation|pseudoreal]]) representation, taking the real component is redundant, because even if the orientation of a Wilson loop is flipped, its contribution to the action remains unchanged.

There are many possible lattice Yang-Mills actions, depending on which Wilson loops are used in the action. The simplest "Wilson action" uses only the 1×1 Wilson loop, and differs from the continuum action by "lattice artifacts" proportional to the small lattice spacing <math>a</math>. By using more complicated Wilson loops to construct "improved actions", lattice artifacts can be reduced to be proportional to <math>a^2</math>, making computations more accurate.

==Measurements and calculations==
[[File:Fluxtube_meson.png|thumb|150px|This result of a [[Lattice QCD]] computation shows a [[meson]], composed out of a quark and an antiquark. (After M. Cardoso et al.<ref>{{cite journal | last1=Cardoso | first1=M. | last2=Cardoso | first2=N. | last3=Bicudo | first3=P. | title=Lattice QCD computation of the color fields for the static hybrid quark-gluon-antiquark system, and microscopic study of the Casimir scaling | journal=Physical Review D | volume=81 | issue=3 | date=2010-02-03 | issn=1550-7998 | doi=10.1103/physrevd.81.034504 | page=034504|arxiv=0912.3181| bibcode=2010PhRvD..81c4504C | s2cid=119216789 }}</ref>)]]

Quantities such as particle masses are stochastically calculated using techniques such as the [[Monte Carlo method]]. Gauge field configurations are generated with [[probability|probabilities]] proportional to <math>e^{-\beta S}</math>, where <math>S</math> is the lattice action and <math>\beta</math> is related to the lattice spacing <math>a</math>. The quantity of interest is calculated for each configuration, and averaged. Calculations are often repeated at different lattice spacings <math>a</math> so that the result can be [[extrapolation|extrapolated]] to the continuum, <math>a \to 0</math>.

Such calculations are often extremely computationally intensive, and can require the use of the largest available [[supercomputer]]s. To reduce the computational burden, the so-called [[quenched approximation]] can be used, in which the fermionic fields are treated as non-dynamic "frozen" variables. While this was common in early lattice QCD calculations, "dynamical" fermions are now standard.<ref>{{cite journal | author=A. Bazavov| title=Nonperturbative QCD simulations with 2+1 flavors of improved staggered quarks | journal=Reviews of Modern Physics | volume=82 | issue=2 | year=2010 | pages=1349–1417 | doi=10.1103/RevModPhys.82.1349 | arxiv=0903.3598 | bibcode=2010RvMP...82.1349B| s2cid=119259340 |display-authors=etal}}</ref> These simulations typically utilize algorithms based upon [[molecular dynamics]] or [[microcanonical ensemble]] algorithms.<ref>{{cite journal | author=[[David Callaway|David J. E. Callaway]] and [[Aneesur Rahman]] | title=Microcanonical Ensemble Formulation of Lattice Gauge Theory | journal=Physical Review Letters | volume=49 | year=1982 | issue=9 |pages=613–616 | doi=10.1103/PhysRevLett.49.613 | bibcode=1982PhRvL..49..613C}}</ref><ref>{{cite journal | author=[[David Callaway|David J. E. Callaway]] and [[Aneesur Rahman]] | title=Lattice gauge theory in the microcanonical ensemble | journal=Physical Review | volume=D28 |year=1983 | issue=6 | pages=1506–1514 | doi=10.1103/PhysRevD.28.1506|bibcode = 1983PhRvD..28.1506C | url=http://cds.cern.ch/record/144746/files/PhysRevD.28.1506.pdf }}</ref>

The results of lattice QCD computations show e.g. that in a meson not only the particles (quarks and antiquarks), but also the "[[Flux tube|fluxtube]]s" of the gluon fields are important.{{citation needed|date=May 2016}}

==Quantum triviality==
Lattice gauge theory is also important for the study of [[quantum triviality]] by the real-space [[renormalization group]].<ref>{{cite journal | last=Wilson | first=Kenneth G. |author-link=Kenneth G. Wilson| title=The renormalization group: Critical phenomena and the Kondo problem | journal=Reviews of Modern Physics | publisher=American Physical Society (APS) | volume=47 | issue=4 | date=1975-10-01 | issn=0034-6861 | doi=10.1103/revmodphys.47.773 | pages=773–840| bibcode=1975RvMP...47..773W }}</ref> The most important information in the RG flow are what's called the ''fixed points''.

The possible macroscopic states of the system, at a large scale, are given by this set of fixed points. If these fixed points correspond to a free field theory, the theory is said to be ''trivial'' or noninteracting. Numerous fixed points appear in the study of lattice Higgs theories, but the nature of the quantum field theories associated with these remains an open question.<ref>{{cite journal
| author=[[David J E Callaway|D. J. E. Callaway]]
| year=1988
| title=Triviality Pursuit: Can Elementary Scalar Particles Exist?
| journal=[[Physics Reports]]
| volume=167
| issue=5 | pages=241–320
| doi=10.1016/0370-1573(88)90008-7
|bibcode = 1988PhR...167..241C }}</ref>

Triviality has yet to be proven rigorously, but lattice computations have provided strong evidence for this. This fact is important as quantum triviality can be used to bound or even predict parameters such as the mass of [[Higgs boson]].

==Other applications==

Originally, solvable two-dimensional lattice gauge theories had already been introduced in 1971 as models with interesting statistical properties by the theorist [[Franz Wegner]], who worked in the field of phase transitions.<ref>F. Wegner, "Duality in Generalized Ising Models and Phase Transitions without Local Order Parameter", ''J. Math. Phys.'' '''12''' (1971) 2259-2272. Reprinted in [[Claudio Rebbi]] (ed.), ''Lattice Gauge Theories and Monte-Carlo-Simulations'', World Scientific, Singapore (1983), p. 60-73. [http://www.tphys.uni-heidelberg.de/~wegner/Abstracts.html#12 Abstract]</ref>

When only 1×1 Wilson loops appear in the action, Lattice gauge theory can be shown to be exactly dual to [[spin foam]] models.<ref>{{cite journal |author1=R. Oeckl |author2=H. Pfeiffer |year=2001 |title=The dual of pure non-Abelian lattice gauge theory as a spin foam model |arxiv=hep-th/0008095 |doi=10.1016/S0550-3213(00)00770-7 |volume=598 |issue=1–2 |journal=Nuclear Physics B |pages=400–426|bibcode=2001NuPhB.598..400O |s2cid=3606117 }}</ref>

==See also==
*[[Hamiltonian lattice gauge theory]]
*[[Lattice field theory]]
*[[Lattice QCD]]
*[[Quantum triviality]]

==References==
{{reflist|2}}

==Further reading==
* M. Creutz, ''Quarks, gluons and lattices'', Cambridge University Press 1985.
* I. Montvay and G. Münster, ''[https://books.google.com/books?id=NHZshmEBXhcC&printsec=frontcover#v=snippet&q=%22Lattice%20gauge%20theory%22&f=false Quantum Fields on a Lattice]'', Cambridge University Press 1997.
* Y. Makeenko, ''Methods of contemporary gauge theory'', Cambridge University Press 2002, {{ISBN|0-521-80911-8}}.
* [[Jan Smit (physicist)|J. Smit]], ''Introduction to Quantum Fields on a Lattice'', Cambridge University Press 2002.
* T. DeGrand and C. DeTar, ''[https://books.google.com/books?id=r8bICgAAQBAJ&printsec=frontcover#v=onepage&q=%22Lattice%20gauge%20theory%22&f=false Lattice Methods for Quantum Chromodynamics]'', World Scientific 2006.
* C. Gattringer and C. B. Lang, ''Quantum Chromodynamics on the Lattice'', Springer 2010.
* {{cite journal | author = Weisz Peter, Majumdar Pushan | year = 2012 | title = Lattice gauge theories | url = http://www.scholarpedia.org/article/Lattice_gauge_theories | journal = Scholarpedia | volume = 7 | issue = 4| page = 8615 | doi = 10.4249/scholarpedia.8615 | bibcode = 2012SchpJ...7.8615W | doi-access = free }}

==External links==
* [https://web.archive.org/web/20050304091436/http://www.fermiqcd.net/ The FermiQCD Library for Lattice Field theory]
* [http://usqcd.jlab.org/usqcd-software/ US Lattice Quantum Chromodynamics Software Libraries]

[[Category:Lattice models]]