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In , critical phenomena is the collective name associated with the physics of . Most of them stem from the divergence of the , but also the dynamics slows down. Critical phenomena include
relations among different quantities,
divergences of some quantities (such as the
in the ) described by , ,
behaviour,
breaking. Critical phenomena take place in , although not exclusively.
The critical behavior is usually different from the
which is valid away from the phase transition, since the latter neglects correlations, which become increasingly important as the system approaches the critical point where the correlation length diverges. Many properties of the critical behavior of a system can be derived in the framework of the .
In order to explain the physical origin of these phenomena, we shall use the
as a pedagogical example.
Let us consider a
{\displaystyle 2D}
square array of classical spins which may only take two positions: +1 and -1, at a certain temperature
{\displaystyle T}
, interacting through the
classical :
{\displaystyle H=-J\sum _{[i,j]}S_{i}\cdot S_{j}}
where the sum is extended over the pairs of nearest neighbours and
{\displaystyle J}
is a coupling constant, which we will consider to be fixed. There is a certain temperature, called the
{\displaystyle T_{c}}
below which the system presents
long range order. Above it, it is
and is apparently disordered.
At temperature zero, the system may only take one global sign, either +1 or -1. At higher temperatures, but below
{\displaystyle T_{c}}
, the state is still globally magnetized, but clusters of the opposite sign appear. As the temperature increases, these clusters start to contain smaller clusters themselves, in a typical Russian dolls picture. Their typical size, called the ,
{\displaystyle \xi }
grows with temperature until it diverges at
{\displaystyle T_{c}}
. This means that the whole system is such a cluster, and there is no global magnetization. Above that temperature, the system is globally disordered, but with ordered clusters within it, whose size is again called correlation length, but it is now decreasing with temperature. At infinite temperature, it is again zero, with the system fully disordered.
The correlation length diverges at the critical point: as
{\displaystyle T\to T_{c}}
{\displaystyle \xi \to \infty }
. This divergence poses no physical problem. Other physical observables diverge at this point, leading to some confusion at the beginning.
The most important is . Let us apply a very small magnetic field to the system in the critical point. A very small magnetic field is not able to magnetize a large coherent cluster, but with these
clusters the picture changes. It affects easily the smallest size clusters, since they have a nearly
behaviour. But this change, in its turn, affects the next-scale clusters, and the perturbation climbs the ladder until the whole system changes radically. Thus, critical systems are very sensitive to small changes in the environment.
Other observables, such as the , may also diverge at this point. All these divergences stem from that of the correlation length.
As we approach the critical point, these diverging observables behave as
{\displaystyle A(T)\propto (T-T_{c})^{\alpha }}
for some exponent
{\displaystyle \alpha \,,}
where, typically, the value of the exponent α is the same above and below Tc. These exponents are called
and are robust observables. Even more, they take the same values for very different physical systems. This intriguing phenomenon, called , is explained, qualitatively and also quantitatively, by the .
Critical phenomena may also appear for dynamic quantities, not only for static ones. In fact, the divergence of the characteristic time
{\displaystyle \tau }
of a system is directly related to the divergence of the thermal correlation length
{\displaystyle \xi }
by the introduction of a dynamical exponent z and the relation
{\displaystyle \tau =\xi ^{\,z}}
 . The voluminous static universality class of a system splits into different, less voluminous dynamic universality classes with different values of z but a common static critical behaviour, and by approaching the critical point one may observe all kinds of slowing-down phenomena.
is the assumption that a system, at a given temperature, explores the full phase space, just each state takes different probabilities. In an Ising ferromagnet below
{\displaystyle T_{c}}
this does not happen. If
{\displaystyle T&T_{c}}
, never mind how close they are, the system has chosen a global magnetization, and the phase space is divided into two regions. From one of them it is impossible to reach the other, unless a magnetic field is applied, or temperature is raised above
{\displaystyle T_{c}}
The main mathematical tools to study critical points are , which takes advantage of the Russian dolls picture or the
to explain universality and predict numerically the critical exponents, and , which converts divergent perturbation expansions into convergent strong-coupling expansions relevant to critical phenomena. In two-dimensional systems,
is a powerful tool which has discovered many new properties of 2D critical systems, employing the fact that scale invariance, along with a few other requisites, leads to an infinite .
The critical point is described by a . According to the
theory, the defining property of criticality is that the characteristic
of the structure of the physical system, also known as the
ξ, becomes infinite. This can happen along critical lines in . This effect is the cause of the
that can be observed as binary fluid mixture approaches its liquid–liquid critical point.
In systems in equilibrium, the critical point is reached only by precisely tuning a control parameter. However, in some
systems, the critical point is an
of the dynamics in a manner that is robust with respect to system parameters, a phenomenon referred to as .
Applications arise in physics and chemistry, but also in fields such as . For example, it is natural to describe a system of two political parties by an Ising model. Thereby, at a transition between one majority to the other one the above-mentioned critical phenomena may appear. 
, vol. 1-20 (), Academic Press, Ed.: , ,
J.J. Binney et al. (1993): The theory of critical phenomena, Clarendon press.
N. Goldenfeld (1993): Lectures on phase transitions and the renormalization group, Addison-Wesley.
and V. Schulte-Frohlinde, Critical Properties of φ4-Theories, ; Paperback
(Read online at )
J. M. Yeomans, Statistical Mechanics of Phase Transitions (Oxford Science Publications, 1992)
, Renormalization Group in Theory of Critical Behavior, Reviews of Modern Physics, vol. 46, p. 597-616 (1974)
H. E. Stanley, Introduction to Phase Transitions and Critical Phenomena
P. C. Hohenberg und B. I. Halperin, Theory of dynamic critical phenomena , Rev. Mod. Phys. 49 (.
Christensen, K Moloney, Nicholas R. (2005). Complexity and Criticality. . pp. Chapter 3.  .
W. Weidlich, Sociodynamics, reprinted by Dover Publications, London 2006,
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--------------------------Page1------------------------------LatticeGaugeTheoryAChallengeinLarge-ScaleComputing--------------------------Page2------------------------------NATOASISeriesAdvancedScienceInstitutesSeriesAseriespresentingtheresultsofactivitiessponsoredbytheNATOScienceCommittee,whichaimsatthedisseminationofadvancedscientificandtechnologicalknowledge,withaviewtostrengtheninglinksbetweenscientificcommunities.TheseriesispublishedbyaninternationalboardofpublishersinconjunctionwiththeNATOScientificAffairsDivisionALifeSciencesPlenumPublishingCorporationBPhysicsNewYorkandLondonCMathematicalD.ReidelPublishing':ompanyandPhysicalSciencesDordrecht,Boston,andLancasteroBehavioralandSocialSciencesMartinusNijhoffPublishersEEngineeringandTheHague,Boston,andLancasterMaterialsSciencesFComputerandSystemsSciencesSpringer-VerlagGEcologicalSciencesBerlin,Heidelberg,NewYork,andTokyoRecentVolumesinthisSeriesVolume134-FundamentalProcessesinAtomicCollisionPhysicseditedbyH.Kleinpoppen,J.S.Briggs,andH.O.LutzVolume135-FrontiersofNonequilibriumStatisticalPhysicseditedbyGaraldT.MooreandMarianO.ScullyVolume136-HydrogeninDisorderedandAmorphousSolidseditedbyGustBambakidisandRobertC.Bowman,Jr.Volume137-ExaminingtheSubmicronWorldeditedbyRalphFeder,J.Wm.McGowan,andDouglasM.ShinozakiVolume138-TopologicalPropertiesandGlobalStructureofSpace-TimeeditedbyPeterG.BergmannandVenzoDeSabbataVolume139-NewVistasinNuclearDynamicseditedbyP.J.BrussaardandJ.H.KochVolume140-LatticeGaugeTheory:AChallengeinLarge-ScaleComputingeditedbyB.Bunk,K.H.MOtter,andK.SchillingVolume141-FundamentalProblemsofGaugeFieldTheoryeditedbyG.VeloandA.S.Wightman--------------------------Page3------------------------------LatticeGaugeTheoryAChallengeinLarge-ScaleComputingEditedbyB.BunkK.H.MOtterandK.SchillingGesamthochschuleWuppertal,FederalRepublicofGermanyPlenumPressNewYorkandLondonPublishedincooperationwithNATOScientificAffairsDivision--------------------------Page4------------------------------ProceedingsofaNATOWorkshoponLattice
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