Inphysics,critical phenomena is the collective name associated with thephysics ofcritical points. Most of them stem from the divergence of thecorrelation length, but also the dynamics slows down. Critical phenomena includescaling relations among different quantities,power-law divergences of some quantities (such as themagnetic susceptibility in theferromagnetic phase transition) described bycritical exponents,universality,fractal behaviour, andergodicity breaking. Critical phenomena take place insecond order phase transitions, although not exclusively.

The critical behavior is usually different from themean-field approximation which is valid away from thephase 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 therenormalization group.

In order to explain the physical origin of these phenomena, we shall use theIsing model as a pedagogical example.

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 theIsing classicalHamiltonian:

${\displaystyle H=-J\sum _{[i,j]}{S}_i\cdot S_j}$

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 theCurie temperature orcritical temperature,${\displaystyle T_c}$ below which the system presentsferromagnetic long range order. Above it, it isparamagnetic 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 thecorrelation length,${\displaystyle \xi }$ 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 calledcorrelation 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${\displaystyle T\to T_c}$,${\displaystyle \xi \to \mathrm{\infty }}$. This divergence poses no physical problem. Other physical observables diverge at this point, leading to some confusion at the beginning.

The most important issusceptibility. Let us apply a very small magnetic field to thesystem in the critical point. A very small magnetic field is not able to magnetize a large coherent cluster, but with thesefractal clusters the picture changes. It affects easily the smallest size clusters, since they have a nearlyparamagnetic 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 thespecific heat, 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 T_c. These exponents are calledcritical exponents and are robust observables. Even more, they take the same values for very different physical systems. This intriguing phenomenon, calleduniversality, is explained, qualitatively and also quantitatively, by therenormalization group.^[1]

Critical phenomena may also appear fordynamic quantities, not only forstatic ones. In fact, the divergence of the characteristictime${\displaystyle \tau }$ of a system is directly related to the divergence of the thermalcorrelation length${\displaystyle \xi }$ by the introduction of a dynamical exponentz and the relation${\displaystyle \tau ={\xi }^z}$ .^[2] The voluminousstatic universality class of a system splits into different, less voluminousdynamic universality classes with different values ofzbut a common static critical behaviour, and by approaching the critical point one may observe all kinds of slowing-down phenomena. The divergence of relaxation time${\displaystyle \tau }$ at criticality leads to singularities in various collective transport quantities, e.g., the interdiffusivity,shear viscosity${\displaystyle \eta \sim {\xi }^{x_{\eta }}}$,^[3] and bulk viscosity${\displaystyle \zeta \sim {\xi }^{x_{\zeta }}}$. The dynamic critical exponents follow certain scaling relations, viz.,${\displaystyle z=d+x_{\eta }}$, where d is the space dimension. There is only one independent dynamic critical exponent. Values of these exponents are dictated by several universality classes. According to the Hohenberg−Halperin nomenclature,^[4] for the model H^[5] universality class (fluids)${\displaystyle x_{\eta }\simeq 0.068,z\simeq 3.068}$.

Ergodicity 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, 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${\displaystyle T_c}$.

See alsosuperselection sector

The main mathematical tools to study critical points arerenormalization group, which takes advantage of the Russian dolls picture or theself-similarity to explain universality and predict numerically the critical exponents, andvariational perturbation theory, which converts divergent perturbation expansions into convergent strong-coupling expansions relevant to critical phenomena. In two-dimensional systems,conformal field theory 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 infinitesymmetry group.

The critical point is described by aconformal field theory. According to therenormalization group theory, the defining property of criticality is that the characteristiclength scale of the structure of the physical system, also known as thecorrelation lengthξ, becomes infinite. This can happen alongcritical lines inphase space. This effect is the cause of thecritical opalescence that can be observed as a 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 somenon-equilibrium systems, the critical point is anattractor of the dynamics in a manner that is robust with respect to system parameters, a phenomenon referred to asself-organized criticality.^[6]

Applications arise inphysics andchemistry, but also in fields such associology. For example, it is natural to describe a system of twopolitical parties by anIsing model. Thereby, at a transition from one majority to the other, the above-mentioned critical phenomena may appear.^[7]

• Catastrophe theory
• Conformal field theory
• Critical brain hypothesis
• Critical exponent
• Critical opalescence
• Critical point
• Ergodicity
• Ising model
• Rushbrooke inequality
• Self-organized criticality
• Variational perturbation theory
• Widom scaling

• Phase Transitions and Critical Phenomena, vol. 1-20 (1972–2001), Academic Press, Ed.:C. Domb,M.S. Green,J.L. Lebowitz
• 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.
• H. Kleinert and V. Schulte-Frohlinde,Critical Properties of φ⁴-Theories,World Scientific (Singapore, 2001); PaperbackISBN 981-02-4659-5 (Read online at[1])
• J. M. Yeomans,Statistical Mechanics of Phase Transitions (Oxford Science Publications, 1992)ISBN 0-19-851730-0
• M.E. Fisher,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

• Media related toCritical phenomena at Wikimedia Commons

• Critical phenomena
• Physical phenomena
• Conformal field theory
• Renormalization group

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