Critical phenomenon

In physics, critical phenomena are an umbrella term for the characteristic behavior of materials in the vicinity of one of their critical points .

Critical phenomena occur in part - but not exclusively - in second-order phase transitions . The divergence of the correlation length when approaching the critical temperature is particularly characteristic of almost all models : ${\displaystyle \xi }$ ${\displaystyle T_{\text{c}}}$

${\displaystyle \xi \propto |T-T_{\text{c}}|^{-\nu }}$

with a model-dependent value of the critical exponent that is uniform within very large universality classes . ${\displaystyle \nu }$

In quantitative terms, the critical phenomena are also characterized primarily by algebraic divergences in order parameters and scaling relationships between different sizes, fractal behavior and the violation of ergodicity .

Critical phenomena are also considered in sociophysics .

2D Ising model

The 2D Ising model at the critical point (with H  = 0)

The same model at a temperature well below the critical value

The two-dimensional Ising model can be used to illustrate the behavior of critical phenomena . This describes a field of classical spins , which can only assume the two discrete states +1 and −1. The interaction is described by the classical Hamilton operator:

${\displaystyle H=-J\sum _{[i,j]}S_{\text{i}}\cdot S_{\text{j}}}$.

The sum extends over neighboring pairs and is a coupling constant assumed to be constant . If it is positive, the system below a critical temperature , the Curie temperature , ferromagnetic long-range magnetic order on. Above this temperature it is paramagnetic and, when averaged over time, is without order. ${\displaystyle J}$ ${\displaystyle T_{\text{c}}}$

At absolute zero , the expected thermal value can only have one of the values ​​+1 or −1: ${\displaystyle \langle S_{\text{i}}\rangle _{T}}$

${\displaystyle T=0:\qquad \langle S_{\text{i}}\rangle _{T}=\pm 1}$

At higher temperatures, the overall state below is still magnetized: ${\displaystyle T_{\text{c}}}$

${\displaystyle 0<T<T_{\text{c}}:\qquad \langle S_{\text{i}}\rangle _{T}=\pm \sigma (T)}$ With ${\displaystyle 0<\sigma (T)<1,}$

d. H. areas (clusters) now appear with different signs . The typical diameter of these clusters is called the correlation length. As the temperature increases, the clusters themselves consist of smaller and smaller clusters. ${\displaystyle \xi .}$

The correlation length increases with temperature until it diverges at the critical point:

${\displaystyle T\to T_{\text{c}}:\qquad \xi \to \infty }$

This means that the entire system now forms a single cluster and there is no longer any global magnetization .

Above the critical temperature, the system is globally disordered, but consists of ordered clusters, the size of which decreases with increasing temperature. The size of the clusters in turn defines the correlation length. In the limit of very high temperatures, this is zero and the system is completely disordered:

${\displaystyle T\to \infty :\qquad \xi \to 0}$

Critical point

At the critical point, the correlation length diverges. This divergence is the reason why other physical quantities , e.g. B. the specific heat can diverge at this point or go to zero with special power laws . The correlation length represents the length scale on which there is a correlation between events or on which fluctuations extend.

In addition to the correlation length, the magnetic susceptibility is a variable that diverges at the critical point. If the system is exposed to a small magnetic field , implemented in the Hamilton operator by an additional term , this will not be able to magnetize a large coherent cluster. If, however, small fractal clusters exist, the picture changes: the smallest of these clusters are influenced without any problems because they show an almost paramagnetic behavior. However, this change affects the next larger cluster, and the disturbance spreads rapidly and radically changes the entire system. ${\displaystyle -h\sum S_{\text{i}}}$

Critical systems are therefore extremely sensitive to small changes in the environment.

Violation of ergodicity

Ergodicity is the assumption that a system with a certain temperature explores the entire phase space . However , this does not happen in an Ising ferromagnet below . Instead, the system chooses a global magnetization, with positive and negative values ​​occurring with the same probability, so that the phase space is divided into two areas: ${\displaystyle T_{\mathrm {c} }}$

${\displaystyle 0<T<T_{\text{c}}:\qquad \langle S_{\text{i}}\rangle _{T}=\pm \sigma (T)}$ With ${\displaystyle 0<\sigma (T)<1}$

It is not possible to move from one area to the other without applying a magnetic field or raising the temperature above the critical temperature . In the case of Heisenberg magnets , any directions below the critical temperature are even permitted as equivalent-separate "ergozity components", the description of the transition as a "critical phenomenon" (especially with the above-mentioned critical exponents) is still valid. ${\displaystyle T_{\text{c}}}$

In the case of spin glasses - certain disordered spin systems - this no longer applies, at least not in three dimensions, essentially because there they have a continuum of non-equivalently separate ergozity components.

Critical exponents and universality

In the case of critical phenomena, the general rule is that the observables behave as they would with an exponent when approaching the critical point . The exponent above and below generally have the same value. He is ${\displaystyle A(T)\propto (T-T_{\text{c}})^{\alpha }}$ ${\displaystyle \alpha }$ ${\displaystyle \alpha }$ ${\displaystyle T_{\text{c}}}$

• positive on convergence ${\displaystyle \left(\alpha >0\right)}$
• at logarithmic divergence or discontinuous behavior is possible. ${\displaystyle \alpha =0}$
• negative in case of divergence ${\displaystyle \left(\alpha <0\right).}$

The exponents for various physical quantities are called critical exponents  and are characteristic observables that are particularly insensitive to disturbances, provided they do not change the symmetry of the system.

There are various scale relationships between the critical exponents, such as

${\displaystyle \nu ={\frac {\gamma }{2-\eta }}}$

with the critical exponent

• ${\displaystyle \nu }$ for the correlation length
• ${\displaystyle \gamma }$ for the susceptibility
• ${\displaystyle \eta }$for the correlation function .

This phenomenon is known as " scaling ".

In addition, universality applies ; H. the mentioned exponents depend on the dimension of the system and the symmetry present, but have the same value for an infinitely large class of models.

Both the "scaling" and the existence of universality classes can be explained qualitatively and quantitatively by renormalization group theory .

Critical Dynamics

Even with dynamic phenomena there is critical behavior and universality: The divergence of the characteristic time (connected with other characteristic phenomena of the "critical deceleration") is attributed to the divergence of the correlation length by a dynamic exponent : ${\displaystyle \tau }$ ${\displaystyle z}$ ${\displaystyle \xi }$

${\displaystyle \tau =\xi ^{\,z}}$

The generally “very extensive” static university classes split into “less extensive” dynamic university classes, with different but the same critical statics. ${\displaystyle z}$

Critical opalescence

With certain liquid mixtures there is the phenomenon known as critical opalescence of "milky cloudiness": at the critical point of the liquid mixture, more and more microscopic droplets are formed, with the wavelength of the fluctuations constantly increasing , but at the same time the fluctuation dynamics continue to slow down ${\displaystyle (\xi \to \infty ),}$ ${\displaystyle (\tau \to \infty ).}$

Mathematical tools

Many properties of critical behavior can be derived from renormalization group theory. This uses the image of self-similarity to explain universality and to predict numerical values ​​of the critical exponents. The variational perturbation theory also plays a role, which changes divergent perturbation series into convergent developments of strong coupling.

The mean field theory is not to describe critical phenomena, as it is only a short distance from the valid phase transition and neglected correlation effects, the gain in the vicinity of the critical point of importance because there diverges the correlation length.

In two-dimensional systems, conformal field theory is an effective tool. By taking advantage of scale invariance and some other conditions that lead to infinite symmetry groups , a number of new properties of two-dimensional critical systems could be found.

Applications

In addition to physics and chemistry, there are also applications in subjects such as sociology and finance ( econophysics ). It is z. For example, it would seem close to describing a two-party system (approximately) using an Ising model. In passing from one majority opinion to the other, one can then observe the critical phenomena described above.

literature

• James J. Binney, et al .: The theory of critical phenomena - an introduction to the renormalization group. Clarendon Press, Oxford 2001, ISBN 0-19-851393-3 .
• W. Gebhardt, U. Krey: Phase transitions and critical phenomena - an introduction. Vieweg, 1980, ISBN 3528084227 .
• Nigel Goldenfeld: Lectures on phase transitions and the renormalization group. Addison-Wesley, Redwood City 1997, ISBN 0-201-55408-9 .
• Igor Herbut: A modern approach to critical phenomena. Cambridge Univ. Press, Cambridge 2007, ISBN 0-521-85452-0 .
• H. Kleinert and V. Schulte-Frohlinde, Critical Properties of φ ⁴ -Theories. World Scientific, Singapore 2001, ISBN 981-02-4659-5 . ( Online ).

Individual evidence

1. ^ Serge Galam, Yuval Gefen (Feigenblat), Yonathan Shapir: Sociophysics: A new approach of sociological collective behavior. I. mean-behavior description of a strike. In: The Journal of Mathematical Sociology. 9, 2010, p. 1, doi : 10.1080 / 0022250X.1982.9989929 .
2. ^ Sorin Solomon, Gerard Weisbuch, Lucilla de Arcangelis, Naeem Jan, Dietrich Stauffer: Social percolation models. In: Physica A: Statistical Mechanics and its Applications . Volume 277, number 1-2, 2000, pp. 239-247, doi : 10.1016 / S0378-4371 (99) 00543-9 .
3. ↑ PC Hohenberg, BI Halperin: Theory of dynamic critical phenomena , Reviews of Modern Physics, Volume 49, 1977, DOI: 10.1103 / RevModPhys.49.435
4. ↑ W. Weidlich: Sociodynamics . Republication by Dover Publications, London 2006, ISBN 0-486-45027-9 .