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| title | chunk | source | category | tags | date_saved | instance |
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| Critical phenomena | 2/2 | https://en.wikipedia.org/wiki/Critical_phenomena | reference | science, encyclopedia | 2026-05-05T10:54:54.979917+00:00 | kb-cron |
== Mathematical tools == The main mathematical tools to study critical points are renormalization group, which takes advantage of the Russian dolls picture or the self-similarity to explain universality and predict numerically the critical exponents, and variational 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 infinite symmetry group.
== Critical point in renormalization group theory == The critical point is described by a conformal field theory. According to the renormalization group theory, the defining property of criticality is that the characteristic length scale of the structure of the physical system, also known as the correlation length ξ, becomes infinite. This can happen along critical lines in phase space. This effect is the cause of the critical 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 some non-equilibrium systems, the critical point is an attractor of the dynamics in a manner that is robust with respect to system parameters, a phenomenon referred to as self-organized criticality.
== Applications == Applications arise in physics and chemistry, but also in fields such as sociology. For example, it is natural to describe a system of two political parties by an Ising model. Thereby, at a transition from one majority to the other, the above-mentioned critical phenomena may appear.
== See also == 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
== Bibliography == 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); Paperback ISBN 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
== References ==
== External links == Media related to Critical phenomena at Wikimedia Commons