Critical Phenomena

In General > s.a. complexity; correlations [length]; renormalization group.
* Criticality: The behaviour of extended systems at a phase transition where scale invariance prevails.
* Critical point: The set of values of the external parameters of a system at which its behavior changes abruptly; Usually marks a phase transition, and the critical configuration has characteristic scaling properties.
* Approaches: The traditional one uses dimensional analysis; Now one often uses the renormalization group.

Self-Organized Criticality > s.a. complexity.
* Idea: A situation in which a complex system far from equilibrium organizes itself into a configuration (statistically) describable by just a few parameters; Emerges from the dynamics of extended, dissipative systems that evolve through a sequence of meta-stable states into a critical state, with long range spatial and temporal correlations; Central questions are, How does this happen? How do we extract usable information from vast amounts of data?
* Modeling: 2002, The phenomenology is seen in many areas, but it is not well understood theoretically; There are attempts at understanding and modeling it using energy, statistics and information concepts; A tool that may be useful is Paczuski's "metric" used to describe correlations between events in earthquake studies.
* Features: A 1/f noise, as opposed to white noise; Arises from the cooperative phenomena of many degrees of freedom, giving rise to simple phenomena in complex situations (in this sense, opposite to chaos).
@ General references: Kauffman 93 [in evolution]; Maslov & Zhang PhyA(96)ao [percolation, transport model]; Jensen 98; Paczuski & Bak cm/99-in; Alava cm/03-in [as a phase transition]; Cessac et al JSP(04) [thermodynamic formalism]; Dhar PhyA(06) [models].
@ Related topics: Bak & Boettcher PhyD(97)cm [and punctuated equilibrium]; Baiesi & Paczuski PRE(04)cm/03 [metric for earthquakes]; Stapleton et al JSP(04) [sensitivity to initial conditions]; Yang JPA(04) [origin of power-law distributions].

Examples and Phenomena > s.a. geometric phase; phase transitions; sigma models; yang-mills gauge theory.
* Applications: The central paradigm is the sand pile; Other common ones are earthquakes (the Earth's crust may be in a self-organized critical state; & Maya Paczuski), extinctions, economics, coast lines, language; Many can be classified as "stick-slip" or "punctuated equilibrium" phenomena.
@ Matter near criticality: Bernevig et al AP(04) [spectroscopy]; > entanglement [scaling].
@ In general relativity: Loustó PRD(95)gq/94 [black holes, effective 2D description]; > s.a. critical collapse; types of singularities.
@ In quantum gravity: Smolin gq/95 [and cosmology]; Ansari & Smolin CQG(08)-ht/04 [spin network evolution and classical spacetime].
@ Geometry, combinatorics: Lise & Paczuski PRL(02)cm [random graph]; > s.a. dynamical triangulations [surfaces], networks.
@ Non-equilibrium lattice and spin systems: Ruelle mp/00; Ódor 08.
@ Other topics: Grassberger JSP(95) [directed percolation]; Ballhausen et al PLB(04)ht/03 [continuous dim]; Creutz PhyA(04) [sand piles]; Paczuski & Hughes PhyA(04) [solar activity]; Turcotte & Malamud PhyA(04) [examples]; Barmatz et al RMP(07) [in microgravity].

General References
@ Intros, Reviews: Bhattacharjee cm/00-ln; Tobochnik AJP(01)RL [and phase transitions]; Brankov et al 02 [finite systems]; Christensen & Moloney 05 [and complexity].
@ Texts: Ma 76; Stora & Osterwalder ed-86; Stanley 87; Binney 92; Bak 96; Cardy 96; Zinn-Justin 96; Sornette 00; Amit & Martín-Mayor 05; Herbut 07.
@ And renormalization group: Barber PRP(77); Wilson RMP(83); Vicari a0709-in [for multi-parameter ⁴ theories].
@ Field theory techniques: Bagnuls & Bervillier JPS(97)ht, IJMPA(01)ht; Zinn-Justin ht/98; Folk & Moser JPA(06) [critical behavior in equilibrium].
@ Related topics: Robledo PhyA(04) [and Tsallis statistics]; Davatolhagh AJP(06) [scaling laws, critical exponents]; Zanardi et al a0707 [quantum, Bures metric approach].

Main page – Abbreviations – Journals – Comments – Other sites – Acknowledgements
Send feedback and suggestions to bombelli at olemiss.edu – Modified 25 may 2008