Critical Phenomena and ScalingJim Sethna, Physics 562, April 2002Karin Dahmen, Matt Kuntz, Olga Perkovic, Chris Myers …The Ising Model at TcStructure on All Scales• Continuous Phase Transition• Competition Entropy vs. Energy• Thermal Disorder• High Temperature: Random• Low TemperatureLong-Range Order• Critical Point• Tc = 2/log(1+√2) ~ 2.27• Fluctuations on All ScalesPercolationStructure on All Scales• Connectivity Transition• Punch Holes at Random, Probability 1-PPc~ 0.593 Falls Apart(2D, Square Lattice, Site)• Static (Quenched) Disorderhttp://www.ipm.sci-nnov.ru/~demidov/perc/perc.htmLargest Connected ClusterP=0.65P=0.55EarthquakesSpatially Extended Events of All SizesEarthquakes of Many Sizes: 1995Burridge-Knopoff (Carlson & Langer)http://simscience.org/crackling/Advanced/Earthquakes/EarthquakeSimulation.html• Earthquakes of All Sizes• Gutenberg-Richter Law:Probability ~ Size-Power• Simple Block-Spring Model• No disorder• Slow driving rate (cm/year)Magnetic Barkhausen NoiseEvents of All Sizes, Structure on All ScalesBarkhausen Noise in MagnetsMagneticAvalanches Fractalin Time and SpaceEvents of All SizesHysteresis Model for MagnetsT=0 Driven Random-Field Ising ModelH=-Σij nnJ SiSj–ΣiH Si–hiSiSi= ±1, magnetic domainJ coupling between neighboring spinsH external applied fieldhirandom field at site, dirt,chosen from Gaussian width RP(h) = Gaussian RMS width RDynamics• Start all spins down, H=-∞• Increase field slowly • Spin flips when pushed over• Initial spin 13 pushed by H• Pushes neighbors:
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