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Kim Christensen Blackett Laboratory, Imperial College London 05.12.2006 Complexity and Criticality • Criticality – Ising model – Phase transition at (Tc , 0) – Scale invariance and £xed points • Complexity – Earthquakes and rainfall – Rice-pile experiment – Oslo rice-pile model – Self-organised criticality m(T, 0) 1 ↑ ↓ ↑ ↓ ↑ ↑ ↑ ↑ ↑ ↓ ↓ ↑ ↓ ↓ ↑ ↓ ↓ ↑ ↑ ↑ ↓ ↓ ↓ ↑ ↑ 0 −1 0 1 T /Tc De£nition Criticality: Ising Model The simplest model of a ferromagnet consists of N spins s i = ±1 = ↑ or ↓, i = 1, . . . , N with constant nearest-neighbour interaction J > 0 placed in a uniform external £eld H . The energy of microstate {si } = {s1 , s2 , . . . , sN } ↑ ↓ ↑ ↓ ↑ E{si } = spin-spin interaction + spin-external £eld interaction = −J X hiji si sj − H N X si . i=1 The partition function Z(T, H) = X {si } The free energy per spin ¡ The magnetisation per spin m(T, H) = − ¢ exp −βE{si } . µ ↑ ↑ ↑ ↑ ↓ ∂f ∂H ¶ T . 1 f (T, H) = − kB T ln Z. N The susceptibility per spin χ(T, H) = µ ∂m ∂H ¶ T . ↓ ↑ ↓ ↓ ↑ ↓ ↓ ↑ ↑ ↑ ↓ ↓ ↓ ↑ ↑ Phase transition at (T, H) f (T, H) = (Tc , 0) Criticality: Ising Model Free energy per spin in mean-£eld model. H Critical point (T, H) = (Tc , 0) m(T, H) T H T Magnetisation per spin in mean-£eld model. Phase transition at (T, H) Assume H = (Tc , 0) Criticality: Ising Model = 0. In equilibrium, the free energy is minimised F = hEi − T S. T = 0: Energy minimised: spins are aligned: m(0, 0) = ±1. Con£gurations are self-similar. The correlation length ξ(0, 0) = 0. T = ∞: Entropy maximised: spins are randomly orientated: m(∞, 0) = 0. Con£gurations are self-similar. The correlation length ξ(∞, 0) = 0. T = Tc : hEi and T S balanced. Spins are “undetermined”: m(Tc , 0) = 0. Con£gurations are self-similar. The correlation length ξ(Tc , 0) = ∞. T →0 T = Tc T →∞ Scale invariance and £xed points m(T, 0) 1 0 0 Criticality: Ising Model m(T, 0) ∝ ±(Tc − T )β for T → Tc− . 1 T /Tc −1 ξ(T, 0) ξ Scaling factor b ξ/b ξ(T, 0) ∝ |Tc − T |−ν for T → Tc . ξ/b2 0 > 1. 1 T /Tc Scale invariance and £xed points T < Tc Criticality: Ising Model T = Tc T > Tc ξ → ξ/b ξ/b → ξ/b2 Fixed points ξ = ξ/b ξ=0 ξ=∞ ξ=0 Warning Equilibrium vs. Non-equilibrium • No truly isolated natural systems exist. • Most systems have a ¤ux of mass or energy passing though them. • Most systems are in a non-equilibrium steady state. • Take great care not to apply results from equilibrium systems outside their range of validity. Plate tectonics Complexity: Earthquakes Palace Hotel, San Francisco, U.S.A. 5:12AM – 18 April, 1906. World-wide occurrence of earthquakes. Outline plate boundaries. 1.1.1997 − 30.6.1997, M > 4 Earthquake catalogue Scale invariance: Gutenberg-Richter Law Complexity: Earthquakes At a fault, strain builds up slowly. Energy released through earthquakes. Gutenberg-Richter Law: N (S > s) ∝ s−b . Large truck passing by Small atom bomb 100 hydrogen bombs N (S > s) earthquakes/year N(S>s) [earthquakes/year] 10 10 10 10 10 10 5 4 3 b = 0.95 2 1 0 −1 10 −2 10 0 1 2 3 4 5 6 7 Magnitude =loglog Magnitudem m= 10(S) 10 s 8 Rain event Complexity: Rainfall Europe, August 2002. Level of River Elbe = 9.39m. Rain event over Grand Canyon dissipates energy in the atmosphere. Rain gauges Standard rain gauge: Tipping bucket. Resolution of rain rate qmin = 0.25 mm/h. Temporal resolution ∆t = ? min. Radar: Resolution of rain rate qmin = 0.005 mm/h. Temporal resolution ∆t = 1 min. Complexity: Rainfall A rain event is a sequence of successive non-zero rain rates. The event size M = q(t + 1) + · · · + q(t + T ). with event duration T . Complexity: Rainfall Rain rate q(t) [mm/min] Rain event de£nition 1.50 1.25 1.00 0.75 0.50 0.25 0 0 50000 100000 150000 200000 250000 Rain-equivalent of Gutenberg-Richter Law: N (M ) ∝ M −τ M . N (M ) [event/year/mm] Time t [min] 10 10 5 4 10 10 2 10 10 10 τM = 1.4 3 1 0 -1 -2 10 -4 10 10 -3 10 -2 10 -1 10 0 10 Event size M [mm] 1 10 2 Conclusion Complexity: Earthquakes and Rainfall • Rain is a complicated spatio-temporal phenomenon. • Trickles, drizzle, bursts, showers, downpours, and torrents. • Identifying rain events as the basic entities reveals that – The frequency-event size distribution is scale free. This is the rain-equivalent of the Gutenberg-Richter law for earthquakes. – The frequency-drought duration distribution is scale free. This is the rain-equivalent of the Omori law for earthquakes. Rain is “Earthquake in the Sky”. Experimental setup Complexity: Rice-pile experiment L • Reaches statistically stationary state where hin¤uxi = hout¤uxi. • Avalanches dissipate energy. Statistically stationary states Complexity: Rice-pile experiment Avalanche size E E = 294 L = 33 Complexity: Rice-pile experiment E = 2868 L = 33 Energy dissipated by avalanche, E , measured in unit of mgδ = 1.54µJ . Focus on avalanche-size probability density, P (E; L) dE , in system of size L. Avalanche-size probability Complexity: Rice-pile experiment 0.3 1 0.8 0.6 Rescaled avalanche sizes, E/Emax as function of number of additions. E/Emax 0.4 0.2 0.2 0 0 2000 4000 t 6000 10 1000 t 1200 L = 16 L = 33 L = 66 L = 105 P (E; L) ∝ E for E À 1. P (E; L) 10 10 -4 τE ≈ 2 -5 10 10 1400 -2 -3 −τE 10000 0.1 0 800 Avalanche-size probability densities: 8000 -6 -7 10 0 10 1 10 2 10 3 10 E [1.54µJ] 4 10 10 5 Conclusion Complexity: Earthquakes, rainfall, and avalanches System Crust of Earth Atmosphere Granular pile Energy source Convection Sun Adding grains Energy storage Tension Vapour Potential Threshold Friction Saturation Friction Relaxation Earthquake Rain event Avalanche Common basis: • Slowly driven non-equilibrium systems. • Threshold dynamics. • Relaxation event dissipates energy. • Reaches statistically stationary state where hin¤uxi = hout¤uxi. • Relaxation events of all sizes up to a system dependent cutoff. De£nition Complexity: Oslo rice-pile model Lattice with L sites, vertical wall at left boundary and open at right boundary. • The height, hi , is the number of grains at column i, with hL+1 = 0. • The local slopes zi = hi − hi+1 , i = 1, . . . , L. 1 2 L 1 2 L 1 V Train model of earthquakes. 2 L De£nition Complexity: Oslo rice-pile model The algorithm for the Oslo rice-pile model: 1. Initialise the critical slopes zic ∈ {1, 2} and place the system in an arbitrary metastable state with zi ≤ zic for all i. 2. Add a grain at site i 3. If zi = 1: z1 → z1 + 1. > zic , the site relaxes and zi → z i − 2 zi±1 → zi±1 + 1. The critical slope zic is chosen randomly zic ∈ {1, 2} A new metastable state is reached when zi ≤ zic for all i. 4. Proceed to 2 and reiterate. Avalanche-size probability Complexity: Oslo rice-pile model 1.0 Rescaled avalanche sizes, s/smax as function of number of additions. s/smax 0.8 0.6 0.4 0.2 0 0 2000 4000 6000 t 8000 10000 0 10 L = 64 L = 256 L = 1024 L = 4096 -2 10 P (s; L) ∝ s −τs D G(s/L ). P (s; L) Avalanche-size probability densities: -4 10 -6 10 -8 10 -10 10 -12 10 -14 10 0 10 1 10 2 10 3 10 4 10 s 5 10 6 10 7 10 8 10 Data collapse reveals scaling function G Complexity: Oslo rice-pile model 0 Transformed avalanche-size probability. sτs P (s; L) 10 -1 10 10 -2 -3 10 0 10 1 10 2 10 3 10 4 10 s 5 10 6 10 7 10 8 10 9 10 0 sτs P (s; L) ∝ G(s/LD ). Data collapse obtained using exponents τs = 1.55, D = 2.25. sτs P (s; L) 10 -1 10 10 -2 -3 10 -8 10 -7 10 -6 10 -5 10 -4 10 -3 10 s/LD -2 10 -1 10 0 10 10 1 Conclusion Complexity: Oslo rice-pile model • The susceptibility is the average avalanche size: hsi = L. • The slowly driven pile organises itself, without any external £ne-tuning of control parameters, into a highly susceptible state where the susceptibility diverges with system size. • The Oslo rice-pile model displays self-organised criticality. • The Oslo model is the “Ising Model” for self-organised criticality. Collaborators Kim Christensen Per Bak. Leon Danon, University of Barcelona, Spain. Tim Scanlon, Imperial. Vidar Frette, Haugesund Højskole, Norway. Anders Malthe-Sørenssen, University of Oslo, Norway. Jens Feder, PGP, University of Oslo, Norway. Torstein Jøssang, PGP, University of Oslo, Norway. Paul Meakin, PGP, University of Oslo, Norway. Alvaro Corral, Universitat Autonoma de Barcelona, Spain. Gunnar Pruessner, Imperial. Matthew Stapleton, Imperial. Nicholas Moloney, Elte University, Budapest, Hungary. Ole Peters, Santa Fe Institute and Los Alamos Natinal Laboratory, U.S.A. Christopher Hertlein, Germany. Papers/animations available via: www.cmth.ph.ic.ac.uk/people/k.christensen/