Avalanches in Equilibrium and Nonequilibrium In the same Universality Class???
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1 Avalanches in Equilibrium and Nonequilibrium In the same Universality Class??? UIUC: John Carpenter (now Sandia), Amit Mehta, Robert White (now San Diego), Matthew Delgado, Yang Liu, Andrew Missel, Geoffrey Poore, Alex Travesset (now Assis. Prof. Iowa/Ames Lab) REU s: Ma ayan Bresler (Princeton), Sharon Loverde (now Northwestern), Riva Vanderveld (now Cornell) Jim Sethna, M. Kuntz, O. Perkovic (Cornell University), A. Middleton -- theory Yehuda Ben Zion (USC, Earth and Planetary Sciences), (D.S. Fisher (Harvard)) - earthquakes A. Berger, O. Hellwig (IBM/Hitachi) -- exp. Gianfranco Durin (Turino, Italy) -- exp. Andrea Mills, Mike Weissman (UIUC) -- exp. Funding/Equipment: NSF, MCC, SLOAN, UIUC, IBM, SANDIA E. Carlson (Perdue), E. Fradkin (UIUC), S. Kivelson (Stanford), D. VanHarlingen (UIUC), M. Weissman (UIUC), C.Panagopoulos (Cambridge) -- superconductors C. Marchetti (Syr.)-- plastic CDW D. Nelson (Harvard) and N. Shnerb (Israel) - spreading Bacteria Colonies
2 Magnets and Barkhausen Noise- or Martensites and acc. emission Crackling noise Sample Volt H(t)
3 Crackling Noise / Avalanches: Barkhausen Noise (magnets) Acoustic emission (Martensites( Martensites) (Ortin, Vives,... ) Superconductors (P.Adams; Field,Witt,Nori,...) Liquid He invading Nuclepore (Hallock, Lilly,Wooters...) Rupture of fibrous Materials Earthquakes Broad Range of s -τ sizes & durations
4 The Zero Temperature Non-equilibrium Random Field Ising Model Random Field Distribution H ρ(h i ) R= strength of the disorder Each Ηspin systemis = always J aligned S i S j with the direction + S iof the H local " force" ( H(t) hi ) Zero Temperature : (Equilibration time scale) >> (Experimental time scale) n. n. = H ( t ) + i h i + J n. n. S j Jumps = Avalanches = Barkhausen noise
5 The Disorder Induced Critical Point Seen also in experiments: Berger et al, PRL 2000, J.Appl.Phys H Up Down Clean critical Dirty disorder R Log(#) τ+σβδ=2.03±0.03, 1/σ=4.2 ±0.3,... Dirty critical Vary Disorder by 50% of critical amount: still find 2 decades of powerlaw scaling! HUGE scaling region!!! LOG (AVALANCHE SIZE)
6 Self-similarity at critical disorder R c =2.16J (Cross-sections of avalanches during magnetization) CRITICAL POINT: system is at a fixed point under coarse graining transformation (Renormalization Group)
7 Why powerlaw at (H c, R c )? Renormalization Group (Wilson, Fisher, Kadanoff) history dependent, deterministic time dependent description: H=H(t)= Ωt with Ω 0 Random fields probability distribution for paths Average and expand around mean field theory Martin Siggia Rose / Bausch Janssen Wagner Formalismus soft spins: Renormalization group: (6-ε) expansion Langevin dynamics:
8 Generating Functional: Path integral: Narayan, Middleton PRB 1993, Narayan, Fisher, PRB 1992, Zippelius PRB 1984, Sompolinsky, Zippelius PRB 1992, De Dominicis, PRB 1978, Bausch, Janssen, Wagner, ZPB 1976, Martin Siggia Rose, PRA 1973 Product over all times, average over the random fields: Transformation to local fields: Random field averaged probability distribution for path η : P( η )
9 Effective Action: Momentum Shell Renormalization: 1. Integrate over momentum shell 2.rescale New Fixed Point (to O(ε)) a
10 Huge Universality Class!!! (Details don t matter!) H Η system = J S i S j + n. n. ( H(t) hi ) i S i Replace J by random couplings Use different distribution of h i or replace h i by random anisotropies Magnets (Sethna,KD,Myers, Nature 2001), plastic charge density wave depinning (Marchetti, KD PRB 2002), earthquakes (Mehta, BenZion, KD 2005), maybe superconductors (Carlson, KD, Fradkin, Kivelson, PRL 2005), others? H Η system = With nucleation of new domains J Other Universality classes? S is j + n. n. ( H(t) hi ) i + 2 Dynamics S i + long range forces Single domain wall; Nattermann, Robbins, Ji, Zapperi, Ciseau, Durin, Stanley, Urbach et al., Narayan, Sethna,...
11 RESULTS Simulation 6-ε expansion (Renormalization Group) 1/ν = 2 - ε/3-0.1ε ε 3-0.3ε 4 + ε 5 + O(ε 6 ) (PRL 93, 95, 2003 PRB 96, 99, 2002 (R)) Nature 410, 242 (2001) Experiments and Simulations in 3 dim. (Barkhausen Noise): Need Noise Exp. Tuning disorder!!!
12 SURPRIZINGLY SIMILAR: EQUILIBRIUM and NONEQUILIBRIUM RFIM: 1. SAME MEAN FIELD EXPONENTS (β=1/2, ν=1/2, δ=3) τ=3/2 and σ=1/2 (Liu, KD) 2. ABOUT SAME SIMULATION EXPONENTS in 3D and even in 4D (WITHIN ERRORBARS) 3. 6-ε expansion of noneq. mapped to all orders in ε to wrong eq. expansion (KD, Sethna) 4. SAME SIMULATION EXPONENTS for GROUND STATE & DEMAGN. STATE (Zapperi et al.) 5. SAME EXP. AND SCALING FUNCTS FOR DEMAGN. CURVE & SAT. LOOP (Carpenter, KD) 6. Middleton's no-passing rule: flipped spin cannot flip back with increasing field. (Liu, KD)
13 Surprizing Result: same Avalanche Exponents And Scaling Function In Equilibrium and Nonequilibrium Ground state Avalanches : Liu, KD 2006
14 Result: ~Same scaling of Avalanche Surface Area Distribution Integrated avalanche surface distribution for avalanche size S Liu, KD, 2006
15 Demagnetizing curves in the T=0 Random Field Ising Model: John Carpenter, KD, PRB 2003 (R) Same disorder induced critical point as saturation loop: Need: Experiments on Barkhausen Noise for various disorders??? Avalanche Size Distribution ΔM vs R R=3.3 R=
16 Conclusions Avalanche exponents and spatial structures (fractal dimensions and anisotropy measures), etc. all strongly suggest that the equilibrium and non-equilibrium transition of the T=0 RFIM belong to the same universality class!!!??? Thanks to A. A. Middleton and J.P. Sethna (Yang Liu, KD, condmat 2006)
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