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Continuous Phase TransitionsJim Sethna, Physics 653, Fall 2010
Yanjiun Chen, Stefanos Papanikolaou, Karin Dahmen, Olga Perkovi, Chris
Myers, Matt Kuntz, Gianfranco Durin, Stefano Zapperi,
Experiment
Theory
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The Ising Model at TcStructure on All Scales
Continuous Phase TransitionCompetition Entropy vs. EnergyThermal DisorderHigh Temperature:
Random
Low TemperatureLong-Range Order
Critical PointTc = 2/log(1+2) ~ 2.27Fluctuations on All Scales
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PercolationStructure on All Scales
Connectivity TransitionPunch Holes at Random,
Probability 1-P
Pc
=1/2 Falls Apart
(2D, Square Lattice, Bond)
Static (Quenched) Disorder
Largest Connected Cluster
P=PcP=0.51P=0.49
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EarthquakesSpatially Extended Events of All Sizes
Earthquakes of Many Sizes: 1995
Burridge-Knopoff (Carlson & Langer)http://simscience.org/crackling/Advanced/Earthquakes/EarthquakeSimulation.html
Earthquakes of All SizesGutenberg-Richter Law:
Probability ~ Size-Power
Simple Block-Spring ModelNo disorderSlow driving rate (cm/year)
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Magnets
Rice
Krispi
es
Paper
Crump
ling
Crackling
noise
Tearing
Paper
Discrete crackles
span enormous
range of sizes.
Should becomprehensible;
scaling theory.
Analogy withhydrodynamics:
Molecules dont matter for
Navier-Stokes fluid flow
Microscopics wont matter
for crackling
Magnets
Foams
SolarFlares
Fracture
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Magnetic Barkhausen NoiseEvents of All Sizes, Structure on All Scales
Barkhausen Noise in Magnets
Magnetic
Avalanches
Fractal
in Time and
Space
Nucleated
Invasion
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Plasticity
Avalanches in Ice
(Miguel et al.)
Avalanches in Nickel Micropillars
(Uchic et al.)
Dislocation Tangle Structure
Dislocation avalanches when bending forks
Ice crackles when it is squeezed
So, surprisingly, do metals
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Universality: Shared Critical BehaviorIsing Model and Liquid-Gas Critical Point
Liquid-Gas Critical Point
-c ~ (Tc-T)
Ar(T) = A CO(BT)
Ising Critical Point
M(T) ~ (Tc-T)
Ar(T) = A(M(BT),T)
Same critical
exponent
=0.332!
Universality: Same Behavior up to Change in Coordinates
A(M,T) = a1 M+ a2 + a3T + (other singular terms)
Nonanalytic behavior at critical point (not parabolic top)
All power-law singularities (, cv,) are shared by magnets, liquid/gas
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Microscopic Details IrrelevantUniversality in Percolation
Bond Percolation Site Percolation
Statistical morphology
of critical point
independent of
microscopic details:
depends only on
dimension of space,
type of transition
(universality class).
(Note site percolation
lighter: overall scale of
order parameter non-
universal)
Site, bond percolation
look the same for big
clusters!
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Coarse GrainingRemove microscopic
details
Continuum limit averageover details in small regions,
get effective laws for coarser
systemExample: majority-rule block-spin transformation (3x3
blocks)
Renormalization group: findeffective block-spin free
energy: new interactions from
old by tracing over microscopic
variables
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The Renormalization GroupWhy Universal? Fixed Point under Coarse Graining
System Space Flows
Under Coarse-Graining
Renormalization Group
Not a groupRenormalizedparameters
(electron charge from QED)
Effect of coarse-graining(shrink system, remove
short length DOF)
Fixed point S* self-similar(coarse-grains to self)
Critical points flow to S*Universality
Many methods (technical)real-space, -expansion, Monte Carlo,
Critical exponents fromlinearization near fixed point
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Slow Driving / Inhomogeneous /
Long-range Forces
Drives to Critical Point
(Earthquakes, Sandpiles, Front
Propagation, Forest Fires)
Spontaneous CriticalityGeneric Scale Invariance; Self-Organized Criticality
Attracting Fixed Point: Phases!Sometimes still fluctuations
(Polymers, random walks,
surface growth)
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Random Walks:
Self Similarity of an N-step walk is
(statistically) like
shrinking by
Endpoint ~ (N a) half as farFractal: self similarMass ~ radiusfractal dimensionRandom walk dimension = 2
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Self-SimilaritySelf-Universality on Different Scales
Ising Model at TcHysteresis Model atRc
Fixed point S* maps
onto itself, at a longer
scale:self-similar.
Models cross C at
critical point Tc, flow
to S*: also self-similar.
Self-similarity Power
Laws
Expand rulers by B=(1+);
Avalanche size distribution
D[S] = A D[C S]
=(1+a) D[(1+c)S)]
a D = -c S dD/dS
D[S] = D0 S-a/c
Universalcritical exponentsc=df=1/, a/c= : D0 system dependent
Ising Correlation C(x) ~ x-(d-2+)at Tc,random walkx~t1/2
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Scaling Near CriticalitySelf-Universality on Different Scales
RG Flow near Critical Point.
Two points that flow toward one
another must be similar on long
length scales.
f[4](Tc-t, x) = f[3](Tc-Et,x)
so
f(Tc-t,y)=Af(Tc-t,By) ~ f(Tc-Et,y)
at large y: the system is similar to
itself at a different set of
parameters.
M(Tc-t)=AM(Tc-t)=M(Tc-Et)
(1+/) M(Tc-t)=M(Tc-t(1+/))
M ~ (Tc
-t)~ t
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Critical ExponentsCombinations of Greek Letters
: Maximum avalanche size Smax ~ (R-Rc)-
:Correlation Length ~ (R-Rc)-:Probability of AvalancheP(S, Rc, Hc) ~ S
-
=+:Integrated ProbabilityPint(S, R) ~ S-(+)
:Fractal Dimension 1/
z:Duration T~ z~ (R-Rc)-z
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Hysteresis Model for MagnetsT=0 Driven Random-Field Ising Model
H=-ij nn J Si Sj i H Si hi SiSi = 1, magnetic domain
Jcoupling between neighboring spins
Hexternal applied fieldhirandom field at site, dirt,
chosen from Gaussian widthR
P(h) = Gaussian RMS widthR
Dynamics
Start all spins down,H=-Increase field slowlySpin flips when pushed overInitial spin 13 pushed byHPushes neighbors: avalanche!V(t) = number of spins in shell t
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Simulation at the critical disorder(Chris Pelkie)
Avalanches of all scalesEarly small avalanches,growing in size
Infinite (red) avalanche,large jump in magnetizationSmall final avalanches fillin gaps
System tuned to special
critical disorderR
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Phase Transition in Nucleated HysteresisCritical Disorder: First Infinite Avalanche
R=2 R=2.5R=Rc=2.16
Transition in Shape of Hysteresis Loop
AtRc,
M-Mc~(H-Hc)1/
What happens
away fromRc?
Small Disorder
Neighbors Dominate
One Big Avalanche
(First Spin Triggers
All)
Large Disorder
Dirt Dominates
Many Small
Avalanches
(Each Spin to
Itself)
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Scaling FunctionsSelf-Universality away from Criticality
Universal Scaling Function DScaling Collapse: Plot S(+)D[S,R] vs. S/(R-Rc)
-, measure D(inset)
M(H,T)=(Tc-T)M(H/(Tc-T)
); C(x,t,T)=x-(2-d+)C(x/|T-Tc|-,t/|T-Tc|
-)
Avalanche Size DistributionD(S)AtRcget Power Law
D(S) ~ S-S-(+)
Big ones cut off at (R-Rc)-
Scale invariance: write as powerlaw times function of one fewer
variable
D[S,R] = S-(+)D[S/(R-Rc)-]
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Real Barkhausen NoiseMotion of single fronts (Robbins, Fisher, Bouchaud)
http://www.ien.it/~durin/bk_intro.html
Gianfranco Durin 880660m 440330m
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Demagnetizing FieldHow magnets self-organize to front depinning point
Long-range dipolar
fields (+/- = N/S) cost
energy mostly due to
non-canceling regions
(hence ~ M,
net magnetizationM
demagnetization
factor).
Extra
M
Once the external field
depins a front, why does
it ever stop moving?
acts as much like T-Tc, a relevantperturbation
that cuts off the largest avalanches
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Demagnetizing Field Front propagation: limits
sizes of avalanches
=10-5 =10-7
Self-similar at different Rescaling w=b w and h=bh
makes look like b-x
YJ Chen,
Others
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Self-AffineFront propagation: Heights
and Widths Scale Differently
Cut bottom left-hand quarterRescale widths by 2, heights by 2
Effective lower demagnetizing field:larger avalanches
Fronts appear statistically similar: self-affine
YJ Chen,
Others
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Barkhausen Noise Size Distributions3D Universality classes
Different systems, same exponentsExperiment and theory, same exponents
Universality!
Avalanche size
distributions (and
other critical
exponents) cluster into
two families. One isthefront propagation
model, the other is a
mean-field theory due
to long-range forces.
(Our model doesnt describeany of the experiments.)
Gianfranco Durin
CutoffSmax ~
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Beyond Power LawsUniversalScaling Functions
Functions of one variable become power laws at critical points
Functions ofNvariables become power laws times universal
functions ofN-1 scaling variables
P(S) = bx P(S/bdf) = bnx P(S/bndf) = = s-
P(S, H, W |) = by P(S/bdf,H/b,W/b |/bxk)
= = s-P(H1+/S, W1+1//S, /S)
Universal scaling functions forms:
avalanche shape, T1/z-1V(t/T)avalanche energy, S2E(1/zS)avalanche size/duration,
S-P (S/)
Ising model, other critical points
magnetization rM(h/r)correlation length
(T,H)=t-Y (h/t)
finite size scaling
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Avalanche Temporal StructureDuration scaling, fractals, average shapes
Hierarchical structure in timeavalanche almost stops
many times
Average shape for given
duration: dashed green line
Average size S grows with
duration
S ~ ~Tz
Another power law
Stefanos
Papanikolaou,
Others
Duration T
AverageSize
Time t
V(t)[nV]
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Scaling
CollapsesUniversalfunctions
t/T0 1 t/T0 1
V(t/T)/
Vmax
Experiment Theory
Scaling CollapseDivide variables byscale (time T,
voltage Vmax)
Universal scalingform (parabola MF)Demagnetizingfield crossover
flattening
Scaling away from
criticality?
t (s)
(t,T
)(
nV)
Average (t,T) over
avalanches of duration T
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Avalanche Spatial StructureBeyond Critical Exponents
All geometrical features
of large avalanches
should be universal.
Correlation functions,fractal dimensions
Aspect ratiosTopology
(holes,
interconnectedness)(string theory)
Front shapesHeight, widthdistributions
880X660m 440 X330m
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Demagnetizing Field Front propagation: limits
sizes of avalanches
=10-5 =10-7
Self-similar at different Rescaling w=b w and h=bh
makes look like b-x
YJ Chen,
Others
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Self-AffineFront propagation: Heights
and Widths Scale Differently
Cut bottom left-hand quarterRescale widths by 2, heights by 2
Effective lower demagnetizing field:larger avalanches
Fronts appear statistically similar: self-affine
YJ Chen,
Others
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Avalanche heightsScaling away from criticality
h
(h)-
(2-
)(1+)/hA(h)
h
A(h|
)
A(h|) similar to XnA(h /Yn| 10n)
A(h|) = X-log10A(h / Ylog10)
= -log10XA(h log10Y)= -()A(h )
X
Y
A(h ) is a universal
scaling function of the
scaling variableh .
YJ Chen,
Others
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Traditional Equilibrium CriticalityEnergy versus Entropy
Traditional Equilibrium Critical Points
Ising, Potts (N-state), Heisenberg (3D vector)Helium: 3D XY model2D XY Kosterlitz-Thouless Transition, 2D Melting, Hexatic PhasesLiquid Crystals (Nematic to Smectic A)Wetting TransitionsAhlers: Superfluid Density versus T
Five decades oft = |Tc-T|/TcPower law Singular correction to scalingx
s/= k |Tc-T| (1+d |Tc-T|
x)
exp = 0.67490.0007 , xexp=0.50.1th = 0.6690.002 , xth=0.5220.017 Ahlers Rev Mod Phys 52, 489 (1980).
Theory: LeGuillou & Zinn-Justin
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Quantum Phase Transitions vs. Disorder, Field,
Quantum Phase TransitionsMetal-Insulator Transitions (Localization)Superconductor-Insulator TransitionsTransitions between Quantum Hall PlateausMacroscopic Quantum Tunneling
(Quantum Coherence and Schrdingers Cat)Kondo Effect
Goldman & Markovic, Phys. Today, p. 39, Nov. 1998
SC to Insulator with
Film Thickness
Right: Resistance vs. TLeft: Scaling Plot
R/Rc vs. |d-dc|/T1/z
Left Inset: Phase
Boundary (B, d)
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Disordered SystemsFancy Tools, Still Controversial
Disordered Systems (Disorder vs. Temperature)
Spin Glasses: Dilute Magnetic AlloysFrustration, Competing Ferro/Antiferro, RKKYLong-range order in Time lim(t) Replica Theory vs. ClustersNeural Networks, Tweed in Martensites
Random Field Ising ModelsDimensional Reduction, Supersymmetry WrongDiverging Barriers, Analogies to Glasses?
Vortex Glass Transition,
Frustration
Tweed Precursors
Martensites Change Shapehttp://www.lassp.cornell.edu/sethna/Tweed/What_Are_Martensites.html
Form Stripes
But
First
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Dynamical Systems and ChaosCoarse-Graining in Time
Low Dimensional Dynamical Systems
Bifurcation TheorySaddle-Node, Intermittency, Pitchfork, HopfNormal Forms = Universality Classes
Feigenbaum Period DoublingTransition from Quasiperiodicity to Chaos:
Circle Maps
Breakdown of the Last KAM Torus:Synchrotrons and the Solar System
Fixed Points vs. Period Doubling Cascade
High-Dimensional Systems
Turbulence?Spatiotemporal Defect Chaos?Avalanches
Bodenschatz
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Logical Satisfiability and NP-CompletenessSelman, Kirkpatrick, Gomes, Mzard, Montanari, Monasson,
Worst-case problems exponentially hard
Typical problem hard only near phase transition Two phasetransitions!
RG: Coppersmith
Universality?
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Continuous Phase TransitionsJim Sethna, Physics 653, Fall 2010
Yanjiun Chen, Stefanos Papanikolaou, Karin Dahmen, Olga Perkovi, ChrisMyers, Matt Kuntz, Gianfranco Durin, Stefano Zapperi,
Experiment
Theory