Continuous Phase Transitions

8/2/2019 Continuous Phase Transitions

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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

Documents

Continuous Phase Transitions