Daniel Stein

Departments of Physics and Mathematics

New York University

Spin Glasses and Related TopicsSpin Glasses and Related Topics

Banff International Research Station

July 21, 2014

Partially supported by US National Science Foundation Grant DMR1207678

Collaborators: Chuck Newman (NYU), Jing Ye (NYU, Princeton), Jon Machta (UMass, Amherst)

Predictability in Nonequilibrium Discrete Spin DynamicsPredictability in Nonequilibrium Discrete Spin Dynamics

Consider the stochastic process σConsider the stochastic process σt t = σ= σt t (ω) with(ω) with

Dynamical Evolution of Ising Model Following a Deep Quench

corresponding to the zero-temperature limit of Glauber dynamics for an corresponding to the zero-temperature limit of Glauber dynamics for an Ising model with HamiltonianIsing model with Hamiltonian

We are particularly interested in σWe are particularly interested in σ00’s chosen from a symmetric Bernoulli ’s chosen from a symmetric Bernoulli product measure .product measure .

The continuous time dynamics are given by independent, rate-1 Poisson The continuous time dynamics are given by independent, rate-1 Poisson processes at each processes at each x x when a spin flip (σwhen a spin flip (σxx

t+0 t+0 = - σ= - σxxt-0t-0) is ) is consideredconsidered. If the change . If the change

in energyin energy

is negative (or zero or positive) then the flip is done with probability 1 (or ½ or is negative (or zero or positive) then the flip is done with probability 1 (or ½ or 0). We denote by P0). We denote by Pω ω the probability distribution on the realizations ω of the the probability distribution on the realizations ω of the

dynamics and by = x Pdynamics and by = x Pω ω the joint distribution of the σthe joint distribution of the σ00’s and ω’s. ’s and ω’s.

In physics, the time evolution of such a model is known as In physics, the time evolution of such a model is known as coarsening, phase coarsening, phase separation, separation, oror spinodal decomposition. spinodal decomposition.

http://www.youtube.com/watch?v=DdnAOfnU6p4http://www.youtube.com/watch?v=DdnAOfnU6p4

ΔΔ

Two questionsTwo questions

1) For a.e. σ1) For a.e. σ00 and ω, does σ and ω, does σ∞∞(σ(σ00, ω) exist? (Or equivalently, for every , ω) exist? (Or equivalently, for every x x does does σσtt

xx(σ(σ00, ω) flip only finitely many times?) , ω) flip only finitely many times?)

2) As 2) As tt gets large, to what extent does σ gets large, to what extent does σt t (σ(σ00, ω) depend on σ, ω) depend on σ0 0 (``nature’’) and(``nature’’) and to to what extent on ω (``nurture’’)? what extent on ω (``nurture’’)?

Phrasing (2) more precisely depends on the answer to (1). Phrasing (2) more precisely depends on the answer to (1).

We will consider two kinds of models: We will consider two kinds of models:

the homogeneous ferromagnet where Jthe homogeneous ferromagnet where Jxyxy=+1 for all {x,y}.=+1 for all {x,y}.

disordered models where a realization disordered models where a realization JJ of the J of the Jxyxy’s is chosen from the ’s is chosen from the

independent product measure Pindependent product measure PJJ of some probability measure on the real line. of some probability measure on the real line.

Simplest case: Simplest case: d d = 1 = 1

TheoremTheorem (Arratia `83, Cox-Griffeath `86): For the (Arratia `83, Cox-Griffeath `86): For the dd = 1 homogeneous = 1 homogeneous ferromagnet, σferromagnet, σ∞∞

xx(σ(σ00, ω) does not exist for a.e. σ, ω) does not exist for a.e. σ00 and ω and every and ω and every xx..

R. Arratia, R. Arratia, Ann. Prob. Ann. Prob. 1111, 706-713 (1983); J.T. Cox and D. Griffeath, , 706-713 (1983); J.T. Cox and D. Griffeath, Ann. Prob. Ann. Prob. 1414, 347-370 , 347-370 (1986). (1986).

Proof: Proof: The joint distribution is translation-invariant and translation-ergodic.The joint distribution is translation-invariant and translation-ergodic.

Define ADefine Axx++ (A (Axx

--) to be the event (in the space of (σ) to be the event (in the space of (σ00,ω)’s) that σ,ω)’s) that σxx∞∞(σ(σ00,ω) exists ,ω) exists

and equals +1 (-1); denote the respective indicator functions as Iand equals +1 (-1); denote the respective indicator functions as Ixx++ (I (Ixx

--). By ). By

translation-invariance and symmetry under σtranslation-invariance and symmetry under σ00 -> -σ -> -σ00, it follows that for all , it follows that for all xx, , (A(Axx

++) = (A) = (Axx--) = p with 0 ≤ p ≤ ½. So, by translation-ergodicity,) = p with 0 ≤ p ≤ ½. So, by translation-ergodicity,

for a.e. σfor a.e. σ00 and ω. and ω.

ProofProof (continued): Suppose now that p > 0. Then for some (continued): Suppose now that p > 0. Then for some xx < < xx’, σ’, σxx∞∞=+1 and =+1 and

σσx’x’∞∞=-1 with strictly positive probability, and so for some t’, σ=-1 with strictly positive probability, and so for some t’, σxx

tt=+1 and σ=+1 and σx’x’tt=-1 for =-1 for

all t≥t’. But for this to be true requires (at least) the following: denote by all t≥t’. But for this to be true requires (at least) the following: denote by S’ S’ the the set of spin configurations on set of spin configurations on ZZ such that σ such that σx x = +1 and σ= +1 and σx’ x’ = +1. Then one needs = +1. Then one needs

the transition probabilities of the Markov process σthe transition probabilities of the Markov process σ tt to satisfy to satisfy

But this is not so, since for any such σ, we would end up with σBut this is not so, since for any such σ, we would end up with σxxt+1 t+1 = -1 if = -1 if

during the time interval [t,t+1] the Poisson clock at x’ does not ring while those during the time interval [t,t+1] the Poisson clock at x’ does not ring while those at at xx’-1, ’-1, xx’-2, …, ’-2, …, x x each ring exactly once and in the correct order (and all each ring exactly once and in the correct order (and all

relevant coin tosses are favorable). relevant coin tosses are favorable).

What about higher dimensions?What about higher dimensions?

TheoremTheorem (NNS ‘00): In the (NNS ‘00): In the dd = 2 homogeneous ferromagnet, for a.e. σ = 2 homogeneous ferromagnet, for a.e. σ00 and ω and ω and for every and for every xx in in ZZ22, σ, σxx

t t (ω) flips infinitely often. (ω) flips infinitely often.

S. Nanda, C.M. Newman, and D.L. Stein, pp. 183—194, in S. Nanda, C.M. Newman, and D.L. Stein, pp. 183—194, in On Dobrushin’s Way (from Probability On Dobrushin’s Way (from Probability Theory to Statistical Physics) Theory to Statistical Physics), eds. R. Minlos, S. Shlosman, and Y. Suhov, Amer. Math. Soc. , eds. R. Minlos, S. Shlosman, and Y. Suhov, Amer. Math. Soc. Trans. (2) 198 (2000).Trans. (2) 198 (2000).

Higher dimensions: remains open. Older numerical work (Stauffer ‘94) Higher dimensions: remains open. Older numerical work (Stauffer ‘94) suggests that every spin flips infinitely often for dimensions 3 and 4, but a suggests that every spin flips infinitely often for dimensions 3 and 4, but a

positive fraction (possibly equal to 1) of spins flips only finitely often for positive fraction (possibly equal to 1) of spins flips only finitely often for dd ≥ 5. ≥ 5.

D. Stauffer, D. Stauffer, J. Phys. AJ. Phys. A 2727, 5029—5032 (1994). , 5029—5032 (1994).

We’ll return to the question of predictability in homogeneous Ising ferromagnets, We’ll return to the question of predictability in homogeneous Ising ferromagnets, but first we’ll look at the behavior of σbut first we’ll look at the behavior of σ∞∞(σ(σ00,ω) for ,ω) for disordereddisordered Ising models. Ising models.

Will provide proof in one dimension.Will provide proof in one dimension.

In some respects, this case is In some respects, this case is simpler simpler than the homogeneous one.than the homogeneous one.

Theorem (NNS ‘00): If μ has finite mean, then for a.e. Theorem (NNS ‘00): If μ has finite mean, then for a.e. JJ, σ, σ00 and ω, and for every and ω, and for every xx, there are only finitely many flips of σ, there are only finitely many flips of σxx

tt that result in a that result in a nonzerononzero energy change. energy change.

Recall that the JRecall that the Jxyxy’s are chosen from the independent product measure P’s are chosen from the independent product measure PJJ of of

some probability measure on the real line. Let μ denote this measure.some probability measure on the real line. Let μ denote this measure.

It follows that a spin lattice in any dimension with continuous coupling disorder It follows that a spin lattice in any dimension with continuous coupling disorder having finite mean (e.g., Gaussian) has a limiting spin configuration at all sites. having finite mean (e.g., Gaussian) has a limiting spin configuration at all sites.

But the result holds not only for systems with continuous coupling disorder. It But the result holds not only for systems with continuous coupling disorder. It holds also for discrete distributions and even homogeneous models where holds also for discrete distributions and even homogeneous models where

each site has an odd number of nearest neighbors (e.g., honeycomb lattice in each site has an odd number of nearest neighbors (e.g., honeycomb lattice in 2D).2D).

ProofProof ( (1D1D only): Consider a chain of spins with couplings J only): Consider a chain of spins with couplings Jx,x+1 x,x+1 chosen from a chosen from a

continuous distribution (which in continuous distribution (which in 1D1D need not have finite mean). Consider the need not have finite mean). Consider the doubly infinite sequences xdoubly infinite sequences xnn of sites where is a of sites where is a strictstrict local maximum local maximum

and yand yn n in the interval (xin the interval (xnn, x, xn+1n+1) where is a ) where is a strict strict local minimum: local minimum:

That is, the coupling magnitudes are strictly increasing from yThat is, the coupling magnitudes are strictly increasing from yn-1 n-1 to xto xnn and strictly and strictly

decreasing from xdecreasing from xnn to y to ynn..

Now notice that the coupling is a ``bully’’; once it’s satisfied (i.e., Now notice that the coupling is a ``bully’’; once it’s satisfied (i.e.,

, the values of , the values of and and can never change thereafter, can never change thereafter,

regardless of what’s happening next to them. For all other spins in {yregardless of what’s happening next to them. For all other spins in {yn-1n-1+1,y+1,yn-n-

11+2, …, y+2, …, ynn}, σ}, σyy∞∞ exists and its value is determined so that exists and its value is determined so that

JJx,yx,yσσxx∞∞σσyy

∞ ∞ > 0 for x and y=x+1 in that interval.> 0 for x and y=x+1 in that interval.

In other words, there is a In other words, there is a cascade of influence cascade of influence to either side of {xto either side of {xnn, x, xnn+1} until y+1} until yn-n-

11+1 and y+1 and ynn, respectively, are reached., respectively, are reached.

PredictabilityPredictability

Define ``order parameter’’ qDefine ``order parameter’’ qDD = lim = limt->∞ t->∞ qqtt , where, where

TheoremTheorem (NNS ‘00): For the one-dimensional spin chain with continuous (NNS ‘00): For the one-dimensional spin chain with continuous coupling disorder, qcoupling disorder, qDD=½. =½.

ProofProof: Choose the origin as a typical point of : Choose the origin as a typical point of ZZ and define X=X( and define X=X(JJ) to be the x) to be the xnn

such that that 0 lies in the interval {ysuch that that 0 lies in the interval {yn-1n-1+1,y+1,yn-1n-1+2, …, y+2, …, ynn}. Then σ}. Then σ00∞∞ is completely is completely

determined by (determined by (JJ and) σ and) σ00 if J if JX,X+1X,X+1σσXX00σσX+1X+1

00 > 0 (so that <σ > 0 (so that <σ00>>∞∞ = +1 or -1) and = +1 or -1) and

otherwise is completely determined by ω (so that <σotherwise is completely determined by ω (so that <σ00>>∞∞ = 0). Thus q = 0). Thus qDD is the is the

probability that σprobability that σXX00 σ σX+1X+1

0 0 = sgn(J= sgn(JX,X+1X,X+1), which is ½.), which is ½.

Also P.M.C. de Oliveira, CMN, V. Sidoravicious, and DLS, Also P.M.C. de Oliveira, CMN, V. Sidoravicious, and DLS, J. Phys. A J. Phys. A 3939, 6841-6849 (2006). , 6841-6849 (2006).

Numerical workNumerical work

J. Ye, J. Machta, C.M. Newman and D.L. Stein, J. Ye, J. Machta, C.M. Newman and D.L. Stein, Phys. Rev. E Phys. Rev. E 8888, 040101 , 040101 (2013): simulations on (2013): simulations on L L xx L L square lattice with square lattice with

Have to use finite-size scaling approach.Have to use finite-size scaling approach.

`̀``Stripe states’’ occur roughly 1/3 of the time ``Stripe states’’ occur roughly 1/3 of the time (V. Spirin, P.L. Krapivsky, and S. Redner, (V. Spirin, P.L. Krapivsky, and S. Redner,

Phys. Rev. EPhys. Rev. E 6363, 036118 (2000)). , 036118 (2000)).

To distinguish the effects of nature vs. nurture, we simulated a To distinguish the effects of nature vs. nurture, we simulated a pairpair of Ising of Ising lattices with identical initial conditions (i.e., ``twins’’) (lattices with identical initial conditions (i.e., ``twins’’) (cf.cf. damage spreading). damage spreading).

Examine the overlap q between a pair of twins at time t: Examine the overlap q between a pair of twins at time t:

We are interested in the time evolution of the mean We are interested in the time evolution of the mean and its final valueand its final value

when the twins have reached absorbing states. when the twins have reached absorbing states.

Looked at 21 lattice sizes from L = 10 to L = 500. For each size studied 30,000 Looked at 21 lattice sizes from L = 10 to L = 500. For each size studied 30,000 independent twin pairs out to their absorbing states (or almost there). independent twin pairs out to their absorbing states (or almost there).

for the largest sizes gives a ``heritability exponent’’ θfor the largest sizes gives a ``heritability exponent’’ θhh=0.22±0.02.=0.22±0.02.

Look at vs. Look at vs. tt for several for several LL. .

The plateau value decreases from small to large The plateau value decreases from small to large LL. A power law fit of the form . A power law fit of the form

Next look at vs. Next look at vs. L L for sizes 10 to 500. for sizes 10 to 500.

The solid line is the best power law fit for size 20 to 500 and corresponds to The solid line is the best power law fit for size 20 to 500 and corresponds to

Finite size scaling ansatzFinite size scaling ansatz

Use the fact that during coarsening, the typical domain size grows as tUse the fact that during coarsening, the typical domain size grows as t1/z1/z, with , with

z = 2 for zero-temperature Glauber dynamics in the 2D ferromagnet.z = 2 for zero-temperature Glauber dynamics in the 2D ferromagnet.

A.J. Bray, A.J. Bray, Adv. Phys. Adv. Phys. 4343, 357-459 (1994). , 357-459 (1994).

Postulate the finite-size scaling form , where the functionPostulate the finite-size scaling form , where the function

f(x) is expected to behave as f(x) is expected to behave as

So the t->∞ behavior is , giving b = zθSo the t->∞ behavior is , giving b = zθh h = 2θ= 2θhh. .

Summary of 2D Uniform Ferromagnet Summary of 2D Uniform Ferromagnet

For finite For finite L, L, there are limiting absorbing states and overlaps there are limiting absorbing states and overlaps with with bb = 0.46 ± 0.02. = 0.46 ± 0.02.

appears to approach as with θappears to approach as with θhh = 0.22 ± 0.02. = 0.22 ± 0.02.

A finite size scaling analysis suggests that b = 2θA finite size scaling analysis suggests that b = 2θhh, consistent with our , consistent with our

numerical results. numerical results.

Since θSince θhh > 0, the > 0, the 2D2D Ising model displays weak LNE. But given the Ising model displays weak LNE. But given the

smallness of θsmallness of θhh, information about the initial state decays slowly., information about the initial state decays slowly.

Next StepsNext Steps

How does qHow does qDD behave as a function of dimension? (Currently studying with behave as a function of dimension? (Currently studying with

Jing Ye, Jon Machta, and Dan Jenkins)Jing Ye, Jon Machta, and Dan Jenkins)

What is the behavior of the uniform ferromagnet in d>2?What is the behavior of the uniform ferromagnet in d>2?

Other types of spin systems (±J spin glass, Potts models, others)Other types of spin systems (±J spin glass, Potts models, others)

Thank you!Thank you!

Questions?Questions?

Heritability and PersistenceHeritability and Persistence

Persistence: in the context of phase ordering kinetics, persistence is defined as Persistence: in the context of phase ordering kinetics, persistence is defined as the fraction of spins that have the fraction of spins that have notnot flipped up to time flipped up to time tt. This quantity is found to . This quantity is found to

decay as a power law with exponent θdecay as a power law with exponent θpp called the called the persistence exponentpersistence exponent..

Numerical simulations on the Numerical simulations on the 2D2D Ising model yield θ Ising model yield θpp = 0.22 (Stauffer, = 0.22 (Stauffer,

J. Phys. A J. Phys. A 2727, 5029-5032 (1994)) and θ, 5029-5032 (1994)) and θpp = 0.209 ± 0.002 (Jain, = 0.209 ± 0.002 (Jain, Phys. Rev. E Phys. Rev. E

5959, R2493-R2495 (1999)). Within error bars of our θ, R2493-R2495 (1999)). Within error bars of our θhh = 0.22 ± 0.02. = 0.22 ± 0.02.

Moreover, our exponent b = 0.46 ± 0.02 describing the finite size decay of Moreover, our exponent b = 0.46 ± 0.02 describing the finite size decay of heritability can be directly compared to the finite-size persistence exponent heritability can be directly compared to the finite-size persistence exponent

θθIsingIsing = 0.45 ± 0.01 (Majumdar and Sire, = 0.45 ± 0.01 (Majumdar and Sire, Phys. Rev. Lett.Phys. Rev. Lett. 7777, 1420-1423 (1996))., 1420-1423 (1996)).

So, is there something deeper going on? So, is there something deeper going on?

Heritability and Persistence (continued)Heritability and Persistence (continued)

In In 1D1D one can compute analytically both the persistence exponent and the one can compute analytically both the persistence exponent and the heritability exponent. One finds there that θheritability exponent. One finds there that θpp = 3/8, but θ = 3/8, but θhh = ½ and b = 1. = ½ and b = 1.

Currently looking at ferromagnets and disordered models in higher dimensions, Currently looking at ferromagnets and disordered models in higher dimensions, and also Potts models. and also Potts models.

Stay tuned …Stay tuned …

How does one define and study predictability in systems How does one define and study predictability in systems

where σwhere σ∞∞ does does notnot exist? exist?

C.M. Newman and D.L. Stein, C.M. Newman and D.L. Stein, J. Stat. Phys.J. Stat. Phys. 9494, 709-722 (1999). , 709-722 (1999).

NS ‘99: Consider the NS ‘99: Consider the dynamically averaged dynamically averaged measure κmeasure κtt; that is, the distribution ; that is, the distribution

of σof σtt over dynamical realizations ω for fixed over dynamical realizations ω for fixed JJ and σ and σ00. Two possibilities were . Two possibilities were conjectured:conjectured:

Even though σEven though σtt has no limit σ has no limit σ∞∞ for a.e. for a.e. JJ, σ, σ00 and ω, κ and ω, κtt doesdoes have a limit κ have a limit κ∞∞..

κκtt does not converge as t -> ∞. (This has been proved to occur for some does not converge as t -> ∞. (This has been proved to occur for some systems; see Fontes, Isopi, and Newman, systems; see Fontes, Isopi, and Newman, Prob. Theory Rel. Fields Prob. Theory Rel. Fields 115115, 417-, 417-

443 (1999).)443 (1999).)

We refer to the first as ``weak local nonequilibration (weak LNE)’’, and to the We refer to the first as ``weak local nonequilibration (weak LNE)’’, and to the second as ``chaotic time dependence (CTD)’’ .second as ``chaotic time dependence (CTD)’’ .