Topics in coarsening phenomena

in Fundamental Problems in Statistical Physics XII Leuven, Aug 30 - Sept 12, 2009

Leticia F. Cugliandolo

Université Pierre et Marie Curie - Paris VI

Laboratoire de Physique Théorique et Hautes Energies

August 14, 2009

Contents

1 Introduction 2

2 Models 22.1 Geometric and probabilistic models . . . . . . . . . . . . . . . . . 2

2.1.1 Site percolation . . . . . . . . . . . . . . . . . . . . . . . . 22.1.2 Directed percolation . . . . . . . . . . . . . . . . . . . . . 3

2.2 Hamiltonian systems . . . . . . . . . . . . . . . . . . . . . . . . . 32.2.1 The Ising model . . . . . . . . . . . . . . . . . . . . . . . 32.2.2 The Potts model . . . . . . . . . . . . . . . . . . . . . . . 42.2.3 Microscopic dynamics . . . . . . . . . . . . . . . . . . . . 4

2.3 Kinetically constrained models . . . . . . . . . . . . . . . . . . . 52.3.1 The spiral model . . . . . . . . . . . . . . . . . . . . . . . 5

3 Geometric description of phase transitions 73.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2 Number densities . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.3 Fractal exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4 Coarsening 94.1 Critical coarsening . . . . . . . . . . . . . . . . . . . . . . . . . . 124.2 Dynamic scaling hypothesis . . . . . . . . . . . . . . . . . . . . . 12

4.2.1 The growing length . . . . . . . . . . . . . . . . . . . . . . 124.2.2 Scaling functions . . . . . . . . . . . . . . . . . . . . . . . 17

5 Dynamics of (weakly) random systems 175.1 Crossover in the growing length . . . . . . . . . . . . . . . . . . . 175.2 Super-universality . . . . . . . . . . . . . . . . . . . . . . . . . . 18

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6 Statistics and geometry of coarsening 196.1 General picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

7 Coarsening in the spiral model 22

1 Introduction

These notes are preliminary. They should serve as a guideline for my (hope-fully blackboard) lectures. My aim is to discuss some recent developments inthe theory of coarsening phenomena [1]. Coarsening is possibly the simplestexample of macroscopic non-equilibrium relaxation and it has very far-reachingpractical applications. This problem, although pretty old and rather well under-stood qualitatively, is still far from having a close and satisfactory quantitativedescription that goes beyond the pretty successful but somehow deceiving simpleuse of the dynamic scaling hypothesis.

After presenting the models I shall focus on and the coarsening phenomenon,I shall discuss the following aspects: (1) Super-universality; (2) the statistics andgeometry of evolving ordered structures; (3) cooling rate dependencies and theKibble-Zurek mechanism; (4) domain growth in a kinetically constrained spinmodel, the two dimensional spiral model. References to the original articleswill be given in the main text. My results on this field have been obtained incollaboration with the colleagues that I warmly thank in the acknowledgements.

2 Models

In this section we introduce examples of three kinds of problems with phasetransitions: purely geometric models with percolation as the standard example,Hamiltonian models with Ising and Potts spin models as typical instances, andkinetically constrained spin systems focusing on the bi-dimensional spiral model.

2.1 Geometric and probabilistic models

We recall the definition of two purely geometric and probabilistic models:site percolation and directed percolation. These models change behaviour at aspecial value of their control parameter, pc, and this threshold has many pointsin common with usual thermal second-order phase transitions. Percolation the-ory deals mainly with the critical phenomenon and the study of cluster sizesand their geometric properties.

2.1.1 Site percolation

Percolation is the propagation of activities through connected space. Thesite percolation model [2] is defined as follows. For each site on the considered

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lattice one tosses a coin independently. With probability p the site is occupiedand with probability 1− p it remains empty. A configuration is constructed bysweeping the lattice once and filling the sites (or not) independently with thisrule.

Clusters are defined as groups of first-neighbour filled (or empty) sites. Ata (lattice dependent) threshold pc an infinite cluster (on an infinite lattice)appears. Strictly speaking, for p ≤ pc (1−p ≤ pc) there is no infinite connected-component of filled (empty) sites with probability one while for p > pc (1− p >pc) there is a unique infinite connected-component of filled (empty) sites withprobability one. The critical values of pc in some typical bi-dimensional latticesare: pc = 1/2 (triangular), pc ≈ 0.59 (square), pc ≈ 0.62 (honeycomb).

2.1.2 Directed percolation

Directed percolation [3] is percolation with a special direction along whichthe activity propagates. It mimics filtering of fluids through porous materialsalong a given direction. For example, in the percolation of coffee making, thesource is the top surface of the grounded coffee that receives the water andthe activity is to have water flow. Varying the microscopic connectivity of thepores, these models display a phase transition from a macroscopically permeable(percolating) to an impermeable (non-percolating) state. Directed percolationon a 2d square lattice has a transition at pc ≃ 0.705.

More generally, the terms percolation and directed percolation stand fora universality classes of continuous phase transitions which are characterizedby the same type of collective behavior on large scales. A number of criticalexponents, linked to the size of the percolation cluster, correlation functions, andso on can be defined similarly to what is done in thermal critical phenomena [2].

2.2 Hamiltonian systems

Physical models, as well as some mathematical problems such as those re-quiring optimisation, have an energy or cost function associated to each con-figuration. In equilibrium, the configurations are sampled with a probabilitydistribution function that depends on their energy and the external parameterssuch as temperature or a chemical potential. The experimental conditions, thatis to say whether the system is isolated or in contact with thermal or particlereservoirs, dictate the statistical ensemble to be used (micro-canonical, canon-ical or gran-canonical). Out of equilibrium, say when the system is let evolvefrom an initial condition that is not one selected from the equilibrium measure,the system wanders in phase space in a manner that we shall discuss below.

2.2.1 The Ising model

Anisotropic magnets are modelled in a very simple way by using Ising spins,si = ±, to describe the magnetic moments and by choosing an adequate inter-

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action among them. The Hamiltonian of the d dimensional Ising model is

H = −∑

〈ij〉

Jijsisj . (1)

The spin variables are placed on the vertices of a lattice and the sum runs overnearest-neighbours. The exchange interactions Jij are all positive, favouringferromagnetism but they can, in principle, take different values. We shall focuson the usual uniform case Jij = J , that is to say the clean model, and thecase in which the Jij ’s are quenched random variables drawn from a probabilitydistribution with positive support, a dirty case.

A totally random spin configuration in which si = ±1 with probability a halfis a realization of a site percolation configuration with p = 1/2 and, in d = 2,neither positive nor negative clusters percolate. The identification is achievedby defining site occupation variables ni = (si + 1)/2 = 0, 1.

The static properties of the Ising model (1) are more easily analyzed in theexperimentally relevant canonical ensemble in which the system is coupled to athermal reservoir at temperature T . A second-order continuous phase transitionat a finite critical temperature, Tc, separates a high-temperature disorderedparamagnetic phase from a low-temperature ferromagnetic ordered one. Tcdepends on d and the statistics of the interaction strength. The nature of thetwo phases is understood since the development of the Curie-Weiss mean-fieldtheory while the critical phenomenon has been accurately described with thehelp of the renormalization group, that allows for the computation of all criticalexponents [4].

2.2.2 The Potts model

A natural generalization of the Ising model is the Potts model [5] in whichthe spins σi take q integer values from 1 to q. The Hamiltonian is given by

H = −∑

〈ij〉

Jijδσiσj (2)

where the sum is over nearest-neighbours on the lattice. Either uniform, Jij = J(pure case), or bimodal P (Jij) = pδ(Jij−J1)+(1−p)δ(Jij −J2) (random case)interactions are usually studied. In d = 2 the transition, discontinuous for q > 4and continuous for q ≤ 4, occurs at (eβcJ1 − 1)(eβcJ2 − 1) = q in the randomcase. Tc for the pure limit is recovered by setting J1 = J2.

Soap films and general grain growth [6] are two physical systems mimickedby Potts models. In 2d these physical systems are made of polygonal-like cellsseparated by thin walls endowed with line energy; different ‘colour’ domainscorrespond to different cells.

2.2.3 Microscopic dynamics

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Once the Hamiltonians have been defined, we need to assign updating rulesto the microscopic variables. Classically, the spins do not have an intrinsicdynamics. Quite generally, we are interested in the evolution of systems incontact with an environment that provides thermal agitation and dissipation.Thus, the evolution is assumed to be stochastic [7] and it does not conserve thetotal energy. Since we want to let the system reach equilibrium at sufficientlylong times we use dynamic rules that satisfy detailed balance:

W (C → C′)Peq(C) = W (C′ → C)Peq(C′) , (3)

with W (C → C′) the transition probability from configuration C to configu-ration C′ and Peq(C) = e

−βH(C)∑

C′ e−βH(C′) the Boltzmann weight. Finally,

one has to decide whether there are conserved quantities, for instance the globalmagnetization, M =

∑Ni=1 si, or a local magnetization, e.g. mloc = si+sj with i

and j nearest-neighbours. The existence of conserved quantities put constraintson the microscopic updates. In the context of Ising and Potts models two typesof dynamics are common:Non-conserved order parameter; there are no constraints on the stochastic movesand two examples are Monte Carlo or Glauber dynamics for Ising spins [8] .Conserved order parameter; it mimics particle-hole exchanges in lattice gas mod-els obtained from a mapping of the Ising model and the local magnetization isconserved. Exchanges of up and down spins are proposed and these are acceptedwith a probabilistic law (Kawasaki dynamics [8]).

2.3 Kinetically constrained models

Spin kinetically constrained models capture many features of real glass-forming systems (see [9] for reviews). These models display no thermodynamicsingularity: their equilibrium measure is simply the Boltzmann factor of inde-pendent spins and correlations only reflect the hard core constraint. Bootstrappercolation arguments allowed C. Toninelli et al. to show that, when defined onfinite dimensional lattices, these models do not even have a dynamic transitionat a particle density that is less than unity in the thermodynamic limit [10].On Bethe lattices instead a dynamical transition similar to the one predictedby the mode coupling theory might occur [11].

2.3.1 The spiral model

A finite dimensional kinetically constrained model with an ideal glass-jamm-ing dynamic transition at a particle density that is different from one, the two-dimensional spiral model, was introduced in a series of papers by C. Toninelliet al. [12]. A binary variable nij = 0, 1 is defined on the sites (i, j) of an L× Lperiodic bi-dimensional square lattice. nij = 1 if a particle occupies the site andnij = 0 otherwise. One defines the couples of neighbouring sites (see Fig. 1):

(i, j + 1),(i+ 1, j + 1), north east (NE) couple;

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NE

ES

SW

WN(i,j) (i,j+1)

(i,j-1)

(i+1,j)

(i-1,j)

0 50 1000

50

100

p=0.99 , t=105

0 50 1000

50

100

p=0.99 , t=107

Figure 1: Left: The neighbouring sites determining the frozen or free to movecharacter of the center site (i, j). Centre and right panel: Configuration of thesystem at two times (t = 105 and t = 107) in a quench to p = 0.99. Black andwhite sites are frozen particles and vacancies, respectively. Red and green sitesare particles and holes that can be updated. Figure taken from [13].

(i+ 1, j),(i+ 1, j − 1), south east (ES) couple;(i, j − 1),(i− 1, j − 1), south west (SW) couple;(i− 1, j),(i− 1, j + 1), north east (WN) couple.

Whether the variable nij may be updated or not depends on the configurationsat these pairs: if the sites belonging to at least two consecutive couples (namely,NE+SE or SE+SW or SW+NW or NW+NE) are empty site (i, j) can be up-dated (either emptied or filled). Otherwise it is blocked. Each site is coupled to aparticle reservoir in such a way that particles can enter or leave the sample fromits full volume. Defining Mij = 0 if site (i, j) is frozen andMij = 1 otherwise theupdating rule can be expressed in terms of the transition ratesW (nij |n′ij) to addor remove a particle as Wp(0|1) = Mij p and Wp(1|0) = Mij (1−p). The modelcan be regarded as a two-level system described by H [s] = −∑ij(2nij − 1)at the inverse temperature β = (1/2) ln[p/(1 − p)]. When p → 1/2 the in-verse temperature vanishes β → 0 whereas for p → 1 it diverges β → ∞. TheBernoulli measure implies ρ = 〈nij〉 = p for the equilibrium density of particles.In equilibrium at zero temperature (p → 1), the lattice is full while at infi-nite temperature (p → 1/2) it is half-filled. The existence of a blocked clusterat p ≥ pc < 1 was shown by studying the critical p above which an equilib-rium configuration cannot be emptied and this coincides with the threshold for2d directed percolation (pc ≃ 0.705). At the transition an infinite cluster ofblocked particles exists and the density of the frozen cluster is discontinuous atpc. Many aspects of the super-cooled liquids slowing down are reproduced close(but below) pc [12].

In Sect. we shall summarize results in [13] where we studied, in particular,

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the dynamics following a quench from p0 < pc to p > pc.

3 Geometric description of phase transitions

Geometric domains are ensembles of connected nearest-neighbour sites withspins pointing in the same direction. Their area is the number of spins belong-ing to the domain. By lumping together with a certain temperature-dependentprobability neighboring spins in the same spin state, spin models can be mappedonto percolation theory. The resulting Fortuin-Kasteleyn spin clusters [14] builtusing pij = 1− e−βJij percolate at the critical temperature, while their percola-tion exponents coincide with the thermal ones. In this way, a purely geometricaldescription of the phase transition in these models is achieved. In the rest ofthis Section we sum up the definitions of the main geometric objects used insuch a static description of critical phenomena.

3.1 Definitions

A number of linear dimensions of a cluster can be defined. The radius ofgyration is

R2g = s−1

s∑

i=1

|ri − rcm|2 (4)

with the centre of mass position given by rcm = s−1∑

i=1 ri.Take the cluster site with the largest (smallest) y component and among

these the one with the largest (smallest) x component. These sites are thetwo ‘end-points’ of the cluster. The end-to-end length of the cluster, ℓ, is theEuclidean distance between these two points.

There are also several ways of defining the surface of a cluster [2]. Differentdefinitions are relevant to different applications that have to do, for example,with the adsorption of particles on the surface. Without entering into all thezoology let us recall some of these definitions.

The external border of a cluster of occupied sites is the set of vacant sitesthat are nearest-neighbours to sites on the cluster and are connected to theexterior by (a) nearest and next-to-nearest neighbours or (b) just next nearestneighbours. Figure 2 (a) and (b) show the external borders of a cluster ofoccupied sites thus defined. Note that in (a) there are two ‘inner’ sites thatbelong to the external border and that are connected to the outside by a narrowneck of width

√2a with a the lattice spacing while in (b) these two sites do not

belong to the border and the ‘fjord’ has been excluded.The hull is the envelop of a cluster, meaning the ensemble of cluster sites

obtained by using a biased walk that encircles the cluster on its left and on itsright linking the lowest lying site to the left and the highest lying site to theright, see Fig. 3. The hull of a cluster of occupied (vacant) sites lies on occupied(vacant) sites.

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The hull-enclosed area is the total area within the hull (including the siteson the hull). It therefore ignores the type of site (occupied or vacant, spin upor spin down) that lies within and counts them all on equal footing.

3.2 Number densities

We call n(s) the number of objects (hulls, clusters, etc.) with s sites perunit number of lattice sites. Scaling theory, some exactly solvable cases (e.g.the Bethe lattice), renormalization group arguments [2], Coulomb gas mappings,conformal field theory calculations, stochastic Loewner evolution techniques [20]and numerical experiments [21] suggest

n(s) ∼ s−τ f [θsσ] for large s , (5)with τ and σ two control parameter and lattice-independent exponents that dodepend on the space dimensionality and θ measuring the distance from critical-ity, say θ = |g−gc| with g the control parameter. The function f(z) approachesa constant for |z| ≪ 1 and it falls-off rapidly for |z| ≫ 1; it thus provides acut-off with a unique cross-over size sξ ≡ |p − pc|−1/σ such that clusters withs < sξ are effectively critical while those with s > sξ are not. At criticality(g = gc) the factor f [θs

σ] = f [0] that suppresses large objects is absent andclusters of all sizes are present.

The exponent σ and the so-called Fisher exponent τ determine all criticalexponents of percolation through scaling relations. See table for their valuesin d = 2. As regards thermal phase transitions, once the relevant Fortuin-Kasteleyn clusters are constructed and analysed their two independent expo-nents σ and τ determine the entire set of thermal critical exponents (see, how-ever, [15]).

3.3 Fractal exponents

Geometric objects at criticality have a fractal behaviour, that is to say [2]

s ∝ ℓD . (6)s is the mass of the object (cluster area, its external border, its hull), ℓ isthe linear size of the cluster, say its radius of gyration, and D is the fractaldimension. The fractal dimension D is related to the exponent τ in eq. (5) viaτ = d/D + 1 where d is the space dimension.

The fractal exponents depend strongly on the precise definition of the ge-ometric object. For instance, Saleur and Duplantier showed that 2d criticalpercolation clusters have DE = 4/3 and DH = 7/4 where H stands for hull andE for external border connected to the exterior by nearest-neighbour vacantsites only [16] (case (b) in Fig. 2).

In [17] we studied the relation between areas and perimeters

A ∝ pα (7)

8

τAG τAH τ

pH D

GA DH

Percolation 18791 [2] 2 [19]157 [16]

9148

74

Ising q = 2 379187 [23] 2 [19]2711 [22]

18796

118

Potts q = 3 15380 [21]1712

Potts q = 4 158 [21]32

Table 1: Geometric exponents. σ = 36/91 in d = 2 percolation. Note thatDAG = 2/τ

AG − 1 and DH = 2/τ

pH − 1.

x x

x

x

x

x

xx

x

x

x

xx

x

x

x

x

x

x

(a)x x

x

x

x

x

xx

x

x

x

xx

x

x

x

x

(b)

Figure 2: Square bi-dimensional lattice. Filled (empty) circles represent occu-pied (empty) sites. Red crosses indicate external border of the cluster of filledsites including (a), or nor (b), vacant sites that are connected to the exterior bya next nearest neighbour (a ‘diagonal’).

with p defined as the number of broken bonds on the external border or, equiva-lently, with a small variant of the Grossman-Aharony construction [2] includingfjords). Note that if A ∝ ℓDA and p ∝ ℓDp with ℓ the linear size of the object,say the radius of gyration, then α = DA/Dp.

4 Coarsening

Take a system in equilibrium in the symmetric (positive g − gc, high tem-perature) phase and quench it into the symmetry breaking (negative g− gc, lowtemperature) phase through a second order phase transition. For concreteness,we focus on problems with a discrete symmetry. Once set into the ordered phasethe system locally selects one (among the possible) equilibrium configurations.

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(c)

Figure 3: Same cluster as in Fig. 2 Encircled are the sites on the cluster thatbelong to its hull. The hull-enclosed area is everything that lies within the hull.

However, different ‘states’ are picked up at different locations and topologicaldefects in the form of domain walls are created. In the course of time thepatches of ordered regions tend to grow while the density of topological defectsdiminishes. In an infinite system this coarsening process goes on forever.

An example of the above process is given by a magnetic system or a Pottsmodel. An initial condition in equilibrium at, for instance, infinite temperatureis just a random configuration in which each spin takes one of its possible valueswith probability 1/q. After a quench below the critical temperature, T < Tc,the ferromagnetic interactions tend to align the neighbouring spins in ‘parallel’direction (same value of the spin) and in the course of time domains of theordered phases form and grow. At any finite time the configuration is such thateach type of domains exist. Under more careful examination one reckons thatthere are some spins reversed within the domains. These ‘errors’ are due tothermal fluctuations and are responsible of the fact that the magnetization ofa given configuration within the domains is smaller than one and close to theequilibrium value at the working temperature (apart from fluctuations due tothe finite size of the domains). At each instant there are as many spins of eachtype (up to fluctuating time-dependent corrections that scale with the squareroot of the inverse system size). As time passes the typical size of the domainsincreases in a way that we shall discuss below.

Coarsening from an initial condition that is not correlated with the equi-librium state and with no bias field does not take the system to equilibriumin finite times with respect to a function of the system’s linear size, L. Moreexplicitly, if the growth law is a power law [see eq. (10)] one needs times of theorder of Lzd to grow a domain of the size of the system. This gives a roughestimate of the time needed to take the system to one of the two equilibriumstates. For any shorter time, domains of the two types exist and the system is

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Figure 4: Snapshot of the 2d Ising model at a number of Monte Carlo stepsafter a quench from infinite to a sub-critical temperature. Left: the up anddown spins on the square lattice are represented with black and white sites.Right: the domain walls are shown in black. Figure taken from [17].

out of equilibrium.We thus wish to distinguish the relaxation time, tr, defined as the time

needed for a given initial condition to reach equilibrium with the environment,from the decorrelation time, td, defined as the time needed for a given config-uration to decorrelate from itself. At T < Tc the relaxation time of any initialcondition that is not correlated with the equilibrium state diverges with thelinear size of the system. The self-correlation of such an initial state evolving atT < Tc decays as a power law and although one cannot associate to it a decaytime as one does to an exponential, one can still define a characteristic timethat turns out to be related to the age of the system.

In contrast, the relaxation time of an equilibrium magnetized configurationat temperature T vanishes since the system is already in equilibrium while thedecorrelation time is finite and given by td ∼ |T − Tc|−νzeq .

The lesson to learn from this comparison is that the relaxation time and thedecorrelation time not only depend on the working temperature but they alsodepend strongly upon the initial condition. Moreover, the relaxation time de-pends on (N,T ) while the decorrelation time depends on (T, tw). For a randominitial condition one has

tr ≃

finite T > Tc ,

|T − Tc|−νzeq T >∼ Tc ,Lzd T < Tc .

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4.1 Critical coarsening

After a quench to the critical point ordered structures grow with a lawRc(t) ≃ t1/zeq and zeq the equilibrium dynamics exponent. This and otherdetails of the critical dynamics can be computed with renormalization grouptechniques [24].

4.2 Dynamic scaling hypothesis

The dynamic scaling hypothesis states that at late times and in the scalinglimit [1]

r ≫ ξ(g) , R(g, t) ≫ ξ(g) , r/R(g, t) arbitrary , (8)

where r is the distance between two points in the sample, r ≡ |~x − ~x′|, andξ(g) is the equilibrium correlation length that depends on all parameters (T andpossibly others) collected in g, there exists a single characteristic length, R(g, t),such that the domain structure is, in statistical sense, independent of time whenlengths are scaled by R(g, t). Time, denoted by t, is typically measured from theinstant when the critical point is crossed. In the following we ease the notationand write only the time-dependence in R. This hypothesis has been provedanalytically in very simple models only, such as the one dimensional Ising chainwith Glauber dynamics or the Langevin dynamics of the d-dimensional O(N)model in the large N limit.

In practice, the dynamic scaling hypothesis implies that the real-space cor-relation function should behave as

C(r, t) ≡ 〈sisj〉|~ri−~rj |=r ≃ F [r/R(t)] . (9)

See Fig. 5 for a test of the scaling hypothesis in the MC dynamics of the 2dPotts model (see the caption for details). In order to fully characterise thecorrelation functions one then has to determine the typical growing length, R,and the scaling functions, fc, F , etc.

4.2.1 The growing length

It turns out that R can be determined with semi-analytic arguments andthe predictions are well verified numerically – at least for clean system. In pureand isotropic systems the growth law is

R(t) = λ t1/zd (10)

with zd the dynamic exponent (see [1]) and λ a material/model dependent pref-actor that weakly depends on temperature and other parameters. In curvaturedriven Ising or Potts cases with non-conserved order parameter the domains aresharp and zd = 2. For systems with continuous variables such as rotors or XY

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0

0.2

0.4

0.6

0.8

1

1 2 3

100

101

102

103

100 101 102 103

r/R(t)

C(r/R

(t))

t

R2(t

)

t

Figure 5: Check of dynamic (super-)scaling in the space-time correlation ofthe Potts model quenched from T0 → ∞ to Tf = Tc(q)/2. Data taken atseveral times (from t = 24 MCs to 211 MCs) for q = 2, 3 and 8, with andwithout weak disorder are shown. Distance is rescaled by the R(t) obtainedfrom C(R, t) = 0.3. Inset: R2(t) against t in a double logarithmic scale forq = 2, 3 and 8 (from top to bottom). The characteristic length, related tothe average domain radius, depends weakly on q and T for the pure model(through the pre-factor). Within the time window explored there are still somesmall deviations from the R(t) ≃ t1/2 expected law in the case q = 8. Whenweak disorder is introduced (not shown), the growth rate is greatly reduced anddeviates from the linear behavior, see Sect. . Figure taken from [35].

models and the same type of dynamics, a number of computer simulations haveshown that domain walls are thicker and zd = 4. The effects of temperatureenter only in the parameter λ and, for clean systems of Ising type, growth isslowed down by an increasing temperature since thermal fluctuation tend toroughen the interfaces thus opposing the curvature driven mechanism. Let uslist some special cases below and sketch how zd can be estimated.Clean one dimensional cases with non-conserved order parameter

In one dimension, a space-time graph allows one to view coarsening as thediffusion and annihilation upon collision of point-like particles that representthe domain walls. In the Glauber Ising chain with non-conserved dynamics onefinds that the typical domain length grows as t1/2 while in the continuous casethe growth is only logarithmic, ln t.Non-conserved curvature driven dynamics (d > 2)

The time-dependent Ginzburg-Landau model allows us to gain some insighton the mechanism driving the domain growth and the direct computation of

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0.1

1

0.1 1 10 100

101102103104

100 101 102 103

r/R(t)

C(r,t

)

tR

2(t

)

r−4/15

t

Figure 6: Test of dynamic scaling in the q = 3 Potts model space-time correla-tion after a quench from equilibrium at T0 = Tc to T = Tc/2. Data at severaltimes (t = 4, . . . , 210 MCs) are displayed. The slope r−η with η = 4/15 is shownas a guide to the eye. Inset: R2(t) for q = 2 (top) and 3 (bottom) togetherwith the law R2 ≃ t. The growth law follows the curvature driven expectationR(t) ≃ t1/2 but the correlation function remembers the fact that the systemwas quenched from a critical initial state. Figure taken from [35].

the averaged domain length. This is a stochastic partial differential equationon a coarse-grained order parameter field with a deterministic force that is phe-nomenologically proposed to derive from a Ginzburg-Landau type free-energy.In clean systems temperature does not play a very important role in the domain-growth process, it just adds some thermal fluctuations within the domains, aslong as it is smaller than Tc. In dirty cases instead temperature triggers thermalactivation.

We focus first on zero temperature clean cases with Ising-like symmetry. AtT = 0 the GL equation is just a gradient descent in a (free-)energy landscape,F . Two terms contribute to F : an elastic energy (∇φ)2 which is minimizedby flat walls if present and a bulk-energy term that is minimized by constantfield configurations, say φ = ±φ0 in an Ising-like case. If the walls are sharpenough, interactions between them can be neglected and the minimization pro-cess implies that regions of constant field, grow and get separated by flatterand flatter walls. Within this scenario one can easily derive the Allen-Cahnequation [25] that states that the local wall velocity is proportional to the localgeodesic curvature:

v ≡ ∂tn|φ = −~∇ · n̂ ≡ − λ2πκ , (11)

14

in all d and points in the direction of reducing the curvature. This equationyields an intuition on the typical growth law in such processes. Take a sphericalwall in any dimension. The local curvature is κ = (d−1)/R where R is the radiusof the sphere within the wall. Equation (11) is recast as dR/dt = −λ(d− 1)/Rthat implies R2(t) = R2(0) − 2λ(d − 1)t and R decreases as t1/2. This calcu-lation implies that all hull-enclosed areas decrease in time with the same law,independently of their own size. This does not imply that all domains shrinkas well since some gain size from the disappearance of inner objects. Tem-perature effects are simply taken into account by a material and T -dependentproportionality constant λ in the Allen-Cahn equation.

In d = 2 the time dependence of the area contained within any finite hullon a flat surface is derived by integrating the velocity around the envelope andusing the Gauss-Bonnet theorem:

dA

dt=

∮

vdl = − λ2π

∮

κdl = −λ(

1 − 12π

∑

i

θi

)

, (12)

where θi are the turning angles of the tangent vector to the surface at the npossible vertices or triple junctions between domains of different colour (we arenow generalizing the discussion to a Potts-model situation with q ≥ 2). In theIsing q = 2 model,

∑

i θi = 0 since there are no such vertices and we obtaindA/dt = −λ for all hull-enclosed areas, irrespective of their size. If, instead,like in soap froths, there is a finite number of such vertices with an angle of2π/3, that is, θi = π/3, ∀i (for highly anisotropic systems, as the Potts modelthe angles are different from 2π/3). The above equation thus reduces to the vonNeumann law [26] for the area An of an n-sided hull-enclosed area:

dAndt

=λ

6(n− 6). (13)

Whether a cell grows, shrinks or remains with constant area depends on itsnumber of sides being, respectively, larger, smaller or equal to 6.Conserved order parameter: bulk diffusion

A different type of dynamics occurs in the case of phase separation (thewater and oil mixture ignoring hydrodynamic interactions or a binary allow).In this case, the material is locally conserved, i.e. water does not transform intooil but they just separate. The main mechanism for the evolution is diffusion ofmaterial through the bulk of the opposite phase. After some discussion, it wasestablished, as late as in the early 90s, that for scalar systems with conservedorder parameter zd = 3 (see [27, 8]).Role of disorder: thermal activation

The situation becomes much less clear when there is quenched disorder inthe form of non-magnetic impurities in a magnetic sample, lattice dislocations,residual stress, etc. Qualitatively, the dynamics are expected to be slower thanin the pure cases since disorder pins the interfaces. In general, based on an

15

Figure 7: Time evolution of a spin configuration; three snapshots of a 2dIM ona square lattice with locally conserved order parameter (Kawasaki dynamics).Figure taken from [18].

argument due to Larkin (and in different form to Imry-Ma) one expects thatin d < 4 the late epochs and large scale evolution is no longer curvature drivenbut controlled by disorder.

A hand-waving argument to estimate the growth law in dirty systems isthe following. Take a system in one equilibrium state with a domain of linearsize R of the opposite equilibrium state within it. This configuration could bethe one of an excited state with respect to the fully ordered one with absoluteminimum free-energy. Call ∆F (R) the free-energy barrier between the excitedand equilibrium states. The thermal activation argument yields the activationtime scale for the decay of the excited state (i.e. erasing the domain wall)

tA ∼ τ0 e∆F (R)/(kBT ) . (14)

For a barrier growing as a power of R, ∆V (R) ∼ Υ(T, J)Rψ (where J representsthe disorder) one inverts (14) to find the linear size of the domains still existingat time t, that is to say, the growth law [28]

R(t) ∼(

kBTΥ(T ) ln

tτ0

)1/ψ

. (15)

All smaller fluctuation would have disappeared at t while typically one wouldfind objects of this size. The exponent ψ is expected to depend on the dimen-sionality of space but not on temperature. In ‘normal’ systems it is expectedto be just d − 1 – the surface of the domain – but in spin-glass problems, itmight be smaller than d − 1 due to the presumed fractal nature of the walls.The pre-factor Υ is expected to be weakly temperature dependent.

16

To extend this result to the actual out of equilibrium coarsening situa-tion one assumes that the same argument applies out of equilibrium to there-conformations of a portion of any domain wall or interface where R is theobservation scale.

However, not even for the (relatively easy) random ferromagnet there isno consensus about the actual growth law [29]. We shall discuss a possibleway out in Sect . In the case of spin-glasses, if the mean-field picture witha large number of equilibrium states is realized in finite dimensional models,the dynamics would be one in which all these states grow in competition. If,instead, the phenomenological droplet model applies, there would be two typesof domains growing and R(t) ∼ (ln t)1/ψ with the exponent ψ satisfying 0 ≤ψ ≤ d− 1. Some refined arguments that we shall not discuss here indicate thatthe dimension of the bulk of these domains should be compact but their surfaceshould be rough with fractal dimension ds > d− 1 [30].

4.2.2 Scaling functions

The scaling function are harder to obtain. For a much more detailed dis-cussion of these methods see the review articles in [1]. We shall discuss theirsuper-universality properties in Sect. .

5 Dynamics of (weakly) random systems

In this Section I explain that the existence of a static typical length yieldsa natural crossover between the clean growth law, say R ≃ [λ(T )t]1/2, andthe activation-ruled one, R ≃ Υ(T )[ln t]1/ψ, and how these two regimes mightnot be well separated in numerical and experimental measurements being easilyconfused with a disorder and temperature dependent power [31].

I also discuss the super-universality hypothesis [30] and list some checks ofit.

5.1 Crossover in the growing length

Numerical simulations of dirty systems tend to indicate that the growinglength is a power law with a disorder and T -dependent exponent. This can bedue to the effect of a disorder and T -dependent cross-over length, as explainedin [31]. For concreteness, let us assume that the width of the quenched randomdistribution is characterised by a parameter ǫ and that the cross-over lengthdepends on ǫ/T . We call the latter LT by absorbing the ǫ dependence in T .The proposal is that below LT the growth process is as in the clean limit whileabove LT quenched disorder is felt and the dynamics is thermally activatedabove barriers that are usually taken to grow as a power of the size:

R(t) ∼{

t1/zd for R(t) ≪ LT ,(ln t)1/ψ for R(t) ≫ LT .

(16)

17

These growth-laws can be first inverted to get the time needed to grow a givenlength and then combined into a single expression that interpolates between thetwo regimes:

t(R) ∼ e(R/LT )ψRzd (17)where the relevant T -dependent length-scale LT has been introduced. Now, bysimply setting t(R) ∼ Rz(T ) one finds

z(T ) − zd ≃ z/LψT for times such that tψ/zd ln t ≃ ct . (18)

Similarly, by equating t(R) ∼ exp(Rψ(T )/T ) one finds that ψ(T ) is a decreasingfunction of T approaching ψ at high T .

5.2 Super-universality

The super-universality hypothesis states that in cases in which temperatureand quenched disorder are ‘irrelevant’ in the sense that they do not modify thenature of the low-temperature phase (e.g. it remains ferromagnetic in the caseof ferromagnetic Ising models with quenched random interactions) the scalingfunctions should not be modified [30]. Only the growing length changes from the,say, curvature driven t1/2 law to an asymptotically slower law due to domainwall pinning by impurities. Tests of quenches from T > Tc into the orderedphase in the 3d RFIM [32] and the 2d RBIM [33, 34] (perhaps excluding thepossibility of having zero bonds, see the discussion in Henkel & Plaimling) givesupport for this hypothesis.

We recently investigated two aspects of the super-universality hypothesis.On the one hand we studied the space-time correlation function of the Pottsmodel with different values of q after quenches from T0 → ∞ and T0 = Tc [35].We found that

(i) The scaling function of all q ≥ 2 clean and weak disordered Potts modelsis the same after quenches from T0 → ∞. See Fig. 5.

(ii) The scaling functions of the 2 ≤ q ≤ 4-Potts models decay algebraicallyin contrast to the ones in (i), with the exponents η of the critical initial conditionthat depends on q. See Fig. 6.

(iii) We are currently studying the scaling functions of the 2 ≤ q ≤ 4-Potts models from its critical point. Since the η exponent changes under weak-disorder [36] we expect that the scaling functions will change too.

(iv) Quenches from the critical temperature of the q > 4-Potts model arealso under study. Note that for these values of q the transition of the (weak)random model becomes second order [37].

On the other hand, we analysed the geometric properties of areas and perime-ters in the 2d RBIM and we discuss these results in the next section.

18

6 Statistics and geometry of coarsening

In [17] we studied the distribution of domain areas, areas enclosed by do-main boundaries, and perimeters for curvature-driven two-dimensional Ising-likecoarsening, employing a combination of exact analysis and numerical studies,for various initial conditions. We showed that the number of hulls per unit area,nh(A, t) dA, with enclosed area in the interval (A,A + dA), is described, for adisordered initial condition, by the scaling function

nh(A, t) = 2ch/(A+ λht)2 , (19)

where ch = 1/8π√

3 ≈ 0.023 is a universal constant and λh is a material pa-rameter. For a critical initial condition, the same form is obtained, with thesame λh but with ch replaced by ch/2. For the distribution of domain areas, weargued that the corresponding scaling function have the form

nd(A, t) = (2)cd(λdt)τ−2/(A+ λdt)

τ , (20)

where cd and λd are numerically very close to ch and λh respectively, and theexponent τ is the one characterising the distribution of initial structures, criticalpercolation or critical Ising, see table . These results were extended to describethe number density of the length of hulls and domain walls surrounding con-nected clusters of aligned spins. These predictions were supported by extensivenumerical simulations. We also studied numerically the geometric properties ofthe boundaries and areas.

The derivation of eq. (19) is very easy. The number density of hull enclosedor domain areas at time t as a function of their initial distribution is

n(A, t) =

∫ ∞

0

dAi δ(A−A(t, Ai))n(Ai, ti) (21)

with Ai the initial area and n(Ai, ti) their number distribution at the initialtime ti. A(t, Ai) is the hull enclosed/domain area, at time t, having startedfrom an area Ai at time ti. The number density of hull-enclosed areas in criticalpercolation and critical Ising conditions was obtained by Cardy and Ziff [19]

nh(A, 0) ∼{

2ch/A2 , critical percolation,

ch/A2 , critical Ising.

(22)

These results are valid for A0 ≪ A≪ L2, with A0 a microscopic area and L2 thesystem size. Note also that we are taking an extra factor 2 arising from the factthat there are two types of hull enclosed areas, corresponding to the two phases,while the Cardy-Ziff results accounts only for clusters of occupied sites (and notclusters of unoccupied sites). nh(A, 0) dA is the number density of hulls perunit area with enclosed area in the interval (A,A + dA) (we keep the notation

19

10-12

10-10

10-8

10-6

10-4

100 101 102 103 104 105

n(A

,t)

A

t = 4 8

16 32 64

128 256

10-4

10-3

10-2

10-1

10-2 10-1 100 101

(λ t)

2 n

h(A

,t)

A / λ t

t = 4 8

16 32 64

128 256

Figure 8: Left: number density of hull-enclosed areas at different times aftera quench from T) → ∞ to T = 0 in the 2dIM; numerical results are shownwith points and the analytic prediction with lines. There is only one fittingparameter, λ, that has been independently determined by a study of the space-time correlation. Right: comparison between the scaled number-density after aquench from T0 → ∞ (above) and T0 = Tc (below). Figure taken from [17].

to be used later and set t = 0). The adimensional constant ch is a universalquantity that takes a very small value: ch = 1/(8π

√3) ≈ 0.022972. The time

dependent area is given by the Allen-Cahn result, A(t) = Ai − λh(t − ti). Bysimple integration of eq. (21) one obtains eq. (19).

It is interesting to note that the infinite temperature initial condition is notcritical percolation on the square lattice. Still, it is very close to it an aftercoarse-graining p = 1/2 becomes critical continuous percolation. The dynamicsof the discrete model, thus, very quickly settles into critical percolation ‘initial’conditions and this determines the distribution of large structures dynamically.This observation was used by Barros et al to interpret freezing of Ising modelsat very low temperatures [38].

The calculation of the number density of domain areas cannot be done ana-lytically but a mean-field-like approximation that uses an expansion in powersof ch yields eq. (20), a result that is very convincing since it agrees very wellwith numerical simulations [17].

The discussion on the geometric properties of critical objects suggests tostudy the fractal properties of growing structures. In [17] we pursued this lineof research by analysing the relation between areas and perimeters during coars-ening. We summarize the results below.

Several extensions of these results are: to the Ising model with conservedorder parameter [18], the random bond Ising model [34], the Potts model [35]and experiments in liquid crystals [39].

20

6.1 General picture

The summary of results in, and picture emerging from, the study of coars-ening in 2d systems with Ising symmetry [17, 34, 18, 39] is the following.

(i) We proved scaling for the number density of hull-enclosed areas in bi-dimensional zero-temperature curvature-driven coarsening in the Ising univer-sality class.

(ii) We argued that in these systems temperature effects are two-fold: theyrenormalize the pre-factor in the growth law and they introduce thermal do-mains that are distributed as in equilibrium. Their contribution to the totaldistribution of structures can be safely subtracted.

(iii) We obtained approximate expressions for the number density of domainareas and perimeters in curvature-driven coarsening in the 2d Ising universalityclass by using a mean-field-like analytic argument

(iv) We derived the scaling functions of the number density of hull-enclosedareas, domain areas and perimeters with an approximate analytic argument in2d Ising model class with conserved order-parameter dynamics by assuming thatsmall structures behave roughly independently of each other.

(v) We analysed the area number densities in the RBIM and we found thatsuper-scaling holds once the growing length is modified to incorporate the slow-ing down due to pinning.

All these results are described by the following conjecture for the numberdensity of domains and hull-enclosed areas [18]

R2τ (t) nd,h(A, t) =

(2)cd,h a2(τ−2)

[

A

R2(t)

]1/2

{

1 +

[

A

R2(t)

]zd/2}(2τ+1)/zd

(23)

(the is exact for hull-enclosed areas in curvature-driven coarsening). zd is thedynamic exponent (in the RBIM case, it is the effective temperature-dependentone). τ characterises the initial distribution, see table . A way to derive eq. (23)is to assume that each time-dependent area is independently linked to its initialvalue by

Azd/2(t) ≈ Aizd/2 −R(t) , (24)In all these cases the expressions that we obtained have two distinct limiting

regimes.(vi) For areas that are much smaller than the characteristic area, R2(t), the

distributions ‘feel’ the microscopic dynamics and thermal agitation while forareas that are much larger than R2(t) the distributions are simply those of theinitial condition. For conserved order parameter dynamics the Lifshitz-Slyozov-Wagner [40] behaviour is recovered after a quench from T0 → ∞ and evolving atsufficiently low T . For critical Ising initial conditions the distribution of small

21

areas does not approach the Lifshitz-Slyozov-Wagner. We conjectured that thereason is that our starting assumption, independence of domain wall motion forsmall domains, is not valid due to strong correlations in this case.

(vii) The distribution of the time-dependent areas that are larger than R2(t)are, in all cases, the ones of critical continuous percolation (for all initial condi-tions in equilibrium at T0 > Tc) and critical Ising (for T0 = Tc).

We also studied the geometric properties and fractal dimensions of differentobjects during coarsening. We found that

(viii) Small structures are compact and tend to have smooth boundaries,that are pretty close to circular (A ∼ p2) in the conserved order-parameter caseand little bit less so for non-conserved order-parameter dynamics.

(ix) The long interfaces retain the fractal geometry imposed by the equilib-rium initial condition.

The geometrical analysis of coarsening cells in the Potts model will be pre-sented in [35].

7 Coarsening in the spiral model

Among other features, in [13] we studied the dynamics of the spiral modelafter a sudden quench, in which a completely empty configuration is evolvedfrom time t = 0 onwards with the dynamic rule specified by a different valueof p > p0. This procedure is similar to the temperature quench of a liquid.For p < pc, after a non equilibrium transient the system attains a non-blockedequilibrium state. The relaxation time for attaining such a state diverges asp → pc. In the critical case with p = pc, the system approaches the blockedequilibrium state by means of an aging dynamics similar to that observed incritical quenches of ferromagnetic models. Freezing is not observed because thedynamic density ρ(t) is always smaller than the critical one ρc = pc, at anyfinite time. Something different happens for filling at p > pc. The system keepsevolving but a blocked state is never observed despite the fact that the densityof particles exceeds ρc = pc at long enough times. This is no surprise since thedynamics are fully reversible: for any p, each configuration reached dynamicallycan always evolve back to the initial empty state by the time reversed process,although with a very low probability. Therefore, a blocked state cannot bedynamically connected to the initial empty state. The dynamics of the spiralmodel at p > pc strongly resembles coarsening in ferromagnets, as illustratedin the centre and right panels in Fig. 1. In the p ≃ 1 limit in which thedynamics can be analyzed semi-analytically the system coarsens by forminglonger and longer one-dimensional objects of vacancies that are almost frozenbut not completely since they could be destroyed by the boundaries. The densityof these objects is irrelevant in the large size limit and the density of particlescan asymptotically approach 1 although the system is never blocked.

22

Acknowledgments I wish to thank J. J. Arenzon, A. Aron, G. Biroli, A. J.Bray, S. Bustingorry, C. Chamon, F. Corberi, J. L. Iguain, A. B. Kolton, M.P. Loureiro, M. Picco, Y. Sarrazin and A. Sicilia for our collaboration on phaseordering problems that lead to the new results discussed in these notes.

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25