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arXiv:0902.0282v2 [cond-mat.dis-nn] 9 Feb 2009

Correlations in avalanche critical points

Benedetta Cerruti1and Eduard Vives1, ∗

1Departament d’Estructura i Constituents de la Mat` eria, Universitat de Barcelona

Mart´ ı i Franqu` es 1 , Facultat de F´ ısica, 08028 Barcelona, Catalonia

(Dated: February 9, 2009)

Avalanche dynamics and related power law statistics are ubiquitous in nature, arising in phe-

nomena like earthquakes, forest fires and solar flares. Very interestingly, an analogous behavior is

associated with many condensed matter systems, like ferromagnets and martensites. Bearing it in

mind, we study the prototypical 3D RFIM at T = 0. We find a finite correlation between waiting

intervals between avalanches and the previous avalanche size. This correlation is not found in other

models for avalanches, such as the standard BTW model, but it is experimentally found in earth-

quakes and in forest fires. Our study suggests that this effect occurs in critical points which are at

the end of an athermal first-order transition line separating two behaviors: one with high activity

from another with low activity.

PACS numbers: 05.70.Jk, 05.40.-a, 75.60.Ej,, 75.40.Mg, 75.50.Lk

In the last few years much experimental and theoreti-

cal effort has been devoted to the study of avalanche pro-

cesses. Deep understanding of the statistical correlations

in such stochastic processes is needed in order to make

advances towards predictability. The importance of the

subject is beyond discussion due to the many implications

in natural disasters and social crises.

cesses are characterized by extremely fast events whose

occurrences are separated by waiting intervals without

activity. The magnitudes characterizing avalanches (en-

ergy, size, duration) are, in most cases, statistically dis-

tributed according to a power law p(s)ds ∼ s−τds char-

acterized by a critical exponent τ: extremely large events

hardly occur, whereas small events are very common.

This is the famous Gutenberg-Richter law for earth-

quakes. Power-law distributions have been found not

only for other large-scale natural phenomena ranging

from solar flares [1] to forest-fires [3], but also in labora-

tories associated with many condensed matter systems:

condensation [2], ferromagnets [4], martensitic transi-

tions [5], superconductivity [6], etc.

tics of waiting intervals δ are not often studied, espe-

cially in condensed matter. The distribution p(δ)dδ has

been described by different laws including exponentials

and also power-laws. For solar flare statistics and earth-

quake models, it has been found [7, 8] that the unavoid-

able threshold definition (separating activity from inac-

tivity) alters the distribution p(δ) and that this thresh-

olding effect is a signature of the existence of correla-

tions [9, 10]. Direct measurement of the correlations

between waiting intervals and avalanche sizes has been

obtained by measuring the constrained interval distribu-

tions pprev(δ|s > so) and pnext(δ|s′> so). These are the

probabilities of having a waiting interval δ, given that the

previous (s) or the next (s′) avalanche is larger than s0.

For earthquake and forest-fire statistics, while pnexthas

been found to be independent of s0, pprev does exhibit

significant changes when varying s0[11, 12].

Avalanche pro-

However, statis-

In this letter we study some statistical correlations

for the 3D-RFIM at T = 0 with metastable dynamics

based on the local relaxation of single spins. This model

was introduced [13] for the study of Barkhausen noise

in ferromagnets [4] and acoustic emission in martensitic

transitions [5], and has been used as a prototype for the

study of crackling noise and other avalanche phenomena

[14, 15]. The model is defined on a cubic lattice with

size N = L3. At each lattice site there is a spin variable

Si = ±1 (i = 1,...,N) that interacts with its nearest

neighbors (n.n.) according to the Hamiltonian (magnetic

enthalpy):

H = −

?

n.n.

SiSj−

N

?

i=1

Sihi− H

N

?

i=1

Si. (1)

The local random fields hiare Gaussian distributed with

zero mean and standard deviation σ. This parameter not

only allows the critical behavior (σ = σc≃ 2.21±0.01) to

be studied but also subcritical (σ > σc), and supercritical

(σ < σc) regimes [16, 17]. This tuning is absent in so-

called Self-Organized Criticality models like the original

BTW model [18], in which the critical state is reached

after waiting for a certain time without any parameter

adjustment. In the RFIM, the time variable is replaced

by the external field H, which is adiabatically increased

from −∞ to ∞. The system responds by an increase of

the order parameter m ≡?N

spin) from −1 to 1. The spins flip according to the dy-

namical rule Si= sign(?

runs over the n.n. of spin Si), which corresponds to a

minimization of the local energy. This non-equilibrium

dynamics leads to hysteresis and avalanches when many

spins flip at constant field. The avalanche size s is de-

fined as the number of spins flipped until a new stable

state is reached. Avalanches are separated by field wait-

ing intervals δ without activity. When σ = σcand H is

close to Hc≃ 1.43±0.05 [16, 19], the avalanche size dis-

tribution becomes approximately a power law p(s) ∼ s−τ

i=1Si/N (magnetization per

jSj+ hi+ H) (the first sum

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(with τ ≃ 1.6 [16]). The properties of large (percolating)

avalanches at the critical point have been previously dis-

cussed [19]. Critical avalanches have been found to be

fractal so that ?s?c∼ Ldfwith df∼ 2.88. When σ > σc,

the avalanche distribution is exponentially damped, i.e.

all avalanches are negligible compared to system size, and

the magnetization m evolves continuously when the sys-

tem is infinite. For σ < σc, avalanches are also infinitesi-

mally small compared to L3, except for a unique infinitely

large and compact (s ∼ L3) avalanche corresponding

to a first-order phase transition between a phase with

low m and a phase with high m. The first-order tran-

sition line can be linearly approximated by the equation

Ht(σ) = Hc[1 − B′(σ − σc)/σc] with B′= 0.25. [19]

In this letter we will concentrate on the analysis of the

field waiting intervals δ, and their correlation with the

previous and next avalanche sizes. We have first checked

that intervals are exponentially distributed. Fig. 1 shows

the distribution of waiting intervals for σ = 2.21, L = 30

and three different H-ranges. One can observe that be-

fore and after the transition region the distributions are

very well described by exponentials (continuous lines),

whereas in the transition region the distribution becomes

a linear mixture of exponentials. The mean value ?δ?,

which is the only parameter characterizing such distribu-

tions, depends on σ and H, and below σc it exhibits

a discontinuity ∆δ when H increases and crosses the

first-order transition line from the region of high activity

(small ?δ?) to the region of low activity (large ?δ? ), as

shown in the inset of Fig. 1. We can, therefore, compare

the behavior of the discontinuity ∆δ to that of an order

parameter. Note that, given the finite size of the system,

the pseudo-critical point where the mixture of exponen-

tials becomes a single exponential distribution will be

located at σ > 2.21. For this reason, although data in

Fig. 1 correspond to σc, one can still observe a range of

fields with the two-peak distribution.

Besides the histogram analysis, we have numerically

checked that for all σ and H, ?δ? ∼

which is more evidence of the exponential character of

p(δ). The continuous line in the inset of Fig. 1 shows

this agreement, except for some deviations close to the

transition line due to finite-size effects.

For the following discussions it is interesting to ana-

lyze the finite-size dependence of ?δ?. A simple argu-

ment can be used to state that far from the critical point,

since the correlation length is finite, the probability for

an avalanche to start when the field is increased by dH is

proportional to the number of triggering sites and thus to

L3. This implies that ?δ? ∼ L−3. Fig. 2 shows examples

of this behavior for different values of σ and H. At crit-

icality, the correlation length diverges and the triggering

argument may be too naive. Nevertheless, as shown in

Fig. 2, numerical simulations indicate that the exponent

is always very close to 3. Uncertainties in Hcand σcdo

not allow for an accurate enough finite-size scaling anal-

??δ2? − ?δ?2,[20]

10-5

10-4

10-3

10-2

10-1

100

101

-180-160 -140-120-100

20 log10 δ(dB)

-80 -60-40-20 0

p(δ) (arbitrary units)

1.35<H<1.40

1.45<H<1.50

1.55<H<1.60

0

0.2

0.4

0.6

1.2 1.4 1.6 1.8 2

< δ >, [< δ2>-< δ>2]1/2

H

FIG. 1: Log-log plot of the histograms corresponding to the

distribution of waiting intervals δ for L = 30, σ = 2.21 and

three different field ranges, indicated by the legend. Note that

the horizontal scale is in dB and bins are logarithmic. The

dashed lines show fits corresponding to the exact exponential

behavior. Data have been obtained by averaging over 50000

realizations of disorder. The inset shows the behavior of ?δ?

(symbols) and

p?δ2? − ?δ?2(continuous line) as a function of

H for σ = 1.95 and L = 30.

ysis to determine small variations. For the discussions

here the exact value of this exponent is not needed and

we will assume that ?δ? ∼ L−zwith z ∼ 3.

10-4

10-3

10-2

10-1

100

10 100

< δ >

L

σ=2.21, 1.425<H<1.430

σ=2.21, 0.400<H<0.405

σ=2.21, 1.800<H<1.805

σ=2.40, 1.425<H<1.430

σ=2.40, 0.400<H<0.405

σ=2.40, 1.800<H<1.805

σ=1.95, 1.425<H<1.430

σ=1.95, 0.400<H<0.405

σ=1.95, 1.465<H<1.470

100

101

102

103

104

10 100

[< s2 >-<s>2]1/2

L

FIG. 2: Log-log plot of the average waiting interval ?δ? as a

function of the system size L for different values of σ and H

as indicated by the legend. The continuous line shows the

behavior L−3and the discontinuous lines are fits to the last

2-3 points for each series of data. Data are averaged over

50000 disorder realizations. The inset shows the behavior of

p?s2? − ?s?2for the values of σ and H indicated. The contin-

uous lines show the behavior Ldf(above) and L3/2(below).

Let us now focus on the study of correlations and con-

sider a sequence of two avalanches, the first with size

s, then a waiting interval δ, and the next avalanche with

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size s′. We define the following two correlation functions:

ρs,δ =

?sδ? − ?s??δ?

??s2? − ?s?2??δ2? − ?δ?2

?s′δ? − ?s′??δ?

??s′2? − ?s′?2??δ2? − ?δ?2

(2)

ρδ,s′ = (3)

Fig. 3 shows examples of the behavior of the correlation

functions for different system sizes and σ = σc≃ 2.21 as a

function of the scaling variable v = [(H − Hc)/Hc]L1/µ

that measures the distance to the critical point.

exponent µ = 1.5 has been taken from the literature [19].

(Note that this scaling variable only takes three values in

the thermodynamic limit v = ±∞,0).

For the understanding of the behavior of the correla-

tion functions for increasing system size, one must first

discuss what the expected behavior of a generic correla-

tion function ρ(H,σ = σc,L) close to Hcis likely to be.

It should be borne in mind that correlations are, by defi-

nition, bounded between −1 and 1. Therefore no critical

divergences can occur with increasing L. Consequently,

any critical correlation either goes to zero or tends to

a constant value. In this second case, it should exhibit

scaling behavior ρ(H,σc,L) ∼ ˆ ρ(σc,v).

The first observation from Fig. 3 is that ρδ,s′ is much

smaller than ρs,δ not only at the critical point (v = 0),

but also for other values of the field. In addition, the

data shows that ρδ,s′ systematically decreases in abso-

lute value with increasing system size. Therefore the nu-

merical data is consistent with a vanishing ρδ,s′ in the

thermodynamic limit.

The second important observation is that correlation

between an avalanche size and the next waiting time, ρs,δ

exhibits a constant value ∼ 0.4 at the critical point. The

overlap of the curves is rather good, specially if one takes

into account the fact that even at the critical point there

are non-critical avalanches which may slightly perturb

the scaling function behavior. The fact that the scaling

function in Fig. 3(a) goes to zero for v → ±∞ indicates

that the finite correlation only survives exactly at the

critical point, but vanishes for fields both above and be-

low. We would like to note that the result of a finite cor-

relation ρs,δ is not in contradiction with the Poissonian

character for the triggering instants of the avalanches.

A second way to go deeper into the understanding of

the nature of the correlations is the direct measurement

of the restricted distribution of intervals pprev(δ|s > so)

and pnext(δ|s′> so). Fig. 4 shows, as an example, the de-

pendence of these two distributions for L = 30, σ = 2.21

and 1.40 < H < 1.45.The distribution of intervals

pprev(δ|s > so) clearly exhibits a dependence on the pre-

vious avalanche size, whereas pnext(δ|s′> so) is indepen-

dent of s0. Thus, the larger the size of an avalanche, the

larger the probability that the following waiting inter-

val is large. We should note at this point that a similar

same causal dependence has been found for the statis-

The

-0.2

0

0.2

0.4

0.6

-20-10 0 10 20 30

(H-Hc)/Hc L1/µ

(a) ρ(s,δ)

L=18

L=24

L=30

L=36

-0.15

-0.1

-0.05

0

0.05

-20-10 0 10 20 30

(H-Hc)/Hc L1/µ

(b) ρ(δ, s’)

FIG. 3: Correlation functions defined by Eq. (2) and (3) as

functions of the scaling variable for σ = 2.21 and different

system sizes as indicated by the legend. Data correspond to

averages over 50000 disorder realizations and field intervals

with size ∆H = 0.005. Note the different vertical scale in the

two figures.

10-4

10-3

10-2

10-1

100

101

-220 -180 -140 -100 -60-20

pnext(δ) , pprev(δ) (arbitrary units)

20 log10 δ (dB)

pnext

pprev

all

s>2

s>5

s>10

s>20

s>50

s>100

s>200

all

s’>2

s’>5

s’>10

s’>20

s’>50

s’>100

s’>200

FIG. 4: Log-log plot of the restricted distribution of intervals

δ given that the previous (next) avalanche is larger than s0.

pprev has been displaced two decades up in order to clarify

the picture.

tics of earthquakes and forest fires [11, 12], but with a

different sign. The larger the size of an earthquake, the

smaller the waiting time to the next event.

The origin of such correlations in our RFIM can be

understood by noting that, for a finite system below σc,

after an avalanche with a large size (of the order L3),

one can “guess” that the system has jumped to the low-

activity region and, therefore, one can “predict” that the

next interval δ will be large. We can provide an heuris-

tic argument of why such finite-size correlations vanish

in the thermodynamic limit, everywhere except at the

critical point. Let us analyze the behavior of the nu-

merator and the two square roots in the denominator

in Eq. 2. First note that, given the fact that p(δ) is

exponential, the fluctuations of

??δ2? − ?δ?2behave as

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δ ∼ L−z. The fluctuations of s behave differently below

σcand at σc. If we consider an interval ∆H that crosses

the first-order transition region, below σc we will have

a number of avalanches proportional to L3contributing

to the averages. Among these, most will display a small

size ∼ L0but one will have a size ∼ L3. When comput-

ing ?s? we get a ?s? ∼ L0behavior, but when computing

??s2? − ?s?2we will get L3/2, as shown in the inset of

Fig. 2 for σ = 1.95 and the interval 1.465 < H < 1.470

crossing the transition line. Exactly at criticality, the dis-

tribution of avalanches becomes a power-law. This means

that the averages ?s? and ?s2? should be computed by in-

tegrating the distribution from 1 to the largest avalanche

size which has a fractal dimension smax∼ Ldf[19]. By

integration, one trivially gets ?s? ∼ Ldfand ?s2? ∼ L2df.

The fluctuations therefore will go as??s2? − ?s?2≃ Ldf,

as shown in the inset of Fig. 2.

To study the numerator in Eq. (2) one should note that

?sδ?−?s??δ? = ?s′δ′?−?s′??δ? = ?s′(δ′−δ)?. This average

measures the avalanches that carry an associated change

in δ. For σ < σc, among the set of L3avalanches in a

interval ∆H, there is one avalanche with size L3associ-

ated with a change ∆δ ∼ L−z. The rest of the avalanches

have a size L0and carry no change in δ. Therefore the

numerator in Eq. 2 goes as L−zand consequently, below

σc, ρ(s,δ) ∼ L−z/(L3/2L−z) ∼ L−3/2→ 0.

For σ = σcthe behavior of the numerator in (2) is much

more intricate since different kinds of critical avalanches

exist close to the critical point. Apart from a number

of non-critical avalanches, there is an infinite number

(∼ Lθ) of spanning avalanches and another infinite num-

ber ∼ Lθnscof critical non-spanning avalanches. It is

difficult to argue which of these avalanches have an asso-

ciated ∆δ. We cannot provide a definite argument, but,

at least, the product ?s??δ? (appearing in the numerator

of Eq.2) behaves as LdfL−z. Therefore it is plausible that

at σc, ρ(s,δ) ∼ Ldf−z/(LdfL−z) ∼ 1, which justifies the

finite value of the correlation ρ(s,δ) found numerically,

as shown in Fig. 3.

In order to compare with the avalanche models based

on SOC, we have performed an analysis of the same prob-

ability densities and correlations for the 2D-BTW model.

In this case, the waiting times are discrete since they are

identified with the number of grains added before a new

avalanche starts. For large enough systems, these wait-

ing intervals are distributed according to the geometric

distribution which is the discrete version of the exponen-

tial distribution and all the two correlation functions (2)

and (3) clearly vanish.

In summary, we have numerically studied the T = 0

RFIM with metastable dynamics as a prototype for

avalanche processes in condensed matter systems which

display an underlying first-order phase transition. The

sequence of avalanches and waiting times can be con-

sidered as a compound Poisson process since waiting in-

tervals tend to be exponentially distributed. Neverthe-

less, correlations between the avalanche size and the next

waiting time exist at the critical point in the thermo-

dynamic limit.Such a causal correlation ρ(s,δ) ?= 0

has been found experimentally in earthquakes and forest

fires, although with a different sign. This difference could

easilly be explained by considering a distonintuity ∆δ <

0. An experimental challenge for the future is to look

for these effects in laboratory experiments on condensed

matter systems exhibiting avalanches (Barkhausen noise,

acoustic emission in martensites, etc).

This work has received financial support from CICyT

(Spain), project MAT2007-61200, CIRIT (Catalonia),

project 2005SGR00969, B.C. acknowledges the hospital-

ity of ECM Department (Universitat de Barcelona) and a

grant from Fondazione A. Della Riccia. We also acknowl-

edge fruitful discussions with A.Corral, J. Vives and A.

Planes.

∗Electronic address: eduard@ecm.ub.es

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line where weexpect