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  to forest-fires , but also in labora-
tories associated with many condensed matter systems:
condensation , ferromagnets , martensitic transi-
tions , superconductivity , 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].
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  for the study of Barkhausen noise
in ferromagnets  and acoustic emission in martensitic
transitions , 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
H = −
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 , 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
(with τ ≃ 1.6 ). The properties of large (percolating)
avalanches at the critical point have been previously dis-
cussed . 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. 
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,
-180 -160-140 -120-100
20 log10 δ(dB)
-80 -60-40 -20 0
p(δ) (arbitrary units)
1.2 1.4 1.6 1.8 2
< δ >, [< δ2>-< δ>2]1/2
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?
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.
< δ >
[< s2 >-<s>2]1/2
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
size s′. We define the following two correlation functions:
?sδ? − ?s??δ?
??s2? − ?s?2??δ2? − ?δ?2
?s′δ? − ?s′??δ?
??s′2? − ?s′?2??δ2? − ?δ?2
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 .
(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
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-
-20-10 0 10 20 30
-20-10 0 10 20 30
(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
-220-180 -140 -100-60-20
pnext(δ) , pprev(δ) (arbitrary units)
20 log10 δ (dB)
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
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. 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.
∗Electronic address: email@example.com
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 Except on thetransition
p?δ2? − ?δ?2∼
p?δ?2− k0∆δ2with k0 = constant.
R.S´ anchez, and