We perform numerical simulations of the sandpile model for non-vanishing
driving fields $h$ and dissipation rates $\epsilon$. Unlike simulations
performed in the slow driving limit, the unique time scale present in our
system allows us to measure unambiguously response and correlation functions.
We discuss the dynamic scaling of the model and show that
fluctuation-dissipation relations are not obeyed in this system.Comment: 5 pages, latex, 4 postscript figure

A. Chessa

A. Díaz-Guilera

A. Giacometti

A. Montakhab

A. Vespignani

Alain Barrat

Alessandro Vespignani

B. Kutnjak-Urbanc

D. Dhar

H.-J. Xu

J. B. Rundle

L. F. Cugliandolo

L. Pietronero

M. De Menech

M. Paczuski

O. Narayan

P. Bak

P. Grassberger

R. Dickman

S. Lübeck

S. S. Manna

S. Zapperi

Stefano Zapperi

T. Hwa

V. B. Priezzhev

English

arXiv

We perform numerical simulations of the sandpile model for nonvanishing driving fields h and dissipation rates \u3b5. Unlike simulations performed in the slow driving limit, the unique time scale present in our system allows us to measure unambiguously the response and correlation functions. We discuss the dynamic scaling of the model and show that fluctuation-dissipation relations are not obeyed in this system

A. Barrat

AIR Universita degli studi di Milano

VOLUME 83, NUMBER 10 P H Y S I C A L R E V I E W L E T T E R S 6 SEPTEMBER 1999
1962Fluctuations and Correlations in Sandpile Models
Alain Barrat,1 Alessandro Vespignani,2 and Stefano Zapperi3
1Laboratoire de Physique Théorique,* Bâtiment 210, Université de Paris-Sud, 91405 Orsay Cedex, France
2The Abdus Salam International Centre for Theoretical Physics (ICTP), P.O. Box 586, 34100 Trieste, Italy
3PMMH-ESPCI, 10 rue Vauquelin, 75231 Paris Cedex 05, France
(Received 11 March 1999)
We perform numerical simulations of the sandpile model for nonvanishing driving fields h and
dissipation rates e. Unlike simulations performed in the slow driving limit, the unique time scale
present in our system allows us to measure unambiguously the response and correlation functions. We
discuss the dynamic scaling of the model and show that fluctuation-dissipation relations are not obeyed
in this system.
PACS numbers: 45.70.Ht, 05.70.LnThe sandpile automaton [1] is one of the simplest
models of avalanche transport, a phenomenon of growing
experimental and theoretical interest. In the model,
introduced by Bak, Tang, and Wiesenfeld (BTW) [1],
grains of “energy” are injected into the system. Open
boundary conditions [1] or bulk dissipation ensure a
balance between input and output flow and allow for
a nonequilibrium stationary state. In the limit of slow
external driving and small dissipation, which corresponds
to an infinite time scale separation between driving
and response, the model displays a highly fluctuating
avalanche behavior, indicative of a critical point. Despite
the impressive theoretical effort devoted to understanding
the critical behavior of the model [2–4], several important
issues still remain to be addressed.
Numerical simulations are usually performed under
slow driving and boundary dissipation, since the limit
of infinite time scale separation is easily implemented
in the computer and provides a simple way to access
the avalanche critical behavior [5–9]. However, due to
the presence of two infinitely separated time scales, an
unambiguous definition of dynamic response and correla-
tion functions is not possible [10]. This hinders a clear
characterization of the nonequilibrium stationary state
in terms of static and dynamic response and correlation
functions. Evaluation of these quantities helps to eluci-
date the nature of the critical point and provides a test of
fluctuation-dissipation relations, at least in some weaker
sense. Recently, it has been proposed to interpret the
behavior of sandpile models in analogy with other
nonequilibrium critical phenomena, such as absorbing
phase transitions [11], driven interfaces in random media
[12–14], and branching processes [15]. These theo-
retical studies suggest new ways to perform numerical
simulations in which a unique time scale is considered
[11,14,16].
In this Letter, we present numerical simulations of the
sandpile model for different driving rates h and study how
the system approaches the critical point when h ! 0. In
this way, we are able to measure quantities that are not0031-90079983(10)1962(4)$15.00accessible in the time scale separation regime. The local
density of active sites, which can be identified as the order
parameter of the model [16], is homogeneous only in
the case of bulk dissipation. For boundary dissipation,
it displays a marked curvature that was anticipated in
Refs. [11,14] and could explain several scaling anomalies
found in the BTW model. The energy landscape is instead
homogeneous in both cases and its statistical properties do
not depend on the dissipation rate e in the limit h ! 0.
We measure correlation and response functions in time
and space domains and observe the scaling of the related
characteristic lengths and times. We find two different
characteristic times, implying that fluctuation-dissipation
relations are not obeyed. We observe, however, a well-
defined scaling behavior and the value of the critical ex-
ponents are in agreement with recent large scale numerical
simulations of slowly driven sandpiles [7–9]. Finally, the
present numerical analysis opens the road to future studies
to resolve some long-standing problems such as the precise
identification of universality classes for these models [8].
In sandpile models [1], each site i of a d-dimensional
lattice bears an integer variable zi $ 0, which we call
energy. At each time step an energy grain is added on
a randomly chosen site. When a site reaches or exceeds
a threshold zc it topples: zi ! zi 2 zc and zj ! zj 1 1
at each of the g nearest neighbors (nn) of i. Each
toppling can trigger nn to topple and so on, generating
an avalanche. The original BTW model is conservative
and energy is dissipated only at the boundary, i.e., energy
grains from toppling boundary sites flow out of the
system. Infinitely slow driving is implicitly built into
the model: during the avalanche the energy input stops,
until the system is again quiescent (no active sites are
present), so that we can identify two distinct time scales
Td and Ta, for driving and activity, respectively. A single
driving time step can, in principle, be followed by an
infinite number of avalanche time steps and TaTd ! 0.
For this reason, there are two possible definitions for the
correlation function, depending on the choice of the scale
used to measure time (slow or fast) [10,17].© 1999 The American Physical Society
VOLUME 83, NUMBER 10 P H Y S I C A L R E V I E W L E T T E R S 6 SEPTEMBER 1999Here we simulate the BTW sandpile model for a non-
vanishing driving field: each site has a probability h per
unit time to receive an energy grain also if active sites
are present in the system. This defines a unique time
step for both driving and activity updating. The parame-
ter h sets the driving rate, and in the limit h ! 01 we
recover the slow driving limit; i.e., during the evolution
of an avalanche the system does not receive energy. We
consider two possible mechanisms for energy dissipation,
(i) usual boundary dissipation and (ii) bulk dissipation,
simulated by introducing the probability a that a toppling
site loses its energy without transferring it to the neigh-
bors, which corresponds to an effective average dissipation
e  azc. In case (ii), we impose periodic boundary con-
ditions. We use two-dimensional lattices with linear sizes
ranging from L  64 to L  300, and parameters in the
range 1026 , h , 1023 and 1023 , e , 1022.
The order parameter in sandpile models is the density
ra [18] of active sites, whose energy is equal to or larger
than zc [1,11,12,14,16]. The dependence of the order
parameter on the control parameters h and e is readily
obtained by means of conservation arguments [16]: since
energy is conserved in the stationary state, the incoming
energy flux Jin  hLd must be equal to the dissipated
energy Jout  eraLd . By equating the two fluxes we
obtain ra  he. In systems with boundary dissipation,
the effective dissipation rate scales with the system size as
e  L2m, with m  2 [16], yielding ra  hL2. The
model is critical just in the double limit e ! 0 and
he ! 0 [16].
We study the behavior of the density of active sites
in the system and measure the stationary average den-
sity of local energies, i.e., the density ri of sites with i
energy grains. In Fig. 1 we report the behavior of the
densities as a function of he. For small values of he
we find ri  r0i 1 cihe 1 O he2, where r0i are
the values extrapolated from the limit h ! 01 and are
given by r00  0.0751, r01  0.1761, r02  0.3071,
and r03  0.4421. These values are in excellent agree-
ment with the exact values obtained for the slowly driven
sandpile (with boundary dissipation) [3] and are indepen-
dent of the dissipation rate. For i . 3 we obtain r0i  0
and for small h we observe ra  r4, while for larger h
higher energy levels become populated and ra has non-
negligible contributions coming from ri with i . 4. Fi-
nally, we confirm that ra  he for the whole range of
parameters. In the case of boundary dissipation we recover
ra  hL2.
To elucidate the differences between bulk and boundary
dissipation, we measure the local density of active sites
rar. In the case of bulk dissipation the density profile
is flat rar  ra, while in the case of boundary dissi-
pation we obtain a surface that can be well approximated
by a paraboloid (see Fig. 2). This is due to the highly in-
homogeneous dissipation which imposes a zero density of
active sites on the lattice boundary and corresponds to an
elastic interface pinned at the boundaries as discussed inFIG. 1. Mean densities ri of sites with i energy grains vs
he. Inset: mean density ra of active sites vs he. Values
of h range from 1026 to 1023, and e is between 1023 and 1022.
We observe that ra  he and that the various ri depend
on h and e only through the ratio he (see text).
Refs. [11,14]. An inhomogeneous order parameter could
give rise to anomalies in the avalanche exponents [7,9] or
persistent deviations from simple scaling [19].
In order to obtain a quantitative description of the
stationary state, we study the effect on the stationary
density of a small perturbation in the driving field
Drar , t 
Z
xh,er 2 r 0, t 2 t0Dhr 0, t0 dr 0 dt0,
(1)
where xh,er , t is the local response function. In the
limit h ! 01 the integrated susceptibility x  R dt dr 3
xh,er , t scales as the average avalanche size x  s
and the time integrated response function scales as [16]
x¯h!0,er 
Z
xh!0,er , t dt ~
1
rd22
e2rj , (2)
where j is the characteristic length. Since x  ≠ra≠h
and ra  he, the response function diverges in the
FIG. 2. Local density of active sites rax, y, in the case
of boundary dissipation; the linear lattice size is L  100, and
h  1024.1963
VOLUME 83, NUMBER 10 P H Y S I C A L R E V I E W L E T T E R S 6 SEPTEMBER 1999limit of vanishing driving and dissipation as x  1e.
By noting that x  j2, we obtain that j  e2n with
n  12 [16]. These results hold in all dimensions due
to conservation [11,14,16].
To measure the response function, we drive the sys-
tem in the stationary state with a given h and we then
add n energy grains (i.e., Dh  nL2) on a given lat-
tice site [20]. The time integrated response function is
equivalent to the average difference of activity x¯h,er 
Drar  rarh1Dh 2 rarh, where r denotes the
distance from the perturbed site. We observe that this
function decays exponentially as predicted by Eq. (2) and
we measure the correlation length j (see Fig. 3). In the
case of bulk dissipation, for small driving fields the j de-
pends only on the dissipation rate and scales as j  e2n ,
with n  0.50 6 0.01 (see Fig. 4). In the case of bound-
ary dissipation, to evaluate x¯h,Lr we have to consider
explicitly the spatial inhomogeneity of the stationary den-
sity: rar ﬁ he. We also observe in this case that the
integrated response function decays exponentially and de-
fines a correlation length increasing linearly with the lat-
tice size, i.e., j  L. This result does not agree with the
anomalous scaling found in a continuous energy sandpile
[17]. We perform analogous simulations in d  3 and find
that Eq. (2) is still verified [21].
Furthermore, we study the response function in the
time domain defined as x˜h,et 
R
dr xh,er , t after a
small variation Dh of the driving field. Also, in this
case, we obtain a clear exponential behavior defining the
characteristic time scale t. For small driving field, t
scales as a function of the dissipation rate as t  e2D,
with D  0.75 6 0.05 (see Fig. 4). We then evaluate
the dynamical exponent z  Dn  1.5 6 0.1 relating
time and spatial characteristic length: t  jz . In the
limit h ! 01, we expect that the critical exponents n and
z express the divergence of avalanche characteristic size
FIG. 3. Time integrated response function x¯h!0,e to a con-
stant perturbation as a function of r; the linear lattice size is
L  200. The lines are exponential fits.1964and time, respectively. The numerical results confirm this
observation [7–9]. For increasing values of h, the driving
field enters the scaling form, which, however, goes to a
constant for h ! 0 [21].
We now turn to the analysis of the correlation function
defined as Cr , t  rar , tra0, 0 2 ra2. In previ-
ous simulations, performed in the slow driving limit, cor-
relation functions were usually measured with respect to
the slow time scale [1,17,22], and the fast time scale was
explored studying the avalanche propagation. The intro-
duction of nonvanishing driving and dissipation allows us
to bridge the gap between the two regimes. We study the
correlation function in time and space domains and find an
exponential decay at long times and distances [23], defin-
ing the correlation lengths jc and tc for space and time,
respectively. The scaling of these correlation lengths is in
agreement with the one obtained analyzing the response
functions (i.e., jc  e2n , with n  0.5, and tc  e2D,
with D  0.75) and confirms the existence of a unique
critical behavior in time and space (see Fig. 4).
In order to clarify the interplay between slow and
fast dynamical modes, we analyze fluctuation-dissipation
relations. In equilibrium phenomena, the fluctuation-
dissipation theorem ensures that the response of the
system to a small perturbation is related to the correlation
function. In particular, the response function is given by
xt  2
1
T
dCt
dt
, (3)
where T is the temperature. Equation (3) is strictly veri-
fied only in equilibrium systems, but it has been recently
generalized to some classes of nonequilibrium systems,
namely systems displaying “aging” [24]. In those
examples the fluctuation-dissipation relation provides an
FIG. 4. Characteristic length j and characteristic time
limh!01 th, e estimated from the spatial response and
correlation functions with bulk dissipation and for various
system sizes and dissipation rates. For small dissipation rates
j ! `, and larger lattice sizes must be used. The straight
lines represent the best fits with slope n  0.5 and D  0.75
for j and t, respectively.
VOLUME 83, NUMBER 10 P H Y S I C A L R E V I E W L E T T E R S 6 SEPTEMBER 1999information on an effective nonequilibrium temperature
that rules the dynamical evolution of the system.
We test Eq. (3) and we find that the usual linear be-
havior does not hold. On the contrary, we show that the
parametric plot of xt versus Ct defines a power law be-
havior, as shown in the double logarithmic plot of Fig. 5.
This is striking evidence that the fluctuation-dissipation re-
lation does not hold in these systems. Since we are in the
presence of two exponential functions, the linear behavior
on the logarithmic scale is the signature of two different
values for characteristic times tc and t for the correla-
tion and response functions, respectively. The slope indi-
cates that the ratio between the two time scales is given by
tct  0.4 and does not depend on driving and dissipation
rates. This observation reflects the fact that the correlation
and the response times scale with the same exponents with
respect to dissipation and define unambiguously the critical
behavior of the model. In particular, it implies that the dy-
namical exponent z  1.5 is unique and can be estimated
either by measuring avalanche distributions or the correla-
tion functions. Previous simulations [7,17] revealed two
different dynamical exponents in the fast and slow time
scales. These differences are probably due to the ambigu-
ous definition of time in the infinite time scale separation
limit.
Finally, it is interesting to compare our results with a re-
cent work [25] showing that the stationary state of nonequi-
librium threshold models, similar to the one studied here,
can be described by Boltzmann statistics in the mean-field
limit. The validity of the claim of Ref. [25] has been
debated in the literature [26]. We measure fluctuation-
dissipation relations in a random neighbor sandpile model,
which is described by mean-field theory, and find that
fluctuation-dissipation relations are not satisfied [21], in
disagreement with the conclusions of Ref. [25].
We thank A. Chessa, D. Dhar, R. Dickman, S. Franz,
K. B. Lauritsen, E. Marinari, M. A. Muñoz, R. Pastor-
Satorras, and A. Stella for comments and discussions.
FIG. 5. Double logarithmic plot of xt vs Ct for various
values of h and e; the slope of the straight lines is tct  0.4.A. V. and S. Z. acknowledge partial support from the
European Network Contract No. ERBFMRXCT980183.
*Unité Mixte de Recherche UMR 8627.
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