Particle-hole symmetry in a sandpile model

0.31 MB·English

by  R. Karmakar

Particle-hole symmetry in a sandpile model
R. Karmakar and S. S. Manna
Satyendra Nath Bose National Centre for Basic Sciences Block-JD, Sector-III, Salt Lake, Kolkata-700098, India
In asandpile modeladdition ofa holeis definedastheremoval ofa grain from thesandpile. We
5
show that hole avalanches can be defined very similar to particle avalanches. A combined particle-
0
hole sandpile model is then defined where particle avalanches are created with probability p and
0
hole avalanches are created with the probability 1−p. It is observed that the system is critical
2
withrespecttoeitherparticleorholeavalanchesforallvaluesofpexceptatthesymmetricpointof

n pc =1/2. Howeveratpcthefluctuatingmassdensityishavingnon-trivialcorrelationscharacterized
a by1/f typeof power spectrum.
J

2 PACSnumbers: 05.65.+b05.70.Jk,45.70.Ht05.45.Df
2

] The dynamics of a large number of physical processes
h
c are characterized by bursts of activity in the form of 2.0
e avalanches. For example, the mechanical energy release
m
during earthquakes[1], rivernetworks[2], forestfires [3],
1.8
- landslidesonmountainsorsandavalanchesonsandpiles >
t L)
a etc. Bak, Tang and Wiesenfeld in 1987 proposed that p,
st these systems may actually be exhibiting signatures of <n(t,1.6
t. a critical stationary state [4]. More precisely they sug-
a
m gested that as an indication of the critical state, long 1.4
rangedspatio-temporalcorrelationsmay emergeinsome
- systems governed by a self-organizing dynamics, in ab-
d
1.2
n sence of a fine tuning parameter. This is in essence the 0 200000 400000 600000 800000 1000000
t
o basic idea of Self-organized Criticality (SOC). Sandpile
c modelsaretheprototypemodelsofSOC[4,5,6,7,8,9].
[
In the sandpile model an integer variable n represent- FIG.1: Thefluctuationofthemeannumberofparticlesper

1 ing the number of particles (sand grains) in a sand col- site with time in a system of size L = 64 for the probabili-
v umnisassociatedwitheverysiteofasquarelattice. The ties p = 0.60 (top) 0.52 (middle) and 0.50 (bottom). Both
0 the width of fluctuation and the correlation increases as p
system is driven by adding a particle at a randomly se-
4 approaches pc =1/2.
lectedsite i: n →n +1. Athresholdnumbernp forthe
5 i i c
1 stability of a sand column is pre-assigned. If at any site
0 the particlenumber n >np the columntopples andthis
i c
5 site looses 4 particles and allfour neighboring sites j get state is called a transient state and never appears in the
0 one particle each [4]. stationary state.
/
t The main question we would like to ask in this paper
a n →n −4 and n →n +1 (1)
m i i j j is, for an arbitrary sandpile model to attain the BTW
As aresultsomeofthese neighboringsites mayalsotop- critical behaviour is it absolutely necessary that the sta-
-
d plewhichcreatesanavalancheofsandcolumntopplings. tionary states should only be the recurrent states of the
n The extent of such cascading activity measures the size BTWmodel? Canithappenthattheneighbouringtran-
o
of the avalanche. sient states which are very close to the recurrent states
c
: Sand particles drop out of the system through the of the BTW model are also acceptable in the stationary
v
boundary of the lattice so that in the steady state the statestoachievetheBTWcriticalbehaviour? Inthefol-
i
X fluxes ofin-flowingandout-flowingparticlecurrentsbal- lowingweintroducetheconceptofholesandonaddition
r ance. Inastablestatenumberofparticlesatallsitesare ofholes to the systemthe stationarystates ofthe result-
a lessthannp. Additionofaparticletakesthesystemfrom ing sandpile model cannotbe anymorestrictly restricted
c
onestablestatetoanotherstablestate. Dharhadshown totherecurrentstatesoftheBTWmodelsincetheFSCs
that under this sandpile dynamics, a system evolves to can very well be present in the stationary states of this
a stationary state where all stable states are restricted model. In fact the recurrent and stationary sets coin-
to a subset of all possible stable states. These states are cide in this model. However the distribution of weights
called recurrentstates and they are characterizedby the ofthesestatesmaybequitenon-trivialandthisquestion
absence of forbidden sub-configurations (FSCs) [10]. All remains open. Our numerical results show that even in
recurrent states occur with uniform probabilities in the such a case the critical behaviour is very similar to that
stationary state. A stable state which is not a recurrent of the BTW model.
2
0.12
2.0
1.8 0.10
>
L)1.6
p,
<n(1.4 46 0.08
1.2 3.
-L
1.0 L)0.06
p,
0.45 0.47 0.49 p 0.51 0.53 0.55 (
τ 0.04
2.0
1.8 0.02
>
L)1.6
p, 0.00
n(1.4 -60 -40 -20 0 20 40 60
< 1.5
1.2 (p-p )L
c
1.0
-50 -30 -10 10 30 50
1.68 FIG. 3: Variation of the correlation time τ(p,L) of fluctu-
(p-p )L
c ation of the particle density hn(p,L)i with the probability p
for system size L. The data is shown for two system sizes:
L= 32 and 64.
FIG. 2: The time averaged number of particles per site
hn(p,L)i in the stable stationary states as a function of the
probability of adding a particle p. This variation is symmet-
In this paper we study a combination of particle and
ric about the mid point pc =1/2 and hn(pc,L)i =3/2. The
hole avalanches. We probabilistically add either a parti-
data is for L=32 (circle), L=64 (square) and for L=128
(triangle). cle with a probability p or add a hole with a probability
1−p. Therefore when p = 1, the situation is identical
to the ordinaryBTW model of sandavalancheswhenno
A ‘hole’ may be defined as the absence of a particle. hole is added. On the other hand for p = 0 only holes
Therefore adding a hole to a lattice site implies taking are added to the system and no particle. Therefore for
out one particle from that site: ni → ni−1. Repeated p > 1/2 more particles are added to the system than
addition of holes at randomly selected sites may reduce the number of holes and therefore the net particle cur-
the number of particles at a site less than another pre- rent is 2p−1 and it flows from the bulk of the system
assigned threshold nhc. Therefore if ni at a site is less to the boundary. Howeverfor p<1/2 holes are dropped
than nh, the site losses four holes i.e., four particles are more than particles and the net particle current flows in
c
added to this site and each neighboring sites gets one the opposite direction. At p = 1/2 there is no net cur-
hole (looses one particle): rentinthesystem. Weconsiderp=p =1/2isacritical
c
probabilityandstudythebehaviorofthissystemaround
n →n +4 and n →n −1 (2)
i i j j this critical probability. The time t is measured by the
number of particles and holes dropped in the system.
We call this event as a ‘reverse toppling’. Consequently
at some of the neighboring sites particle numbers may If ni(t,p,L) is the generalised notation for the num-
alsogobelowthenh whichagainreversetoppleandthus ber of particles at site i, then the total number of par-
an avalanche of revcerse topplings take place in the sys- ticles in the system is: n(t,p,L) = ΣLi=21ni(t,p,L). The
tem. Addition of a particle creates a particle avalanche mean number of particles per site is then hn(t,p,L)i =
where as the addition of a hole creates a hole avalanche. n(t,p,L)/L2. In the stationary state hn(t,p,L)i fluctu-
We assign np =3 and nh =0. ates rapidly around its time averaged value hn(p,L)i.
c c
Inverse avalanches were introduced before to get back These fluctuations are shown in Fig. 1 for the system
the recursive configurationcorresponding to the particle size L = 64 and for p = 0.60, 0.52 and for 0.50. It is
deletion operator [13]. observed from this figure that both the width as well as
During the particle avalanches particle current flows the correlation of fluctuation increases as p approaches
intothesystembyadditionofparticlesinthebulkofthe pc from either side of it.
systemandthenthey flowoutofthesystemthroughthe The time averaged number of particles per site
boundary. On the contrary in hole avalanches particle hn(p,L)i is a function of p and the system size L. At
current flows into the system through the boundary and p = 1 it is equal to the average number of particles per
flows out of the system through the bulk of the system. site in the ordinary BTW model which is n1 = 2.125 in
3
1.0 This autocorrelation is observed to decay exponentially
(a)
as: C(t,p,L)∼exp(−t/τ(p,L))where τ(p,L)is the cor-
0.9 relation time. On a semi-log plot of C(t,p,L) vs. t the
)
L
,c slope of the plot gives the value of the correlation time
p
t, 0.8 τ(p,L)whichismeasuredfordifferentprobabilitiespand
(
C
for different L values. For a given system size the corre-
lation time is maximum at p and then decreases mono-
c
0.7
tonically with increasing |p−p |. Also τ(p,L) increases
c
0 2 4 2 6 8
t/L with L at a given p. A scaling plot of the data collapses
103 very nicely as (Fig. 3):
(b)
102 τ(p,L)L−3.46 ∼F((p−p )L1.5) (5)
) c
L
,c
f,p 101 At pc, τ(pc,L) increases as Lµ where µ is estimated to
(
S be 3.45±0.10.
100 Fourier transform of the autocorrelation function
C(t,p,L)isknownasthespectraldensityorpowerspec-
10-1 100 101 f 102 103 trum S(f,p,L) defined as
∞
S(f,p,L)= e−iftC(t,p,L)dt (6)
FIG.4: (a)TheautocorrelationC(t,pc,L)ofthetimeseriesof Z−∞
thefluctuatingmeannumberofparticlespersitehn(t,pc,L)i
for the system size L=64 at pc =1/2. C(t,pc,L) is plotted InFig. 4(a)weshowtheplotoftheautocorrelationfunc-
2
with thescaled timeaxist/L on asemi-log scale. In(b)the tion C(t,p ,L) with scaled time t/L2 for a system size
c
power spectrum S(f,pc,L) is plotted with the frequency f
L=64 andexactly atp . A straightline plot ona semi-
on a double logarithmic scale showing a power law decay of c
log scale implies an exponential decay of the correlation
the spectrum with the spectra exponent being nearly equal
to one. function. In Fig. 4(b) we show the Fourier transform
of the autocorrelation function plotted in Fig. 4(a) gen-
erated by the plotting routine ‘xmgrace’. On a double
the asymptotic limit of large system sizes [9, 14]. As p logarithmic scale the power spectrum S(f,p ,L) vs. the
c
decreases hn(p,L)i slowly decreases but near pc =1/2 it frequency f plot gives a very good straight line for the
decreasesvery fast to a value of hn(p,L)i =3/2. When p intermediate range of frequencies implying a power law
decreases from 1/2 even further, hn(p,L)i decreases fast decay of the spectral density: S(f,p ,L)∼f−β. We es-
c
buteventuallysaturatestoavalueofn0 =3−n1 =0.875. timate β ≈ 1 showing the existence of 1/f type of noise
In Fig. 2(a) we show this variation. To see if the steep in the power spectrum.
riseofhn(p,L)iaroundpcisassociatedwithsomecritical The avalanche size distributions for both particle as
exponent,we makea scalingplotofhn(p,L)i withp−pc wellashole avalanchesare measured. Itis observedthat
foranumberofdifferentsystemsizesLinFig. 2(b). The in the range of p>p the particle avalanche sizes are of
c
data collapse shows: widelyvaryingmagnitudesandofalllengthscaleswhere
astheholeavalanchesizesareverysmallandoftheorder
hn(p,L)i∼G((p−pc)L1.68). (3) of unity. Opposite is the situation for the range p < pc.
Atp howeverboththeparticleaswellasholeavalanche
c
The width of fluctuation is calculated as: w(p,L) = sizedistributionsaresimilarandtheyarefoundtofollow
2 2
hn (p,L)i−hn(p,L)i . For a given L the width is max- a stretched exponential distribution like:
imum at p = p and then monotonically decreases as
c
|p−p | increases. On the other hand for a given p the P(s)∼exp(−asγ) (7)
c
width also decreases with increasing L. It is observed
from numerical estimation that at p = p , w(p ,L) de- where γ is estimated to be around 0.4. Away from this
c c
creases with system size as w(pc,L) = wo + w1L−1/2 critical point pc, the particle avalanches have power law
where wo = 0.076 and w1 = 0.527 are estimated. Be- distribution for p>pc and hole avalanches follow power
yond pc the width decreases as: w(p,L) = |p − pc|−ν law distributions for p < pc. Particle avalanche size dis-
with ν ≈0.82 is estimated. tributionsarecalculatedatp=0.51andforsystemsizes
The time-displaced autocorrelation of the fluctuating L=256,512,1024and2048. Thesedistributionsarevery
mass per site is defined as: similartotheavalanchedistributionsinBTWmodel. For
small avalanche sizes they do follow a power law distri-
hn(t +t,p,L)n(t ,p,L)i−hn(p,L)i2 bution P(s) ∼ s−τ where the exponent τ slowly varies
o o
C(t,p,L)= (4)
hn2(p,L)i−hn(p,L)i2 withthesystemsizeandgraduallyincreasestowards1.2.
4
The large avalanches have multi-fractal distribution and Disorder : Concepts and Tools, Springer; J. M. Carlson
simple scaling does not work for the full distribution and J. S. Langer, Phys. Rev. Lett. 62, 2632 (1989); Z.
[11, 12]. Also the average avalanche size, area and life Olami, H. J. S. Feder and K. Christensen, Phys. Rev.
Lett. 68, 1244 (1992).
timeshavesystemsizedependancesverysimilartothose
in the BTW sandpile: hs(L)i ∼ L2, ha(L)i ∼ L1.72 and [2] I. Rodriguez-Iturbe, A. Rinaldo, Fractal River Basins :
Chance and Self-Organization, Cambridge, 1997.
ht(L)i∼L. [3] P. Bak and K. Chen, Physica D 38, 5 (1989).
To summarize, we have studied a new sandpile model [4] P.Bak,C.TangandK.Wiesenfeld,Phys.Rev.Lett.59,
wherebothparticleaswellasholeavalanchesarecreated. 381 (1987); P. Bak, C. Tang and K. Wiesenfeld, Phys.
Their relative stengths are tuned by a parameterp vary- Rev. A 38, 364 (1988).
ing between 0 and 1 which is the probability for adding [5] D.Dhar,StudyingSelf-OrganizedCriticalitywithExactly
Solved Models, arXiv:cond-mat/9909009.
aparticle andconsequently1−pbeing the hole addition
[6] L.P. Kadanoff, S.R. Nagel, L. Wu and S.M. Zhou, Phys.
probability. Specifically at p = 1 the system is identical
Rev. A 39, 6524 (1989).
totheordinaryBTWmodelforonlyparticleavalanches. [7] P.GrassbergerandS.S.Manna,J.Phys.(Paris)51,1077
Similarlyatp=0thereisonlyhole avalanchesandtheir (1990).
distribution are very similar to the avalanche size distri- [8] A. Corral and M. Paczuski, Phys. Rev. Lett., 83, 572
bution for the BTW model. In the range 1/2 < p < 1 (1999).
there are particle as well as hole avalanches, but the net [9] D. V. Ktitarev, S. Lu¨beck, P. Grassberger, and V. B.
Priezzhev, Phys.Rev.E 61, 81 (2000).
currentisduetotheparticleswhichflowsintothebulkof
[10] D. Dhar, Phys.Rev.Lett. 64, 1613 (1990).
the system. Critical behavior of the particle avalanches
[11] M. De Menech, A. L. Stella, and C. Tebaldi, Phys. Rev.
are observed to have multi-scaling behavior and is very E 58, 2677 (1998).
similar to those of the BTW model. Opposite situation [12] C. Tebaldi, M. De Menech and A. L. Stella, Phys. Rev.
happens in the range 0 < p < 1/2 where net current is Lett. 83, 3952 (1999).
duetoholeswhichflowsintothebulkofthesystem. The [13] D.DharandS.S.Manna,Phys.Rev.E.49,2684(1994).
hole avalanche sizes also have multi-scaling distributions [14] S. S. Manna, J. Stat. Phys., 59, 509 (1990).
very similar to the BTW model.
We thankfully acknowledge S. M. Bhattacharjee for
useful discussions.
[1] D. Sornette and D. Sornette, Critical Phenomena in
Natural Sciences: Chaos, Fractals, Selforganization, and