Damage Spreading in the One-Dimensional
Hinrihsen-Domany Model
Everaldo Arashiroand J.R. Drugowihde Felio
DepartamentodeFsiaeMatematia
FauldadedeFilosoa,Ci^eniase LetrasdeRibeir~ao Preto
Universidadede S~aoPaulo
Av. dosBandeirantes, 3900-14040-901-Ribeir~aoPreto, SP,Brasil
Reeivedon1November,2000
Inthispaperwestudydamagespreadinginaone-dimensionalmodelundertwodynamisintrodued
by Hinrihsenand Domany. Inpartiular,westudythe eetsof synhronousandasynhronous
updatingonthespreadingproperties. Weshowthatthedamagedoesnotspreadwhentheseond
dynami is implementedina synhronous way. We ndthat the rules for updating the damage
produed bythisdynamis,as thetemperaturegoestoinnityandaertain parameteriszero,
areequivalenttothoseofGrassberger'swell-knownmodelAellularautomaton.
I Introdution
The damage spreading tehnique was proposed by
Kaumanthirtyyearsagointheontextofbiologially
motivated ellular automata [1℄. More reently, this
oneptwasextendedtoanalysekinetiIsingandPotts
models whih are desribed bywell-known
Hamiltoni-ans and exhibit ritial properties [2℄. The tehnique
requires followingthe behaviorof twosimilar samples
(slightly dierent initial onditions) under same
ther-mal noise. Usually,the evolutionis done bymeans of
Glauber,Metropolisorheat-bath updating. However,
in order tounderstandtherole ofthedynamis, other
kindsof updatinghavebeenstudied. Infat,this was
the goalof areentpaperby Hinrihsenand Domany
[3℄ whih desribesthe spreading properties in a
one-dimensional modelsubmittedtotwokindsofupdating
(heneforth named I andII). Therst ase(dynamis
I) isverysimilartoGlauberdynamisbut dependson
thesignofthetwoneighboursinsteadofthesiteitself.
Thenewvalueofthevariableat thesiteiis:
i (t+1)= +sign(q i
(t) z) if
i 1 i+1 =+1 sign(1 q i
(t) z) if
i 1 i+1 = 1 (1) where i
(t) = 1, z is arandom number (between
zeroand1)andtheprobability q
i
(t) isgivenby
q i (t)= e h i (t) hi(t) hi(t) (2) with h i (t)= J kT X j j ( t)
and J the exhange oupling onstant (H =
J P <i;j> i j
). DynamisIIombinestheGlauber
up-dating with information about three sites (two rst
neighbours and the site itself). In this ase, the new
valuefor isobtainedby
i (t+1)= +sign(q i
(t) z) if
i 1 i i+1 =+1 sign(1 q i
(t) z) if
i 1 i i+1 = 1 (3a)
ifarandomnumberrisgreaterthanagivenparameter
. Ifr<,theupdatefollowsGlauberdynamis
i
(t+1)=
+sign(q
i
(t) z) if
i
(t)=+1
sign(1 q
i
(t) z) if
i
(t)= 1
(3b)
Thetworulesaboveanbeombinedinoneupdating,
byintroduinganotherparameter,y. Thus,
i
(t+1)=
+sign(q
i
(t) z) ify
i =+1
sign(1 q
i
(t) z) ify
i = 1 (4) where y= i [(1+ i 1 i+1
)+(1
i 1
i+1
)sign( z)℄ ;
zisarandomnumberbetween0and1. Inbothases
thereisadamagespreadingtransition,buttheritial
propertiesaretotallydierent. Whereasthetransition
lass,dynamisIIleadstoa\parity-onserving"phase
transition. Intherstase,theevolutionofthedamage
(dierene between two samples) an bedesribed by
Domany-Kinzelellularautomata rules, at leastwhen
T =1[3℄. Suh amappingisnotknownfor the
se-ondase. Inthispaperweaddresstwoquestions: rst,
Whatistheeet ofsynhronizationonthespreading
of the damage; seond, Is there a known model that
dynamisIIanbemappedto?
II The role of updating
Toimplement thedynamis presentedabove,we have
at least two possibilities: updating one siteat atime
(asynhronousor\ontinuousdynamis")asisusualin
simulationofIsing-likemodels,orhangingallthesites
at the same time (parallel or synhronous updating),
usuallydoneinellularautomata. Oursimulationsfor
dynamisIshowthatdamagespreadingoursinboth
ases. We heked this onlusion by simulating the
Domany-Kinzelellularautomaton(relatedto
dynam-isI,atT =1,viaamappingfoundbyHinrihsenand
Domany)with ontinuousdynamis. Evolutionofthe
damageisthesameinbothases(seeFig.1). However,
for dynamis II, the situation is ompletely dierent.
Dependingonthekindoftheupdating(synhronousor
asynhronous)damagespreadingmayormaynotour
(seeFig. 2). IntheirstudyHinrihsenandDomany[3℄
usedonlyparallelupdating[4℄. Iftheyhadused
ontin-uous dynamis they would not haveobserved damage
spreading,aswehaveshownin oursimulations. Inthe
nextsetion, wewill obtainamappingof dynamisII
to the model A ellular automaton of Grassberger et
al.[5℄. This will allow a better understanding of our
resultsondamagespreading.
III Mapping to Grassberger's
model A
Webeginbysetting=0(whihimpliesworkingonly
withtheequation(3.a)forupdatingthespins)and
tak-ing high temperature limit (T ! 1). In this ase,
equation(3.a)beomes
i
(t+1)=
i 1
i
i+1
sign(0:5 z) (5)
whihmeansthattheevolutionofthedamage()
fol-lowsdeterministiruleswhihdepend onthesiteitself
andthetwonearestneighbours:
t: 111 110 101 100 011 010 001 000
t+1: 0 0 0 1 0 1 1 0
(6)
Nowweobservethattheserulesorrespondtotheone-dimensionalellularautomatonobeyingrule22(inWolfram's
lassiation),whihis knownto exhibithaotibehavior. Ontheotherhand,wenotethat therule22isexatly
thelimitof theGrassberger'smodelAbelowwithp=1. Thusthedamagespreads.
t: 111 110 101 100 011 010 001 000
t+1: 0 1 p 0 1 1 p 1 1 0
(7)
Inthissense,weanonludethatHinrihsen-Domany'sdynamisII exhibitsdamagespreadingatleastwhen
=0andT =1. Ontheotherhand,for=1,andT =1;equation4anbesimpliedto
i
(t+1)=
i
sign(0:5 z) (8)
whihimpliesanotherkindofupdatingforthedamage. Wendthattheloaldamageinthisaseevolvesexatly
astheone-dimensionalellularautomatongovernedbytherule204(againinWolfram's notation).
t: 111 110 101 100 011 010 001 000
t+1: 1 1 0 0 1 1 0 0
(9)
We remark that damage does not spread in this limit beause the value of is onserved in the evolution
Figure1.PatternsreatedwithDomany-Kinzelautomaton
rules (withp1 =0.83ep2 =0). (a)Synhronousupdating
and(b)Asynhronousupdating.
Figure2.PatternsreatedwithmodelAellularautomaton
rules (with p=0.3) frominitialstates ontaininga single
kink. (a)Synhronousupdatingand(b)Asynhronous
up-dating.
IV Conlusions
We studied damage spreading in a one-dimensional
modelevolvingundertwodierentdynamis,proposed
by Hinrihsen and Domany[3℄. We showed that
syn-hronous updating andestroy the damage spreading
when the dynamis II is used. In addition, we
ob-tainedamappingofdynamisII tothemodelA
ellu-larautomatonintrodued byGrassbergeret al.[5℄. In
thislight, it is notstrangethat ritial exponents
ob-tainedbyHDwereintheparityonservinguniversality
lass[6℄.
Referenes
[1℄ S.Kauman,J.Theor.Biol.22,437(1969).
[2℄ H.E.Stanleyetal., Phys.Rev.Lett.59,2326(1987)
[3℄ H. Hinrihsen and E. Domany, Phys. Rev. E56, 94
(1997).
[4℄ H.Hinrihsen,privateommuniation.
[5℄ P.Grassberger,F.KrauseandT.vonderTwer,J.Phys.
A:Math. Gen.17,(1984).
[6℄ G.
