International Journal of Modern Physics B, Vol. 10, No. 10 (1997) 1245-1255@ World Scientific Publishing Company

JOINT EVOLUTION OF TWO DOMANY-KINZELCELLULAR AUTOMATA

EMÍLIO P. GUEUVOGHLANIAN and TÂNIA TOMÉ

Instituto de Física, Universidade de São Paulo,Caixa Postal 63618, 05389-970 São Paulo, SP, Brazil

Received 6 May 1996

We analyze the joint dynamics of two systems each one evolving in time according to theDomany-Kinzel mies. Two prescriptions for the joint evolution are studied. We makeanalytical approaches, based on dynamical mean-field approximations at the levei ofthree-site correlations, to obtain the phase diagram. AIso, the susceptibilities associatedwith the order parameters are calculated by using a pair approximation.

\

1. Introduction

An important approach in the study of nonequilibrium phase transitions is theone related to the construction of lattice models that evolve in time according toirreversible stochastic rules.1-3 One of the main purposes of this approach is thestudy of models that are simpIe from the mathematical point of view, but that cangive relevant contributions to the understanding of the intrinsic aspects of the irre-versible phenomena. On the other hand, it has been verified that such formalismcan be very useful in the description of biological, chemical and physical systems.In this case, lattice models are constructed in order to mimic the microscopic localinteractions related to a specific macroscopic phenomenon. In the context of suchapproach there are the cellular automata,4 which are systems that evolve in discretetime steps according to a synchronous update of the sites in the lattice. The up-dating of each site is performed according to a set of local rules. Among the varietyof lattice models that have been devised to study nonequilibrium phenomena theDomany-Kinzel cellular automaton5,6 appears to be one of the simplest irreversiblemodels that displays a phase transition in one dimension. Moreover the modeland its time evolution are themselves the realization of the directed percolationproblem.5 As it is known, the directed percolation is dosely related to the propaga-tion of epidemics.2,7 Therefore the Domany-Kinzel cellular automaton is insertedin the class of lattice models which intend to explain some of the aspects relevantto the study of simple epidemics8: a problem coming from the epidemiology andthat has been also studied from the point of view of stochastic lattice models whichpossesses absorbing states.

1245

I

1246 E. P. Gueuvoghlanian & T. Tomé

The recent developments on the study of the Domany-Kinzel cellular automatonrelated to the spreading of damage9 evidenced new important features in this model,as the detection of a chaotic phase. The spreading of damage is, here. studied inan analytical way by considering the joint evolution of two automata subjected tothe same noise. The dynamics associated with the joint evolution of two automataevolving iIi. time subjected to the same noise has irreversible aspects. The phasetransition due to the damage spreading in this modeI raised another problem relatedto the study of universality in nonequilibrium critical phenomena.2 The active-chaotic transition is in fact related to an absorbing subspace of the configurationalspace associated to the joint dynamics. According to Grassberger2 this damagespreading transition belongs in the directed percolation class.

Here we pedorm analytical studies of the automaton itself and the spreading ofdamage in the automaton (that is the joint evolution of the set of two automata) byusing dynamical mean-field approximations.lO The paper is organized as fo11ows.lnSeco2, it is shown the analytical definition of the Domany-Kinzel cellular automa-tono ln Seco3, we set up the analytical formulation for the study of the spreading ofdamage. We devise the evolution equations for the joint evolution of two automataand study them by using two different prescriptions. ln Seco4, a three-site meanfield approximation is used to get the phase diagrams. An extrapolation of the re-sults obtained from the one, two and three-site approximations is made to comparewith the results from numerical simulations. The susceptibilities associated withthe order parametersl1-13 of this modeI are calculated in Seco5, by using a pairapproximation. Conclusions and discussions are presented in Seco6.

..

..

2. The Automaton

The Domany-Kinzel cellular automaton5,6 is represented by a one-dimensionallat-tice with N sites. The configurations of the system are given by a set of occupationvariables U = (Ul,U2,.. . ,UN), with Ui = Oor 1. The system evolvesat discretetime steps and at each step alI the sites are updated simultaneously according tothe following rule: the transition probability in which the site i acquires, at timef + 1, the value Ui = O or 1 is conditioned to the Ui-l and Ui values at time f:

(1) ..

The transition probabilities that define the model are

w(O,O ~ 1) = po,

w(O,1 ~ 1) = Pl,

w(l,O ~ 1) = P~,

(2)

(3)

(4)

and

w(l, 1 ~ 1) = P2 . (5)

Joint Evo/ution 01 Two Domany-Kinzel Cel/u/ar Automata 1247

The other four remaining probabilities are obtained by taking into account that

(6)

....

for any values of 0-;_1 and 0-;.

Here, we will consider only the case P~ = P1 and po = O, except in Seco 5 where

we allow Po =I O. At stationary states, the system can find itself in two differentsituations:

(a) All sites in state O, that is, o-= (0,0,... ,0).(b) A fraction p of the sites are in state 1, with P =IO.

The first situation corresponds to an absorbing state. Once the system entersthat state, it cannot leave it, since po = O. The points (P1,112)ofthe phase diagramfor which the system always reaches the absorbing state define the frozen phase.5,6The points (P1,P2) of the phase diagram for which the system can have a nonzerofraction of sites in the state 1 define the active phase.

The time evolution of the automaton is given by

(7)17'

where o- is given by the vector o- = (0-1,0-2,...,o-N),Pi(o-)is the probabilityoffinding the system in the state o- at time f, and W(o-' --> 0-) is the probabilityassociated to a transition at which the system passes from state 0-' to state 0-,giventhat at the previous time the system was in state 0-'. The transition probabilityW (o-' -->0-) is given by

N

W(o-' --> 0-) = IIW(0-;_1'0-; --> o-i)i=1

(8)

obeying the normalization condition

L W (o-'-> 0-)= 1. (9)17

which follows from Eq. (6).The active phase order parameter is identified as Pi(o-i = 1) = Pi(l), which is

the probability that a given site is occupied at time f. Using Eqs. (7) and (8) oneobtains the evolution equation for this quantity,

Pi+1(o-i)= L W(0-;_1'0-;-> o-i)Pi(0-;_1,o-D. (10)O'~_l,q~

The Eq. (10) relates P(o-i) to the probabilities involving two neighbor sites. Theselast quantities obey a time evolution equation given by

(11)

,

1248 E. P. Gueuvoghlanian & T. Tomé

In general, the probabilities Pi+1 (O"i,O"i+1," o, O"i+M) are related to the proba-

bilities Pi(O"i,. oo, O"i+M+1)o Thus, if one wishes to know the value of P(O"i), oneneeds to work with an infinite number of equations which constitutes a hierarchyof equationso One way of dealing with this prablem is to use truncation schemes aswill be done in Seco 40

3. Prescriptions for the J oint Evolution

The active state af the Domany-Kinzel automaton can behave in two different

ways when ane looks to the spreading of damage09,13-15The meaning of spreadingof damage can be understood by considering the simulation of the automaton usingpseudorandom numberso When a stationary state is reached, a replica of the originalsystem is created, with a fraction of the sites having their occupation variableschangedo The two systems, original and replica, are allowed to evolve under thesame noiseo The Hamming distance is measured. This quantity is defined as thefraction of sites in the replica that have different occupation variable values thanthe corresponding sites in the original systemo If the Hamming distance is differentfrom zero, ane says the damage spreads throughout the system and the systemis sensitive to the initial damageo In this case the system is in a chaotic phaseoOtherwise, it is in a non-chaotic phase and there is no spreading of damageo

The complete analysis of the phase diagram requires the obtention of equationsto evaluate the spreading of damage, which is done by considering the problem ofthe joint evolution of two systems.13,15 Consider a system and its replica given byO"= (0"1, oo. , O"N) and r = (rI, ooo, rN). The time evolution of the joint probabilityPi(O";r) is given by

Pi+1(O";r) = L Pi(O"',r')W(O"',r' -+ 0",r)U',T'

(12)

whereN

W(O"',r' -+ 0",r) = IIW(O"~-I,O"~,rLl,rI -+ O"i,ri),i=1

(13)

which obeys the normalization condition

L W (O"' , r' -+ 0",r) = 1 , (14)a,r ..

by requiring that

(15)

A relation between w(O"~_I,O"~,rLl,T: -+ O"i,Ti)and W(O"~_I'O"~-+ O"i)is deter-mined by requiring that each system evolves in time independently of the other,which implies that the following equations are satisfied:

Joint Evolution 01 Two Domany-Kinzel Cellular Automata 1249

" (I I I I

) (I I

)~ W 17;_1'a;, Ti-I' T; ---+ai, Ti = W T;_I' T; ---+Ti , (16)Ui

" ( I I I I . ) _ (I I

)~ W ai-I' a;, Ti-I' Ti ---+ai, Ti - W ai-I' a; ---+ai . (lí)Ti

In addition, we should impose the condition that the two systems evolve intime under the same noise. The following table gives the transition probabilities

(I I I I

) b.

d.

th ' .W ai-I' ai' Ti-I' Ti ---+ai, T; o tame m IS way.

(18)

The other transition probabilities are obtained using the symmetries inherent tothis mode!. For example:

w(OO,l1 ---+0,1) = w(l1, 00 ---+1,O). (19)

In the table given above there are four quantities, a, b, c and d, not yet specified.Two different prescriptions for these quantities have been suggested. They are:

(1) Prescription A, introduced in Ref. 13 and given by

a = Pl, b = P2- Pl, C= O, d = 1 - P2, if Pl::; P2 , (20)

a=P2, b=O, C=PI-P2, d=I-Pl, if P2::;Pl, (21)

(2) Prescription B, introduced in Ref. 15, given by

Prescription B implies that one must use, in certain cases, two different randomnumbers to update the original and the replica.

Now turning back to the question of the spreading of damage, we need an ex-pression for the chaotic phase order parameter. This is given by the Hammingdistance

Dl = (ai - Ti)2}l = Pl(ai = 1,Ti = O)+ Pl(ai = O,Ti = 1). (23)

Since the two quantities in the right hand side are equal, the Hamming distance isjust 2P(I, O). Using Eqs. (12) and (13) we obtain the time evolution equation forthis quantity

Pl+1(l,O)= L L w(a~_I,a~,T:_l,T: ---+ai,Ti)Pl(a~_l,a~,TLl,TD. (24)(7'~_1,O'~T:_1 ;r:

W (11,11) (10,10) (00,00) (11,00) (10,00) (11,10)

(1,1) P2 Pl O O O a

(1,0) O O O P2 Pl b

(0,1) O O O O O c

(0,0) 1- P2 1- Pl 1 1- P2 1- Pl d

1250 E. P. Gueuvoghlanian fj T. Tomé

It can be verified that the probabilities P((Ji,"', (Ji+M;Ti,. .., Ti+M) are relatedto the probabilities P((Ji,... ,(Ji+M+1;Ti"", Ti+M+d which means that we need aninfinite number of equations to evaluate the Hamming distance.

4. Three-Site Approximation

Our aim, in this section, is to obtain the phase diagram of this automaton inthe stationary regime using a three-site dynamical mean-field approximation. One-site13,15and pair mean-field approximations13,16 have already been used to obtainthe phase diagram of this model. Here, we will use a three-site approximation,in which the probabilities of clusters of four or more sites are written in terms ofprobabilities of three or less sites, that is,

P( )_ P((Ji, (Ji+1, (Ji+2)P((Ji+1, (Ji+2, (Ji+3)

(Ji,(Ji+1,(Ji+2,(Ji+3 -P( )(Ji+1, (Ji+2

(25)

and

_ P( (Ji, (Ji+1, (Ji+2, Ti, Ti+1, Ti+2 )P( (Ji+1, (Ji+2, (Ji+3, Ti+1, Ti+2, Ti+3)

- P((Ji+1, (Ji+2, Ti+1, Ti+2)(26)

Moreover, we have used many relations that come from considerations of the

symmetry between the original system and its replica, from the homogeneity of theindividual systems and from the occurrence of complementary events.

Using the analytical formulation presented in this section, we obtained in thethree-site approximation a nonlinear system with twenty equations and twenty vari-ables, namely:

P(I), P(l1), P(lOl), P(111),

P(I, O),P(10, 01), P(l1, 00), P(l1, 10),

P(OOO,001), P(OOO,011), P(OOI, 011), P(OOI, 101),P(OOI,111), P(OlO, 001),

P(OlO, 011), P(Ol1, 111), P(100, 001), P(100, 011), P(101, 011), P(110, 011) .

1.We will not write down the equations here because they are toa cumbersome. Nu-

merically, we verified that the solutions of these equations are of three types:

(a) P(I) = P(I,O) = o, characterizingthe frozenphase.(b) P(I) =Io, P(I, O)= o, corresponding to the activephase.(c) P(I) =Io, P(I,O) =IOwhich characterizes the chaoticphase.

Figures 1 and 2 show the phase diagrams obtained, in the three-site approxi-mation, for the prescriptions A and B, respectively. It is convenient to give thevalues of some special points. When P1 = 1 the transition active-chaotic occurs at

Joint Evo/ution 01 Two Domany-Kinze/ Cel/u/ar Automata 1251

1.0

~0.5FROZEN

ACTIVE

CHAOTIC

0.00.0 0.5

P,1.0

Fig. 1. Phase diagram of the Domany-Kinzel cellular automaton according to prescription Aobtained by using three-site dynamical mean-field approximation.

1.0

FROZEN

~ 0.5

CHAOTlC

0.00.0 0.5

p,1.0

Fig. 2. Phase diagram of the Domany-Kinzel cellular automaton according to prescription Bobtained by using three-site dynamical mean-field approximation.

1252 E. P. Gueuvoghlanian & T. Tomé

P2= 0.427. For P2 = O the transition frozen-chaotic is given by Pl = 0.709. Thetwo prescriptions give the same results for these points because the prescriptionsbecome identical when PI = 1,or when P2 = O. The point where the three phasesmeet each other is the point pi = 0.696 and P; = 0.251, for the case of prescriptionA and given by pi = 0.678 and P; = 0.466for the case of prescription B.

Below we write down the values of pí:, P2, pi, and P; for both prescriptionsobtained by various mean-field approximations together with an extrapolation andMonte Carlo results2,16

The extrapolated values for Pl' P2, pi(A), and pi(B) where obtained by a linearfitting of these values versus l/n where n is the degree of mean-field approximation.The points are fairly aligned so that the extrapolation values are sensible. For thecase of p;(A) and p;(B) it is not possible to use the same procedure since thepoints are not aligned. However, if one plots p;(A) against n-I/4 the points arefairly aligned, yielding the extrapolated value p; = o.

5. Conjugate Fields

The models studied in the scope of the equilibrium statistical mechanics, a conjugatefield H associated with a quantity M is introduced by adding, in the Hamiltonianthat defines the system, a term linear in H. In this cellular automaton this proceduremust be modified, since the system is defined by a set of irreversible dynamic rules.In fact, it has been suggested that one can consider conjugate fields related to thedynamic rules.u-13 Following the analytical formulation given in Ref. 13, we studyconjugate fields to the transition probabilities and the associated susceptibilities.Here we calculate these susceptibilities by using the pair approximation.

The order parameter associated with the active phase is given by the occupiedsites density, P(I). One can consider the conjugate field as the transition probabilityw(O,O ~ 1) = Po, given by Eq. (2), because if Po i= O, then P(I) must havestationary values different from zero. The associated susceptibility is given by Xl =ôP(I)/ôpo.

Using Eq. (10) and consideringPoi= Owe obtain the followingexpression

(27)

approximation Pl P2 pi( A) p;(A) pi(B) p2(B)

one-site 0.5 0.667 0.5 0.333 0.5 1

two-site 0.667 0.482 0.650 0.287 0.619 0.617

three-site 0.709 0.427 0.696 0.251 0.678 0.466

extrapolation 0.80 0.33 0.80 0.80

Monte Carlo 0.810 0.312 0.810 O 0.810 O

Joint Evo/ution of Two Domany-Kinze/ Cellu/ar Automata 1253

We derive the evolution equation for Pe(ll) and evaluate this equation by using theapproximation P(O"i,O"i+1.O"i+2)= P(O"i,O"i+dP(O"i+1,O"i+2)jP(O"i+d.As the otherpair correlation functions in Eq. (27) are defined as functions of P(ll) and P(1).this Eq. (2'/) with the evolution equation for P(ll) form a closed set of equations.which we use to derive the susceptibility at the leveI of pair approximation.

We find that, approaching the critical points in the frozenjactive transition fromboth sides. the susceptibility calculated at Po = Ois

(28)

where

(29)

and

G(. ) _ [2pl(P2 - pd - 1](2Pl - P2) + 2(2Pl - 1)(Pl - P2)2 (30)PbP2 -

(1 )(1 )? .

- P2 - Pl -

The susceptibility Xl diverges at

which defines the frozen-active transition line (within the pair approximation).Moreover, the susceptibility at Po = O diverges as Àl '" !P2 - F(pdl-"Y with aclassical critical exponent "I = 1, characterizing a second-order transition. Theseresults agree with the one-site approximation results.13

The other order parameter of this problem, the Hamming distance. which isrelated to the spreading of damage is given by P(1, O). A possible choice to theconjugate field h, in this case, is

where H' (O"'~ 0") and W (O"',r' ~ 0",r) are given, respectively. by Eqs. (8) and(13) and n-J1(O"'.r' ~ O".r) is a modified joint transition probability. One canverify that, when h gets the value one, the two systems, original and replica, arecompletely disjoint. Thus, when h i=O, the Hamming distance attains a non-zerostationary value. The associated susceptibility is defined by \2 = ÔP(l. O)jôh.

The joint time evolution equation is now,

(32)q',T'

Using Eq. (32) we obtain:

PHl (1,O)= 2p1Pe(10, 00) + p2Pe(1l, 00) + 2(b + c)Pt(ll, 10)

+ h{2Pl(1- pd[Pe(10,0l) + Pt(lO, 10)] + P2(1- P2)Pt(1l, 11)

+ 2[Pl + P2 - 2PIP2 - (b + c)]Pt(ll, 1O)}. (33)

1254 E. P. Gueuvoghlanian & T. Tomé

Evaluating the evolution equations for the correlations in the pair approximationand considering the conditions where we are very dose to the active-chaotic tran-sition line we obtained an expression for X2 at h = O. Using the expression for the

active-chaotic line given in the pair approximation we have obtained, numerically,that X2 at h = O diverges at the criticalline active-chaotic with a dassical critical

exponent equal to 1.

6. Conclusions

ln this paper we obtained the phase diagram of the Domany-Kinzel cellular automa-ton using a three-site approximation. Our extrapolated results are in agreementwith simulations. We have calculated the susceptibilities using a pair approxima-tion and confirmed the one-site approximation results obtained previously.13 Ourcalculations were done for two prescriptions (A)l3 and (B).15 ln prescription A theevolution of the system and its replica is subjected to the same noise. ln prescrip-tion B the system is updated with the same noise in some cases and in others it isnecessary to generate two random numbers to update the system and the replica. Itis important to observe that the joint dynamics of the two systems must be under-stood as a third dynamics with more states per site. ln this sense, distinct (joint)prescriptions define different dynamics. 80, a priori, two different prescriptions cangive distinct results for the active-chaotic line. This is what really happens for thecase of prescriptions A and B here presented. However if a numerical simulationof the spreading of damage is made with one sequence of random numbers, as per-formed in the pioneering work by Martins et al.,9 then the analytical prescriptionto be considered must contain the element of joint evolution subjected to the samenoise. A prescription where this fact is taken into account is given in Ref. 13, andit is here stated in Eqs. (20)-(21). It does not mean that the prescription A isthe only prescription capable to describe the joint evolution. One can devise otherdynamics as long as one indudes the element of the joint evolution subjected to thenoise.

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Joint Evolution of Two Domany-Kinzel Cell1/./a,' Automnln 12.55

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