ELSEVIER Physica A 234 (1996) 427-434

I l l

Critical phenomena in an one-dimensional probabilistic cellular automaton

Pratip Bhattacharyya Low Temperature Physics Section, Saha Institute of Nuclear Physics, 1/AF Bidhanna#ar,

Calcutta 700 064, India

Received 15 May 1996

Abstract

A one-dimensional probabilistic cellular automaton that models a transition from elementary rule 4 to elementary rule 22 (following Wolfram's nomenclature scheme) is studied here. The evolution of the automaton follows rule 4 with probability 1 - p and rule 22 with probability p. In course of the transition the system shows two critical points, a trivial pc~ = 0 and a nontrivial pc2 "~ 0.75, at which the relaxation time of the system is observed to diverge in the form of a power law z ,,~ ( p - p c 1 ) -z~ and z ~ ( p c 2 - p ) -z2 with zl --~ 0.86 and z2 ----- 0.92. The point pc2 is also a point of phase transition with the density of occupied sites in the equilibrium state as the order parameter; the order parameter goes to zero as n ~ (p - pc2) #, fl _~ 0.32 for p---~pc2+. The possible cause of the observed behaviour is discussed.

Adding very specific nonthermal noise to one-dimensional systems with short-range

interactions can bring about critical phenomena. This was shown to occur in certain

one-dimensional elementary cellular automata [1] characterised by the presence o f two

absorbing states. These systems however never show critical behaviour in the presence

o f thermal noise. A one-dimensional elementary cellular automaton rule [2] is composed

o f 23-- 8 components corresponding to the transition o f eight distinct three-site neigh-

bourhoods (Xi- l ,Xi ,Xi+l) . Specific noise can be added to an originally deterministic rule

by making one or more (not all) o f the eight components occur with a probabili ty p

and their conjugates occur with probabil i ty 1 - p [1]. Varying the parameter p from

0 to 1 leads to a transition from one deterministic rule (corresponding to p = 0) to

another (corresponding to p = 1) [3]. Previous studies o f the effect o f specific noise in

cellular automaton models involved the transition from rule 94 to rule 22 and from rule

50 to rule 122 [1]. In each case two symmetric components o f the rule were made to

occur with a probabil i ty p while the deterministic nature o f the other six components

were retained. The models showed a nontrivial (nonzero) critical point that marked

a phase transition with respect to the stability o f kinks (boundary between domains

0378-4371/96/$15.00 Copyright (~) 1996 Elsevier Science B.V. All rights reserved PH S0378-4371 (96)00273-7

428 P. Bhattacharyya/Physica A 234 (1996) 427-434

of ordered states); the density of kinks in the equilibrium state served as the order parameter.

Here I report the behaviour of a one-dimensional probabilistic cellular automaton that models a transition from the simple elementary rule 4 at p = 0 to the complex elementary rule 22 at p = 1 (following Wolfram's nomenclature scheme). In course of this rule transition the system exhibits two critical points: the first point p c l = 0

is trivial for one-dimensional systems; the second point Pc2 is different from zero and marks a phase transition with the equilibrium density of occupied sites as the order parameter.

The model is defined as a linear array of sites with a discrete variable x assuming two states per site x i E {0, 1}. A site is said to be occupied if x i = 1 and unoccupied otherwise. All sites are updated simultaneously following the evolution rule:

t : 111 1 1 0 0 1 1 101 010 100 001 0 0 0 (l)

t + l : 0 0 0 0 1 0 • J

I with pr~ability p, 0 with probability 1 - p

The evolution rule therefore is equivalent to a linear superposition of elementary rule 4 (following Wolfrarn's nomenclature scheme) occuring with probability 1 - p and elementary rule 22 occuring with probability p.

To study the behaviour of this one-dimensional cellular automaton model the follow- ing computer experiment was performed. A one-dimensional lattice of 5000 sites was set to a random initial condition: at t---0 each site assumes value 1 with probability 0.5 and 0 with probability 0.5. Periodic boundary conditions were used. This random initial state was then allowed to evolve following the model rule (1). Typically evolu- tion from 50 independent initial states were studied for each value of p . The following observations were made:

(1) For p = 0 the system has a constant density of occupied sites from t = 1 onward; this density is found to be 0.128 compared to the theoretically expected value of I" For p = 0 the evolution of the automaton takes place by the rule component 0 10--~ I only; this represents evolution by elementary rule 4. Therefore, all ( 010) neighbourhoods in the initial state evolve unchanged. The pattern formed on the space-time plane consists of linear stripes of occupied sites parallel to the time-axis (Fig. 1).

(2) For p > 0 the rule components 001 --. 1 and 100 --+ 1 become active with probability p each. For small p the density of occupied sites n was observed to decay to zero exponentially (Fig. 2(a)):

n ,.~ e - t /~ . (2)

As the noise level p is increased the decay process is enhanced. Looking in the other direction, as p ~ 0 from above the relaxation time r of the system increases and diverges at p = 0 (this corresponds to a constant density of occupied sites). Therefore, the first critical point of the model system is given by

pd = 0 , (3)

P. BhattacharyyalPhysica A 234 (1996) 427-434 429

I llll II 1111 !

(a) - !._~_.[_11[[_.

(d)

I

Cb)

(e)

L 1. .1. , I. 1,. l

(e)

(f)

Fig. 1. Part of the one-dimensional lattice evolving in time according to the model rule (1) for different noise levels p: (a) p = 0 (elementary rule 4), (b) p = 0.01, (c) p = 0.5, (d) p = 0.65, (e) p = 0.8, (f) p = 1.0 (elementary rule 22).

which is trivial for one-dimensional systems with short-range interactions. The critical behaviour observed is characterised by a dynamical exponent z l (p ~ Pc~ +) (Fig. 2(b)):

z ~ ( p - Pct )-z~, zl ----- 0.86. (4)

(3) As the noise level p is increased above a certain value (p >~0.5), the relaxation time of the density of occupied sites n is found to increase with p and it diverges at a second critical point:

p c 2 ~ 0 . 7 5 . (5)

As p approaches Pc2 from below the density of occupied sites n decays exponentially only at large times (Fig. 3(a)). At p = Pca the decay of n becomes critically slow and shows power law behaviour:

n ~ t -~, ~ 0 . 1 6 . (6)

The point p = Pc2 marks the point of transition from a phase with zero equilibrium density of occupied sites (P<~Pc2) to a phase with nonzero equilibrium density of occupied sites (p > Pc2) (Fig. 4(a)). The behaviour of the relaxation time z and the

430 P. Bhattacharyya/Physica A 234 (1996) 427-434

0.1

0.01

0.001

0.0001

1 e -05

(a)

p = 0.00002

p = 0.00005

p = 0.0001

. p = 0,0002

p = 0.0003

2000

p = 0.002

l

4000

p = 0.001

6000 8000 t

- p = 0 .0005

10000 12000

0,01 . . . . . . . . I

(b)

0,001

/ , /

P

0.0001 ,~"

,,/

,7"

1@.05 . . . . . . . . J " 1 e-05 0.0001

P

O i 1

,/ 7"

/ /

. . . . . . . I . ,

0,001

Fig. 2. The behaviour near the first critical point Pci =0: (a) The decay of the density of occupied sites n with time near Pcl. (b) Near Pcl the relaxation time r follows a power law of the form r = 0.485 ( p - P c l )0.86 shown by the dashed line (---).

0.1

0.01

(a)

P. Bhattacharyyal Physica A 234 (1996) 427-434

i 1 t i ' '

P = p = 0.743

I I I I I 0 2000 4000 6000 8000 100(30

t

431

12000

0,001

'~ 0.0001

. . . . i

(b)

/ J f " @

/ , "

~ / , "

7 " f / "

. /

le -05 . . . . ~ . . . . . . . . 0,001 0.01

p - 0,75

Fig, 3. The behaviour near the second critical point Pc2 = 0.75: (a) The decay of the density of occupied sites n with time near pc2. For large t, data averaged over many time-steps have been plotted to suppress fluctuations. (b) Near Pc2 the relaxation time z follows a power law of the form ~ = 0.024 (Pc2 - p)0.92 shown by the dashed line (---).

432 P. Bhattacharyya/ Physica A 234 (1996) 427-434

0.5

0.45

0.4

0.35

0.3

" 0.25

0,2

0.15

0.1

0.05

0 0.6

(a)

0.65

~ C ¢ ¢..~_a f I I t

0.7 0.75 0.8 0.85 0.9 0.95 P

0.1

(b) ,~

/ H

j 7 I

j z

0.001 0.01 0.1 p - 0.75

Fig. 4. (a) The variation of the density of occupied sites n with the noise level p. p -~ 0.75 marks a point of phase transition. (b) Near Pc2 ~ 0.75 n shows a power law behaviour. The dashed line (---) shows the function n =0.67 ( p - Pc2) 0-32.

P. BhattacharyyaI Physica A 234 (1996) 427-434 433

equilibrium density of occupied sites n ~ near the second critical point are characterised

by the exponents z z ( p ~ P c 2 - ) and fl respectively (Figs. 3(b) and 4(b)):

T ~ (p¢.2 - P)-~'~, z2 -~ 0.92, (7)

nc¢ ~ ( p - Pc2) [t, fi ~- 0.32. (8)

The simulations were not accurate enough to determine the dynamical exponent for

P---~ Pc2+. With the rule 22 cellular automaton showing long-range correlations [5], it is ex-

pected that correlations will build up as the model makes a transition toward rule 22

and a mean-field scheme [2] for the equilibrium density of occupied sites fails com-

pletely. Due to large fluctuations in the simulations with the present system size it was

not possible to study the spatial and temporal correlations in the model.

The observed behaviour of the cellular automaton model may be explained as fol- lows. The deterministic rule component 0 1 0 ~ 1 keeps the occupied central site of

only (0 1 0) neighbourhoods occupied forever; this characterises the ' s imple ' behaviour

[2] o f elementary rule 4. The peculiar behaviour of the model is due to the probabilis-

tic components 0 0 1 ~ 1 and 1 0 0 ~ 1. These components give rise to two competing processes: (1) creation of occupied sites, x~ t) = 0 ~ x~ t+l) = 1; (2) annihilation of occupied sites, x~t)= 1 ~ x ~ t + l ) = 0 . To illustrate the point I choose an initial state with

a single occupied site, x) °) =6ji:

.. i - 3 i - 2 i - 1 i i + 1 i + 2 i + 3 ..,

.. 0 0 0 1 0 0 0 . . .

Evolving according to the model rule (1) four configurations are possible at t = 1:

. . . i - 3 i - 2 i - 1 i i + 1 i + 2 i + 3 . . .

(1) . . . 0 0 0 1 0 0 0

(2) . . . 0 0 1 1 0 0 0 (3) . . . 0 0 0 1 1 0 0 (4) . . . 0 0 1 1 1 0 0

with probabilities (1 - p)2, p(1 - p) , (1 - p ) p and p2, respectively. The first con-

figuration is due to inactivity of the probabilistic rule components whereas the other three configurations show creation of new occupied sites; the probability that at least

one occupied site is created at t = 1 is 2p - p2. Once new occupied sites are created at t - -1 , neighbourhoods of the form (0 1 1), (1 1 0) and (1 1 1) appear and all o f these forms deterministically evolve to an unoccupied central site at t = 2 according the model rule (1); this defines the process of annihilation. Thus the process o f creation directly leads to the process of annihilation. I f a nonzero number of occupied sites is to be maintained at least one of the (00 1) and (1 00 ) neighbourhoods at the periphery of the occupied regions at t = 1 has to evolve to a new occupied site at t = 2. It

434 P. Bhattacharyya/Physica A 234 (1996) 427-434

is the competition between the creation-induced annihilations and further creation of occupied sites that governs the evolution of the system. In any disordered state such creations and annihilations take place in the vicinity of occupied sites. At p = 0 all (0 1 1), (1 1 0) and (I 1 1) neighbourhoods are eliminated in the very first time-step of evolution; thereafter no further creation or annihilation of occupied sites occur and the (0 1 0) neighbourhoods maintain a constant density of occupied sites. For p close to zero, creation of occupied sites are few but once a new occupied site appears in the neighbourhood of an existing one it leads to the annihilation of both. The low proba- bility of further creations fails to compensate for the annihilations and the density of

occupied sites decays to zero with time. The smaller the value of p (i.e. the closer it is to zero), the fewer are the creation-induced annihilations and the longer it takes for the density of occupied sites to relax to zero. This results in the occurence of the first critical point Pc1 = 0. An increase in p enhances the process of creation of occupied sites which in turn increases the creation-induced annihilations; that causes a rapid decay of the density of occupied sites. The probability of creation of occupied sites increases with p and overcomes the annihilation process at the second critical point Pc2. For p > Pc2, dominance of the creation process over the annihilation process leads to a nonzero density of occupied sites in the equlibrium state ( t=c~) .

Though all the simulations were done starting with an initial density of occupied sites no--0.5, the same critical behaviour occurs for any initial density. This is expected from the fact that all complex one-dimensional deterministic rules evolve to a equilibrium density of occupied sites irrespective of their initial density [2]. It only takes a longer time to reach the equilibrium state from lower initial densities; thus the rule 22 cellular automaton shows critical slowing down of the process of relaxation to the equilibrium state as the initial density of occupied sites approaches zero [4]. However this may not be true for two-dimensional cellular automata where not all rules lead to a unique equilibrium density of occupied sites and phase transitions are observed with respect to the initial density of occupied sites [6].

Acknowledgements

I thank Professor Bikas K. Chakrabarti, Dr. Krishna Kumar and Dr. Sujit Sarkar for discussions and a critical reading of the manuscript. I am grateful to Professor D. Stauffer for pointing out reference [6]. This work was supported by C. S. I. R., India.

References

[1] P. Grassberger, J. Phys. A: Math. Gen. 17 (1984) L105. [2] S. Wolfram, Rev. Mod. Phys. 55 (1983) 601. [3] W. Kinzel, Z. Phys. B 58 (1985) 229. [41 J.G. Zabolitzlcy, J. Statist. Phys. 50 (1988) 1255. [5] P. Grassberger, J. Statist. Phys. 45 (1986) 27. [6] D. Stauffer, Physica A 157 (1989) 645; S.S. Manna and D. Stauffer, Physica A 162 (1990) 176.