The Domany-Kinzel cellular automaton revisited

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The Domany-Kinzel cellular automaton revisited

G. Kohring, M. Schreckenberg

To cite this version:

G. Kohring, M. Schreckenberg. The Domany-Kinzel cellular automaton revisited. Journal de Physique

I, EDP Sciences, 1992, 2 (11), pp.2033-2037. �10.1051/jp1:1992264�. �jpa-00246683�

Classification Physics Abstracts

05.70F 02.50 89.80

Short Communication

The Domany-Kinzel cellular automaton revisited

G-A-

Kohring

and M.

Schreckenberg

Institut fur Theoretische Physik, Universitit _zu K61n, Ziilpicher Strafle 77, D-5000 K61n, Germany

(Received17August

1992, accepted 24 August

1992)

Abstract. New analytic and numerical results for the one-dimensional Domany-Kinzel cel- lular automaton are presented. Analytically _we show that the qualitative features of the phase diagram, including the _new phase recently found by Martins et al., _can be predicted by going one-step beyond the _mean field approximation. Numerically _we find unusual finite size effects which make the extrapolations to the thermodynamic limit difficult.

In recent years cellular _automata

(CA)

have attracted interest from _many researchers because

of their

practicality

in

simulating

differential

equations

and because of their

fascinating

intrinsic

dynamics

[1, 2].

Normally

these automata

obey

deterministic

updating

rules. However, there exist _a class of automata known _as

probabilistic

^cellular automata

(PCA)

in which the

updating

rules _are chosen

randomly

from _a

given probability

distribution _at each time step. ^One of the best known of these PCA is the sc-called

Domany-Kinzel

PCA [3].

The

Domany-Kinzel

PCA consists of _a I-dimensional chain of N

spins,

_n;,

taking

on the values

(0,1) (empty, occupied).

All of the

spins

are

updated simultaneously

at discrete time steps and the _state of each

spin

at time t + I

depends only

_upon the _state of the two nearest

neighbor spins

at time t

according

to the

following

rule:

P(ni

_ni+i>

n;-i)

₌

Ii (2ni

1)11

2Pi(ni+i

+

ni-1)

+

2(2Pi P2)ni+i

_n;-

ill (1)

where

P(ni n;+ini-i)

is the conditional

probability

that site I takes _on the value _n;

given

that its

neighbors

have the values n;+i and _n;-i at the

previous

time step. _pi

(p2)

^{is the}

probability

that site I is

occupied

if

exactly

one

(both)

of its

neighbors

is

(are) occupied.

If neither

neighbor

is

occupied,

the site I will also become empty, ^therefore the state with all sites empty ^{is the}

absorbing

state of the PCA.

The

original

work of

Domany

and Kinzel demonstrated the existence of two

phases,

a frozen

phase

and _an active

phase

_[3]. In the frozen

phase

all sites become empty in the

long

time limit

2034 JOURNAL DE PHYSIQUE I N°11

6

5

co3w o o o

o

4 o

Cl ₃

I

,oo. °°~o

2

o9q~

^" .

o o o

~~~~~

~ 9

. Q

&&

10~ 10~ 10~ 10~ 10~

N

Fig-I-

Plot of the order parameters a and D _{as a} function of system ^size at p2 = 0. The _curves flom bottom to top correspond to _{pi =} 0.5, 0.75, 0.781, 0.797, 0.813, 0.938.

(I.e.,

all initial states lead to the

absorbing state),

^while in the active

phase,

for all

times,

_a

macroscopic

fraction of the sites will be

occupied.

Their work _was based _upon the

technique

of

solving

the transfer matrix

equations

and then

extrapolating

the results to the limit N _{- oo.}

Since this

technique

_was

only applied

to small systems, ^N _~w 10, the

extrapolations

_were not very reliable and in fact their _paper is

ambiguous

with

regard

to the line _{p2 "} 0.

Recently,

Martins et al. [4]

reinvestigated

this model via numerical simulations _on systems _up

to N ₌

3200,

and found that the active

phase

_can be

split

into two

phases,

_one chaotic and _one non-chaotic- The existence of the chaotic

phase

_was found

by taking

two initial

configurations

which dilser in _a finite fraction of

spins

and

simulating

them

subject

to the _same noise. In active

(and

of _course the

frozen) phase,

the two initial

configurations

will become identical after _some

time, however,

in the chaotic

phase

^the two initial

configurations

^will not become

identical.

The purpose of the present work is to

develop

_a better

understanding

of this _new

phase

via

larger

numerical simulations and witfi the

help

of _a novel

analytical

treatment. As _a first step we have

performed

simulations

along

the line p2 "

0,

in order to resolve the

ambiguity

mentioned in the

Domany-Kinzel analysis. Furthermore,

_as will be

explained later,

the

activity,

a, and the normalized

Hamming distance,

D,

(concentration

of

spins

which differ in the t~vo

configurations

n and

n') along

this line _are

equivalent,

thus

simplifying

the

analysis. Initially,

n and n' dilser

only

at _one site.

Figure

I shows the value of _a, D _as function of system

size,

N, for

periodic boundary

_con-

ditions. In contrast to the work of Martins et al., _we simulate the finite PCA

only

_up to

N/2

time steps,

hence,

the data shown

correspond

to that of _an infinite system simulated _up to t ₌

N/2

time steps. ^It can be _seen that the order parameters are not monotonic functions of the system

size,

rather there is _a maximum for _a and D in the range 100 < N < 200. For

this _reason,

investigations

_on smaller systems may have led to wrong conclusions

regarding

the

location of the transition

point. Figure

2 shows _our

extrapolated

values of the order parameter

as a function of _pi The data indicates _a second order

phase transition, however,

at the present

1.

8

~

~d

o o o 2

0

0

2

Pi

2036 JOURNAL DE PHYSIQUE I N°11

o

active

01

Qq frozen

chaotic

~0

_{2 4 6 8 0}

Pi Fig.3.

Phase diagram in the simple ^{mean-field limit.}

Expectation values, f(n, n'))t,

at time t _are

simply

defined

by:

lfIn>n'))t

=

~ f(n>n')P<(In>n'l) (4)

where the summation extends _over all

configurations

of the _n; and

n(

with I ₌ I,.., N.

In the

simplest approximation

all correlation functions _are

decoupled (I.e. (nin;

₌

(ni) in; ))

and _one gets the iteration

equations

for the

expectation

values _{at "}

(n)t

and Dt _"

(l#)t

at+1 "

2piat (~Pi P2)a~ (~)

D,+1

₌

2piDt (p2 4pi(1 p2))D)

+

2p2(1 2pi)at

Dt.

(6)

2

The result for the critical lines is shown in

figure

3. As _can be _seen the chaotic

region

extends

up to the line _{p2 "}

yielding

a

triple point

on that line in contradiction to the work of Martin

et al, and _{our own}

computer

simulations.

Therefore _we went _one step

beyond

the mean-field

approximation

and

applied

the sc-called two tree

approximation,

known from the studies of directed

polymers

_[5]. In this scheme the

correlation functions _are

decoupled only

after _every second time step. We do _not show the iteration

equations

themselves because

they

_are to

involved, especially

for flit. This

improved approximation gives

rise to _a dilserent behavior _as shown in

figure

4. The tricritical

point

now is located _on the _p2

" 0 line _as found in the simulations. It should be

pointed

out

that this

approximation gives

reentrant behavior both in _pi and _p2

direction,

_a

phenomenon

which is

completely

absent in the numerical simulations where both critical lines _are monotonic functions of pi and p2

Nevertheless,

these

approximations

support ^{the existence} of the chaotic

phase. Numerically

_we have confirmed that the

phase diagram

is

qualitatively

the _{same as}

found in references [3,

4].

As _{a summary we} shown that the numerical simulations need to be done _on

larger

systems in order to overcome the finite size elsects. And _we found that the

simple

mean-field

approach

o

active

01

Qq frozen

chaotic

0 2 4 6 8 1-o

Pi FigA.

Phase diagram in the 2-tree approximation.

does not

give

the

right qualitative

behavior of the

phase diagram. However,

it is

possible

to

systematically improve

the

approximation

which is closer to the numerical simulations.

Acknowledgements.

The authors _are indebted to S. Nicolis for many

stimulating

discussions. This work _{was per-} formed within the research _program of the

Sonderforschungsbereich

341 K61n-Aachen-Jfilich

supported by

the Deutsche

Forschungsgemeinschaft.

We thank the

University

of

Cologne

for time _on the NEC-SX3

Ill

and the HLRZ at the KFA Jfilich for time _on the

Cray-YMP /832.

References

ill

Wolfram S., Theory and Applications of Cellular Automata

(Singapore:

World Scientific,

1986).

[2] Staufler D., Phys. Scr. T35

(1991)

66.

[3] Domany E. and Kinzel W., Phys. ^Rev. Left. 53

(1984)

311;

Kinzel W., Z. Phys. B 58

(1985)

229.

[4] Martins Ml., Verona de Resende H-F-, Tsallis C, and de Magalh£es A-C-N-, Phys. Rev. Lett. 66

(1991)

2045;

Tsallis C., to appear in "Condensed Matter Theories", Vol. 8, L. B lum Ed.

(New

York: Plenum,

1992).

[5] Cook J, and Derrida B., ^J. Phys. A: Math Gen. 23

(1990)

1523.