Ising cellular automata : universality and critical exponents

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Ising cellular automata : universality and critical

exponents

Naeem Jan

To cite this version:

201

LE

JOURNAL

DE

_PHYSIQUE

Short Communication

Ising

cellular

automata :

_universality

and critical

_exponents

Naeem Jan(*)

HLRZ,

c/o KFA 5170 Julich

1,

F.R.G. and

Cavendish

_Laboratory,

_Madingley

Road,

Cambridge University, Cambridge,

CB3

OHE,

G.B.

(Reçu

le 31 octobre

1989,

accepté

le 16 novembre

₁₉₈₉₎

Résumé . 2014 Des simulations

_numériques

sont effectuées sur des réseaux d’automates bidimension-nels de

_type

_Ising

en

_champ

extérieur nul _pour des tailles allant de

242

à

16002.

Au seuil

_{d’apparition}

du chaos

_{(instabilité}

d’étalement

_d’erreur)

la dimension fractale

_df

est

_1,9

et celle associée au _temps

_dt

est

1,3.

Ces valeurs sont indiscernables de celles mesurées _{récemment pour} les automates cellulaires de

Kauffman,

et

_proches

de celles observées dans la

_percolation.

Abstract. 2014 Two-dimensional

Ising-like

cellular automata are simulated in zero field for 24 x 24 to

1600 x 1600

_systems.

We find that the mass fractal

_{dimensionality}

at the onset of "chaos"

_(instability

of

"damage

spreading")

is 1.9, and

_dt

= _1.3 for the time. These values are

_{indistinguishable}

from those

recently reported

for the Kauffman cellular automata and close to those observed for

_percolation.

1

_Phys.

France 51

_{( 1990)}

201-204 ler FÉVRIER

_1990,

Classification .

Physics

Abstracts 05.50

Cellular automata are fast

_becoming

a

_{high powered computational technique}

for

simulat-ing

complex

systems,

for

_{example genetic}

_switching

_{cycles, hydrodynamics, immunology,}

neural networks and chemical

_{reactions[l, 2].}

_Ising-like

cellular automata which

_incorporated

the

Boltz-mann

_probability

distribution of the

Ising

model but with

quenched

random numbers were

re-cently

introduced

_by

Mac Isaac et aL

_{[3, 4].}

Consider the two-dimensional

_square

_Ising

lattice in

a

_magnetic

_field,

h. The _energy at site i is

(* )

Permanent address :

_{Physics Department,}

St. Francis Xavier

_{University, Antigonish,}

Navo

_Scotia,

Canada B2G 1CO.

202

and the

_probability

of

_finding

the

_spin

at site i _up is

where k is the Boltzmann constant, T the

_temperature

and

_{E; ( 1) = -E¡(!).}

In a usual heat bath simulation an

_independent

random number is considered at the

_attempted

transition at each and every lattice site but if the random numbers are

_{quenched initally}

for the sixteen

_possible

configu-rations for the nearest

_neighbours

arround site i then we recover an

_Ising

cellular automata

_(ICA)

where now deterministic rules _govern the time evolution of the

_system.

The ICA is constructed as

follows : a

_température

is

_stipulated

and the Boltzmann

_{probabilities}

calculated for the 5 distinct energy values for the 16

possible neighbour configurations ;

a random number is

_compared

with the Boltzmann

_probability

for a

_{given configuration}

and if it is less than this

_probability

then the rule is that the

_spin

will be

_up

at that site at the next time

_step

or otherwise the

_spin

will be down

at the next time

_{step ;}

the

_procedure

is continued for the 15 other

_{configurations}

at the site in

question ;

thèse

_procédures

are then

_repeated

for ail the other sites of the

_system.

We have de-fined deterministic "rules" which

_govern

_completely

the

_subsequent

evolution of the

_system,

and

we now start the simulation

_keeping

these rules fixed.

This ICA at

_high

_temperatures

in a

_magnetic

field is the Kauffman

_[5]

model which was

in-troduced to

_study

the

_{switching properties}

of _genes. A

_{phase diagram comparable}

to the

_Ising

model and

_percolation

is

_expected

for the ICA and the Kauffman model also. At low

tempera-tures a local

_perturbation

will remain localised while at

_high

_temperatures

the

_perturbation

will

propagate

throughout

the

_system.

We shall refer to the former case as "frozen" and the latter

case as "chaotic". At low

_temperatures

the

_system

is

_non-ergodic

in the sense that the ensemble average for

example

leads to the Boltzmann distribution of

_energies

whilst the time

_average

is

quite

différent,

especially

at low

_temperatures

_[3,4].

In this note we

_investigate

the critical

_properties

at the onset of "chaos" for L x L

_square

lat-tices in the

_range

24 L 1600.

_{Equilibrium configurations}

are

_generated

[6]

with the standard

heat bath

_algorithm,

a

_replica

of the

_system

made and a line of

_{spins along}

the central row is

kept premanently

in the

_opposite

state to the

_spins

in the

_equivalent

row of the

_original

_system.

Both

_systems

are

_subjected

to the same set of rules and should evolve in the same manner

_apart

from the effects introduced

_by

the

_{perturbation.}

Are these effects localised or do

_they

_propagate

throughout

the

_{system ?}

A measure of

_séparation

of the two

_systems

is the

_Hamming

distance

- the number of

spins

in

_opposite

states. These sites are referred to as

_damaged.

The

_large

lat-tices considered i.e. L > 64 were simulated on a CRAY YMP

_using

the da Silva Herrmann

_[7]

algorithm

which allowed 190

_{spin flips}

_per

microsecond

_{per processor.}

We now consider the

_spatial

and

_temporal

evolution of the

_damage.

For T

_Tk(L)

the

damage

is localised to the

_vicinity

of the

_{permanently damaged}

line whilst for T >

_Tk(L)

the

damage

propagates

freely

to the

_edge

of the

_system.

Figure

1 shows the variation

_{of Tk (L) v s. 1 IL,}

the

_{asymptotic value}

_appears

to be

_Tk(oo)

= 5.6

in units of

_J/kT,

more than twice the

_Ising

Curie

_{température Tc!}

This behaviour is similar but

more dramatic than that observed for some

_Ising

model studies

_[8]

where the transition to chaos does not coincide with

Tc.

We have monitored the variation of the

_{magnetisation}

with

_temperature

of the ICA and this

goes

to zero within a few

_percent

of the Curie

_temperature.

_Figure

2 shows the variation of the

expected scaling

behaviour

is observed with a fractal

_{dimensionality}

_{d f}

of 1.9 and

dt

of 1.3.

Fig.

1. - Variation

ofTk(L) vs. 11L. Tk(oo)

is 5.6.

Fig. 2

Fig.

3

Fig.

2. - ln

_Damage

vs. In L at

_Tk(L).

A least _squares fit

_gives

a

_df

of 1.9.

204

The

_equivalent

_exponents

for the Kauflman model remained elusive until the recent careful

investigation

by

Stauffer

_[9]

who resolved earlier inconsistences and

_reported

1.8

df

1.9 and 1.3

dt

1.4. These values are

_compatible

with our data and there exists a line of critical

points

between this zero-field critical

_point

and the critical

_point

at infinite

_temperature

and non-zero field. This latter

_point

is the critical threshold of the Kauffman model. It is most

_likely

that

the critical

_exponents

are

_unchanged

_along

this line and hence these ICA all

_belong

to the same

universality

class. There remains the

_tantalizing

_prospect

that the critical

_exponents

measured

by damage

spreading

for Kauffman and ICA may also

belong

to the same

_universality

class as

percolation.

Acknowledgements.

,

It is a

_pleasure

to thank Dietrich Stauffer and the

_hospitality

of HLRZ for

_making

this

_project

possible.

References

[1]

Ed. S.

_Wolfram,

_Theory

and

_Application

of Cellular Automata

_(World

Scientific

_Press)

1986.

[2]

TOFFOLI T. and MARGOLUS

N.,

Cellular Automata Machines : A New Environment for

_Modelling

(MIT Press)

1987.

[3]

MAC ISAAC A.B. and NAEEM

JAN,

Poster at Stat

_Phys

17

Conference,

Riuo de Janeiro Brazil

_(1989).

[4]

MAC ISAAC

A.B.,

CORSTEN

M.J.,

HUNTER D.L. and NAEEM

JAN,

St Francis Xavier

_{University preprint}

(1989).

[5]

KAUFFMAN

S.A.,

J. Theor. Biol. 22

₍₁₉₆₉₎

437.

[6]

MOUKARZEL

C.,

preprint (1989).

[7]

DA SILVA L.R. and HERRMANN

H.J., J.

Stat.

_{Phys. 52 (1988) 463.}

[8]

COSTA

U.M.S.,

J.

_Phys.

A 20

₍₁₉₈₇₎

_{L583 ;}

LE CAER

G., J.

Phys.

A 22

₍₁₉₈₉₎

_{L647 ;}

_Physica

A159

₍₁₉₈₉₎

329.