Damage spreading in Ising model with increased range of interaction

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Damage spreading in Ising model with increased range

of interaction

S.S. Manna

To cite this version:

1261

LE

JOURNAL DE

_PHYSIQUE

Short Communication

Damage spreading

in

_Ising

model

with increased

_range

of interaction

S.S. Manna

HLRZ,

c/o

Forschungszentrum (KFA)

Jülich,

D-5170 Jülich

_1,

F.R.G.

(Reçu

le 30 mars

_1990,

_accepté

le 4 avril

₁₉₉₀₎

Abstract. 2014 We have studied the

spreading

of

_damage

in

_Ising

models with increased _range of interaction.

_Simulating

a _square lattice of linear size 1000 we find that the

_spreading

_temperature

and the Curie

_temperature

of

_magnetic

transition are the same for both nearest and next nearest

neighbour

models whereas for third nearest

_neighbour

model

_they

_may not be the same.

1

_Phys.

France 51

₍₁₉₉₀₎

1261-1265

Classification

Physics

Abstracts 05.50

How the effect of a small

_perturbation

_grows,

_particularly

near the critical

_point

in a

_system

showing

a

_phase

transition is the

_object

of

_{"Damage Spreading"}

studies.

_{They yield}

information

for the

_dynamical

behaviour of the

_system

concerned. In this method one starts with two

_systems

which are identical

_except

for a small localised

_{region (initial damage)}

and then both

_systems

are allowed to evolve under identical

_dynamics.

It is seen that the effect of initial

_perturbation

depends

on the

_{temperature, dynamical}

_process

of evolution and the

_geometry

of the

_system.

Damage spreading

has been studied in

_Ising

_systems

_[1-4]

and also in other

_systems

like cellular

automata

_[5,6]

and

_spin

_glasses

_[7,8].

Consider a

_square

lattice of size L with

_{Ising spin}

variables 1 at each site.

_Starting

from

a

_{spin configuration}

in which all

_spins

are

_up

the

_system

is allowed to evolve under a

_particular

dynamical

process

and after a

_long

time

_equilibrium

is obtained

_{corresponding}

to a fixed

_assigned

temperature.

A

_replica

of this

_{spin configuration}

is constructed and

_only

a small number of

_spins

(for

site

_damage

it is

_only

the central

_spin,

for line

_damage

it is a line of

_spins

at the centre

_parallel

to one

_edge)

of this lattice are

_flipped.

Now both

_spin

_systems

are allowed to evolve

_obeying

the same

_dynamics

with the same set of random numbers at the same

_temperature.

_Damage

is

defined as the fraction of sites in which

_spins

of the two lattices are different. This is called the

"Hamming

distance" in neural network

_language.

With time evolution the

_damage

may

_spread

1262

to the whole

_lattice,

it can be

_fixed,

or it can vanish

_("healing")

in two dimensions.

_{Stanley et}

al.

_[1]

observed that there exists a critical

_temperature

_Ts

which is the demarcation line for the finite and infinite

_damage

but from their medium size numerical

_{study they}

were unable to _say

very

precisely

whether

7s

is the same or different from the Curie

_{temperature Te}

of

_{ferromagnetic}

transition. For

_{Metropolis dynamics damage}

_spreads

for T >

_Ts

and ceases to _{grow below} this

temperature.

For heat bath

_dynamics

the behaviour is

_opposite

_[7].

In this

_paper

with

_large

scale numerical simulation we

_investigate

whether

_Ts

is

_equal

to or

different from the Curie

_{temperature Te}

of

_{ferromagnetic}

transition. For cubic lattices

_Ts

_Te

[2,3],

and thus one can

_suspect

_Ts/Tc

to be lower for a

_larger

number of

_neighbours.

Thus it was

suggested

[11]

to increase the _range of interaction for the two-dimensional

_Ising

model.

We have studied here

_damage

_spreading

in the

_Ising

model with

_increasing

_range of interac-tion

_{using Metropolis algorithm.}

We find that the

_spreading

_temperature

2§

is

_{indistinguishable}

from

_Te

for both the nearest

_(nn)

and next nearest

_{neighbour (nnn)}

interactions of

_Ising

model but for the third nearest

_{(nnnn) neighbour}

interaction

_they

are found to be

_slightly

different

_(see

however the discussion

_later).

We also

_study

how _average

_spreading

time scales with lattice size. In a recent

_study

Poole and Jan

_[4]

checked that the

_exponent

_{characterising}

this

_scaling

in heat bath

_dynamics

is

_nothing

but the

_dynamical

_exponent

of the

_Ising

model. Their simulation results

support

this idea for three dimensions but not

_very

well in two dimensions.

In

_{Metropolis dynamics spins}

are

_updated

with a Boltzmann

_weight

factor

where DE is the

_change

in _energy of the

_spin

when

_flipped.

At each site a random fraction is

called and

_compared

with the

_weight

factor.

_Spin

is

_flipped

when the fraction is

_greater

than or

equal

to the

_weight

_factor;

otherwise it remains the same.

We

_study

the nearest

_{neighbour Ising}

model on a

_square

lattice with the helical

_boundary

condition

_using

a

_single

_array for the whole lattice. For

vectorisation,

we divide the whole lattice

into two sublattices as in a checker board

lattice,

of dimensions odd

_{(L -1)}

_by

even

_(L).

One site k

has its nearest

_neighbours

at

_{(k+ 1),}

_{( k - 1 ) ,}

_(k-L+1)

and at

_(k+L-1)

sites and

_{two shadow layers}

are used

_parallel

to the odd direction for the

_boundary

conditions. We

_update

first the

_spins

on one

sublattice

_using

the

_spins

on the other sublattice and then the second sublattice is

_updated

with the

spin configuration

of the first sublattice. In this _way one

_gets

two

_{non-interacting}

sublattices. For the next nearest

_{neighbour Ising}

model we need to devide the whole lattice into nine sublattices

to maintain the condition that the

_updating

of

_spins

on one sublattice is to be done

_using

the

_spins

of other sublattices. In

addtion,

we made simulations with less sublattices. For

_example,

to

_study

the next nearest

_{neighbour Ising}

model on the checker board

lattice,

on

_updates

one

_{spin using}

the four

_spins

on the same sublattice and four on the other sublattice. We studied how far this

sublattice

_mixing

affects the result. For that we

_compared

the

_{magnetisation}

below

_Te

with the

result obtained from

_using

nine sublattices. The difference is small and it becomes smaller for

bigger

lattices. Therefore we used checker board lattice for

_{studying large}

next nearest

_neighbour

Ising

models. For third nearest

_{neigbour Ising}

model we

_update

the whole lattice in three

_separate

scans. In each scan we

_update

_every

third site. The condition for vectorisation of the inner most

loop

is that one cannot

_update

one

_spin

which is used to

_update

the

_{previous spin.}

All these methods of

_updating

are vectorised and we

_get

a

_speed

of 8.4

_spins

_per

microsecond for nn and 7.0 for nnn

_Ising

models on one

_Cray-YMP

_processor.

For all three

_ranges

of interactions we

_study

initial line

_damage

and find the fractions of

sites

_damaged

when the

_damage

touches the

_boundary

with a site

_by

site

_comparison

of the

whole lattice. For critical

_temperatures

we use the values

_Tc(nn)

=

2/ln

1

+

vf2-),

_{Tc(nnn) =}

1/0.19019269(5) [9]

and

_Tc(nnnn)

=

1/0.1135

[10].

We

study

damage

spreading

at six

999 x 1000 for nn and nnn cases and 1000 x 1000 for nnnn case. For this lattice size we wait

10000 Monte Carlo time

_steps

_per

_spin

for

_temperatures

below

Te

and

100000,

30000 and 15000

steps

for

_temperature

ratios

1.005, 1.01

and 1.02

_{respectively.}

We

_go

to

_{equilibrium only}

once and successive

_damages

are considered one after the other. We _average over 20 such

_{configurations}

for each

_temperature.

We

_plot

in

_figure

1 the _average

_damage

as a function of the

_temperature

ratio

_T/Tc.

We find that for nn and nnn interactions there is finite and zero

_damage

at 1.005 and 0.995 of

_T/Tc.

Therefore we conclude

_Ts/Tc

= 1.000±0.005 much more

_accurately

than estimated earlier

_[1]

for

nn

_only.

However for nnnn interaction we

_get

15

_percent

_damage

at 0.995 of

_{T /Tc.}

It seems that this

_may

be due to an inaccurate value of

_{Tc(nnnn) = 1/0.1135. To}

see if this

_suspicion

is true we

studied how the _average

_{magnetisation}

falls to zero as we

_approach

the critical

_temperature

for L = 300

(see

Fig.

2).

We see that for nn and nnn models

_they

are still finite at

_T/Tc

= 1.005 but for

nnnn it is zero.

_Again

for

_T/Tc

= 0.995 the

magnetisation

for both nn and nnn are

_considerably

higher

than nnnn model. From this

_study

it seems that the actual value of

_Tc

in nnnn model is not at

_1/0.1135

but

_slightly

less and with an accurate value of

_{Tc, Ts}

_may be

_equal

to

_Te.

Fig.

1. -

Average damage

as function of the

_temperature

ratio

_{T/Tc :}

_{(A) (nn), (p) (nnn)}

and

_{(o) (nnnn).}

Next we

_study

how the _average

_spreading

time Te scales with the lattice size as

(r,) -

L2

in nearest

_neighbour

_Ising

model.

_Recently

Poole and Jan

_investigated

this

_[4]

with heat bath

dynamics.

They

conclude that this

_{exponent z}

is the

_dynamical

_exponent

of the

_Ising

model.

Their numerical calculations _{agree very} well in three dimensions but it is

_slightly

different in two

dimensions from the

_accepted

value of the

_dynamical

_exponent.

We

_repeat

their calculation with somewhat

_bigger

lattice size

_(up

to L =

₁₂₈₎

with

_greater

_accuracy to see if this difference is

really

due to the

_inaccuracy

of finite simulation. In

_figure

3 we

_plot

_average

value of the

_spreading

time

_(r,)

with lattice size L in a

_{log log}

scale. We consider

_only

four lattice sizes L =

32, 64,

88 and 128. A least

_square

fit of these four

_{points gives}

the value of the

_{slope z}

= 2.27. Thus we

1264

Fig.

2. -

Average magnetisation

as function of the

_temperature

ratio

_{TITC :}

_{(à) (nn), (p)}

_(nnn)

and

_(o)

(nnnn).

Fig.

3. -

We do the similar

_study

also for

_{Metropolis dynamics.}

In the same

_{plot (Fig. 3)}

we see that the _average

_spreading

times in this case are much smaller

_compared

to the heat bath

_dynamics

and calculate a value for z = 1.15. This result

gives

another

_support

of the conclusion that the

damage

spreading depends

on the

_{particular dynamical}

_process.

The whole

_project

was done

_using

around 100 hours of CPU in

_Cray-YMP

Acknowledgements.

’

1 like to thank J. Adler for

_suggesting

this work. 1 also like to thank D. Stauffer for _many useful

suggestions

and comments.

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