Surface Growth with Power-Law Noise in 2+1 Dimensions

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Surface Growth with Power-Law Noise in 2+1 Dimensions

R. Bourbonnais, J. Kertész, D. Wolf

To cite this version:

R. Bourbonnais, J. Kertész, D. Wolf. Surface Growth with Power-Law Noise in 2+1 Dimensions.

Journal de Physique II, EDP Sciences, 1991, 1 (5), pp.493-500. �10.1051/jp2:1991183�. �jpa-00247533�

1

l+~s.

IT 1

(1991)

493-510 _MM 1991, PAGE 493

Classification

lfi~s~sAbs~ttcts

5.70L-61.50C-05.40

Sham Communication

Surface Growth with Power-Law Noise in 2+1 Dimensions

R.

Bourbonnaisl'),

J.

Kert6sz(~<* ),

D. E.

WOT(3)

(~

Hbchstleistungsrechenzentrum (HLRZ), Forschungszentrum

KFA JUlich Postfach 1913, D-5170 JUlich I, Federal Republic of

Germany

(2)

InstitutfurTheorischePhysik,UniversithtzuKbln,ZulpicherStmm77,D-5000Koln41,Federal Republic

of Germany

(3) ^{Institut far}

Festk6rperfoschung

(IFF~,

Forschungszentrum

KFA Jolich Postfach 1913, D-5170 Jblich I, Federal Republic of

Germany

(Received ii March 1991, ttccepted 22 March 1991)

Abstract. Large scale simulations of stochastic growth of 2+1 dinJensional surfaces _were carried out _{on a} 16K processor Connection Machine. We introduce _a

family

^of models for which _we could

reproduce

the known scaling behavior of kinetic

roughening

in the presence of bounded _noise. For

noise amplitudes _q distnbuted

according

to

P(q)

_~- q~"~~, the growth exponents depend on p and

they are well descnbed by the recently proposed formula based _on the scaling of _rare events.

1. Intrtduction

Many

recent studies have focused _on the

problem

^of the

dynamics

of

non-equilibrium

surface

growth [Ii.

It is well established that for _a

variety

of

models,

the surface

roughness w(L, t)

₌

(<

h2 _{> <} h >~

)~/~,

^scales with time _t and linear system ^{size L}

according

_to

_[2]:

w(L,i)

₌ Lx

j(i/Lx/fl), _(i)

where the

scaling

function

f(z

bellaves _as

~~~~

~~nstant ^~

~~~

A broad class of these models is believed to be described

by

the

Kardar-Parisi-Zhang ~KPZ)

_equa-

tion

[3]:

fith

= uT72h ₊

(T7h)~

+

q(x, t), (3)

(* ^Permanent address: Institute for'Ibchnical

Physics,

HAS,

Budapest

H-1325, H

494 JOURNAL DE PHYSIQUE II N°5

for the time-evolution of the

height

variable

h(x,

t

describing

_a

growng

surface on a d-dimensional substrate

(grovth

in _a d+I dimensional

space).

It b assumed that the noise

q(x, t)

is uncorrelated white noise.

In the KPZ

equation

with this

type

^{of noise the} two

exponents

X and

p

are related in all dimen- sions

by

the

scaling

relation

[4]:

x(i

+

j)

₌ 2.

(4)

This theoretical result has been confirmed

by

a

large

number of numerical simulations.

The

universality

of rite evolution of

rough

surfaces has

recently

been

questioned by

_some

quasi

I+I dimensional

experiments

on

wetting

iJ~ porous ^{media and} on

grovth

^{of bacterial colonies} [5]

[6]

f.

^These

eTperiments

_{gave eTponents}

significantly larger

than the theoretical or numerical

predictions.

In this context,

Zhang recently proposed

tltat

power-law

distributions of rite noise

amplitude

_q as

P(q)

~-

q~(I+")

results in

p-dependent

_eTponents

_[8].

A

simple theory

_[9]

taking

into account the

scaling

of _rare events and

equation (4) predicts (for

d+I dimensional

systems)

_:

These

expressions

are

suggested

to hold for the range of _{d + I < p < pg} where the upper bound p~ is defined such that for _{p >} p~

(5)

would

imply exponents

^{smaller than the} ones obtained for bounded

("gaussian")

noise. Hence _one

expects

^{that above} pg ^the

roughness

^is no

longer

determined

by

the _rare events.

In I+I

dimensions,

_{x~ =}

1/2

_{and fl~ =}

1/3

is _{an exact}

result[3].

^{In 2+I dbnensions the}

most

precise

numerical result

today

is fig = 0.240 + 0.00i

[10]

^while a

conjecture by

Kim and Kosterlitz

[11] predicts

_{fl~ =}

1/4.

So far numerical

investigations

of the effect of

power-law

noise have been carried out in I+ I dimensions

only [12] [13].

Amar and

Fbmily

found

reasonably good

agreement

^with

(5)

for small _p but

they

contend that pg is

larger

than the

predicted

value. In _a different related model

Buldyrev

et al.

[13]

^{found better} agreement ^with

(5).

A crucial

step

iJ~ the direction of

testing

the

validity

of the

assumptions leading

to the eTpres- sions

(5)

^h to carry out simulations of models with unbounded noise in dimensions

higher

than I+I. The present paper

reports

^{results of 2+1} dimensional simulations of _a

growth

model very close to

Zhang's

one with

power-law noise[8].

2. The model

Zhang's

model in its 2+1 dimensional version consists of

updatiJ~g altemafively

two sub-lattices A and B in the checkerboard

geometry.

^It can be descried

by

^the

following

rule _:

h(x,

t +

I)

₌

maxjh(x

_{+ e;,}

t)

+

q(x

+ e;,

t)j, (6)

where _e; E

( (1, 0), (0, 1), (-1, 0), (0, -1)).

The rule

(6)

^is

applied

at sites _{x on} the A-sublattice for odd times and _on the B-sublattice for _even times.

h(x, t)

and

Q(x, t)

are continuous variables.

The initial condition is h ₌

0,

and

periodic boundary

conditions _are used

perpendicular

to the

growth

^direction.

N°5 SURFACEGROWTHWnMLOWER-LAWNOISE 495

Our

mode(

can be described

by

the _same rule

(6),

but the set for _e; b _now

((0, 0), (1, 0), (0,1), (-1, _{0), (0,} -1)),

and the rule

applies

for all _x and t

(no sublattices). Simply

stated this says ^that at every time

step:

1. the local uncorrelated noise

q(x, t)

is added to every

site,

2. each site then takes _{a new} value

equal

to the maximum

(MAX-) (7)

or minimum

(MIN model)

of itseT and its four nearest

neighbors

The deviation from

Zhang's

model is motivated

by computational

reasons

only

and, in view of

universality,

is not

expected

to

change

the

global scaling

behavior. The model

proposed

here is

fully parallel

in that the rule _can be

applied

to all _x

simultaneously.

It

corresponds

to a

simpler

program ^than

Zhang's

model. These simulations have been done _on the Connection Maclline of the HLRZ in Jiilich

(a

CM-2 with 16384

processors).

The

performance

reached

by

_{our program}

for _most of the simulations

presented

^{in thb} paper ^{is 5} 106

updates/sec.

We _were able to simulate square ^{lattices of size}

2048~

for 4300 time

steps averaged

_over about 20 _runs. The total

computing

time involved in thb work is about 250 hours of

computer

^time.

As in other recent work _on the

subject[8][12][13],

the uncorrelated white noise _q used in the simulation _was taken from the dbtribution _:

Since _our simulations involve

picking

on the order of 10~~ numbers from this distribution

great

care must be taken in

doing

so

[14]. lbchnically

_a number q, is obtained

by

first

generating

_a

number _z,

uniformly

distributed _on the interval

(0,1).

The

power-law

distribution _can then be achieved

by simply computing

n, = z, ~~"

(9)

Because of the way random number generators

operate

on real numbers

(floating-point),

a

single-prechion

32 bit real number

generated randomly

will have

only

its mantissa

(usually

24

bits)

selected

randomly.

^This means that the lowest number that _can be

generated

in the

(0,1)

interval is

1/224 (m

6

.10~~ ).

^We

emphasize

that this is_a limitation that comes from the way ^{random numbers}

are

generated

and not from the way

they

are

represented

in computers. ^{The smallest}

positive single precision

real number that can be

represented

is about 7.5 10~46. lb avoid

problems

with

cuto$

we have used a

double-prechion

random number generator ^{for the uniform}

(0,1)

distribution. This has _a lower cutoff of 2. 10~ ^~6 which is

suflidently

small for _our purposes. ^The number is then converted to

single-prechion

before the

power-law (9)

is

computed.

This method is about twenty ^times as efficient _as

computing

all of the

power-law

noise in

double-precision.

We did _some calculation _on the CRAY-YMP 832

using

the standard RANF random number generator

implemented

there. On a

fully

vectorized program, we found _one CRAY processor ^{to be slower} than the 16k CM-2

by

a factor of 3

[15].

^{We also observed that this random number} generator

has _{a resonance} at

multiples

of128 which lead _to dramatic correlations _m the

growth

process.

3. Results

al

MIN. and MIX- Model

The MIN-model is rule

(7) taking

the minimum. It _was

recently pointed

out

[16]

to be

equivalent

to the directed

polymer problem

at zero

temperature

^{and it h} a

significant

modification of

Zhang's

4% JOURNALDEPHYSIOUEII N°5

maximum-model- One _can

easily

understand that the MIN-model is insensitive to rare events because

large jumps

_are

immediately suppressed by

the rule. The MIN model h thus

expected

to behave in standard

gaussbn-none

fashion and we use it in order to get

insight

into the

scaling

behavior of this class of models.

Throughout

this

study

_we concentrated on

determining

the time

development

^of the surface width

w(L,t)

for fixed

(large)

system ^{sizes. First} we studied the _case of bounded noise. two

impairments

make the

interpretation

of the data difliculL The first _one b the intrinsic width _wo

leading

to a correction of the time

scaling

of the form

[l~

:

'°~(~)

_"

'°l

₊ A~~~'

(lo)

where A is _a constant and _wo is assumed to be

independent

oftime. This

problem

can be solved

[10]

by plotting,

instead of

w(t),

the corrected width W _:

W =

(W~(t) W~(t/7))~'~, (ii)

for _a fixed value of 7

(we

used 1.2 for _our

data).

The other

problem

is that itis known from KPZ

theory

and other studies that, for small

coupling

values, very

long

_crossovers exist

[18].

^It _appears that for the MIN-model the effective

coupling

constant is too small and therefore the time and system ^{sizes needed for the} observation of the universal exponents are

extremely large.

In order to

improve

thb

situation,

_we introduce _{a new} model in which _a parameter can be tuned _so as to increase the effective

coupling

constant. We propose ^the

following rule,

which _we call the MIX-model :

h(x,t

+

i)

_=a

n~in(h(x

⁺

e;,t)

+

n(x

+

ei,t))

+(1 a)(h(x,t)

₊

n(x,t))

^~~~~

where _a is _a

parameter

in the interval

(0,1)

and _e; is in

((0, 0), (1, 0), (0,1), (-1, 0), (0, -1))

for all _z and t.

For _{a =} I this is identical to the MIN-model but _{as a} is decreased, the effect of the noise is increased

compared

to that of the

smoothing

rule

(taking

the

min).

For _a

= 0 the model

corresponds

to random

deposition

for which

fl

₌ 0.5.

As _a test case we have simulated thts model in bounded noise

uniformly

distributed between 0 and I.

Figure

I shows the corrected width W for three values of _a

(0.1,

0.5 and

I).

In order to

analyse

the evolution of the effective

fl,

_we have

computed

the

quantity

fin

(t)

defined _as the

local

slope

on n consecutive data

points leading

_up to time t. The inset of

figure

I shows fl20 so that each

point

of the

plot

involves _a

neighborhood

of1.5 decades

(log~~(72°),

recall that 7 =

1.2 is the

logarithmic spacing

^between our data

points).

The dashed line is _at

fl

₌ 0

24,

the

expected gausshn-noise

^{value. We} see that the _{a =} I data

(U)

do not

quite

reach the correct value because of the slow _crossover. However both

mixing strengths

_{a =}

0.I(o)

and _a

=

0.5(+) perform

the desired effect and lead _{to an} effective

fl

^{value between 0.24 and 0.25 in} agreement

with

references[9][11].

N°5 SURFACE GROWTH WrrH LOWER-LAW NOISE 497

lo

3

1i20

25 ,

+'I$$I@$f~i~

~

~

+++++

o ~~o~ooo~

.2

++~~

g88°°°

°

~4~

@8

o

°B@ oo°

.15

~oo

~°° 1000 ioooo

~~oO°°

^{a = 0.I}

oo ^~+

i

~~oo°°

^~++

oo°°°°~ +++~~~~

a # 0.5

coo°°°°° ^~++~

^area

o°° ~+++

~~oo

o

++ cc

~+++~ ~aa°°

^{am I}

° +++ ^co

+++

ao°D

+++ ~aD

+++ ~~o°

+ ~oa°

+ door°

~

cc

o

.i

i lo loo loco ioooo

Time

Fig.

I. Corrected width W eq. (ll) ^for three values of

a

(0.1,

0.5 and

I)

in the MIX-model _in uniform bounded _noise. The inset shows the consecutive slopes fl20 as well _as the value fl

= 0.24

(dashed line).

The a = I data

(D)

which corresponds to the MIN-model does not reach the correct value because of slow

crossovers. Troth mtxing strength _{a =}

0.I(o)

and _a

= 0

5(+)

lead to 0.24 < fl < 0 25 in a reasonable time.

For the case of

power-law noise, figure

2 illustrates the effectiveness of

using

the corrected width.

Figure

2a shows the consecutive

slopes

fl20 ^of the "raw"

roughness w(t)

from simulation of the MIX-model with _a

= 0.5 in for _{p E}

(3, 4,

5,

6,

7,

9, 10, 20).

The

top

curve is for _{p =}

3,

and _p increases towards the lower curve which

corresponds

to p = 20. The dashed line is set at

fl

₌ 0.24.

From this

analysis,

one could conclude that _{we are} still in the _crossover

regime.

^However

figure

2b

presents

a

completely

different

picture.

The consecutive

slopes

are

computed

on the corrected wtdth W of the data _as

opposed

to the "raw"

roughness

used in

figure

2a. For _{all p}

good scaling

behavior h found with exponents ^{close to the}

expected

value.

b)

MAX- Model

For

symmetric

bounded noise the MAX-model

(taking

the maximum in

(7))

can be

mapped

onto the MIN-model

by

the transformation h

- -h. However in _contrast to the MIN-model the MAX-model _was shown to be sensitive to

power-law

noise in I+ I dimensional systems

[12].

Therefore _we restrict the

study

to the case of

power-law

no~e

(8).

In

figure

3 _we have

plotted

the corrected width _{as a} function of time. A clear variation of the

slope

as a function of _p is observed. Quite

good scaling

behavior is found up ^{to a}

point

where saturation effects _come into

play.

The effective values of

fl

for each p were

computed

^bom a

least-square

fit _on the bracketed data and

reported

on

figure

4. The _error bars _were evaluated

498 JOURNAL DE PHYSIQUE II N°5

~

~~~~~~~

~)

~~~

~ j j ⁾

)

~ ~ ~

~

~

~

^~

~ ~ ~

.15 ~ j j

( I

[ I

S

~ ~ ~ ~ ~ ~ ~

~

.25

/J~~ ~~

~~ ^~~~

~ j

,

~

~ o

15

j

~

o

.i

o.05

loo

Fig.

Z

a)

Consecutive slopesfl20 of the "raw"

roughness w(t)

for the AIIX.model with _a

= 0.5 in power- law noise for _{p E}

(3,

4, 5, 6, 7, 9,10,

20).

The top curve shows _p

= 3 and _{p increases} towards the lower

curve which corresponds to p = 20. The dashed line is at the

expected

fl

= 0.24.

b)

Consecutive slopes computed ^from the corrected width. For all _p good scaling behavior is found with exponents close to the

expected value: fl

= 0.24.

from the observed fluctuation of

rho.

We can compare ^{these results} to the

predictions

bom the

theory (5)

^which

gives

for d=2 _:

fl

₌ ^~

(13)

For small values of _p the agreement ^is very

good,

and the

validity

of

(13)

cannot be excluded in the

vicinity

_{of pg.}

Results for the mixed MAX-model

((12)

with MIN

replaced by MAX)

where also

computed

with _{a =} 0.5 and the results _are similar _to the pure case. This

meins

_{that the MAX-model in}

power-law

^noise has, m

itself,

_a

sufficiently large coupling

constant so that the nobe-enhancement

effect of

introducing

some

mbing parameter

^becomes irrelevant.

In conclusion _we have shown tha~

similarly

to I+I dimensions,

power-law

noise _can

change

the surface

growth

_exponents in 2+1 dimensions. The results

support

^the theoretical

prediction

(13)

based _on the

scaling

of _{rare events.}

N°5 SURFACE GROWTH WITH LOWER.LAW NOISE 499

1000

o°

o °

jO° O ^°

O°O °

O°

100

~o°

₊₊

O

~ ++

O°

g++~~

^~~~

fl~

O° _+~

O ++

O ++ _o oDa

O°

~+~ ~Od~D~

^{~ ~}

~°

_{++ ~O~ XX~XXXX}

10 _O ~+ DO _X ^~

~O + oO~ _~XX

&&&&A

O +~ ~O~ ~XXX ~~~~&

[O°

++~

O~~ _XXX~ ~""" _~*W~

O ~+ ~a~

~~x~

~aA~ _~WWW*~OOOO

~+ CD XX ~A" ~WW ~O~

O j+ ~a ~XX ~A~~ ~~W** ~OOO

+~

~pO°~XX(~~&" ~~~~W(~OO°°°

1

~

~

O~x~~~~"~~W**~~OO°°°

₊₊₊

O x~@" ~W*~OOO _~g ++++

+ x& ~@W*~OO _{~~++ +}

"W~pO°° ~~+++~

& WO ₊₊₊

~ O ~++++

~ +++

~+~~

~+++

+

.i

~~~ loco ioooo

Time

Fig.

3. Corrected width for the MAX-model in power-law noise for (bom top curve to bottom one)

p E

(3,

4, 5,6

,

7

,

9,10,

20).

A clear variation of the slope as a function of _p is otserved.

1

9

p

^.8

7

6

5

.4

'~ 0 25

I

.2

2 4 6 8 10 12 14 16 18 20

v

Flg.

4.

fl(p) computed

bom _a least-square fit on the bracketed data _in

figure

3. The _error bars _were eval- uated bon1the observed fluctuation ofrho. The values offl _are

(bom

small _{p to}

large): (0.9,

0.69, 0 49, 0.41, 0.35, 0.30, 0.28,

0.255).

tile solid line _is the theoretical prediction

(13).

51XJ JOURNAL DE PHYSIQUE II N°5

Acknowledgements.

Thb

project

_grew out of work done in collaboration with T Wcsek and H. J. Herrrnann who de-

serve our

spechl

thanks. The

partial support

of the Deutsche

Forschungsgemeinschaft (SFB341)

is

acknowledged.

References

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(World

Scientific,

Singapore

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(1985).

[3] ^M. Kardar, G. Parisi, and Y.-C.

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(1986).

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