Crossover scaling in surface growth with truncated power-law noise

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Crossover scaling in surface growth with truncated power-law noise

Jacques Amar, Fereydoon Family

To cite this version:

Jacques Amar, Fereydoon Family. Crossover scaling in surface growth with truncated power-law noise.

Journal de Physique I, EDP Sciences, 1991, 1 (2), pp.175-179. �10.1051/jp1:1991123�. �jpa-00246312�

Classification

Physics

^Absiracts

47.55M 05.40 64.60A

Short Communication

Crossover scaling in surface growth with truncated power-law

noise

Jacques

^G. AJnar and

Fereydoon Family

Department

of

Physics, Emory University,

Atlanta, GA

30322,

U.SA.

(Received

14 November199fl

accepied

21

November1990)

Abstract. We demonstrate the existence of _a

simple scaling

form which describes the _crossover from anomalous to Gaussian exponents as a function of cutoff in the

Zhang

model of surface

growth

for _{p >}

2,

where _p is the

exponent characterising

the

power-law

noise distribution. For _{p =} 2 we find

a novel

scaling

form with

logarithmic

corrections. Our results for cutoff

scaling

at p = 2 are

supported by

the

scaling

^{of the} saturation

growth velocity.

Zhang [Ii

^has

recently proposed

a model of surface

growth

in which the noise ~ has ^a

power-law

distribution of the

form,

P(~)

+~

l/~~+" It

for

large

_j From simulations of this model as well as of the related directed

polymer

model

[1, 2]

ⁱⁿ d ₌

2,

he obtained

anomalously large

values for the surface

scaling exponents,

^which were found

to

depend

on p. In

addition,

he

suggested

that the

presence

^of

power-law

noise may

explain

the results of recent

experiments

_on

_porous

media

[3-6]

^{and bacterial}

growth [~

in d ₌

2,

for which

anomalously high

values of the

growth _exponents

have also been

reported. Recently,

_we have carried out extensive simulations [8] of ^several different surface

growth

models with

power-law

noise which

strongly support

^{the existence of anomalous}

exponents. However,

there still exists

some

controversy

over whether the observed anomalous

exponents represent

a new

universality

class for the

Kardar-Parisi-Zhang (KPZ) equation

_[9]. In order _to resolve _some of these

questions

we have conducted _a

study

of the

Zhang

model

[Ii

^with truncated noise.

In our simulations _we used the

power-law

noise distribution

given by equation ii ),

with _a cutoff

~c such that

P(~)

= 0 for ~ > ~c. The noise ~ was determined _at each odd

(even)

^site at each odd

(even)

time

step

t,

by generating

an

independent

random number _r at each site such that

~/"

_{< r}

< I and

calculating

the

quantity

_{~ =}

^r-I/".

Simulations _were

performed

for _a range ^{of values} of the

cutoff,

with

~p

^"

ranging

from

2-~~

_to

10~~

_in

_powers

_{of10 for each value of p. The saturation}

value of the surface

width,

as measured

by

the _r.m.s. fluctuation of the interface

heigh~

was

determined for each value of the cutoff _~c, for

system

^{sizes L} = 16-1024 in

powers

^{of 2.}

Averages

were taken _over times of the order of several million time

steps.

176 JOURNAL DE PHYSIQUE I N°2

The surface width

w(L, t)

_on

length

scale L at time t h

expected

to

satisfy

the

scaling

form

[10], w(L, t)

₌ L°

f (t/L~),

where _{z =}

alp

is the

dynamic exponent, f(z)

~

zP

_for

z < I and

f(z)

_- const for _z »

I,

and _cy has the value of1/2 [9] ^for Gaussian noise. In order to scale

our results for the saturation width _w

(L,t

= m,

~c)

as a function of cutoff _{~c, we} propose ^the

following

crossover form for _p

>2,

w

(L,

_m,

_~c)

~

L°

f _{(L~~C ~c)} (2)

where

f(u)

_~

^u("-

^~/~)/~C for _u « I and

f(u)

_- const for _u » I and _cy and _xc

depend

_{on p.} This h

equivalent

to the

scaling form,

w

(L,

_m,

_~c)

~

Ll/~~i"~~~~~~~~g _(L~C ~pl) (3)

with

g(u)

~

u(°-1/~)/~C

_for

u < I and

g(u)

- const for _u » I.

Figure

I shows

scaling plots

of

the form of

(2)

for _p

= 3-5. For _{a we} have used the values

(cy= 0.75, 0.62, 0.56)

obtained from _a

previous study [8],

while the crossover

exponents

xc ^were obtained from best fits

(xc

m

0.62,

0.375 and 0.2

respectively).

We find

good scaling

for all three values of _p,

except

^{at small L for which the}

asymptotic region

has _not

yet

^been reached. We note that with

increasing

_p the crossover

scaling exponent

xc decreases towards zero.

However,

the

quantity (a 1/2) /zc

_m 0.33-0.4

appears

to

remain

approximately

constant for _p >2.

~00

~ = 3

I

~

U

~

~

~

~~ 0

log

appears

to show

asymptotic

behavior for very small ^L

[1, 8].

^{This led} us to consider an altemate form with

logarithmic

corrections at p =

2,

w

(L,

_m,

~c)

~

L(In L)P

F

((L(In ^L)~P)

^~~~

~c) (4)

p=2

<

-l, hwf'q~i

-2 -1 0 1 2 3 4

log _(q~ iL(jog

_L)~~^~~

Fig.

2.

Scaling plot

of form of

equation (4)

for _{~ =}

2,

with p = 0.20 and _{rc =} 1. L ranges ^{from 16 to 1024} and _~c varies from 31.6 to 46340. Inset shows

scaling plot

^{of form of}

equation (3).

where

F(u)

~-

ul/2rC

_for

u < I and

F(u)

- const for _u » I.

Using

this

scaling

form with _xc

= I

and _{p m} 0.20-0.25 _we find very

good scaling

^for the entire range ^of

system

^{sizes L.}

Thus,

there

appears

^{to exist} a

logarithmic

correction at p = 2. The existence of such a correction may

explain why

values of_cy

slightly higher

than I _{cy m}

1.04)

were obtained for several different

growth

models

with

power-law

noise at p = 2 in reference

[8].

In order to obtain further

support

^for our

results,

we also studied the

scaling

of the saturation

velocity V(L)

which is

expected

to have a finite-size correction of the form

[12],

V(L) V(m)

~- <

1(T7h(~

> _~

A(I)L~°-~ ₍₅₎

where

A(I)

is

proportional

to the

nonlinearity parameter

I in the KPZ

equation. Very good

fits of the form of

equation (5)

were obtained for _p

=

3-5, using

the values of _cy

previously

obtained from the

scaling

of the interface width. For _{p =}

2, however,

the saturation

velocity

does not

approach

a constant value _as

expected

for _cy

= I and in fact appears ^{to be}

diverging

with L. We

conjecture

a

logarithmic

correction ofthe form

V(L)

~

(ln _L)~P Figure

3 shows _a

plot ofln(V(L))

versus

In(In(L)).

The

slope

of the

straight-fine

fit is

2p

_m 0.58. This

supports

^{the existence of} a

logarithmic

correction at p =

2,

which is _most

likely

due to the fact that p = 2 is the critical value for the transition from non-invariant to self-invariant _noise.

Our results for cutoff

scaling

have

implications

for

experiments

as well as for simulations of the

Zhang

model with

large _system

^sizes. The _reason is that

typically

32-bit _or 46-bit

integer

178 JOURNAL DE PHYSIQUE I _N°2

,t§

i'

/

a'

"

'

p ~'

2i

$'

'~~ '

' ' k1 ' '

6. '

'

~° l.0 1.2 1.4 1.6 1.8 2.0

in in L

Fig.

3. Plot of In

V(L)

_ve~sus In In L

showing logarithmic divergence

of

V(L)

at ~ = 2. Dashed line fit has

slope 2p

₌ 0.58.

random number

generators,

^{for which the cutoff} rc is 2- ^3~

(4.65

^{x 10~}

_~°)

^and

^2-45 (2.8

^{x 10~}

⁴⁾ respectively,

are used.

Equation (2) implies

that the

scaling region

for which the surface width is not affected

by

the cutoff _occurs for _{u =}

~c/L~C

₌

rc-I/" _/L~C

_>

uc where _uc » I. For _a 32-bit random number

generator, using

the value _{uc ci} 40

(log(uc

₌

1.6,

see

Fig. I),

_we find for p =

3, 4,

5 that

Lc

₌

257, 88,

21

respectively.

^For a 46-bit random number

generator

for which rc = 2.8 _x

10~~~,

we obtain

substantially larger

values for

Lc (of

the order of

10~). However,

on the order of 10~~ random numbers _must be

generated

to

sample

the full range ^of the random number

generator,

^while a reasonable estimate of the number of random numbers accessible in current

computer

simulations is

only

of the order of

10~-10~~. Thus,

the effects of _a cutoff should

begin

to show up ^for

systems

^{of order 10~} even in

large-scale

simulations. Crossover to Gaussian

exponents, ^however, (u

_<

I)

should

only

occur for much

larger system

^sizes

(10~

or

larger).

Summary.

We have demonstrated the existence of _a

simple scaling

form which describes the _{crossover to} Gaussian

exponents

as a function of cutoff in the

Zhang

model of surface

growth

for _{p >} 2. This

crossover form

strongly _supports

the existence of anomalous

exponents

in thin model. In

addition,

for _p

=

3,

4 and 5 _we have obtained

approximate

^{values for the} crossover

exponents zc(p).

With

increasing

_{p, xc} decreases towards zero while the

quantity (a 1/2) /zc

remains

approximately

constant.

For _p

= 2 _we find a novel

scaling

form with

logarithmic

corrections. Our results for cutoff

scaling

at p = 2 are further

supported by

the

scaling

of the saturation

growth velocity

V

(L)

which scales _as

(In _L)~P

with _{p m} 0.29. For _{p >}

2,

our results for the finite-size correction to the

growth

velocity

_agree with

previously

obtained estimates for the saturation

exponent

a as a function of p, thus

verifying

the

validity

^{of the} nonlinear KPZ

equation

_{as a}

description

of the

Zhang

model.

Due to the natural existence of

cutoffs,

our results may ^be useful in the

analysis

of

experiments

with

power-law

noise.

Acknowledgements.

This work _was

supported by

the Office of Naval Research and the Petroleum Research Fund Ad- ministered

by

the American Chemical

Society.

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