Kinetic roughening by exceptional fluctuations

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Kinetic roughening by exceptional fluctuations

Joachim Krug

To cite this version:

Joachim Krug. Kinetic roughening by exceptional fluctuations. Journal de Physique I, EDP Sciences,

1991, 1 (1), pp.9-12. �10.1051/jp1:1991112�. �jpa-00246306�

J

Phys.

I 1

(1991)

9-12 JANVIER1991, PAGE 9

Classification

Pliysics

^Abstracts

05.40 61.50C 05.70L

Show Communication

Kinetic roughening by exceptional fluctuations

Joachim

Krug(*)

Theoretische

Physik, Ludwig

Maximilians Universitlt, TheresiensIrasse 37, D-80W Miinchen 2, FR.G.

(Receii,ed

26

September

199fl

accepted

30 October1990)

Abstract.

Using simple scaling

argumenIs ^{it is shown that} a

sufficiently slowly decaying

power law noise distnbuIion leads Io nonuniversal

scaling properties

for a

kinetically roughened

interface.

Good agreement ^{is found with recent} simulations of

Zhang (J Phys.

France 51

(1990) 2129).

In _a recent paper

Zhang [I] presented

numerical evidence that the

scaling properties

of _an interface driven

by

an uncorrelated random local

growth

rate

~(x,

t with _a power ^law distribution

PI??)

"

~ll~~"~~~,

_ll

2

1

(1)

depend continuously

on the noise

exponent

j While the effect of power ^law

spatiotemporal

noise correlations h well understood in the framework of various renormalization group ^{treatments of} the

problem [2],

^{the influence of the noise} disnibufion _{comes as a}

surprise,

since such

microscopic

details are

traditionally expected

to be irrelevant for the

large scale, long

time

properties.

^{It is} the

purpose

^{of this} note to

propose

a

simple physical

^{mechanism that could} account for the observed

nonuniversal behavior.

Specifically,

I _use

scaling _arguments

in the

spirit

^{of Villain}

[3,

4] ^{to derive} the

following expression

for the interface

roughness exponent

x ^{as a} function of _~,

x =

) 12)

where d denotes the

dimensionality

of the interface. For small ~

(2

< ~ <

3)

this formula is in

good

_agreement with the numerical results

ill

and the derivation suggests ^{that it}

gives

a

lower bound _{on ;} for all _p. The _essence of my

approach

^is to concentrate on the

largest

noise fluctuations that occur on a

gi,,en length

and time scale [5]. ^{This is motivated}

by

the numerical

[I]

^and

experimental

_[6] observation that the interface advances in occasional

large

^thrusts which

then

rapidly spread

in the lateral direction.

(*)

Present address IBM Thomas J. Watson Research CenIer, P O. Box 218, Yorktown

HeighIs,

New York 10598, U-S-A-

lo JOURNAL DE PHYSIQUE I N°1

The model

investigated by Zhang

is _a discrete version of the

Kardar-Parisi-Zhang (KPZ)

_equa-

tion

[7,

8]

~~ ^"~~~

~

~~~~~

~ _~' ~~~

The

height

of the interface above _a d-dimensional reference

plane

at time _t is

given by

a

single

valued

height

function

h(x, t).

The first term on the

right

hand side of

(3)

^describes

smoothing

due _{to an} interfacial tension _v and the

second,

nonlinear term reflects the lateral

displacement dynamics [7, 8].

The stochastic

growth

rate

~(x, t)

has _{a nonzero mean} which

gives

^the "bare"

interface

velocity

in the absence of

fluctuations,

and short range correlations in space ^{and time. I} have in mind _a discretization of

(3)

^{such that} the random

height

increments _at different

(discrete) positions

and times _are drawn

independently

from the distribution

(I).

All

lengths

^and times _are

measured in units of the discretization.

The

quantity

of interest h the

height

^difference correlation function

Gllx x'l,t)

₌

ilhlx,t) h(x',t)li 14)

where the average ^extends over all realizations of the process ^with a flat initial

configuration, h(x, 0)

₌ ^0. Scale invariant

roughness implies

the

scaling

^form _[8]

Glr, t)

=

r~g lr~ /t) ₁₅₎

for _{r, t} >

I,

with _g

ix

-

0)

₌ const.,

g(x

-

co)

~w

x~X'~.

_{This means that the interface is}

rough,

I.e. the

typical amplitude ii

^of transverse fluctuations increases

algebraically

with the

distance, ii

~w

rX,

_on

length

scales below _a

dynamical

correlation

length _(jj (t)

~w

t~'~.

_{The nonlinear term}

in

(3)

enforces the

scaling

relation [9] X + z = 2. For a one-dimensional interface the continuum

equation (3)

satisfies detailed balance

[7],

^which

implies

_x

=

1/2

^and hence _z

=

3/2.

The

question

at stake is the

degree

of

universality

of the continuum results. The

predictions

_x

=

1/2,

_z

=

3/2

have been verified [8]

by

many simulations of dhcrete

models,

where the noise h

typically

drawn from _a distribution of bounded

support.

^In contrast, the simulations of

Zhang [I]

^{for the noise}

distribution

(I)

indicate that _x increases

continuously

with

decreasing

_jt,

reaching

^the

limiting

value _x

= I at the

point

_{~ =} ² where the distribution _{ceases to} have _a finite second _moment and hence any ^continuum

description

must break down.

Zhang

_argues that these results may ^be

relevant to several recent

experiments _[6, lo,

_{11] on} one-dimensional

moving

interfaces where

roughness exponents

_x >

1/2

have been observed. In the

following

I

attempt

to

explain

^his

findings.

We consider _a

patch

^{of the interface Of linear extension} r and assume that the

roughness

is

stationary

on the scale _r, I-e- _r «

(jj it ).

The number of random events within the

patch

in _a time interval Of duration _T is N

=

r~T.

_{We want to estimate} the size

~max(N)

of the

largest

of these events.

Clearly

Prob

(~max(N)

<

~)

=

(Prob(~

_{< x)16'~}

=

(1- x~"16'~ (6)

which

yields,

^for

large

^N

(llmaxlN))

_'~

N~'" _17j

This corrects

Zhang's equation (3).

The time

required

to

spontaneously

create a fluctuation of transverse

amplitude ii

^{is thus}

Tc ~w

( [ /r~. (8)

Once created the fluctuations

sprcads laterally

across the

patch,

driven

by

the nonlinear term in

(3). Balancing

this _term

against

the timc dcrivative _we obtain the

spreading

time scale

[4,

8]

Ts ~

r~ Iii. (9)

N° I KINETIC ROUGHENING BY EXCEPTIONAL FLUCTUATIONS II

In _a

Stationary

situation the

largest

fluctuations that contribute _to the

roughness

are those with Tc ~w T~.

Larger

fluctuations will

occasionally

_appear, but

they spread

too

rapidly

to have a

signifi-

cant statistical

weight.

Hence the

spontaneously

created

large

fluctuations

give

rise to a transverse

roughness 11

~

rX with the

roughness _exponent given by (2).

As stated the

argument completely neglects

the cumulative effect of small

fluctuations,

which becomes all

important

for noise distributions of finite

support.

^{It is therefore} not

surprhing

that x ^as

given by (2)

vanishes for jt - co instead of

tending

to some

positive

limit

(e,g.,

_{x =}

1/2

for d ₌

I).

For a power ^{law noise} distribution the

roughness generated by exceptional

fluctuations

superimposes

the cumulative

roughness

and dominates it for small _jt. I

consequently expect (2)

to be _{more accurate} for smaller values of _~ and _to

provide

a lower bound on the _true

roughness exponent

^{for all} _p.

This is borne _out

by

a

comparison

with

Zhang's

simulations of one-dimensional interfaces

(d

₌

I).

For 2 < p < 3 my ^{formula is within the} error bars of the numerical results. In

particular,

the

limiting

behavior _x

- I for _jt

- 2 is

reproduced,

^and

x(jt

=

3)

₌

3/4

_as

compared

to the numerical value _x

= 0.75 + 0.01. For d ₌ I the

roughening exponent

^{for bounded noise} is _{X =}

1/2

and hence

(2)

cannot be valid for _{jt >} 5. In fact the

prediction x(jt

₌

4)

₌

3/5

is

already significantly

lower than the numerical result _x

= 0.66 + 0.02.

Zhang

also observes that the

exponent identity

_{[9] x + z =} 2 holds

independently

of _jt. Here this relation is inherent in the

expression (9)

for the

spreading time,

which

implies

_Ts

~w

r~ with _z

= 2 _x.

As _a check of the

present scaling approach

^I now consider the linearized KPZ

(or

Edwards- Wilkinson

[12]) equation,

I,e.

(3)

with 1

= 0. It is

trivially solvable,

with the result that _{x = xo =}

(2 d)/2

and _{z = zo =} 2 for any noise distribution with _a finite second moment. The linear

problem

^{thus has} a

higher degree

of

universality

than the nonlinear _one, lb _see how this _comes

about,

_we

repeat

our

argument

^for the linear case. In the absence of lateral

growth

fluctuations

spread diffusively,

so

(9)

is

replaced by

_Ts

~w

r2.

More

importantly,

the

integrated height

of _a

spreading

fluctuation is conserved under the diffusive

dynamics fih/fit

₌

vv~h,

and hence in contrast to the nonlinear case the

amplitude

of the fluctuation decreases

during

the

spreading

process. After the

spreading

^time Ts the

amplitude

of _a fluctuation created

by

a noise event of size _~max is

(i

_+~

lJmax/Td.

^As a

consequence (8)

is

replaced by

_{T~ +~}

( (r("~

^~)~ and

balancing

with Ts +~

r2 yields

_{x =}

(2 (~ l)d)/~.

This

exponent

^{is smaller than the continuum}

prediction

Xo =

(2 d)/2

whenever _jt >

2, I-e-,

whenever the distribution

(I)

has a finite second _moment.

We therefore recover the

(obvious)

conclusion that the linear

theory

is insensitive to the none distribution. The same

argument applies

to linear

equations

_[4] with _a

general

relaxation term

V2~h.

Summarizing,

I have demonstrated how

continuously varying

kinetic

roughness exponents

can arise from the

exceptionally large

fluctuations that _are characteristic of the power ^law noise dis- tribution

(I).

It would be of interest to extend the simulations of

Zhang [I]

to

higher

substrate dimensions in order to test the

dimensionality dependence predicted by (2). Concerning

^the ex-

periments [6, lo, 11]

^{that motivated}

Zhang's original study

^the

question

remains of how _to relate the noise distribution

exponent

^to some

physical _[6,

_11] or

biological _[10] properties

of these sys-

tems.

Acknowledgements.

thank Y.-C.

Zhang,

T Vicsek and J. Villain for

communicating

their results

prior

to

publication.

This work _was

supported by

Deutsche

Forschungsgemeinschaft.

12 JOURNAL DE PI.IYSIQUE I N° I

References

[Ii

ZHANG Y.-C., J Pl~ys. ^{France Sl}

(1990)

2129.

[2] ^MEDINA

E.,

HWA

T,

KARDAR M. and ZHANG

Y.-C., Phys.

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J.,

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J.,

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