An Exactly Solvable Lattice Model for Inhomogeneous Interface Growth

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An Exactly Solvable Lattice Model for Inhomogeneous Interface Growth

Gunter Schütz

To cite this version:

Gunter Schütz. An Exactly Solvable Lattice Model for Inhomogeneous Interface Growth. Journal de

Physique I, EDP Sciences, 1996, 6 (11), pp.1405-1410. �10.1051/jp1:1996153�. �jpa-00247253�

Short Communication

An Exactly Solvable Lattice Model for Inhomogeneous Interface Growth

Günter M. Schütz

(*)

Department of Physics, University of Oxford, Theoretical Physics, 1 Keble Road, Oxford OXI 3NP, UK

and

Laboratoire de Physique du Solide (**), Université Henri Poincaré, B-P. 239,

54506 Vandceuvre-lès-Nancy Cedex, France

(Received

6 June 1996, received in final form 25 July 1996, accepted 27 August

1996)

PACS.05.40.+j Fluctuation phenomena, random processes, and Brownian motion

PACS.02.50.Ey Stochastic _processes

PACS.68.35.Fx Diffusion; interface formation

Abstract. We study the dynarnics of _an exactly solvable lattice model for inhomogeneous

interface growth. The interface grows deterministically with constant velocity except along _a

defect fine where trie growth _process is random. We obtain exact expressions for the average

height and height fluctuations as functions of _space and time for

an initially flat interface. For

a given defect strength there _{is a} critical angle between the defect fine and the growth direction above which _a cusp in the interface develops. In the mapping ^to polymers _in random media this is _an example for the transverse Meissner effect. Fluctuations around the _mean shape of the

interface _are Gaussian.

The

problem

of nonlinear,

KPZ-type

^interface

growth

in 1+1 dimensions and the

closely

related

problem

of

polymers

in random media has been the

subject

of _many studies _over the past decade

iii.

Trie

underlying

assumption ^{in this}

approach

is that

growth

occurs

stochastically

normal _to the interface. The

dynamics

^of the interface _are then

given by

^{trie KPZ}

equation ôth

=

vi7~

h+À(i7h)~+~

_[2]. The deterministic version of the equation

(without

the _{noise ~ can} be solved

explicitly. Among

other

things

_one ^finds that in the infinite time limit the interface becomes flat.

However,

m the _presence of noise, _even an

initiàlly

flat interface

roughens

and the derivation of

quantities

^related to the

roughemng

process and other fluctuation

phenomena

become _an

interesting

and

challenging problem, usually

tackled

using

^either the renormalization group or the

study

^of lattice mortels. In this context the

asymmetric

exclusion process [3], a

driven lattice _gas

mortel,

has

played

_a

special

^{role. Both numerical and exact}

analytical

results have been obtained from this

microscopically

motivated

exactly

solvable

mortel,

which _maps to _a lattice

growth

mortel in trie

universahty

Mass of trie

noisy

^KPZ

equation

[4, 5].

(* ^Present address: IFF, Forschungszentrum Jülich, 62425 Jülich, Germany

(e-mail: [email protected])

(**) URA 155 CNRS

@ Les Editions de Physique 1996

1406 JOURNAL DE PHYSIQUE I N°11

h

= 5

...=;.;=...

:.' j ^"..._

..

j

".. _:.'

"i'

h=0

_{1. o o . o .}

. o1

y

Fig. 1. The mapping between the restricted interface and the partiale exclusion _process:

we show

a possible interface configuration and the corresponding particle occupancies _{on a} lattice _with sites labeled by_y. The indicated flips in the interface correspond to partiales hopping _on the lattice, marked

by horizontal

arrows.

More

recently,

various

growth

mortels with

spatially

localized

inhomogeneities

_were investi-

gated.

Kallabis and

Làssig

studied trie KPZ equation with

spatially

localized noise [6]. Fo-

cussing

_on trie properties of trie

corresponding polymer

mortel

(a homogeneous

_{system at non-}

zero temperature with _a random defect

line) they

found _an

interesting phase diagram

_even in

one dimension.

Subsequent

work elucidated _some

properties

of _a related

growth

mortel which

is

fully

deternnnistic and where instead of trie

spatially

localized noise _a constant columnar defect _was considered

iii. Among

other

things

it _was noted that in _one dimension trie inter- face in trie

steady

_state bas _{a cusp} with _constant

slope

for

arbitrarily

small defect

strength.

Such behaviour bas been known for _{some time to}

occur in yet another related

model,

_{an ex-}

clusion _{process with}

a defect which _{mimics trie situation of}

homogeneous

and deterministic

growth

_except

along

_one defect line where

growth

is stochastic [8j. ^In trie

steady

state of this

non-equilibrium

_system

(describing

trie zero-temperature behaviour of _a

homogeneous polymer

system with _a random defect

line)

_one finds _{agam such}

a cusp, for

arbitrarily

small _noise if the defect line is parallel to the _average

growth

direction _{as assumed in the continuum models}

studied in references [6,

îj.

This makes _a

study

of trie

dynamical

_{properties of trie lattice model} desirable. _{It is} trie aim of this paper to

study

trie time evolution of _an

initially

flat interface

using

this model.

Trie model introduced in reference [8j is an

exactly

solvable _version of trie determmistic asymmetric exclusion _process

on a

ring

with _a defect _line. We _use trie well-known RSOS mappmg from _an interface _{to a} lattice _gas [4, 5j: In trie interface model trie

(integer) height

variables

h(y, t)

_may dilfer _on

neighbouring (integer)

_{sites y, y +1}

_{only by}

_{+1. As}

a result of this

restriction, growth

_{can occur}

only

_m local _minima,

no

overhangs

_can

develop.

If _a growth

event occurs, the

height

at site y mcreases

by

_{two units}

(see Fig. l).

In lattice _gas

language

the

height

dilferences between

neighbouring

sites _are

mapped

to _a

partiale occupation

number

n~(t)

= 0,1 with the _presence of _a partiale _{on x}

corresponding

to

slope

_-1 between _{sites y -1}

and _y in the interface _mortel and _a vacancy ^{at site} x

corresponding

to

slope

₊₁

(see Fig. I).

Growth at site y in the

growth

mortel then

corresponds

to _a

partiale hopping

from site _x to

a vacant site _x +1. Note that each lattice site _can be

occupied by

at most _one

partiale.

In _a

finite system ^of L ₌ 2N sites for the

partiale

model _one needs N

partiales

in order to _ensure

periodic boundary

conditions in the

height

model. In _an infinite system, an average

density

p

# 1/2

_{over some} finite range L'

corresponds

to _{a non-zero} average

slope

1-

2p

in that _range.

So far this mapping

keeps

track

only

of the

height

dilferences between

neighbouring

^{sites. In}

order to make the two models

equivalent,

_one has to introduce _an additional

integer

random variable ho E Z in the

partiale

system ^which

gives

the absolute

height

of the interface at _some

arbitrary,

but

specified point,

_{e-g- y =} 0. In the lattice _gas

mapping

the value of this random variable is increased

by

two units each time _a

particle hops

from site 0 to site 1. From this _one may

reproduce

the

height

at any point since

h(y)

= ho +

£(=i Il 2n~).

The deterministic time evolution is realized

by

a

parallel

sublattice

updating

scheme in which in _a first step all odd pairs of sites

(2x

1,

2x)

_are

updated according

to the

following

rules:

If there is _a

partiale

_on site 2x -1 and _{a vacancy on} site 2x, then the

particles hop

with

probability

1 to site 2x. If the pair ^{of sites is in} one of the three

remaining configurations, nothing changes.

^{These rules} are

applied

in

parallel

to all such

pairs.

^In a next step one shifts the

pairing by

one lattice unit and

applies

the _same rules to the _even pairs

(2x,

2x

+1).

This

completes

_one full time step. ^{Note that} so far this model is

fully

deterministic, and any initial

configuration

will

eventually

evolve into _a flat interface characterized

by

_a

partiale configuration

(.

,

1, 0,1, 0, 1, 0, 1, 0,

).

The

dynamical properties

of _a similar system with

slightly

dilferent deterministic

updating

rules _were studied in [9j.

In order _to introduce _a defect line at y = 0 _we

change

the rules such that _a

partiale

_on

site _x

= 0 hops

only

with

probability

_p < 1, ^{if site 1 is} vacant. This randomness results in non-trivial behaviour of trie system. The

stationary properties

of this mortel _{on a} finite ring with N

particles

_were studied _{in [8]}

(N arbitrary).

Here _we shall consider trie

dynamics

of trie infinite system ^{with flat initial}

configuration (.

,1, 0,1, 0,1, 0,1, 0,. .) where trie

partiales

_are

on trie odd sites while trie vacancies _{are on} trie _even sites. Because of reflection symmetry one

has

h(y)

=

h(-y).

It is useful to note that the time evolution may be written in terms of _a transfer matrix T =

pTo

+

(1- p)Ti acting

on states

corresponding

to _some given

configuration

^{of the}

system'[8].

Any

state _is

fully

characterized

by

the

partiale occupation

numbers _R~ and the integer

height

variable ho To is the transfer matrix for the system without

defect,

while Ti _is the transfer matrix of the system with full

blockage

^where a

particle

at site 0 _{can never move} to site 1, _even if it is vacant.

Acting

with To _on the state (.

,

1, 0,1, 0, 1, 0, 1, 0,.

reproduces

this state

by shifting

the whole

configuration by

two lattice units to the

right

after _one full time step. ^This is in accordance with the remark that this

configuration

is stationary if _p

= 1 and means that

the interface has grown

everywhere by

two units. The

height

^variable ho has increased

by

two

units _since a

partiale

crossed trie bond between sites 0 and 1. On trie other

hand,

the action of

Ti _on this state results in _a

configuration

(.

,

0, 0,1, 0,1, 0,1, ^{0. where trie first}

occupation

number in this set

(which

is

0) corresponds

to the

occupation

on site 1. The

occupation

numbers _on site 0 and _on the

negative

^{sites follow from symmetry. In this} case ho has not

increased, only h(y)

_{for )y)} > has grown

by

two units.

Taking

the t~~ _power of T

yields

all

possible configurations

the interface _may take after _t time steps. ^Each realization _{acquires a} factor of the form

pk il _p)~~k

which _gives the

probability

that this

particular growth history

has taken

place.

Using

the

decomposition

of T into To and Ti it becomes

straightforward

to calculate to

respective

probabilities

for all these

configurations: il

The odd sublattice _remains empty ^for aJl tiInes.

(2)

The consequence of 4he defect for the _even

positive

^{sublattice is the}

injection

^of

a

partiale

_on site 0 with

probability

_{p in} each time step.

(3)

If _a

partiale

bas been

injected

it

1408 JOURNAL DE PHYSIQUE I N°11

will _move in each

subsequent

time step

by

two lattice units without _any interaction with other

partiales.

As _an

example

for what this

corresponds

to in

height language,

consider ho which

we take to be _zero at time t

= 0. With trie

dynamical

rules _as

explained

above _one finds after

one step ho ₌ with

probability

_p and ho _" 0 with

probability

1- p. After _two steps _one finds

ho

₌ 2 with

probability p~,

ho

= 1 with

probability 2p(1- p)

and

ho

₌ 0 with

probability il p)~,

and _{so on.}

By construction,

trie

height

_on the _even sublattice _will

aJways

be _even,

while _on the odd sublattice it will

always

be odd. Let _us denote

h(2y, t)

=

21(2y, t).

Then _one gets for the

probability P~,t(h)

of

finding

the

height

h at site y at time t

(for

_y >

0)

P2y,tlh)

₌

1( _{_(} Ii P)~~~p~~~ la

1 _y

1tj

"

ôi,t Ill

2

t) il

On the odd

(positive)

lattice sites _one has

always,

1-e- for aJl times and all realizations of the

randomness, h(2y -1,t)

=

h(2y,t)

1. For the

negative

half _{space y <} 0 _one gets

P[h(-y,t)]

=

P[h(y,t)] by

symmetry. This result

gives

_a

complete description

of the time evolution of the local

heights

of _an interface which is

imtially (macroscopically)

flat _(~

).

Note that there _{are no}

height

fluctuations _{at all for y > t. This must} be _so since trie

perturbation

of trie interface caused

by

trie defect

spreads

with finite

velocity

_~

= l which is trie bulk

growth velocity.

In what follows _we

study only

trie non-negative even sublattice _{up to site y}

= t, since all

quantities relating

trie odd

sublattice,

trie

negative

half _space and trie

region

_y > t _are then

trivially given.

From the distribution

(1)

_one obtains the

generating

function for

height

fluctuations

(e°~~~~'~l) ₌

e~°~(l

_{p +}

pe~°)~~~ (0

< y ^<

t)

" ~~~~ (Y ^~ ~)

(2)

and in

particular

the _average

height (h(2Y, t))

= 2t

2(1 P)(t

_Y)

(o

_{1 y} <

tj

= 2t

(y

>

t). (3)

The

height

fluctuations _can be obtained

by taking

the second

logarithmic

derivative of the

generating

^function. Une finds

(h~(2Y,t)1 (h(2Y, t)l~

₌

4P(1 Pl(t

_Y)

(° 1Y1t)

" ° (Y 2

t) (4)

From

equation (3)

_one ^finds that the defect _causes the

development

of _{a cusp with constant}

slope 2(1 p).

At _{site y}

= 0 the interface _grows with constant

speed

_~o

=

2p,

while _in the

bulk,

at distances

larger

than t, it grows with

velocity

_~b

= 2

(Fig. 2).

From

(1)

one finds that the fluctuations of the interface round its _mean value

(3)

_are Gaussian

(for

t

large)

with

a

space-dependent

_vanance

given by (4).

These _are the main results of this _paper.

We conclude

by briefly discussing

the situation in which the defect line is trot

parallel

_to

the

growth

direction of the interface. This _can be realized within this mortel

by taking

_a

(macroscopically)

_non-zero initial

slope

_s

= 1- 2p of the interface. Since

growth

_occurs

(~jEquation

(1) is _easy to verify _e-g- by applying T _two

or three times _to the initial _state This makes the structure of the interface after t steps already quite clear.

h(y,t)

h = 2t

h=2pt

h(y, 0)

h = 0

§

Fig. 2. Macroscopic _average interface shape at time t ₌ o

(flat

interface with height o) and at _some later time t. The position of the defect at y = o is indicated by the vertical dashed fine. The defect fine is parallel to trie bulk growth direction.

normal to the interface trie net result is indeed _a defect line tilted relative to trie main

growth

direction. In

partiale language

_one has to choose _an initial state with _a

density

_p

# 1/2.,

_e.g. a

configuration (.

,1, 0, 0, 0, 1, 0, 0, 0, .). ^{From the}

analysis

of the

steady

state of this system

one finds that _a

phase

transition takes

place

at p~ =

2p

^for _p <

1/2 (positive

initial

slope)

and _p~

=

2(1 p)

for _p >

1/2 (negative

initial

slope) respectively

_[8] For the

dynamics

^of

an

initially

flat interface trie

steady

state

properties

of the system suggest ^{that for} _p > p~ the interface remains flat for all times except ⁱⁿ a finite

region

close to trie defect. For _p < p~ a cusp will evolve. Unlike in trie situation described

above,

trie

boundary

of this

cùsp

will not be

sharp,

but fluctuate. In _a finite system ^these fluctuations scale in trie system ^{size like}

Ll/~

_and

one would therefore expect a power

growth ^tl/~

in time of these domain

boundary

fluctuations in the infinite system. For fixed _p this is _a

phase

transition that takes

place

at critical

slopes

s~ =

+(1 p).

For

il p)

< _s <

il p)

the system

develops

the _cusp.

This situation is also of interest in the well-known

mapping

^of this

problem

to a directed

polymer

at temperature ^{0 in} a medium with _a random defect line, as studied in reference [6] for

general

temperature, but

only

^{for the}

special

_{case p}

=

1/2

where the defect line and the average direction of the

polymer

_are

parallel.

The

development

of the _cusp

corresponds

to

having

a

bound state where the

polymer

_is bound to the

(attractive)

defect line. At _zero temperature and for _p

=

1/2

it is _no surprise that this

happens

at

arbitrarily

small defect

strength.

This

simply

reflects the fact that in _one dimension _a delta function

potential (the

^defect

line)

with

arbitranly

smaJl

amplitude

^has a bound state. Translated into

polymer language

the

phase

transition in trie interface mortel for _p

# 1/2

is _a transition from _a bound state of the

polymer

for _a tilt of the defect litre

against

^trie

(average)

direction of trie

polymer

below trie critical

angle

_s~ to an unbound _state for _a tilt above _s~. This is _a

simple example

for the _occurrence

of the transverse Meissner elfect

[loi.

A macroscopic cusp with constant

slope

_appears to be _an universaJ feature of

growth

mortels with _a defect fine where trie

growth

_process is

impeded.

In this _{sense some} features of the mortel studied in this paper may be of _more than

purely

theoretical interest. Normal

growth,

which

1410 JOURNAL DE PHYSIQUE I N°11

is the

underlying assumption

that leads _to the KPZ

equation,

has been

reported

to _occur e-g-

m experiments _on

wetting

of tapes

plunged

in _a

liquid il ii.

It would therefore be

interesting

to

adapt

the mortel to this situation

by

_an appropriate choice of

boundary

conditions. On the other

hand,

_our

simple

mortel has _some clear limitations. The strict

discontinuity

of the

slope

of trie interface at y = t

(for

trie half-filled

system)

is

obviously

_an artefact of trie

dynamics

chosen here and _cannot be

expected

to be found in any real system. Une would rather expect

a smeared out

discontinuity

_as in trie system with _a tilted defect _{or as} in trie

fully

stochastic version of trie mortel with _a defect [12].

Another,

intrinsic, limitation of trie mortel is that

growth

at trie defect is

always

slower than in trie bulk. It would be _very

interesting

to

study

a deterministic mortel where trie defect

growth

coula be made faster than the bulk

growth velocity.

Acknowledgments

The author would like to thank M.

Làssig,

H. Kallabis and U. Tàuber for useful discussions.

This work _was supported

by

_an EC

Fellowship

under the Human

Capital

and

Mobility

_program.

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