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Universal behavior of the coefficients of the continuous equation in competitive growth models

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a r X i v : c o n d - m a t / 0 4 0 2 3 7 6 v 3 [ c o n d - m a t . d i s - n n ] 2 N o v 2 0 0 4
UNIVERSAL BEHAVIOR OF THE COEFFICIENTS OF THE CONTINUOUSEQUATION IN COMPETITIVE GROWTH MODELS
D. Muraca
1
, L. A. Braunstein
1
,
2
and R. C. Buceta
1
1
Departamento de F´ısica,Facultad de Ciencias Exactas y Naturales Universidad Nacional de Mar del Plata Funes 3350,
7600
Mar del Plata, Argentina
2
Center for Polymer Studies and Department of Physics Boston University, Boston, MA 02215, USA
The competitive growth models involving only one kind of particles (CGM), are a mixture of twoprocesses one with probability
p
and the other with probability 1
−
p
. The
p
−
dependance producecrossovers between two diﬀerent regimes. We demonstrate that the coeﬃcients of the continuousequation, describing their universality classes, are quadratic in
p
(or 1
−
p
). We show that thesrcin of such dependance is the existence of two diﬀerent average time rates. Thus, the quadratic
p
−
dependance is a universal behavior of all the CGM. We derive analytically the continuous equa-tions for two CGM, in 1+1 dimensions, from the microscopic rules using a regularization procedure.We propose generalized scalings that reproduce the scaling behavior in each regime. In order toverify the analytic results and the scalings, we perform numerical integrations of the derived analyt-ical equations. The results are in excellent agreement with those of the microscopic CGM presentedhere and with the proposed scalings.
PACS numbers: 81.15.Aa, 05.40.-a, 05.10.Gg
Evolving growing interfaces or surfaces can be foundin many physical, chemical and biological processes. Forexample, in ﬁlm growth either by vapour depositionor chemical deposition [1, 2], bacterial growth [3] andpropagation of forest ﬁre [4]. The resulting interfacehas a rough surface that is characterized through scal-ing of the interfacial width
W
deﬁned as
W
(
L,t
) =
[
h
i
−
h
i
]
2
1
/
2
, where
h
i
is the height at the position
i
,
h
i
=
L
d
i
=1
h
i
is the spatial average,
L
is the linearsize,
d
is the spatial dimension and
{}
denote conﬁgura-tion averages. The general scaling relation [1] for thesegrowing interfaces that evolves through a single modelcan be summarized in the form
W
(
L,t
)
∼
L
α
f
(
t/L
z
),where the scaling function
f
(
u
) behaves as
f
(
u
)
∼
u
β
(
β
=
z/α
), for
u
≪
1 and
f
(
u
)
∼
const for
u
≫
1. Theexponent
α
describes the asymptotic behavior where thewidth saturates due to ﬁnite sizes eﬀects, while the expo-nent
β
represent the early time regime where ﬁnite-sizeeﬀects are weak. The crossover time between the tworegimes is
t
s
=
L
z
.The study of growth models involving one kind of par-ticles in competitive processes (CGM) has received littleattention, in spite of the fact that they are more realisticdescribing the growing in real materials, where usuallythere exist a competition between diﬀerent growing pro-cesses. As an example, in a colony of bacteria growingon a substrate, a new bacteria can born near to anotherand stay there, move into another place looking for foodor died. This “bacteria” can be thought as a particle un-dergoing either a deposition/evaporation process or de-position/surface relaxation.The processes involved in the CGM could have diﬀer-ent characteristic average time rate. Recently Shapir etal. [5] reported experimental results of surface rough-ening during cyclical electrodeposition dissolution of sil-ver. Horowitz et al. [6] introduced a competitive growthmodel between random deposition with surface relax-ation (RDSR) with probability
p
and random deposition(RD) with probability 1
−
p
, called RDSR/RD. The au-thors proposed that the scaling behavior is characteris-tic of an Edward Wilkinson (EW) equation, where thecoeﬃcient associated to the surface tension
ν
dependson
p
. The dependance of
ν
on
p
governs the transitionfrom RDRS to RD. Using a dynamic scaling ansatz forthe interface width
W
they found that the results areconsistent provide that
ν
∝
p
2
. Also Pellegrini and Jul-lien [7] have introduced CGM between ballistic deposi-tion (BD) with probability
p
and RDSR with probability1
−
p
, called BD/RDSR. For this model Chame and Aar˜
a
oRies [8] presented a more carefully analysis in 1+1-
d
andshowed that there exist a slow crossover from an EW toa Kardar-Parisi-Zhang (KPZ) for any
p >
0. They alsofound that the parameter
p
is connected to the coeﬃcient
λ
of the nonlinear term of the KPZ equation by
λ
∼
p
γ
,with
γ
= 2
.
1.In this letter, we show that the srcin of such depen-dance is the existence of two diﬀerent average time rates.Thus, the quadratic
p
−
dependance is an universal fea-ture of all the CGM. To our knowledge this is the ﬁrsttime that the
p
-dependance on the coeﬃcient of the con-tinuous equations is obtained analytical from the micro-scopic dynamics.In order to test our hypothesis, we derive the an-alytical continuous equations for the local height forthe RDSR/RD and BD/RDSR models. The proce-dure chosen here is based on regularization and coarse-graining of the discrete Langevin equations obtainedfrom a Kramers-Moyal expansion of the master equation
2[9, 10, 11].Lets introduce ﬁrst the general treatment of this prob-lem. Let us denote by
h
i
(
t
) the height of the
i
-thgeneric site at time
t
. The set
{
h
i
, i
= 1
,...,L
}
de-ﬁnes the interface. Here we distinguish between twocompetitive processes: A with probability p and aver-age time of deposition
τ
A
, and B with probability 1-pand average time of deposition
τ
B
. In deposition pro-cesses with
p
= 1 the average rate of deposition is givenby
τ
−
10
=
{
dh
i
/dt
}
(
p
=1)
. If the process is made withprobability
p
the average rate of deposition is given by
τ
−
1
A
=
{
dh
i
/dt
}
(
p
)
=
p
{
dh
i
/dt
}
(
p
=1)
. The same hold fora process with probability (1
−
p
). Thus, the particlesare deposited at an average rate
τ
A
=
τ
0
p , τ
B
=
τ
0
1
−
p .
(1)In the average time of each process, the height in the site
i
increases by
h
i
(
t
+
τ
A
) =
h
i
(
t
) +
a
⊥
p R
Ai
,h
i
(
t
+
τ
B
) =
h
i
(
t
) +
a
⊥
(1
−
p
)
R
Bi
,
(2)where
R
Ai
and
R
Bi
are the growing rules for processes
A
and
B
respectively and
a
⊥
is the vertical lattice spacing.Expanding
h
i
(
t
+
τ
A
) and
h
i
(
t
+
τ
B
) to second order inTaylor series around
τ
A
and
τ
B
, we obtain
h
i
(
t
+
τ
J
)
−
h
i
(
t
)
≈
dh
i
dt τ
J
,
(3)for the process
J
=
A,B
. Thus, the evolution equationfor the height (in the site
i
) for this CGM is given by
dh
i
dt
=
K
(1
,A
)
i
+
K
(1
,B
)
i
+
η
i
(
t
)
,
(4)where the ﬁrst moments of the transition rate for eachprocess [12] are
K
(1
,A
)
i
=
a
⊥
τ
A
p R
Ai
,K
(1
,B
)
i
=
a
⊥
τ
B
(1
−
p
)
R
Bi
,
(5)and the Gaussian thermal noise
η
i
(
t
) has zero mean andcovariance
{
η
i
(
t
)
η
j
(
t
′
)
}
=
a
⊥
K
(1
,A
)
i
+
K
(1
,B
)
i
δ
ij
δ
(
t
−
t
′
)
.
(6)In order to test our analytical result, we use two mod-els. The ﬁrst model RDSR/RD considers a mixture [6]of RDSR (process A) with probability
p
and RD (processB) with probability 1
−
p
. Lets introduce the growth rulefor each process for the ﬁrst model. In the RD growthmodel one chose a column of a lattice, at random, among
L
and a particle is launched until it reaches the top of the selected column. The RDSR is a variant of the RD:a particle is released from a random position but whenit reaches the top of the selected column is allowed torelax to the nearest neighbor (nn) column if their heightare lower that the selected one. If the height of bothof the nn are lower than the selected one the relaxationtakes place with equal probability to one of them. ForRD,
W
(
L,t
) does not depend on
L
, this means that thewidth
W
does not saturate due to the lack of lateral cor-relations. Thus, in this model:
W
(
t
)
∼
t
β
RD
. Moreover,the RDSR model generates lateral correlations, thereforeone has
β
RDSR
= 1
/
4 and
α
RDSR
= 1
/
2. The ﬁrst mo-ment of the transition rate for these processes are
K
(1
,A
)
i
=
a
⊥
τ
A
p
ω
(2)
i
+
ω
(3)
i
+1
+
ω
(4)
i
−
1
,K
(1
,B
)
i
=
a
⊥
τ
B
(1
−
p
)
ω
(1)
i
,
(7)where the rules for both processes can we written as
ω
(1)
i
= 1
,ω
(2)
i
= Θ(
H
i
+1
i
) Θ(
H
i
−
1
i
)
,
(8)
ω
(3)
i
=
12
1
−
Θ(
H
i
+1
i
)
+ Θ(
H
i
+1
i
)
1
−
Θ(
H
i
−
1
i
)
,ω
(4)
i
=
12
1
−
Θ(
H
i
−
1
i
)
+ Θ(
H
i
−
1
i
)
1
−
Θ(
H
i
+1
i
)
.
where
H
i
±
si
±
k
= (
h
i
±
s
−
h
i
±
k
)
/a
⊥
, and Θ(
z
) is the unitstep function deﬁned as Θ(
z
) = 1 for
z
≥
0 and Θ(
z
) = 0for
z <
0. The representation of the step function canbe expanded as Θ(
z
) =
∞
k
=0
c
k
z
k
providing that
z
issmooth. In any discrete model there is in principle an in-ﬁnite number of nonlinearities, but at long wavelengthsthe higher order derivatives can be neglected using scal-ing arguments, since one expect aﬃne interfaces over along range of scales, and then one is usually concernedwith the form of the relevant terms. Thus, keeping theexpansion of the step function to ﬁrst order in his argu-ment and replacing the expansion Eq. (8), Eq. (4) can bewritten as
dh
i
dt
=
a
⊥
(1
−
p
)
τ
B
+
a
⊥
pτ
A
1 +
c
1
∆
2
h
i
a
⊥
+
η
i
(
t
)
,
(9)where ∆
2
h
i
=
h
i
+1
−
2
h
i
+
h
i
+1
≃
a
2
∂
2
h/∂x
2
⌋
h
i
, and
a
is the horizontal lattice spacing. Replacing the ratesgiven by Eq. (1) in Eq. (9) and using a standard coarse-grain approach [10, 11] the continuous equation for thisCGM is
dhdt
=
F
(
p
) +
ν
(
p
)
∂
2
h∂ x
2
+
η
(
x,t
)
,
(10)where
h
=
h
(
x,t
) and
F
(
p
) =
a
⊥
τ
0
(1
−
p
)
2
+
p
2
,
(11)
ν
(
p
) = 2
c
1
a
2
τ
0
p
2
.
The noise covariance is given by
{
η
(
x,t
)
η
(
x
′
,t
′
)
}
=
D
(
p
)
δ
(
x
−
x
′
)
δ
(
t
−
t
′
)
,
(12)
3where
D
(
p
) =
a
a
⊥
F
(
p
)
.
(13)Equations (11) and Eq. (13) shows that the quadratic de-pendance on the coeﬃcients of the continuous equation,arises naturally as a feature of the CGM and is due tothe existence of diﬀerent average time rates.The second model is a mixture of RDSR with prob-ability 1
−
p
(
B
process) and ballistic deposition (BD)with probability
p
(
A
process) [8]. The evolution rulesfor RDSR are
ω
ji
, with
j
= 2
,
3
,
4 [see Eq. (8)]. In the BDmodel, the incident particle follows a straight trajectoryand sticks to the surface at the column
i
. The height inthe column
i
is increased in max[
h
i
+ 1
,h
i
+1
,h
i
−
1
]. If this process is done with probability
p
(A process), therules can be summarized as:
ω
(5)
i
= Θ(
H
ii
+1
) Θ(
H
ii
−
1
)
,
(14)
ω
(6)
i
=
H
i
+1
i
1
−
Θ(
H
ii
+1
)
1
−
Θ(
H
i
−
1
i
+1
)
,ω
(7)
i
=
H
i
−
1
i
1
−
Θ(
H
ii
−
1
)
1
−
Θ(
H
i
+1
i
−
1
)
,ω
(8)
i
=
12
δ
(
H
i
+1
i
−
1
,
0)
H
i
+1
i
1
−
Θ(
H
ii
+1
)
+
H
i
−
1
i
1
−
Θ(
H
ii
−
1
)
,
where
δ
(
z,
0) = Θ(
z
)+Θ(
−
z
)
−
1 is the Kronecker delta.Following the steps leading to Eq. (10) the evolutionequation for this process can be written as:
dhdt
=
F
(
p
) +
ν
(
p
)
∂
2
h∂x
2
+
λ
(
p
)
∂h∂x
2
+
η
(
x,t
) (15)where
F
(
p
) =
a
⊥
τ
0
(1
−
p
)
2
+
c
20
p
2
,ν
(
p
) =
a
2
τ
0
12
p
2
(1
−
c
0
−
2
c
0
c
1
) + 2
c
1
(1
−
p
)
2
,
(16)
λ
(
p
) =
a
2
τ
0
a
⊥
p
2
c
1
(5
−
4
c
0
−
c
1
)
.
The covariance of noise and
D
(
p
) is given by Eq. (12) andEq. (13), respectively. Notice that we have to change
p
by1
−
p
in all the above equations for RDSR, because in theﬁrst model RDSR is a kind
A
process and now is a kindB process. Equation (16) shows again that quadratic de-pendance on the coeﬃcients of the continuous equation.The quadratic dependance of
λ
on
p
, found by Chameand Aar˜
a
o Reis [8], is a general feature of the CGM.As both models have an EW behavior, it is expectedthat in that regime the following generalized scalingansatz [6, 14],
W
2
(
p,L,t
)
∼
L
2
α
[
D
(
p
)
/ν
(
p
)]
f
(
ν
(
p
)
t/L
z
)
,
(17)where
f
(
u
)
∼
u
2
β
for
u
≪
1 and
f
(
u
)
∼
const for
u
≫
1.Moreover, the second model is represented by a mixtureof EW and KPZ universality classes. In the early timeregime
W
(
t
)
∼
t
β
RDSR
, while a crossover to a KPZ, with
β
KPZ
= 1
/
3 and
α
≡
α
KPZ
= 1
/
2, is expected in theintermediate regime before the saturation. Thus, for theKPZ regime we propose the following generalization [15]of the scaling behavior of the width
W
2
(
p,L,t
)
∼
L
2
α
[
D
(
p
)
/ν
(
p
)]
f
λ
(
p
)
D
(
p
)
/ν
(
p
)
t/L
z
,
(18)where
z
= 3
/
2, and
f
(
u
)
∼
u
2
β
KPZ
for
u
≪
1 and
f
(
u
)
∼
const for
u
≫
1.In order to test our analytical result and the proposedscalings, we perform a numerical integration of Eq. (10)and Eq. (15), and compute
W
2
for both models. Notice
10
-4
10
0
10
4
ν
(p) t
10
-3
10
-2
10
-1
10
0
10
1
10
2
10
3
W
2
ν
( p ) / D ( p )
(a)
10
0
10
1
10
2
10
3
10
4
ν
(p) t
10
0
10
1
10
2
10
3
W
2
ν
( p ) / D ( p )
(b)
FIG. 1: (a) Log-log plot of
W
2
ν
(
p
)
/D
(
p
) for the RDSR/RDmodel as function of
ν
(
p
)
t
for
L
= 128. The diﬀerent symbolsrepresent diﬀerent values of
p
,
p
= 0
.
04 (
◦
),
p
= 0
.
08 (
✷
),
p
= 0
.
016 (
△
),
p
= 0
.
32 (+), and
p
= 0
.
64 (
∗
). Here weused
C
= 2
.
58 and
b
= 0
.
2 as parameters of the Θ-functionrepresentation. The dashed lines are used as guides to showthe RD regime with 2
β
= 1 and the EW regime with 2
β
= 0
.
5.(b) Log-log plot of
W
2
ν
(
p
)
/D
(
p
) for the BD/RDSR model asfunction of
ν
(
p
)
t
for
L
= 1024. The symbols represent thesame as in Fig.(1a). Here we used
C
= 0
.
18 and
b
= 0
.
5. Thecollapse of the curves at the earlier stage clearly shows theEW behavior (2
β
= 0
.
5). After this stage the curves split andundergoes a slow crossover to the KPZ behavior (2
β
= 0
.
66).The dashed lines are used as guides to show the EW regimewith 2
β
= 1
/
2 and the KPZ regime. The slope showed hereis 2
β
= 0
.
61.
that in order to numerically integrate the continuousequation, we do need a continuum representation of the
4
10
-4
10
0
~t
10
-3
10
-2
10
-1
~W
2
FIG. 2: Log-log plot of ˜
W
2
as function of ˜
t
as deﬁned in thetext, for
p
= 0
.
16 (
◦
),
p
= 0
.
32 (
✷
) and
p
= 0
.
64 (
△
). Theempty symbols correspond to
L
= 512 and the ﬁlled onesto
L
= 1024 . The collapse of the curves on the saturationregime using
z
= 3
/
2 shows that the curves saturates with aKPZ behavior as expected.
Θ-function to numerically compute the coeﬃcients
c
0
and
c
1
related to the ones of the continuous equations. Toperform the numerical integration, we chose the shiftedhyperbolic tangent [16] as the continuous representationof Θ-function deﬁned as Θ(
x
) =
{
1+tanh[
C
(
x
+
b
)]
}
/
2,where
b
is the shift and
C
is a parameter that allows torecover the Θ in the limit
C
→ ∞
. The numerical in-tegration was made in short lattices using a discretizedversion of the continuous equations Eq.(10) and Eq. (15).The results in large systems and the details of the inte-gration are beyond the scope of this letter and will bepublished elsewhere.For the ﬁrst model, Horowitz et al. [6] presentedtheir data from simulations plotting the scaling rela-tion
W/L
α
p
−
δ
vs
t/L
z
p
−
y
. Clearly, their
δ
= 1 and
y
= 2 is related to our
ν
(
p
) and
D
(
p
) [see Eq. (11) andEq. (13)]. In Fig. 1(a) we plot
W
2
ν
(
p
)
/D
(
p
) as func-tion of
ν
(
p
)
t
for the that model for diﬀerent values of
p
and
L
= 128. This ﬁgure represent the same as in [6]after coarse-graining. The agreement with the results of our numerical integration, the numerical simulation [6]and the scaling presented in Eq. (17) is excellent. Onthe other hand, for the second model, Chame and Aar˜
a
oReis [8] did not present the result for
W
. They studiedthe crossover from EW to KPZ using an indirect methodbecause of the slow convergence of the discrete model toKPZ behavior. The crossover is well represented in ourFig. 1(b), where we plot the same as in Fig. 1(a) but forthe second model. It is clear the collapse of the curves inthe EW regime. In the intermediate regime the KPZ be-havior appears thus, it is expected that Eq. (18) holds inthat regime. In Fig. 2 we plot ˜
W
2
=
W
2
ν
(
p
)
/
[
L
2
α
D
(
p
)]as function of ˜
t
=
λ
(
p
)
D
(
p
)
/ν
(
p
)
t/L
z
for three diﬀer-ent values of
p
using
z
= 3
/
2. As
p
increases, the KPZbehavior appears earlier, but independent of
p
all thecurves saturate as a KPZ. The agreement with Eq. (18)is excellent in the saturation regime. The departure inthe intermediate regime is due to a slow crossover to theKPZ and to ﬁnite size eﬀects.Finally, notice that the quadratic dependence of thecoeﬃcients of the continuous equation on
p
is indepen-dent of the CGM considered, because it is due to twodiﬀerent rates of deposition given by Eq. (1). This de-pendence is totally generally, as shown from Eq. (1) toEq. (7).In summary, we demonstrate that the coeﬃcient of the continuous equation have quadratic dependance on
p
.This feature is universalfor all the CGM model and is dueto the competition between diﬀerent average time rate.We propose generalized scaling for the model that repro-duce the scaling behavior in each regime. The numericalintegration of the continuous equation are in excellentagreement with the propose scalings and the numericalsimulation of the models.Acknowledgements: We thanks ANPCyT and UNMdP(PICT 2000/1-03-08974) for the ﬁnancial support.
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