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J. Phys. A: Math. Gen. 27 (1994) L81-L85. Printed in the UK

LETTER TO THE EDITOR

Scaling behaviour in extended coalescingmodel

Takashi NagataniCollege of Engineering, Shizuoka University, Harnamatsu 432, Japan

Received 22 November 1993

Abstract A simple aggregation model on one-dimensional lattice is presented to study thescaling behaviour. The model is an extended version of the coalescing random walker model totake into account the dependence of transition probability upon mass _ of particle. A particlemoves ahead one step with transition probability T and is stopped with probability 1 — T whereT = a + bs~a (or > 0, a, b "£ 0 and a + b < 1). It is shown that the mean mass (_) of particlescales as (_) « tp where t is time. The scaling relation p = 1/(1 + a) is satisfied for a = 0.0.For a > 0, the scaling relation p = max[0.5, 1/(1 + or)] is satisfied. We discuss th_ relationbetween our model and the extended kpz equation.

Recently, there has been increasing interest in the scaling structures of growth processessuch as the cluster-cluster aggregation (CCA) model, the rough surface model, the diffusion-limited aggregation (DLA) model, and the river network model [1—3]. Considerable workhas already been performed on the scaling properties of non-equilibrium fractal growth.Most attempts to develop a theoretical understanding of fractal growth focused directlyor indirectly on the screening process. Even for very simple models such as DLA thishas proven to be a formidable challenge. For the case of a growing surface, someanalytical attempts have succeeded in deriving the scaling exponents. The main approachfor describing the growth of surfaces and interfaces is based on coarse-grained Langevin-type equation [3]. Kardar, Parisi and Zhang (KPZ) [4] have presented a nonlinear interfaceequation. In two dimensions, the scaling of the interface width w(L, t) on length scale L attime t for the KPZ equation have been shown from a renormalization-group analysis to bew(L, t) = Ll/2f(t/L3/2) where f(x) « xi/3 for x «; 1 and f(x) « constant for x » 1[].

The scaling exponent of the cluster-size distribution in a simple aggregation model withinjection has analytically been derived by Takayasu [5]. However, the governing equationsuch as the KPZ equation has been unknown until now for aggregation models. The scalingexponents for aggregation models have rarely been derived from analytical method.

In this letter, we present a simple aggregation model on one-dimensional lattice to bedescribed by the diffusion equation with generalized nonlinearity. We study the scalingbehaviour of the aggregation process. In order, to obtain the lattice model, for the nonlineardiffusion equation, we extend the coalescing random walker model to take into accountthe dependence of transition probability upon mass s of a particle. When two particlescollide with each other, they coalesce. Each cluster (or particle) is infinitesimal but hasa finite mass. We show that the mean mass (s) of a particle scales as (s) «- t&, thescaling relation ft = 1/(1 -f- a) is satisfied for a = 0.0, and for a > 0, the scaling relation#6 = max[0.5, 1/(1 -fa)] is satisfied. In the limit of b = 0 and a = 1/2, this model reduces

0305-4470/94/030081+05$07.50 © 1994 IOP Publishing Ltd L81

Letter to the Editor

Ftau-1. The log-log plot of the mean mass (s) against Figure 2. The plot of the scaling exponent B againsttiSTror - -0 3.0 5.0.9.1.5 and 2.0 in mo case of a in the case of - = 0. The solid curve represents the

_ 0 r e l a t i o n B = 1 / ( 1 + o ) .

to the well known irreversible one-species coagulation model [6]. This model is closelyrelated to the diffusion-limited reaction [7]. Also, the aggregation model with injection inthe limit is consistent with the Scheidegger's river-network model [8].

Our aggregation model is defined on a one-dimensional lattice of L sites with periodicboundary conditions. Each site is occupied by one particle or it is empty. We extendthe coalescing random walker model to take into account the mass-dependent transitionprobability T. In our model, transition probability T depends only on the cluster mass s.For an arbitrary configuration, one update of the system is performed in parallel for allparticles. A particle moves ahead one step with transition probability T = a + bs andis stopped with probability 1 - T. When two particles with masses ., and s2 collide witheach other, they coalesce and the resultant particle has the mass ., + s2. Each cluster (orparticle) is infinitesimal but a finite mass. Let.(«, t) be the mass of the particle on the siteat the l time step. The aggregation can be represented by the stochastic equation for .(/. t)

_•(,-, t + i) _. [i - _.(., r)]_(i, t) +1-0' - 1.0-O -LO (l)where w(i t) is a stochastic variable which is equal to 1 with mass-dependent probabilityT = a + bs" (- > 0, a, b _J 0 and a + b ^ 1) when the particle on the /th site jumps tothe (I + nth site and which is equal to 0 with probability 1 - T when the particle on the /thsite does not jump to the (/ + l)th site. In the limit of a = 1/2 and b = 0, the aggregateis consistent with the ordinary coalescing random walker model. We restrict ourselvcstothe case of a > 0 since the mean cluster size (.) does not scale for the case of or < 0. Theaggregation process with a < 0 cannot be simulated by our aggregation model.

We perform simulations of our aggregation model according to (1) for the system sizeL = 105. Each run is calculated until 105 time steps. We study the scaling behaviour ofthe cluster-mass distribution. We define the mean mass {_) of particles as

(2)eo / / oo

1 = 1 ' 1 = 1

_n_

where n, is the cluster-mass distribution with mass .. Figure 1 shows the log-log plot ofthe mean mass {.) against time t for a = 0.3.0.5,0.9, 1.5 and 2.0 where a = 0. The meanmass (.) scales as

Letter to the Editor L83

Figure 2 shows the plot of the scaling exponent /3 against a. The solid curve representsthe relation

P= 1/(1 + _). (4)The data agree with the scaling relation (4).

The scaling relation (4) is derived from a simple scaling argument as follows. Theincrement of the typical mass by the coalescence per unit time is proportional to the transitionprobability. Then, the following equation is satisfied

d„/dr (5)

The particle mass s scales as . «« ''/(l+o). We calculate the case of a = b = 1/2. Figure 3shows the log-log plot of the mean mass (s) against time / for a = 0.3,0.5, 1.0, 1.5 and2.0. Figure 4 shows the plot of the scaling exponent f) against a. For 0 < a < 1. thescaling exponent /? is given by (4). For a ~? 1, the exponent f) becomes the constant value0.5. The scaling exponent /) is represented by

/} = max[l/2. 1/(1+or)]. (6)

The value 0 = 1/2 of the scaling exponent ft agrees with that obtained by the coalescingrandom walker model. When the power a is larger than 1, the aggregation process isgoverned by the ordinary coalescing random walk.

a=0.3I0> — / ' O S

/C/' 1 0<s>

_ * _ _ ^ " "102

^ ^10

1 1 1 1

\ H i- 11=0.5

. 1 , 1 i , , 1

Figure 3. The log-log plot of the mean mass (t) againsttime ( for a = 0.3.0.5. 1.0. 1.5 and 2.0 in the case ofo=„ = 1/2.

Figure 4. The plot of the scaling exponent /) against ain the case of a = h = 1/2 The solid curve representslhe relation B = max[ I/2, I/(1 + cr)).

We calculate the cluster-mass distribution n,> The cumulative mass distribution iV,is defined as iV_ = __.__,'**'• Pig-te 5 shows the semi-log plot 0. the cumulative massdistribution /V, against the mass s for / = I03. 3 x 103. IO4 and 3 x 104 where a = 0.5.In order to study the scaling form of the cumulative mass distribution, we plot lhe rescaledcumulative distribution against the rescaled mass. Figure 6 shows the semi-log plot of therescaled cumulative distribution tu652N, against (he rescaled mass r"°-6525 for the data infigure 5. The data collapses on a curve. We find that the cumulative mass distribution isdescribed in terms of

«V,«(.p/Cr/M) (7)

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