Review of Fundamental Problems and Solutions in Fractal Surface Growth

Analytical and Computational

Solutions for Deposition Processesin Disorderly Surface Growth

Boris (Bo Wen) Lin, Christian Lau

University of Southern CaliforniaPhys 316 - Thermodynamics and Statistical Mechanics

Dr. Christoph A. HaselwandterMay 5th, 2015

Abstract

In this paper we present a concise introduction to the general methodologiesof disorderly surface growth. The dynamics of far from equilibrium surfacegrowth systems can be investigated from a scaling approach making heavyuse of fractal geometry. The information related to the complex behaviorof these systems is therefore collapsed in a few quantities, known as criticalexponents, that regulate their scaling properties. An introduction to thefoundations of this theoretical framework is presented through the exam-ple of the Ballistic Deposition model and the relevance of correlation forsaturation phenomena is discussed. Armed with the necessary theoreticalknowledge, both the Random Deposition model and the Random Deposi-tion with Surface Relaxation model are solved exactly, providing the valuesof their critical exponents. To that aim several new methods are presentedand applied to the individual models, especially the devising of continuumstochastic growth equations through symmetry considerations and all therelated techniques adopted to solve exactly for the critical exponents.

Introduction

Far from complying with the wishful rules of conventional geometry, natureabounds with examples of rough morphologies. Materials science offers abewildering array of phenomena characterized by disorderly surface growth,from the everyday experience of fluid flow in porous media and flame frontspropagation - that everybody can recreate by spilling coffee on a tableclothor setting fire to a paper sheet - to microscopic deposition processes, thatplay a fundamental role in today’s technologies like Molecular Beam Epitaxy(MBE). Further examples can be found in the biological sciences, as in the

1

case of bacterial growth in petri dishes or DNA sampling, and even in thestudy of coastlines, cloud formations and botany.

The study of the fundamental properties of the rough shapes arisingfrom such phenomena began to be seriously tackled by mathematicians onlyin the 1960’s, with the pioneering works of Benoit Mandelbrot [1], who laidthe foundations of what later became known as fractal geometry. Advancesin physics have been somewhat slower, hampered by the fact that mostphenomena in disorderly surface growth operate far from the thermodynamicequilibrium (deposition processes, for example, don’t allow energy nor thenumber of particles to be conserved at all, almost by definition), thereforethe methods of equilibrium statistical mechanics and thermodynamics areinadequate and largely unsuccessful. Nowadays, however, disorderly surfacegrowth is a thriving and expanding multidisciplinary field that can count onseveral different and mutually consistent investigation methodologies.

Scaling methods, building on the fractal concept of self-affinity, allowto reduce the apparently unassailable complexity of such systems by iden-tifying a set of simple quantities, called critical exponents, that completelydescribe the dynamics and morphology of the resulting surface. In addition,numerous experimental methods have been developed for the study of in-terface motion and deposition processes in the comfort of our laboratories,with resolutions previously inconceivable, as in the case of the ScanningTunneling Microscope (STM).

Computers are also playing an unprecedented role, being the naturalinstruments for the implementation of discrete models in simulations, thuscomplementing and integrating the insights obtained in traditional experi-mental settings.

Lastly, the theoretical advances in the fields of stochastic differentialequations and renormalization group theory have provided scientists witha wealth of techniques for deriving continuum growth equations displayingthe same properties of most microscopic discrete models and unifying thedisparate phenomena under analysis through the introduction of universalityclasses.

In this research paper we will touch upon all these methods, apart fromthe experimental approach, through the exposition of a topical class of phe-nomena in disorderly surface growth: deposition processes. One of the mostfamous models, that of Ballistic Deposition (BD), will lead us to the presen-tation of basic scaling concepts (Section1), like that of critical exponents. Arapid exposition of elementary notions in fractal geometry follows in Section2, to offer a deeper theoretical understanding of the subject at hand.

Building on the knowledge exposed, in Section 3 we finally present themodel of Random Deposition (RD), for which analytical solutions for thediscrete system and its continuum analog are both provided.

Nevertheless, as a rule of thumb, not always it is possible to find solutionsin closed form in the discrete formulation of the problem, therefore a section

2

on the Random Deposition with Surface Relaxation (RDSR) model willfollow (Section 4), exemplifying a case in which the use of stochastic growthequations, obtained through symmetry considerations, is both necessary andpractical.

A brief conclusion at the end of the paper will help the reader to put theresults previously exposed into the more general context of fractal surfacesand disorderly surface growth, summarizing the lessons that can be learnedfrom this booming field and offering some suggestions on its future direction.

1 Scaling concepts and the Ballistic Deposition

model: a motivating example

The Ballistic Deposition (BD) model was first introduced for the descriptionof colloidal porous aggregates [2, 3] and only successively, with the study ofvapor deposition, the attention of most scientists was caught by the non-trivial properties of the surface it generates [4]. In this paper we will studythe two-dimensional on-lattice version of the model, in which rigid particles,represented by squares, fall perpendicularly from an infinite height alongone of L columns, chosen randomly, on top of an initially flat surface, asshown in Figure 1.

Figure 1

The particles, upon reaching the surface, stick either to the bottom oftheir column or at an height equal to the nearest neighbor they encounteralong their trajectory. The unit time is chosen to correspond to the de-position of L particles and L is taken to represent the system size. Somequantities of interest are introduced, as the mean height

3

h̄(t) =1

L

L!

i=1

h(i, t)

where h(i, t) is the height of the i-th column at time t, and the interfacewidth

w(L, t) =

"

#

#

$

1

L

L!

i=1

(h(i, t) − h̄(i, t))2

which is a measure of the roughness of the surface at time t and systemsize L.

We have implemented the model to study computionally the evolutionof these two quantities. The surface obtained is shown in Figure 2, whichis a snapshot of the dynamic evolution of the system during a real-timesimulation. The code for the implementation in python can be found inAppendix 3.

Figure 2

The behavior in time of the mean height is straightforward: if the de-position rate is constant h(t) increases linearly with time, h(t) ∼ t. Quitemore perplexing is the behavior of w(L, t), which can be obtained compu-tationally, and is as follows

4

w(L, t) ∼

%

tβ if t ≪ tx

wsat if t ≫ tx

Initially w(L, t) increases as a power law of time, with exponent β, knownas the growth exponent. When a given time tx is reached, known as crossoveror saturation time, the width reaches the saturation value wsat. These twolast quantities, tx and wsat, can be shown to depend on the system sizethrough power laws

wsat ∼ Lα

tx ∼ Lz

where the two exponents α and z take the name of roughness and dy-namic exponent respectively.

Figure 3

Due to these functional relations, plotting w(L, t) as a function of timeyields different curves for different values of L, as can be seen in our com-putational results shown in Figure 3, but a simple process can be devised toconflate them:

- first, the width is divided by the saturation value wsat, with the resultthat all the curves reach the same plateau at their respective crossover times,

- secondly, the time is rescaled by t/tx, with the result that all curvessaturate at the same time.

5

Mathematically we obtain the following scaling relation:

w(L, t) = Lαf

&

t

Lz

'

where f(u) is known as scaling function and satisfies the following asymp-totic behavior

f(u) ∼

%

uβ if u ≪ 1

const if u ≫ 1

An additional important result is that the critical exponents α, β andz, are not unrelated. By imposing the continuity of the width function atcrossover time we obtain

tβx = Lα =⇒ z =α

β

The scaling relation we just saw is known as the Family-Vicsek scalingrelation, but of course alternative scaling relations exist. For example thefollowing functional expression for w(L, t) is perfectly viable

w(L, t) ∼ tβg

&

L

tϕ

'

in which the scaling function satisfies

g(u) ∼

%

const if u ≪ 1

uα if u ≫ 1

Furthermore, by imposing the continuity of w(L, t) at tx, the coefficientϕ can be shown to be equal to ϕ = 1/z = β/α.

Clearly the most puzzling characteristic of the BD model is the veryexistence of a saturation value for w(L, t). As can be shown from simpleconsiderations on the functional dependence of wsat on L, saturation mustbe a finite-size effect, in other words it occurs because the system size Lis finite. But this condition is not sufficient, as will be seen in Section 3,while discussing the RD model. Another requirement that is necessary forsaturation is the presence of correlation. More specifically, due to the factthat the height at which a falling particle stops depends on the height of theneighboring columns, there is a correlation length given by

ξ ∼

%

t1/z if t ≪ tx

L if t ≫ tx

that represents the capacity of each column to influence the height of itsneighbors and thus communicate, given sufficient time, with a large portionof the surface. As soon as ξ reaches the value L, corresponding to completecommunication along the surface, saturation kicks in as a finite-size effect.

6

2 Basic notions of fractal geometry

To better understand the significance and nature of the critical exponentswe found in the study of the BD model, we now turn to a brief expositionof the fundamental insights and concepts at the heart of fractal geometry.

Fractals can be surprising and puzzling mathematical objects at a firstglance, but their basic defining properties are few and intuitive, namely self-similarity or self-affinity (depending on the scale invariance they obey) andfractional dimension.

2.1 Self-similarity

We will start by presenting self-similar fractals. The fractal shown in Figure4 is known as the Vicsek’s fractal and is a typical example of a self-similardeterministic fractal. Its construction is performed in numbered steps and,as is evident from a quick inspection, if a part of the structure at the k-thstep is subject to an isotropic scale transformation of the form

(x1, x2) &→ (bx1, bx2)

with a suitable coefficient b, it can be perfectly juxtaposed to the wholestructure at the (k− 1)-th step. This property of invariance under isotropicscale transformations is known as self-similarity. In the limit in which thestructure has evolved for an infinite number of steps a fractal object isobtained.

Figure 4

There are several definitions of the dimension of a fractal: the box count-ing dimension, the information dimension and the correlation dimension areonly some of them. In this paper we will employ the box counting definition,which is in all likelihood the most intuitive one.

Let dE denote the so called embedding dimension, that is the smallestEuclidean dimension of the space in which the object can be embedded. Inour example dE = 2, since the Vicsek fractal is defined on a Euclidean plane.Imagine to cover the object with N(l) dE-dimensional balls of linear size l.The dE-dimensional volume enclosed by this covering is then V (l) = N(l)ldE

7

. For non-fractal objects the number of boxes needed scales as N(l) ∼ l−dE ,as should be expected. Fractals, on the other hand, enjoy the property ofbeing fully covered by N(l) ∼ l−df boxes, where df < dE is called fractaldimension. df can be calculated by the relation

df = liml→0

ln(N(l))

ln(1/l)

In the case of Vicsek’s fractal, for example, we can cover the entire surfaceat step k with 5k squares of linear size 3−k, as a quick inspection of Figure5 will readily confirm. Therefore its fractal dimension is

df = limk→+∞

ln(5k)

ln(3k)=

ln(5)

ln(3)= 1.4649 . . . < 2 = dE

therefore this structure, despite living in a 2-dimensional Euclidean space,is neither a line nor a surface, but an object of fractional dimension, i.e. afractal [5].

2.2 Self-affinity

Not all fractals are self-similar. Indeed in the case of the fractal shown inFigure 5, a portion at the k-th step can be juxtaposed exactly to the wholestructure at the (k − 1)-th step only if it’s scaled different along the twoaxes’ directions. In another words what is needed is an anisotropic scaletransformation of the form:

(x1, x2) &→ (bx1, bαx2)

where the exponent α is termed Holder exponent or self-affine exponent.It must be noticed that for α = 1 we obtain again the case of self-similarity.In general for a fractal function, h(x̄), defined on a d-dimensional domainthe scaling relation is

h(x̄) ∼ b−αh(bx̄)

Invariance under anisotropic transformations is called self-affinity andthe fractals satisfying this invariance are called self-affine. This property isof fundamental importance since most surfaces in disorderly surface growth- and many other application fields of fractal geometry - are self-affine ratherthan self-similar.

8

Figure 5

It is possible to associate a fractal dimension to self-affine fractals inroughly the same way it is done for self-similar ones. The mathematicaldetails and a brief discussion of the limits of this definition are discussed inAppendix 1.

3 The Random Deposition model: an introduc-

tion to stochastic growth equations

The Random Deposition (RD) model is an extremely simple discrete growthmodel for which it is possible to compute exactly the scaling exponents. Itssolution can be found by working in the setting of the discrete model itself,which is entirely a consequence of the rare simplicity of its formulation, or bydevising an appropriate stochastic growth equation describing its dynamicon a continuum spatial domain. This latter choice is usually the only optionleft in the case of more complex and realistic models, as we will see in thecase of the Random Deposition with Surface Relaxation model presented inSection 4.

Not unlike the BD model, also the RD model is a two-dimensional on-lattice model in which particles are represented by square falling perpendic-ularly along one of L columns, chosen randomly, from an infinite height onan initially flat surface. The only difference with the BD model is that eachparticle stops at the height of the column along which it’s moving, with nointeraction with neighboring particles. In other words, the height in columnsis uncorrelated. As we will see this has profound consequences.

9

3.1 Discrete system exact solution

This model has been implemented in python for simulation purposes. Thecode can be found in Appendix 4 and a snapshot of a real-time simulationis shown in Figure 6.

Figure 6

Given the discrete growth formulation, an exact solution can be foundanalytically without too much effort [6]. We start by noticing that each oneof the L columns has a probability p = 1/L of increasing its height by one.Therefore the probability that a column has height h after the deposition ofN particles follows a binomial distribution of the form

p(h,N) =

&

N

h

'

ph(1− p)N−h

By defining time as the mean number of deposited layers, t = N/L, wereadily obtain

< h >= Np =N

L= t

10

< h2 >= Np(1− p) +N2p2

We see therefore that the mean height scales linearly with time, h ∼ t.The interface width is

w(t) =(

Np(1− p) =

)

N

L(1−

1

L) ∼

√t

therefore the growth exponent is β = 1/2. The behavior of the interfacewidth has been computationally studied and results for different values ofthe system size L are shown in Figure 7.

Figure 7

In the absence of correlation, the correlation length, ξ, is identically zero.This means that a necessary condition to have the phenomenon of saturationis missing in the case of the RD model, therefore the roughness exponent αis not defined (for consistency it is said to be infinite, to allow wsat to beinfinite) and the resulting surface is not self-affine.

3.2 Continuum stochastic growth equation and its solution

Associating a continuum stochastic growth equation to the model is oftenthe only way to directly calculate the scaling exponents of the system. Theobjective is to find an appropriate partial differential equation describingthe growth of the function h(x̄, t) denoting the height of a column above

11

position x̄, where x̄ now lies, in general, in a d-dimensional continuum do-main. Successively the equation must be perturbed with the introduction ofa stochastic term accounting for the randomness of the deposition process.

There are various techniques involved in devising the correct equation,but these will be discussed more thoroughly in Section 4. A suitable equationfor the RD model is the following

∂h(x̄, t)

∂t= F + η(x̄, t)

where F is a constant denoting the average number of particles deposit-ing at site x̄ and η(x̄, t) is a random process satisfying

< η(x̄, t) >= 0

< η(x̄, t)η(x̄′, t′) >= 2Dδd(x̄− x̄′)δ(t − t′)

Suitable distributions for η(x̄, t) are for example the Gaussian distri-bution or the Bernoulli distribution. Solving for h(x̄, t) by integrating theequation with respect to time we obtain

h(x̄, t) = Ft+

* t

0dτη(x̄, τ)

From which we can derive the first and second moment

< h(x̄, t) > = < Ft > + <

* t

0dτη(x̄, τ) >

= Ft+

* t

0dτ < η(x̄, τ) >= Ft

where we have used the fact that the expected value of η(x̄, t) is zero.And

< h2(x̄, t) = < F 2t2 > + < 2Ft

* t

0dτη(x̄, τ) > + <

* t

0dτη(x̄, τ)

* t

0dτ ′η(x̄, τ ′) >

= F 2t2 + 2Ft

* t

0dτ < η(x̄, τ) > +

* t

0dτ

* t

0dτ ′ < η(x̄, τ)η(x̄, τ ′) >

= F 2t2 +

* t

0dτ

* t

0dτ ′2Dδ(τ − τ ′)

= F 2t2 + 2Dt

Substituting these results in the expression of w(t) we find

w(t) =√2Dt ∼

√t

12

which implies that the growth exponent is β = 1/2, in agreement withthe exact solution found in Paragraph 3.1.

4 The Random Deposition with Surface Relax-

ation model

The Random Deposition with Surface Relaxation (RDSR) model [7] is atwo-dimensional on-lattice variation of the basic RD. The crucial differenceis that falling particles, upon reaching the top of their column, are allowedto diffuse locally, i.e. they can move to the top of one of the two neighboringcolumns if its height is lower than the one it’s currently occupying, as shownin Figure 8 . This process, known as surface relaxation, produces a smooth-ing effect on the surface and is introduced in order to mimic the occupationof the locally stable lowest-energy configuration of the system. The couplingof the columns’ heights caused by the discrete system dynamics generates anon-zero correlation length and therefore saturation.

Figure 8

A computer simulation of the model has been implemented, see Figure9 and Appendix 4 for the code, indicating that the values of the criticalexponents are

β = 0.24 ± 0.01

α = 0.58 ± 0.02

13

Figure 9

These values cannot be analytically obtained by solving for the discretesystem dynamics, which is intractable. The only hope is to introduce astochastic growth equation that captures, as best as possible, the relevantproperties of the system shown, for example, in the time evolution of thewidth as a function of system size L obtained through our simulations inFigure 10.

14

Figure 10

4.1 The Edwards-Wilkinson equation

Finding a stochastic growth equation that accurately reproduces the criticalexponents of a discrete system is usually not an easy feat. One method toidentify a suitable candidate for such an equation is that of starting from astochastic differential equation of a very general form, like

∂h(x̄, t)

∂t= G(h, x̄, t) + η(x̄, t)

where G(h, x̄, t) is a generic function and η(x̄, t) obeys the same proper-ties listed in Paragraph 3.2, and successively ruling out all the terms thatare incompatible with the system’s symmetries [8]. For the case at hand, wenotice that the RDSR model displays several different symmetries:

- Invariance under translations in time. The growth equation cannotdepend on the arbitrary definition of time zero. Therefore we deduce thatunder t &→ t + d the equation is left unchanged only if we rule out explicittime dependence of G.

- Translation invariance along the growth direction. Analogously to theprevious case, the equation cannot depend on the level of zero height: underh &→ h + d the equation is left unchanged only if we rule out an explicitdependence of G from h.

- Translation in the direction perpendicular to the growth direction. Theequation must not depend on the actual value of x̄ on which it’s computed.

15

Imposing that invariance under x̄ &→ x̄+ d̄ is respected we see that G cannotexplicitly depend on x̄.

- Rotation and inversion symmetry about the growth direction n. Thisinvariance allows us to rule out all odd order derivatives of h in the coordi-nates.

- Up/Down symmetry for h. Assuming that negative and positive devia-tions from the mean height are equally likely, we must rule out even powersof h and its gradient.

The terms left are all derivatives of h of even order and all their com-binations with even powers of the gradient, so the equation assumes theform

∂h(x̄, t)

∂t= (∇2h)+ . . .+(∇2nh)+(∇2h)(∇h)2+ . . .+(∇2kh)(∇h)2j +η(x̄, t)

In studying the system we are interested only in its scaling properties,that is we are interested in the long-time (t → +∞) long-distance (x̄ → +∞)behavior of the equation. In this limit, known as hydrodynamic limit, severalhigher order terms can be neglected, as is shown in Appendix 2, and thefinal form of the stochastic growth equation is

∂h(x̄, t)

∂t= v∇2h(x̄, t) + η(x̄, t)

where v is the surface tension. This equation is known as the Edwards-Wilkinson equation [9]. There are two ways of investigating the scalingproperties of this stochastic equation: the scaling approach, which is a simpleexploitation of the hydrodynamic limit under self-affine transformations,and the complete analytical solution, which provides an exact solution byintegrating the equation with respect to time and solving for h(x̄, t) in theFourier space.

4.2 Scaling approach

We start with the assumption that the function h(x̄, t) must be self-affine,therefore we impose invariance under the following transformations

x &→ bx

h &→ bαh

t &→ bzt

The equation then becomes

16

bα−z ∂h(x̄, t)

∂t= vbα−2∇2h(x̄, t) + b−d/2−z/2η(x̄, t)

and, after rearranging the terms, we obtain

∂h(x̄, t)

∂t= vbz−2∇2h(x̄, t) + b−d/2+z/2−αη(x̄, t)

We now notice that in the hydrodynamic limit there should be no depen-dence on b, therefore we obtain two relations on the three critical exponentsby imposing that the powers of b are identically zero. The third knownrelation z = β/α provides the missing information to solve for the threeexponents exactly and we thus readily obtain

α =2− d

2

β =2− d

4

z = 2

4.3 Exact solution

Exploiting the linearity of the Edwards-Wilkinson equation we can obtainan exact solution by Fourier transforming the function h(x̄, t). We find

h(x̄, t) =η(k̄,ω)

vk2 − iω

where η(k̄,ω) is the Fourier transform of the noise process. We can thusexploit the known form of the correlation function of the noise to obtain thecorrelation function of the Fourier transform of h(x̄, t)

< h(k̄,ω)h(k̄′,ω′) >=η(k̄,ω)η(k̄′,ω′)

(vk2 − iω)(vk′2 − iω′)

Finally, by operating the inverse Fourier transform and after some alge-braic steps, we obtain the correlation function for h(x̄, t)

< h(x̄, t)h(x̄′, t′) >=D

2v|x̄− x̄′|2−df

&

v|t− t′||x̄− x̄′|2

'

where the function f(u) has scaling properties

f(u) ∼

%

const for u → +∞u(2−d)/2 for u → 0

17

By comparing this function with the general Family-Vicsek scaling re-lation we obtain the same values for the critical exponents that we foundbehavior, in perfect agreement with the scaling approach.

It is also instructive to emphasize how these exponents agree with theexperimental results. This seems to suggest that both the actual RDSRmodel and its continuum growth version display identical scaling behaviors,and thus belong to the same universality class.

5 Conclusion

We have briefly introduced the reader to the methodologies of disorderlysurface growth, showing how to describe the complex behavior of these sys-tems in terms of a few key quantities, the critical exponents, that regulatetheir scaling properties. We have also expounded upon their mathemat-ical significance in the context of fractal geometry. The main models ofBallistic Deposition, Random Deposition and Random Deposition with Sur-face Relaxation have been introduced and solved, whenever possible, eitherwithin the discrete formulation or in the more general context of continuumstochastic growth modeling. We have also discussed the role of correlationin the emergence of saturation phenomena and the importance of symmetryconsiderations in building appropriate stochastic growth equations lying inthe same universality class as the discrete systems analyzed.

From this overview of the field of disorderly surface growth it must beclear that the mathematical challenges posed by the complex behavior of sur-faces are of a unique and unprecedented nature. We have just barely beganto unravel such fundamental principles as self-affinity, universality classesand stochastic modeling. Obviously several central issues remain open, firstof all the ubiquity of fractal properties in nature, which is in itself puzzling.It seems natural to ask whether there are limits to the applicability of thisnew kind of mathematics. In other words, how universal are universalityclasses? The fact that rough surface can be obtained in artificial ways, likein the study of long-range correlations in DNA samples [10] or in the studyof financial time series, seems to indicate that the only limit is our imagina-tion and creativity and this motivates most of the multidisciplinary natureof these studies.

There are principles so deeply hidden in nature that unraveling themmust reward the investigator with insights as precious as those lying at thebottom of every revolution in science. That the study of fractal surfaces isa step in the right direction seems to be beyond doubt. Whether the theo-retical framework being currently built will allow us to reach revolutionarylevels of understanding in the study of nature is, however, still an open ques-tion and, as in the case of any bet with nature, the outcome cannot be safelypredicted without first reaching the end of the investigative trail.

18

Appendix 1

Suppose to have a real function defined on a d-dimensional domain, forexample [0, 1]d. We divide the domain in N boxes of linear size l = 1/N1/d.In the codomain every box is expanded by a factor ∆ ∼ l because of theself-affinity of the function, with a roughness exponent α. To cover thisvolume we need ∆/l ∼ lα−1 more box for each box in the domain. Fromthis we get that the total number of boxes is

N(l) ∼ Nlα−1 = l−dlα−1 = lα−1−d

therefore the fractal dimension is df = d+1−α. Clearly assuming such arelation is always true implies that the fractal is self-similar, not self-affine.This demonstration in fact is true only in the limit of small l. For largeenough l, in fact, ∆ ∼ 1/l and the fractal dimension becomes df = d.

19

Appendix 2

As will be remembered, in the solution of the Edwards-Wilkinson equationtime and height must be scaled by the dynamic and roughness exponentsrespectively:

h &→ bαh

t &→ bzt

In the asymptotic limit we are interested in the behavior of functionsfor t → +∞ and x → +∞, which also implies b → +∞. Keeping this inmind we will show that the second order derivative of h(x, t) in t goes tozero faster than the first order one, thus justifying the absence of this term(known as inertial term) from the Edwards-Wilkinson equation.

Under the self-affine transformation the partial derivatives change in thefollowing way

∂h

∂t&→ bα−z ∂h

∂t

∂2h

∂t2&→ bα−2z ∂h

∂t

In the asymptotic limit the second order derivative approaches zero fasterthan the first order one by a factor b−z, therefore for b → +∞ it is safe toignore the inertial term.

20

Appendix 3

Standard modules are imported, including the submodulematplotlib.animationproviding real-time frame-by-frame animations using the basic framework ofthe matplotlib module.

import glob

import numpy as np

import time

from pylab import cm

from matplotlib import pyplot as plt

from matplotlib import colors

import matplotlib.animation as animation

Global variables for the system are initialized.

#initialize variables

L = 60

t = 1000

h_ceil = 100

h_max = 0

X = np.zeros((h_ceil,L), dtype=np.int)

H = np.zeros((L,),dtype=np.int)

The algorithm is implemented in the updating function of the simulationframes. The function assesses nearest-neighbors and checks for the possibil-ity of side-sticking.

fig = plt.figure()

ax1 = fig.add_subplot(1,1,1)

def updatefig(*args):

global x,H,X,h_max,h_ceil

x = np.random.random_integers(0,L-1)

if x > 0 and x < L-1:

nbhd = [H[x-1],H[x],H[x+1]]

elif x == L-1:

nbhd = [H[x-1],H[x]]

elif x == 0:

nbhd = [H[x],H[x+1]]

local_max = max(nbhd)

if H[x] == local_max:

H[x] = H[x] + 1

else:

H[x] = local_max

h_max = max(H)

21

X[h_ceil-H[x],x] = 1

return plt.imshow(X),

Finally the animation is called in the main body of the program andframes are shown in real-time through the matplotlib viewer.

ani = animation.FuncAnimation(fig, updatefig, interval=10, blit=True)

plt.show()

22

Appendix 4

Standard libraries are imported, including the submodulematplotlib.animation,which provides real-time simulations frame by frame.

import glob

import numpy as np

import time





Global variables for the system are introduced. Notice that the heightis stored as an array, indexed by the position of each column.

L = 50

t = 1000

h_ceil = 100

h_max = 0



Finally the animation is implemented by defining the updating functionwhich contains the core of the system time evolution algorithm.

fig = plt.figure()





H[x] = H[x] + 1

h_max = max(H)

X[h_ceil-H[x],x] = 1



plt.show()

23

Appendix 5

Modules are imported

import glob

import numpy as np

import time





Global variables for the simulation are initialized.

L = 30

t = 1000

h_ceil = 80

h_max = 0

set_epsilon = 5



fig = plt.figure()


As usual, the basic algorithm is implemented in the updating functionfor the animation. Local minima are first searched for, both on the left andon the right of the falling particle. Their values are then compared and theparticle is moved to the column with local lowest-energy level. In case thenearest neighbor columns have same height, one of them is picked randomly.




l_epsilon = set_epsilon

r_epsilon = set_epsilon

#change epsilon to fit within length constraint

if (L-1-x) <= r_epsilon:

r_epsilon = L-x-1

if x <= l_epsilon:

l_epsilon = x

r_local_min = H[x]

r_local_min_x = x

l_local_min = H[x]

l_local_min_x = x

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local_min_x = x

for i in range(x-1,x-l_epsilon-1,-1):

if H[i] < l_local_min:

l_local_min = H[i]

l_local_min_x = i

elif H[i] > l_local_min:

break

for j in range(x+1,x+r_epsilon+1):

if H[j] < r_local_min:

r_local_min = H[j]

r_local_min_x = j

elif H[j] > r_local_min:

break

toss = np.random.random_integers(0,1)

if l_local_min <= r_local_min:

local_min_x = l_local_min_x

elif l_local_min == r_local_min:

if toss == 0:

local_min_x = r_local_min_x

elif toss == 1:

local_min_x = l_local_min_x

else:

local_min_x = r_local_min_x

H[local_min_x] = H[local_min_x] + 1

h_max = max(H)

X[h_ceil-H[local_min_x],local_min_x] = 1


Finally the animation is called in the main body program and the real-time evolution is printed on the screen, frame by frame.


plt.show()

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Bibliography

[1] B. Mandelbrot, The fractalist: Memoir of a Scientific Maverick, (Vin-tage, Reprint edition 2014).

[2] M. J. Vold, A numerical approach to the problem of sediment volume,J. Coll. Sci. 14, 168-174 (1959)

[3] M. J. Vold, Sediment volume and structure in dispersions of anisotropicparticles, J. Phys. Chem. 63, 1608-1612 (1959)

[4] R. Baiod, D. Kessler, P. Ramanlal, L. Sander and R. Savit, Dynamicalscaling of the surface of finite-density ballistic aggregation, Phys. Rev. A38, 3672-3678 (1988)

[5] B. Mandelbrot, The Fractal Geometry of Nature, (Freeman, San Fran-cisco, 1982).

[6] J. D. Weeks and G. D. Gilmer, Dynamics of Crystal Growth, Adv.Chem. Phys. 40, 157-228 (1979)

[7] F. Family, Scaling of rough surfaces: effects of surface diffusion, J.Phys. A 19, L441-L446 (1986)

[8] T. Hwa and M. Kadar, Avalanches, hydrodynamics and dischargeevents in models of sandpiles, Phys. Rev. A 45, 7002-7001 (1992)

[9] S. F. Edwards and D. R. Wilkinson, The surface statistics of a gran-ular aggregate, Proc. R. Soc. London A 381, 17-31 (1982)

[10] S. V. Buldyrev, A.-L. Goldberger, S. Havlin, C.-K. Peng and H.E. Stanley, Fractals in biology and medicine: from DNA to the hearbeat,in Fractals in Science, edited by A. Bunde and S. Havlin (Springer-Verlag,Berlin, 1994)

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Documents

Review of Fundamental Problems and Solutions in Fractal Surface Growth