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BALLISTIC DEPOSITION: GLOBAL SCALING AND LOCAL TIME SERIES
Arne Schwettmann, B.S. equivalent
Problem in Lieu of Thesis Prepared for the Degree of
MASTER OF SCIENCE
UNIVERSITY OF NORTH TEXAS
December 2003
APPROVED:
Paolo Grigolini, Major Professor
Floyd D. McDaniel, Committee Member
and Graduate Advisor
William D. Deering, Committee Member
Floyd D. McDaniel, Chair of the Department of
Physics
Sandra L. Terrell, Interim Dean of the Robert B. Toulouse
School of Graduate Studies
Schwettmann, Arne, Ballistic deposition: global scaling and local time series. Master
of Science (Physics), December 2003, 110 pp., 5 tables, 39 figures, references, 38 titles.
Complexity can emerge from extremely simple rules. A paradigmatic example of this
is the model of ballistic deposition (BD), a simple model of sedimentary rock growth. In
two separate Problem-in-Lieu-of Thesis studies, BD was investigated numerically in (1+1)-
D on a lattice. Both studies are combined in this document. For problem I, the global
interface roughening (IR) process was studied in terms of effective scaling exponents for a
generalized BD model. The model used incorporates a tunable parameter B to change the
cooperation between aggregating particles. Scaling was found to depart increasingly from
the predictions of Kardar-Parisi-Zhang theory both with decreasing system sizes and with
increasing cooperation. For problem II, the local single column evolution during BD rock
growth was studied via statistical analysis of time series. Connections were found between
single column time series properties and the global IR process.
ACKNOWLEDGEMENTS
I would like to thank Prof. Dr. Paolo Grigolini for the many enthusiastic conjectures and for
a great insight into the science of complexity and the philosophy behind it. Special thanks
go to Dr. Massimiliano Ignaccolo and Roberto Failla and the rest of the ”Pisa group” at
UNT for their hospitality and discussions.
ii
CONTENTS
ACKNOWLEDGEMENTS ii
I SCALING IN A GENERALIZED BALLISTIC DEPOSITION MODEL
1
1 INTRODUCTION 3
2 THEORY 5
2.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Scaling in Deposition Growth . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Interface Width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 Correlation Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.5 Height-Difference Correlation Function . . . . . . . . . . . . . . . . . . . . . 18
2.6 Self-Affinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.7 Kardar, Parisi and Zhang Equation . . . . . . . . . . . . . . . . . . . . . . . 21
3 METHODS 26
3.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.2 Effective Exponent β′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.3 Effective Exponent α′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4 RESULTS 31
4.1 Raw Data, Qualitative Behavior . . . . . . . . . . . . . . . . . . . . . . . . 31
4.2 Effective Exponent β′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.3 Effective Exponent α′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
iii
II BALLISTIC DEPOSITION: A TIME SERIES APPROACH 43
1 INTRODUCTION 45
2 CONCEPTS 47
2.1 Ballistic Deposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.2 Random Deposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.3 Waiting Time Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.4 Diffusion Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.5 Diffusion Entropy and Events . . . . . . . . . . . . . . . . . . . . . . . . . . 60
2.6 Many-columns Diffusion Entropy . . . . . . . . . . . . . . . . . . . . . . . . 61
2.7 Two Types of Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3 STICKING EVENTS 63
3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.2 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.4 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4 RECURRENCE EVENTS 70
4.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.2 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.4 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5 HEIGHT FLUCTUATION INCREMENTS 86
5.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.2 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.4 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
iv
6 CONCLUSIONS 94
A PROGRAMS USED FOR PROBLEM I 96
B PROGRAMS USED FOR PROBLEM II 102
REFERENCE LIST 108
v
LIST OF TABLES
4.1 Effective Exponent β′(L) for different B, calculated from linear fits in the w
double-log plots. Empty fields denote cases where no growth regime could
be defined (see Fig. 4.5). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.2 Comparison of this study with results of Ref. [2]: β′ for B = 0. . . . . . . . 37
4.3 Saturation width w∞(L) for different B, calculated from averaging over sat-
uration regime. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.4 Comparison of this study with results of Ref. [2]: w∞ for B = 0. . . . . . . 39
4.5 Effective exponent α′(L) for different B, calculated from linear fits in the w
double-logarithmic plots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
vi
LIST OF FIGURES
PROBLEM-IN-LIEU-OF THESIS I
2.1 Ballistic deposition (BD): Starting with an initially flat substrate, square
particles are dropped from random positions above the substrate one at a
time and stick upon first contact. A′,B′,C ′: sticking positions of particles
A,B,C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Model B for the special case B=1. A’,B’,C’: sticking positions of particles
A,B,C. Note how particle A sticks over a distance S = B + 1. Here it sticks
over a distance of two columns as measured from center to center. . . . . . 7
2.3 Extracts of size 250*250 from bulks grown with model B for different values
of B. The substrate length is L = 250. The bottom line corresponds to a
height of 106. Black dots: particles. Top left: B=0 (BD), top right: B=2,
bottom left: B=4, bottom right B=9. . . . . . . . . . . . . . . . . . . . . . 8
2.4 Evolution of the height profile for BD. Starting at t=0, h=0 (bottom line)
up to a height h=474, the height profile was saved after each period (T=30).
System size L=474. Left: h(x,t) corresponds to the black dots. Right: h(x,t)
corresponds to the border between two colors. . . . . . . . . . . . . . . . . . 9
2.5 Average height vs. time for BD. Numerical simulation data. There is linear
behavior (main) apart from an initial regime where few layers have been
dropped (inset) and the vacancy density has not yet reached a steady state
value. Here L = 1024 is plotted, but curves for other L match this one. . . 10
2.6 Interface width wsingle vs. time for a single system for BD. Numerical sim-
ulation data. Note the double-logarithmic scale. The system size L = 1024
was used. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.7 Interface width w vs. time for an ensemble average for BD. Numerical simu-
lation data. Note the double-logarithmic scale. Different system sizes L were
simulated and averages were taken over 500 to 1000 realizations. . . . . . . 12
vii
2.8 Ideal scaling of the interface width w as described by Eqns. (2.5) to (2.9).
To be compared with actual simulation data as seen in Fig. 2.7 . . . . . . . 14
2.9 Attempted data collapse of BD data from simulations according to the Family-
Vicsek scaling relation. The same data were used as in Fig. 2.7. . . . . . . . 15
2.10 Height-difference correlation function C(l, t) vs. l for an ensemble of 10
systems. L = 50000, t = 7000. αloc ≈ 0.45 ≈ α. The arrow indicates the
crossover length lx ≈ ξ|| < L. Numerical simulation data. . . . . . . . . . . 18
4.1 Average height vs. time for different values of B. Apart from the initial few
layers (inset), the rocks grow linear in time (main). . . . . . . . . . . . . . . 31
4.2 Raw Data example 1: Interface width w vs. havg, L = 2048, different B. . . 32
4.3 Raw Data example 2: Interface width w vs. havg, B = 4, different L. . . . . 33
4.4 Interface width w vs. time for a large system L = 106. Slow crossover to
KPZ scaling with time: the slope in the growth regime approaches the KPZ
value of 1/3 more slowly with larger B. . . . . . . . . . . . . . . . . . . . . . 34
4.5 w(L) for L=1024 and L=128, B=4: No growth regimes could be defined for
some ensembles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.6 Effective exponent β′ vs. L. Note the logarithmic x-axis and the reciprocal
L, both chosen to exhibit asymptotics towards large L. . . . . . . . . . . . . 38
4.7 w∞ vs. L in a double-log plot. Errors are smaller than the pointsize. KPZ
theory predicts a slope of α = 0.5. Note how the B = 19 data saturate at
almost the same value for all considered L. . . . . . . . . . . . . . . . . . . 40
4.8 Effective exponent α′ vs. 1/L. Note the logarithmic x-axis and the reciprocal
L, both chosen to exhibit asymptotics towards large L. . . . . . . . . . . . . 41
PROBLEM-IN-LIEU-OF THESIS II
2.1 Ballistic deposition (BD): Starting with an initially flat substrate, square
particles are dropped from random positions above the substrate one at a
time and stick upon first contact. A’,B’,C’: sticking positions of particles
A,B,C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
viii
2.2 Anomalous diffusion in BD growth: Interface width w vs. time for an ensem-
ble average for BD. Numerical simulation data. Note the double-logarithmic
scale, the anomalous power law growth with w ∼ t1/3 and the times when
saturation sets in tsat ∼ Lz with z = 1.5. . . . . . . . . . . . . . . . . . . . . 49
2.3 Scaling of the interface width w as described in the text, for a single realiza-
tion of BD growth. Numerical simulation data. . . . . . . . . . . . . . . . . 50
2.4 Ideal scaling of the interface width w as described in the text. Drawn ”by
hand”. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.5 Diffusion entropy: A time series of N values is covered with N− l+1 overlap-
ping windows of size l each. Note that it is not limited to integer numbers.
The window size is interpreted as diffusion time t′, while the sum of numbers
in each window give the position of the corresponding walker at time t′. All
the walkers together form a diffusion process. The diffusion entropy is the
Shannon entropy of the resulting pdf. . . . . . . . . . . . . . . . . . . . . . 57
2.6 Many-column diffusion entropy (MCDE): Each column in BD yields one time
series of events. Each of those time series is associated with just one walker.
Starting from the bottom, each walker’s position is the sum of numbers in
his window, up to a maximum point l. l is interpreted as ”diffusion time” t′.
Here t′ corresponds to the real time t = N/L. All the walkers together form
a diffusion process. The Shannon entropy of the resulting pdf is the MCDE. 61
3.1 Waiting time distribution of sticking events. The distribution is exponen-
tial, with no dependence on the regime of analysis or on L. The events are
Poissonian. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.2 Many-columns diffusion entropy of sticking events. The ordinary Brownian
motion scaling of δ = 0.5 is found. Parameters: L = 105, t = 0..105 . . . . . 66
4.1 Time series of height fluctuations Yt. Long term behavior, L=1000. . . . . . 72
4.2 Time series of height fluctuations Yt. Short term behavior, L=1000. . . . . 73
ix
4.3 Waiting time distribution of recurrence events. Inverse power law decay with
no dependence on regime of analysis. Parameters: L = 3000, t = 100..8000;
L = 3000, t = 50000..58000 . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.4 Waiting time distribution of recurrence events for different L. Parameters:
tmin = 100..105. Graphs for L < 8192 are shifted down for clarity. . . . . . . 75
4.5 Same as Fig. 4.4 with logarithmic binning and no downwards shift of curves. 76
4.6 Waiting time distribution of recurrence events. No dependence of µ on the
regime of analysis is detectable. Parameters: L = 100000, tmin = 100,
tmax = 100000; L = 3000, tmin = 5000, tmax = 1000000; logarithmic binning. 77
4.7 Waiting time distribution of recurrence events. Comparison between random
deposition and ballistic deposition. Parameters: L=100000, t = 100..105 . . 78
4.8 Many-columns diffusion entropy of recurrence events. Comparison between
random and ballistic deposition. Parameters: L = 100000, t = 0..105. . . . . 79
5.1 Time series of height fluctuation increments for BD. Note the asymmetric
nature. L=3000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.2 Time series of height fluctuation increments as points. Larger time range than
fig. 5.1. Note the non-uniform distribution of values: They are distributed
in small strips around values 0.86 + n, which is explained in section 5.4. . . 88
5.3 For comparison with Fig. 5.1: Time series of height fluctuation increments
for RD. L=1000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.4 Single-column diffusion entropy of the time series Yt for BD. A time series of
tmax = 106 numbers was used. . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.5 Single-column diffusion entropy of the time series Yt for RD. A time series
of tmax = 106 numbers was used. System size L = 1024. Note how even
random deposition seems to saturate at large t′. This is a numerical artifact
due to a loss of statistics (see text). . . . . . . . . . . . . . . . . . . . . . . . 91
x
PROBLEM-IN-LIEU-OF THESIS I
SCALING IN A GENERALIZED BALLISTIC DEPOSITION MODEL
1
SUMMARY
A numerical model of sedimentary rock growth (model B) was studied with
respect to scaling of the interface width. Model B is a generalization of ballistic
deposition (BD). In both models particles are dropped onto a substrate one at
a time. While in BD particles stick to nearest neighbors only, in model B par-
ticles are allowed to stick over a horizontal distance S := B + 1, with integer
parameter B ≥ 0. Numerical simulations were carried out in (1+1)-D for differ-
ent substrate sizes L. Effective scaling exponents α′(L,B) (roughness exponent)
and β′(L,B) (growth exponent) are shown to depend strongly on L and B and
to be always smaller than the universal exponents (α = 0.5, β = 1/3) predicted
by the celebrated Kardar-Parisi-Zhang (KPZ) equation of random growth of
surfaces. These discrepancies increase both with larger B and with smaller L.
Nevertheless, for large L and large t the exponents of model B approach KPZ
values asymptotically.
2
CHAPTER 1
INTRODUCTION
Kinetic interface roughening processes can be found everywhere in nature. They are pro-
cesses where an interface evolves with time, changing its roughness while being subjected
to random noise. These processes range from burning paper fronts to paper wetting pro-
cesses, from thin-film growth to the growth of sedimentary rock formations, from burning
cigarettes to the growth of coffee spills. Many of these processes are believed to share the
property of at least approximate scale invariance in both space and time. Scale invariance
is indicated by self-affine or self-similar interface profiles. Self-affinity or self-similarity in
turn is reflected in a unique power law behavior in space and time of functions such as the
interface width or other respective correlation functions over the interface. The exponents
of those power laws - the scaling exponents - have been found to be largely independent of
the microscopic rules governing a particular interface evolution process. Rather, the expo-
nents seem to only depend on a few crucial factors, namely the symmetries of the respective
process. This makes them universal quantities. The concept of universality classes is often
used to describe whole sets of microscopically different models or phenomena that share the
same exponents.
In the spirit of universality, for example, the exponents found for slow combustion of
paper seem not to depend on the actual paper used [1] and are indeed the same as those of
the ballistic deposition model of sedimentary growth [2]. Similarly, exponents from paper
wetting processes seem not to depend on the actual liquid used [3]. And perhaps the growth
of certain thin films shares the same universal exponents with a model for the growth of
bacterial colonies [4] and the ballistic deposition model.
The model of ballistic deposition (BD) was first proposed by Vold[5] in 1959 as a model
for sedimentation. It is a stochastic growth process governed by a rather simple cellular
automaton rule. This simple rule however leads to quite a complex behavior in terms of
the interfacial dynamics, the dynamics of the borderline between rock and the surrounding
3
medium. In particular, the roughness of the interface first increases as a power law and
then saturates at a certain value which is system-size dependent.
In BD, particles are dropped from random positions vertically on an initially flat sub-
strate and stick upon first contact. Together with other models of noisy particle aggregation
such as, e.g., the Eden model for biotic growth [4], BD belongs to the so-called far-from-
equilibrium growth processes. Here ”far from equilibrium” means that particles are not
allowed a relaxation to lowest energy states during the whole growth process.
Now, what happens with respect to scaling properties if the correlations in BD are
increased? This is the question I address with this study. To increase correlations in BD
a generalized model is investigated. In this generalized model, subsequently called model
B, particles are allowed to stick to other particles over horizontal distances > 1. I present
numerical calculations of the effective scaling exponents, the roughness exponent α′(L)
and the growth exponent β′(L) of model B for different values of B. The effective scaling
exponents’ convergence behavior for large system sizes is exhibited in appropriate plots. A
remarkable dependence of the effective exponents on B and L is found. All exponents are
found to be significantly lower than the analytical values predicted by Kardar-Parisi-Zhang
(KPZ) theory. However, asymptotic behavior strongly suggests a slow convergence to KPZ
universal exponents α = 0.5, β = 1/3. All simulations are done in (1+1)-D, meaning that
the bulk of the simulated rock is 2-D and the initial substrate is 1-D.
In section 2 model B is explained and basic notation is introduced. Known scaling
properties of surface growth are reviewed and BD is used as an illustrative example of those
properties. The well-known Kardar-Parisi-Zhang theory of growth is sketched and the
concept of universality of scaling exponents is presented. In section 3 numerical methods
are presented. In section 4 results are summarized. Section 4 ends with the conclusions.
4
CHAPTER 2
THEORY
2.1 Model
The model studied in this paper, called model B, is a generalization of the 1+1-D on-lattice
ballistic deposition model (BD). BD [5] was originally proposed as a simple model for sedi-
mentation. It is shown in Fig. 2.1. Here square particles of size 1 ∗ 1 are dropped vertically
onto a flat one-dimensional substrate. The substrate has length L and is partitioned into
L discrete sites (columns). The particles are dropped into random columns above the sub-
strate at a constant rate, one at a time. Each particle falls vertically and sticks upon first
contact with another particle. Sticking might happen on top of a particle or to the side of
a nearest neighboring particle.
The complete dynamics of this growth process are fully determined by the height profile
h(x, t) which is also called the interface. It is a discrete integer-valued function of discrete
integer-valued variables giving the height h of the topmost particle in a given column x at
a certain time t.
The cellular automaton rule for BD is
h(x′, t + 1) = max[h(x′ − 1, t), h(x′, t) + 1, h(x′ + 1, t)
], (2.1)
where h(x, t) is the height-profile and x′ is a randomly chosen column number 1 ≤ x′ ≤ L,
chosen at time t. In this notation a time interval of one corresponds to one particle drop.
In contrast to BD, model B introduces an integer-valued parameter B with 0 ≤ B ≤ L.
Model B is depicted in Fig. 2.2 for B = 1. The idea of B is to increase the amount of
correlation (or cooperation) between distant columns in BD. In BD correlation is low as
only nearest neighbors can exchange information about their height via sticking. To increase
5
Figure 2.1: Ballistic deposition (BD): Starting with an initially flat substrate, square par-ticles are dropped from random positions above the substrate one at a time and stick uponfirst contact. A′,B′,C ′: sticking positions of particles A,B,C.
correlation the horizontal distance over which a particle can stick is increased from 1 to the
value of S := B + 1. As example, for B = 0 we have S = 1 and only nearest neighbors
interact, reproducing BD. In contrast, for B = 1 we have S = 2 which allows particles to
stick to neighbors two columns away.
With these sticking rules model B is driven by the cellular automaton rule
h(x′, t+1) = max[h
(x′ − S, t
), . . . , h
(x′ − 1, t
), h
(x′, t
)+1, h
(x′ + 1, t
), . . . , h
(x′ + S, t
)],
(2.2)
an extension of Eq. (2.1) with the same definitions. The rule depends on the parameter
S = B +1, and thus implicitly on B. To optimize scaling behavior, minimize border-effects
and keep with the tradition of BD simulations, periodic boundary conditions in the lateral
6
Figure 2.2: Model B for the special case B=1. A’,B’,C’: sticking positions of particlesA,B,C. Note how particle A sticks over a distance S = B + 1. Here it sticks over a distanceof two columns as measured from center to center.
direction will be used in a cylindrical wrap-around fashion. The idea behind model B is
to study processes with increased correlation (or cooperation) during sedimentation. An
intuitive idea for this increase might be e.g. the presence of microbial life during sedimentary
rock growth as reviewed in [6]. For completeness, Fig. 2.3 shows typical extracts from bulks
grown with model B for a system size L = 250 and different B values. Note that in this
special case the rocks were actually grown without periodic boundary conditions.
One further notice: The cellular automaton rules are most easily formulated with a time
unit of 1 corresponding to one particle deposited. From now on though it is more convenient
to measure the time in units of layers deposited. So t := N/L where N is the total number
of particles deposited and L is the substrate length.
7
Figure 2.3: Extracts of size 250*250 from bulks grown with model B for different values ofB. The substrate length is L = 250. The bottom line corresponds to a height of 106. Blackdots: particles. Top left: B=0 (BD), top right: B=2, bottom left: B=4, bottom right B=9.
2.2 Scaling in Deposition Growth
Three topics are addressed in the remainder of this chapter. First, the concept of global
scaling, also known as the scaling of the interface width, is sketched using the model of
ordinary BD as example case. Power law growth and saturation of the interface width are
explained, the scaling exponents are defined and the Family-Vicsek relation is presented.
Second, the concept of local scaling as seen in correlation functions is described. Third, a
continuum description of interface evolution is sketched in the form of the Kardar-Parisi-
Zhang theory and the notion of self-affinity of the interface is explained.
8
2.3 Interface Width
A picture of the evolution of the height-profile h(x, t) in a simulation of BD (B = 0) is
shown in Fig. 2.4. For that figure the height profile was saved at time intervals of period
T = 30 while the simulation was running. A system size L = 474 was used, starting at
t = 0 (h = 0) and ending at the time when all columns were higher than h = 474. The
Figure 2.4: Evolution of the height profile for BD. Starting at t=0, h=0 (bottom line) up toa height h=474, the height profile was saved after each period (T=30). System size L=474.Left: h(x,t) corresponds to the black dots. Right: h(x,t) corresponds to the border betweentwo colors.
growth speed is approximately constant as can already be seen from Fig. 2.4. The average
height grows linearly in time after an initial regime (see Fig. 2.5) and is L independent.
Together with the constant deposition rate, this means that the total density of the bulk is
a constant.
9
0
50000
100000
150000
200000
250000
0 20000 40000 60000 80000 100000
h avg
(t)
t
0 0.5
1 1.5
2 2.5
3 3.5
4 4.5
0 0.5 1 1.5 2 2.5 3
Figure 2.5: Average height vs. time for BD. Numerical simulation data. There is linearbehavior (main) apart from an initial regime where few layers have been dropped (inset)and the vacancy density has not yet reached a steady state value. Here L = 1024 is plotted,but curves for other L match this one.
10
The ”roughness” of the interface can be identified with the interface width
wsingle(L, t) =
√√√√ 1L
L∑i=1
(h(x, t)− h(t)
)2. (2.3)
Fig. 2.6 shows the dependence of w on t and L for a single system. It can be seen that the
interface width for a single system fluctuates very strongly.
0.1
1
10
0.01 0.1 1 10 100 1000 10000 100000
w
t
L=1024
Figure 2.6: Interface width wsingle vs. time for a single system for BD. Numerical simula-tion data. Note the double-logarithmic scale. The system size L = 1024 was used.
However, the picture changes when defining the interface width as an average
w(L, t) =
⟨√√√√ 1L
L∑i=1
(h(x, t)− h(t)
)2
⟩(2.4)
with the averaging 〈·〉 done over an ensemble of many different realizations of the growth
process (different random seeds) and (·) done over all columns {x} in a single system. Such
11
ensemble averages over 500 to 1000 single growth processes are shown in Fig. 2.7 for different
L.
0.1
1
10
0.01 0.1 1 10 100 1000 10000 100000
w
t
L=4096L=2048L=1024L=512L=256L=128
Figure 2.7: Interface width w vs. time for an ensemble average for BD. Numerical simulationdata. Note the double-logarithmic scale. Different system sizes L were simulated andaverages were taken over 500 to 1000 realizations.
Each of the curves w(t) can be divided into three main regimes. There is an initial
regime of linear growth for 0 ≤ t ≤ 10. This corresponds to the shot-noise dominated
deposition of the first few layers that is essentially deposition without correlation. This
regime is known as the Poisson regime and will be of no further interest. It is followed by
the growth regime, which has a power law growth (a straight line in the double log plot)
w(L, t) ∼ tβ 10 . t� ts (2.5)
up to a certain t < ts where the growth regime ends. The symbol ∼ should be read as
”scales as” and is equivalent to ”proportional to”, putting an extra emphasis on power law
12
behavior. The time ts is usually defined as the intersect of linear fits of the growth and
saturation regimes. After a transition the saturation regime is reached for a t > ts where
w(L, t) ∼ const t� ts. (2.6)
The constant on the right hand side is known as the saturation width
limt→∞w(L, t) =: w∞(L). (2.7)
Furthermore, the saturation widths are approximately spaced equidistant in the log-log plot
for each doubling of L. This suggests the following power law dependence of the saturation
width on L:
w∞(L) ∼ Lα (2.8)
The time needed to reach the saturation regime also depends on L. Again the plot suggests
a power law:
ts ∼ Lz. (2.9)
The exponents α, β and z macroscopically characterize the dynamics of the surface width
and are called scaling exponents [3]. α is called the roughness exponent and controls how the
saturated interface roughness scales with system size. β is called the growth exponent and
controls how the interface roughness increases with time. z is called the dynamic exponent
and controls the dynamics of correlations spreading along the interface.
In fact, the exponent z is related to α and β. Approaching ts from the right (t+s ) and
from the left (t−s ) we get for the width
limt→t+sw(L, t) = Lα (2.10)
limt→t−sw(L, t) = tβs . (2.11)
13
We obtain the link between z, α and β by setting these two values equal:
tβs ∼ Lα (2.12)
Lzβ ∼ Lα (2.13)
z =α
β. (2.14)
The ideal scaling laws (Eqs. (2.5) to (2.12)) are summarized in Fig. 2.8.
0.1
1
10
0.01 0.1 1 10 100 1000 10000 100000
w
t
Poisson Regime~ t0.5
Growth Regime~tβ
Saturation Regime~Lα
Saturation times
ts~Lz~Lα/β
L=128
L=4096
α
z=α/βSlope=β
Figure 2.8: Ideal scaling of the interface width w as described by Eqns. (2.5) to (2.9). Tobe compared with actual simulation data as seen in Fig. 2.7
It was first discovered by Family and Vicsek [7] that one can summarize Eqns. (2.5),
(2.8) and (2.9) by the finite-size scaling relation
w(L, t) ∝ Lα f(t/Lz) (2.15)
14
with a scaling function f(x) satisfying
f(x) ∼ xβ x� 1 (2.16)
f(x) ∼ const x� 1. (2.17)
Eq. (2.15) is known as the Family-Vicsek scaling relation. This relation implies that plotting
w(t)/Lα vs. t/Lα/β should collapse all w(L, t) curves onto one and the same curve if the
right exponents α and β are used. In fact, the exponents for ordinary BD are known. The
exponents stated in the literature for BD are α = 0.5, β = 1/3 and consequently z = 1.5 [3].
Fig. 2.9 shows an attempted data collapse. While the data for large system sizes collapse
well onto the same curve, small system sizes deviate strongly. This shows the first sign
of a dependence of effective scaling exponents on the system size L, with perfect scaling
happening only in the asymptotic regime of large L.
0.1
1
0.001 0.01 0.1 1 10
w/L
0.5
t/L1.5
L=128L=256L=512
L=1024L=2048L=4096
Figure 2.9: Attempted data collapse of BD data from simulations according to the Family-Vicsek scaling relation. The same data were used as in Fig. 2.7.
15
From Eqs. (2.8) and (2.5) it follows that α can in principle be determined from numerical
data. One has to first determine the saturation widths for different system sizes and then
calculate the slope of log(w∞) with log(L). Likewise, β can be determined directly by linear
fits in the w(L, t) double-logarithmic plots. The dynamical exponent z should then result
from z = α/β.
However, in reality things are not that easy as can be seen by the failed data collapse
of Fig. 2.9. In fact, all of the scaling laws presented here naturally have lower cut-offs and
upper cut-offs due to the lattice spacing (particle size) and the limited total system size
(substrate length). Thus they are only deemed true in the asymptotic limit of small lattice
spacings or (equivalently) large system sizes. Strong corrections to scaling for smaller L
are usually found in discrete models such as BD. These corrections to scaling are generally
attributed not only to the cut-offs mentioned above, but also to the discrete nature of the
height profile. This discreteness allows for large local slopes and an additional cut-off of
high-frequencies. Most certainly there are also other reasons for corrections to scaling [8, 3].
Corrections to scaling effectively change α, β and z with L and thus make them dependent
on system size. These corrections are still poorly understood [8]. Corrections-to-scaling
effects can also be seen by a closer look at Fig. 2.7 where there is a slight dependence of
the slope in the growth regime on L and also a dependence of the saturation width spacing
on L.
Corrections to scaling make applying linear fits and determining the exponents a non-
trivial task. One has to be careful about choosing the methods used. Corrections to scaling
can then be tackled by trying to extrapolate effective exponents, determined from linear
fits, to the asymptotic limit L→∞.
2.4 Correlation Length
To understand why the interface width shows saturation the concept of correlation length
is useful. The dynamics of growth are as follows: At the initial stage only a few particles
are dropped and the columns are essentially independent. As growth proceeds particles
16
start to stick to their nearest neighbors. Due to this horizontal sticking, information can
spread laterally across the interface. As time progresses the correlation across the interface
will grow larger and larger. More and more distant columns begin to ”know” of each other.
After some time the whole interface will be correlated and that is when saturation is reached.
The lateral correlation length can be quantified as [9]
ξ||(t) =L−1∑l=0
l Γ(l, t)dl (2.18)
where
Γ(l, t) = 〈h(x + l, t)h(x, t)〉 − 〈h〉2, (2.19)
is the height-height correlation function [9]. In fact, starting with h(x, 0) = 0, ξ||(0) = 0,
for the growth of initially plane substrates, the correlation length is found to first grow as
a power law [8]
ξ||(t) ∼ t1/z (2.20)
until
ξ||(t) ≈ L, (2.21)
when it stops growing. At the time when ξ||(t) is of the order of magnitude of L, saturation
should be reached. Then, the height profile is maximally correlated. ξ||(t) is related to the
surface width as follows
ξ||(t)α ∼ w(t), (2.22)
which can be seen from Eq. (2.20) and Eq. (2.5). So again
w∞ ∼ Lα (2.23)
in the saturation regime, just as stated before.
17
2.5 Height-Difference Correlation Function
The spatial height-difference correlation function is [8, 10]
C(l, t) := 〈(h(x + l, t)− h(x, t))2〉12 . (2.24)
As before, (·) denotes averaging over the columns in one sample and 〈·〉 denotes the ensemble
average over many realizations of growth. C(l, t) is a local quantity that has been found
to show similar scaling as the global interface width w in many cases of growth of rough
surfaces [8] including BD. It is often used in real world experiments to measure the exponents
(e.g. [1]). An example of C(l, t) for an ensemble of 10 realizations of growth (L = 50000,
t = 7000) is shown in Fig. 2.10. A lower cut-off length can be seen which is due to the
1
10
100
1 10 100 1000 10000
C(l,
t)
l
cut-off length
lx~ξ||
C(l,t), t=7000, L=50000l0.45
Figure 2.10: Height-difference correlation function C(l, t) vs. l for an ensemble of 10 systems.L = 50000, t = 7000. αloc ≈ 0.45 ≈ α. The arrow indicates the crossover length lx ≈ ξ|| < L.Numerical simulation data.
finite lattice spacing that naturally restricts local scaling. If L is large enough to provide
18
good statistics an ensemble is not needed to find scaling of C(l, t).
C(l, t) also has a power law growth regime where [8]
C(l, t) ∼ lαloc l� ξ|| (2.25)
and a saturated regime where
C(l, t) ∼ ξαloc
|| l� ξ||. (2.26)
The crossover length is the length where saturation sets in, similar to the crossover time tx
found for global scaling,
lx ≈ ξ||(t). (2.27)
A scaling relation can also be stated for C(l, t)
C(l, t) = lαlocf(l/t1/z), (2.28)
remembering the scaling of ξ|| from Eqns. (2.20) to (2.22). Here the scaling function f has
the form
f(x) ∼ w l−αloc l� 1 (2.29)
f(x) ∼ const l� 1. (2.30)
And with ξα|| (t) ∼ t1/z this is similar to the Family Vicsek scaling relation for the interface
width, Eq. (2.15).
Scaling of the height-difference correlation function is known as local scaling, contrasted
to global scaling of the interface width w [10, 11]. αloc is known as the local roughness
exponent and sometimes called the Hurst exponent [8]. It is by no means obvious that
the local quantity αloc should be the same as the global quantity α. However, for simple
growth models with uncorrelated noise such as BD, numerical evidence shows [8] that indeed
19
αloc = α. Thus C(l, t) can be used as another way to measure the roughness exponent α.
The distinct case where global and local roughness scale differently (αloc 6= α ) is sometimes
known as anomalous scaling [10].
To obtain good statistics for C(l), a long substrate length L has to be used. A problem
here is that to observe a long growth regime for C(l) the function has to be evaluated at
large times, where ξ|| is accordingly large and saturation happens at large enough values.
However, on the positive side, an ensemble averaging is not as crucial to evaluate the C(l)
function as it is for w, which explains why C(l) is often used in experiments [3].
2.6 Self-Affinity
What is the reason for all of these power laws found both for global and local quantities?
A general reason behind power law behavior is scale invariance. This can be seen from the
symmetry
f(λx) = λαf(x) (2.31)
which holds for any power law f(x) ∼ xα. Power laws are self-affine functions. Self-affinity
here means that the function has the same shape on different scales if and only if the x-axis
is stretched or squeezed by a certain amount and the y-axis is stretched or squeezed by a
corresponding (different) amount. In the above example self-affinity becomes self-similarity
if α = 1. Then both axes have to be scaled by the same amount.
When dealing with growth processes far from equilibrium, stochastic self-affinity of the
height profile in space and time is indeed an important property. This property is usually
conjectured or notioned on the side in the literature. As far as the author knows it is not
yet rigorously proven from the microscopic growth rule for the case of BD [3, 12]. For a
stochastic self-affine interface
h(x, t) ≡ bαh(bx, bzt). (2.32)
Here (≡) means statistical indistinguishability. Starting with a piece of an original height
profile, a squeezing and stretching in different (space and time) directions of that piece gives
20
a new function which is statistically indistinguishable from the original. This can intuitively
explain identical scaling of local and global properties. The conjectured self-affinity is still
limited by the system size L and the lattice spacing for discrete models, resulting in cut-offs.
2.7 Kardar, Parisi and Zhang Equation
An equation to describe kinetic interface roughening processes such as BD growth was first
proposed by Kardar, Parisi and Zhang (KPZ) in 1986 [13]. The KPZ-equation
∂h(−→x , t)∂t
= ν∇2h +λ
2(∇h)2 + η(−→x , t), (2.33)
states that the evolution of the height profile is the sum of (from left to right)
• a surface smoothing process with ν taking the role of a surface tension,
• a lateral growth process. This is a nonlinear term that adds material to the interface
in a direction perpendicular to it. λ is related to the average velocity of the interface∂h(t)
∂t ,
• a random noise term with η(−→x , t) corresponding to Gaussian white noise.
The KPZ equation is nonlinear because of the lateral growth term. It is a stochastic
differential equation (Langevin-type) due to the noise term η.
Kardar, Parisi and Zhang did not directly derive their equation from a microscopic
growth rule such as Eq. (2.1). Rather, they modelled a similar growth process based on
plausibility arguments and physical intuition in quite a different realm. This realm is the
continuum limit, where the height profile h(−→x , t) is taken to be a continuous function of
continuous variables −→x , t. To go from the continuous limit to a discrete height profile, as
present in BD, one simple method would be to use a coarse graining procedure [9]
h(xn, t) =∫ n a−a/2
n a+a/2h(x, t)dx. (2.34)
21
where xn corresponds one of the discrete column numbers1 and a is the lattice spacing. To
go the other way from the discrete profile to the continuous realm is a bit more difficult.
One has to go to the limit L→∞, thus effectively making the columns smaller and smaller,
to approach a continuous independent variable x. Then one also has to smooth the height
profile to get a continuous h.
With this basic assumption of continuity of h and x they then looked for the simplest
possible interface growth equation compatible with the symmetries of the BD growth rule
but still allowing for lateral growth. The symmetries that are obeyed by the KPZ equation
are
• lateral space translation invariance: h(x, t)→ h(x + a, t)
• longitudinal space translation invariance: h(x, t)→ h(x, t) + a
• time translation invariance: h(x, t)→ h(x, t + a)
• invariance under rotations around the direction of growth: h(−x, t) → h(x, t) (in
1+1-D)
The KPZ equation is invariant under all these transformations. In addition to using these
symmetries, higher derivatives on the left-hand-side such as ∂2h∂t2
were dropped. This corre-
sponds to the long-time approximation.
The noise term in the KPZ equation deserves some extra attention. It is the character-
istic term of a Langevin type equation and incorporates the stochastic part into the growth
process. To understand it, it is easier to look at the Langevin equation
∂h(x, t)∂t
= G(h, x, t) + η(t), (2.35)
where G is an arbitrary operator acting on h(x, t) and η(t) is the noise term we are interested
in. η(t) is a random variable. For Gaussian white noise, by definition
〈η(t)〉 = 0 (2.36)1From now on, I will consider only the relevant case of (1+1)-D.
22
〈η(t)η(t′)〉 = δ(t− t′), (2.37)
where in this special case 〈·〉 denotes an average over time and not an ensemble average.
While the Langevin Eq. (2.35) looks like an ordinary differential equation, it is not, due to
the randomness of η. Different realizations of the process will result in different sequences
η, in course resulting in different evolutions of h(x, t). A closer look at the equations for
η reveals that, even though it is called Gaussian, it does not directly obey a Gaussian
probability distribution. This can be seen e.g. from it’s infinite variance. Nevertheless, the
name makes sense if we consider Eq. (2.35) as limit of the difference-difference equation
∆h(x, t) = G(h)∆t + ∆W (t). (2.38)
The definition of Gaussian white noise then means [14] that ∆W (t) is a random variable
drawn from a Gaussian probability distribution
p∆W (t) =1√
2π∆texp
(−∆W 2
2∆t
). (2.39)
So, ∆W (t) corresponds to random numbers picked from a Gaussian distribution with stan-
dard deviation ∝√
∆t. And in the limit of ∆t→ 0, ∆W∆t becomes the η(t) from above2[14].
Kardar, Parisi and Zhang found analytically the exact scaling exponents in (1+1)-D
α = 0.5 (2.40)
β = 1/3 (2.41)
z = 3/2, (2.42)
which are called universal scaling exponents of the (1+1)-D KPZ universality class. They
define the Kardar Parisi and Zhang (KPZ) universality class. While the BD model is2The many mathematical subtleties involved when dealing with processes such as the Wiener process
dW (t), which is inherently non-differentiable and involves building up a stochastic calculus are glossed overhere, due to space constraints.
23
just one of infinitely many possible growth and percolation models, it is widely believed
[3] to belong to the KPZ universality class. Thus, BD belongs to a class of models and
phenomena having different microscopic growth rules but that are nevertheless all sharing
the same universal KPZ scaling exponents.
The notion of universality is somehow restricted, though.
First of all, the universal exponents only refer to the second moment or the width of
the height fluctuations. However, very recent results suggest that the notion of universality
might be less restricted and extend beyond the exponents to include the asymptotic height
distribution (HPDF) function, at least in (1+1)-D. Majumdar and Nechaev [15] found the
exact HPDF function for a quite strongly modified BD model that they call anisotropic BD.
The function they found was the Tracy-Widom distribution function. The same HPDF was
found earlier for three other models believed to be in the KPZ class, a polynuclear growth
model [16], a directed polymer model [17] and a model called ”oriented digital boiling”
[18].3
Second, the KPZ exponents are exact only in the continuum limit, where the KPZ
equation exactly describes the growth process. For any actual realization of discrete growth
processes, h(x, t) is a discrete function. When stating that BD and other models belong to
the KPZ universality class, one means that the effective exponents of the discrete realization
approach the KPZ exponents as L→∞. Now it is clear that the equations and exponents
of the preceding sections describe an ideal growth model with perfect scaling, e.g. the
continuum limit.
In 1997 the KPZ equation has been formally approximated from the microscopic BD
growth rule using a limiting procedure and perturbation theory [19]. This gives further
analytic evidence for the hypothesis that the BD process is indeed a realization of KPZ
growth.
An example of a different universality class is the Edward-Wilkinson universality class,
described by the linear EW equation and having different exponents β = 0.25 and α = 0.5.3My own preliminary calculations of the non-asymptotic HPDF function for BD show a nearly Gaussian
distribution, though. This indicates that universality might not extent to the HPDF
24
For example, the model of random deposition with surface relaxation (RDSR) [20] belongs
to the EW class.
Scaling behavior similar to BD but with different exponents has also been found for other
growth models, suggesting the existence of new universality classes. These were models such
as e.g. BD with multiple species [21], BD with shadowing [22] or recently a model called
symmetric restricted BD [23].
25
CHAPTER 3
METHODS
Due to cut-offs and other corrections-to-scaling, one can only get an estimate of universal
exponents when applying numerical calculations. This can be done by first calculating
effective exponents (α′(L), β′(L)) and then trying to find convergence of those as the system
size L approaches infinity.
In this chapter the dataset is described first. Next, the methods used to calculate effec-
tive exponents and corresponding error-bars are described. Why describe the calculation of
error-bars? The error-bars for β′(L) were obtained in a non-standard way taking a remark
by Meakin [8] seriously:
In most cases, the uncertainties due to corrections to scaling are much larger
and more difficult to assess than the statistical uncertainties. (p. 140)
This corresponds to what I found. The standard errors of the slopes of applied linear fits
were always negligibly small compared to systematic changes of slope throughout the fit
intervals.
A similar study has been performed by Reis in Ref. [2] only for the case of BD (B=0),
and I will compare my results for B=0 with those obtained in that paper.
3.1 Data
Simulations were carried out using the substrate Lengths L = 128, 256, 512, 1024, 2048 and
4096 with the parameters B = 0,1,2,3,4,9,19. For each combination of those, an ensemble of
rocks was grown to a maximum t ∼ 10000 for L < 1024 and to a maximum t ∼ 100000 for
L > 1024. For L < 1024 an ensemble consisted of 1000 realizations (different random seeds)
of the growth process. For L > 1024 only 500 realizations could be used due to prohibitive
computation time. Typically around 15000 data points were saved for each ensemble. An
26
exponential ∆t was used between recording data points as all analysis was carried out on
a logarithmic time-axis. Each saved data point contained the time t = N/L (number of
lattice sweeps), the ensemble average height havg(t) := 〈h〉(t) and the ensemble average
interface width w(L, t).
To settle the point of the asymptotic behavior of β′(L), another simulation was carried
out. In this extra simulation a large L value was used and subsequently the saturation
regime could not be reached. However, the growth exponent could still be evaluated. The
parameters used were a large L = 1000000 and a rather short t ∼ 7000. Only 10 realizations
were created and the same B values were used as for the other sets.
3.2 Effective Exponent β′
The effective growth exponent β′(L) characterizes power law growth of w in the growth
regime and thus the dynamic aspect of the evolution of the roughness of the interface.
To determine β′(L) linear fits were applied to the growth regime in double-logarithmic
plots. However, the growth regime first had to be defined. While there is no universally
accepted standard recipe for definition of the growth regime, Reis [2] proposed the following
method for BD: Starting at t0 = 50 a least squares linear fit is applied to all data in the
interval t0 ≤ t ≤ τ . The time τ is chosen so that the linear correlation coefficient r of the fit
obeys r ≥ r0. If the upper threshold value r0 is chosen too large, only part of the data from
the growth regime will be used in the fit. If it is chosen too small, part of the saturation
regime will erroneously be used. To obtain a reliable threshold value for r0, Reis analyzed
the changes in β′ he got when using different candidates of r0. He also visually inspected
the size of growth regions associated with different r0 candidates. Based on the results of
visual inspection, he chose the value r0 = 0.99995.
This method would work fine for the data I obtained for B = 0, where the behavior of
the curves is close to perfect scaling. However, the departures from exact scaling behavior
for B > 0 in my data are too large to use this method. r0 would have to be changed
manually and somewhat arbitrarily to lower and lower values for larger and larger B to
27
obtain growth regimes that appeal to visual inspection and are larger than 5 datapoints.
To obtain values for β′ a standardized algorithmic method is preferable. I use an it-
erative method of consecutive linear fits, based on Reis’ method and the pragmatic idea
of maximizing r while still keeping a maximum of datapoints from the growth regime. As
start point of the growth regime I choose the first timepoint t0 for which havg ≥ 20. This
criterium is based on visual inspection - the growth regime starts at around havg ≈ 15 for
all the data. I then apply a series of consecutive linear fits to the data, all of them starting
at t0. The first linear fit in the series encompasses all data points with t > t0. This first
fit includes all of the saturation regime in addition to all of the growth regime. Therefore
it has a low r value. Each consecutive fit in the series then uses one datapoint less than
the previous one. The data interval used in the n-th fit becomes smaller and smaller and
fits the growth regime better and better. For each of these fits r is evaluated, resulting in
a series rn. As less and less of the saturation regime is included in the fit, rn is intuitively
expected to grow monotonically until the growth regime is reached. Then it should show a
local maximum at the optimum number of datapoints. When this first local maximum of
rn is reached the process is of course stopped. However, intuition is proven partly wrong
when applying this method to the data. This is due to the heavy fluctuations of w(L, t) in
the saturation regime. Fluctuations create many local maxima of rn right at the beginning
of the process. To avoid the process from stopping right at the start I require the extra
condition rmax > 0.99, which makes sure the process stops in the growth regime. Justifi-
cation of using this consecutive linear fit method is mainly given by a visual inspection of
many of the determined growth regimes, which indeed have reasonable sizes.
In a growth regime thus defined, I then apply a linear regression to obtain the mean
exponent β′ as slope.
For the second dataset of L = 1000000 the saturation regime could not be reached.
Thus, the above recursive method does not work. In addition, a longer transition from the
Poisson regime to the growth regime is observed for these data and thus the growth regime
does not start at ha ≈ 15. However, w clearly approaches a straight line for large t. Thus
28
I used a different method for the L = 1000000 dataset. I applied a simple linear regression
to the last datapoints, starting at havg = 2000 and going up to the last datapoint, which
was at around t = 7000.
For all of those linear fits a reasonable error estimate can not be obtained from the
standard error of the slope. The standard error proves ridiculously small in all cases and
larger errors can be seen by visual inspection. Systematic errors certainly dominate in
this regime. To account for these, I split the growth regime into two halves and then four
quarters and calculate the slopes in those 6 segments separately. The error estimate is then
the standard deviation of this set of slopes from the originally calculated slope. With this
ad-hoc method more realistic errors are obtained which are 2 to 3 magnitudes larger.
3.3 Effective Exponent α′
The effective roughness exponent α′(L) characterizes the scaling of the saturation width of
the interface with the substrate length.
To estimate α from numerical data, the first step is to measure w∞(L), the saturation
width. Even though ensembles of many realizations where used, the data still show fluctu-
ations of w(L, t) with t in the saturation regime. To obtain w∞(L) an average was taken
over as many different points in the saturation regime as possible. The lower L and the
higher B, the earlier saturation sets in and thus the longer the saturation regime. To make
use of as many datapoints from the saturation regime as possible the saturation regime was
defined individually for each dataset by visual inspection. A t value well in the saturation
regime was chosen as starting point and an average was taken over all of the remaining
datapoints. To double-check against systematic increases or decreases in these subjectively
defined saturation regimes a second (and third) averaging was done by just using the last
half (last quarter) of the datapoints. No significant changes in w∞(L) could be found,
indicating that there is no significant systematic error present.
For ideal scaling a double log plot of w∞(L) should show a straight line with slope
α. To obtain the effective exponent α′(L), a simple difference-difference quotient in the
29
double-logarithmic scale can be used [2]. Using systems of size L and 2L,
α′(Leff) :=log(w∞(2 L))− log(w∞(L))
log(2)(3.1)
pertains where
log(Leff) =log(L) + log(2L)
2(3.2)
defines an intermediate effective length.
30
CHAPTER 4
RESULTS
4.1 Raw Data, Qualitative Behavior
The first question I posed of the raw data is how fast the rock grows for different B.
0
100000
200000
300000
400000
500000
600000
700000
800000
900000
0 20000 40000 60000 80000 100000
h avg
(t)
t
B=19B=9B=4B=3B=2B=1B=0
0 2 4 6 8
10 12 14 16 18
0 0.5 1 1.5 2 2.5 3
Figure 4.1: Average height vs. time for different values of B. Apart from the initial fewlayers (inset), the rocks grow linear in time (main).
Fig. 4.1 shows the dependence of the average height havg on t for different B. After the
initial few layers (t . 20) - seen in the inset - the rock grows linearly in t. Also, the higher
B the faster the rock grows. Together with the constant deposition rate this means that
the average density of the bulk is uniform and that it is decreasing with increasing B.
31
0.1
1
10
100
0.1 1 10 100 1000 10000 100000 1e+06
w
havg
B=0B=1B=4
Figure 4.2: Raw Data example 1: Interface width w vs. havg, L = 2048, different B.
Next, I looked at the raw data for the interface width. Some of the raw w(L, t) curves
are shown in figs. 4.2 and 4.3. Note that havg is the independent variable. The following
qualitative observations can be made from the raw data
• As B is increased, the saturation happens earlier, and the growth regime consequently
shrinks, as is intuitively expected from the increased correlation between columns.
• Growth in the growth regime slows as B increases, suggesting a lower β′(L).
• There is no power law growth regime for some of the systems with large B and small L
(see Fig. 4.5). There, saturation coincides with the end of the Poisson regime, making
it impossible to define a growth exponent.
• For larger B, there are stronger corrections to scaling, meaning both a deviation from
equidistant spacing of the saturation widths in the saturation regime, as well as a
deviation from a perfect straight line for the growth regime.
32
0.1
1
10
0.1 1 10 100 1000 10000 100000 1e+06
w
havg
B=4, L=128B=4, L=256B=4, L=512
B=4, L=1024B=4, L=2048B=4, L=4096
Figure 4.3: Raw Data example 2: Interface width w vs. havg, B = 4, different L.
In fact, a B-dependent crossover to KPZ scaling with time was found in the growth
regime of the large system size L = 106 (Fig. 4.4). The growth regime for B = 0 is almost
a perfect straight line. But for higher B the slope increases resulting in a slow crossover effect
with time. A similar change of slope was found by the authors of Ref. [2] for BD in (2+1)-D.
Meakin states in Ref. [8] that stronger corrections to scaling are associated with large local
slopes in the height profile. There are certainly larger local slopes in model B for high B
values, due to the long distance sticking of particles. This could be an explanation of the
observed crossover. However, up to now, there exists no rigorous treatment of corrections
to scaling in BD-like models and they are generally poorly understood [8].
33
1
10
10 100 1000 10000
w
t
Slow crossover to KPZ scaling for growth of interface width, large system L=1e6
L=106, B=19L=106, B=9L=106, B=4L=106, B=0
KPZ
Figure 4.4: Interface width w vs. time for a large system L = 106. Slow crossover to KPZscaling with time: the slope in the growth regime approaches the KPZ value of 1/3 moreslowly with larger B.
34
4.2 Effective Exponent β′
Table 4.1 shows the obtained effective growth exponents β′(L) for the different values of B;
the error estimate was calculated as described in the methods section. In fact, the error
estimate should be considered as a rough estimate and be taken with great care. However,
as stated in the methods section these error estimates do take some systematic errors into
account.
Table 4.1: Effective Exponent β′(L) for different B, calculated from linear fits in the wdouble-log plots. Empty fields denote cases where no growth regime could be defined (seeFig. 4.5).
L \B 0 1 2 3128 0.242 ± 0.016 0.129 ± 0.029256 0.244 ± 0.019 0.182 ± 0.017 0.117 ± 0.028512 0.2658± 0.0060 0.2075± 0.0042 0.156 ± 0.013 0.117± 0.0171024 0.2824± 0.0060 0.2277± 0.0087 0.1859± 0.0084 0.145± 0.0102048 0.2907± 0.0035 0.2445± 0.0095 0.208 ± 0.019 0.176± 0.0194096 0.2954± 0.0072 0.259 ± 0.019 0.227 ± 0.019 0.200± 0.020
100000 0.319 ± 0.019 0.314 ± 0.036 0.327 ± 0.028 0.300± 0.0451000000 0.3275± 0.0042 0.3170± 0.0088 0.3257± 0.0081 0.319± 0.012L \B 4 9 19128256512 0.095 ± 0.0231024 0.122 ± 0.0122048 0.1445± 0.0042 0.076 ± 0.0294096 0.175 ± 0.018 0.089 ± 0.017
100000 0.296 ± 0.069 0.271 ± 0.0761000000 0.319 ± 0.016 0.296 ± 0.032
For large B and/or small L, the growth regime was too small and could not be resolved
using the consecutive fit method (see Fig. 4.5). The same happens for all of the empty
fields in table 4.1. It is clear that for larger B, bigger and bigger systems sizes are needed
to obtain the effective exponents with accuracy.
35
0.1
1
10
0.1 1 10 100 1000 10000 100000 1e+06
w
t
growth regime
no well-defined growth regime
L=1024, B=4L=128, B=4
Figure 4.5: w(L) for L=1024 and L=128, B=4: No growth regimes could be defined forsome ensembles.
36
The values for B = 0 can be compared to those found by Reis [2] in his recent large-scale
study of BD, where 20000 (for L ≤ 1024) and 10000 (for L > 1024) systems per ensemble
were used instead of the 1000 (L < 1024) and 500 (L ≥ 1024) systems used in this study.
Table 4.2 shows the comparison. There is good agreement within the error margins.
Table 4.2: Comparison of this study with results of Ref. [2]: β′ for B = 0.
L this study, B=0 Ref. [2]256 0.244 ± 0.019 0.2545512 0.2658± 0.0060 0.26691024 0.2824± 0.0060 0.27582048 0.2907± 0.0035 0.28824096 0.2954± 0.0072 0.2974
To see whether these results are compatible with the KPZ universality class which
is characterized by β = 1/3, the effective exponent β′(L) should be extrapolated to the
continuous limit L → ∞. I used the plot seen in Fig. 4.6 in order to exhibit the behavior
towards large L.
I plotted β′(L) vs. log(1/L) in order to emphasize the behavior towards large L. I
did not make any assumptions about the actual analytic form of corrections-to-scaling. A
similar method was in fact used in Ref. [24] to gain insight in the asymptotic behavior of
β′(L) for the Eden model.
From Fig. 4.6 a trend can be seen towards the KPZ-value of β = 1/3. Also, this plot
gives a hint to how underestimates of the exponents can arise if one does not investigate
large enough system sizes. By increasing the parameter B one can delay the convergence
to KPZ universality.
37
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
1e-07 1e-06 1e-05 0.0001 0.001 0.01
β’
1/L
B=0B=1B=2B=3B=4B=91/3
Figure 4.6: Effective exponent β′ vs. L. Note the logarithmic x-axis and the reciprocal L,both chosen to exhibit asymptotics towards large L.
38
4.3 Effective Exponent α′
The results obtained for w∞, for different values of B and L, are shown in table 4.3.
Table 4.3: Saturation width w∞(L) for different B, calculated from averaging over satura-tion regime.
L \B 0 1 2 3128 5.829± 0.049 4.350± 0.030 4.075± 0.022 4.134± 0.020256 7.853± 0.056 5.478± 0.039 4.806± 0.027 4.659± 0.022512 10.798± 0.088 7.155± 0.053 5.937± 0.039 5.465± 0.0301024 14.91 ± 0.19 9.64 ± 0.11 7.676± 0.085 6.738± 0.0642048 20.82 ± 0.22 13.21 ± 0.18 10.22 ± 0.13 8.701± 0.0874096 29.55 ± 0.33 18.33 ± 0.20 14.00 ± 0.18 11.66 ± 0.13L \B 4 9 19128 4.312± 0.020 5.443± 0.026 7.138± 0.033256 4.719± 0.020 5.739± 0.021 7.723± 0.028512 5.326± 0.024 6.016± 0.017 8.059± 0.0231024 6.302± 0.052 6.396± 0.025 8.298± 0.0252048 7.851± 0.075 7.009± 0.036 8.543± 0.0224096 10.23 ± 0.12 8.055± 0.058 not sim.
Again the results for B = 0 can be compared with those obtained by Reis [2], which is
done in table 4.4. And again there is good agreement within the error margins.
Table 4.4: Comparison of this study with results of Ref. [2]: w∞ for B = 0.
L this study Ref. [2]256 5.829± 0.049 5.8317± 0.0108512 7.853± 0.056 7.8592± 0.01461024 10.798± 0.088 10.7732± 0.02362048 14.91 ± 0.19 14.9470± 0.02824096 29.55 ± 0.33 29.4140± 0.0610
With ideal scaling, from Eq. (2.8), a double-logarithmic plot of w∞(L) vs. L should be
a straight line with slope α. Fig. 4.7 shows this plot and a straight line of slope 0.5 (the
39
KPZ prediction) for comparison. The B = 19 data show saturation at almost the same
value for all L considered. This is due to the absence of a well defined growth-regime, and
the saturation setting in right after the Poisson regime.
1
10
100 1000 10000
wℜ
∞
L
L0.5
B=0B=1B=2B=3B=4
B=9 (shifted down by 3.0)B=19 (shifted down by 3.5)
Figure 4.7: w∞ vs. L in a double-log plot. Errors are smaller than the pointsize. KPZtheory predicts a slope of α = 0.5. Note how the B = 19 data saturate at almost the samevalue for all considered L.
The α′(Leff) values are shown in table 4.5, computed according to Eqs. (3.1) and (3.2).
I excluded the B = 19 data from this table (see above). In Fig. 4.8, I exhibit the behavior
of α′(L) for large L by plotting α′ vs. 1/L on a logarithmic x-axis. A clear trend can be
seen again. Just as was the case with β′, α′ approaches the KPZ value for larger and larger
L. The transition is slower for larger B.
40
Table 4.5: Effective exponent α′(L) for different B, calculated from linear fits in the wdouble-logarithmic plots.
Leff \B 0 1 2 3181 0.430 ± 0.016 0.332 ± 0.014 0.238 ± 0.011 0.1724± 0.0099362 0.459 ± 0.016 0.385 ± 0.015 0.304 ± 0.012 0.230 ± 0.011724 0.466 ± 0.022 0.431 ± 0.020 0.370 ± 0.019 0.301 ± 0.0161448 0.481 ± 0.024 0.453 ± 0.026 0.413 ± 0.025 0.368 ± 0.0202896 0.505 ± 0.022 0.473 ± 0.025 0.454 ± 0.027 0.423 ± 0.021
Leff \B 4 9 19181 0.1724± 0.0099 0.1301± 0.0091 0.0762 ± 0.0087362 0.230 ± 0.011 0.1746± 0.0089 0.06812± 0.0066724 0.301 ± 0.016 0.242 ± 0.014 0.0881 ± 0.00701448 0.368 ± 0.020 0.316 ± 0.018 0.1320 ± 0.00942896 0.423 ± 0.021 0.382 ± 0.021 0.200 ± 0.013
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.0001 0.001 0.01
α’
1/Leff
0.5B=0B=1B=2B=3B=4B=9
Figure 4.8: Effective exponent α′ vs. 1/L. Note the logarithmic x-axis and the reciprocalL, both chosen to exhibit asymptotics towards large L.
41
4.4 Conclusions
In conclusion, I first summarized the basic concepts of scaling in random growth of surfaces.
The summary included the Kardar Parisi Zhang (KPZ) stochastic differential equation,
which models a class of random growth processes in the continuous limit. The KPZ equation
predicts a universal roughness exponent of α = 0.5 and a universal growth exponent of
β = 1/3 for (1+1)-D. I gave numerical examples for some of the basic scaling relations of
the interface width, using the model of ballistic deposition (BD) as an example case.
Then, a modified version of BD, called model B, was numerically analyzed. In model
B, the amount of lateral correlation is controlled by an integer valued parameter B. For
B = 0 model B reduces to BD. I conducted numerical calculations on ensembles of sys-
tems in (1+1)-D for different values of B and different substrate lengths L. By these
calculations, the average height was shown to increase linearly in time after deposition
of the first few layers of particles and the growth speed was shown to increase with in-
creasing B, being independent of L. Then, for values of B = 0, 1, 2, 3, 4, 9, 19 and L =
128, 256, 512, 1024, 2048, 4096, 1000000, effective scaling exponents α′(L) and β′(L) were
determined numerically. The effective exponents showed remarkable dependence on L and
were found always to be lower than KPZ predictions. However, a ”slow” transition to KPZ
behavior was found, ”slow” both in terms of increasing system sizes needed (α, β, small L),
as well as of increasing times needed (β, L = 1000000) to approach KPZ exponents. The
transition was found to be increasingly ”slower” in these terms for larger and larger B.
Numerical treatments of KPZ related growth models have frequently lead to exponents
lower than KPZ predictions [3] and the ”slow” transition to KPZ scaling found in this
study is another hint at something interesting going on. One might label this state between
random growth and KPZ scaling as an intermediate state of matter, intermediate between
equilibrium and KPZ scaling. Gaining a better understanding of this state by determining
the exact analytical form and thus the exact time- and length-scale associated with the
transition found here is left to further numerical research and could be addressed by larger-
scale supercomputer simulations, involving a larger range of parameters B, L and t.
42
PROBLEM-IN-LIEU-OF THESIS II
BALLISTIC DEPOSITION: A TIME SERIES APPROACH
43
SUMMARY
The ballistic deposition model (BD) is studied numerically in (1+1)-D, adopt-
ing a time series approach with emphasis on single column behavior. The well-
known complex kinetic interface roughening (KIR) process generated by BD is
a multi-column effect. To analyze how it is mirrored in single-column behav-
ior, two types of events and the time series of height fluctuation increments are
analyzed. The horizontal stickings of particles, defined as sticking events, are
shown to have an ordinary exponential waiting time distribution. However, the
changes of sign of height fluctuations Yt := h(x, t) − 〈h〉(t), defined as recur-
rence events, are shown to have an anomalous waiting time distribution that
is a truncated inverse power law with a decay exponent µ ≈ 1.74. µ and the
truncation time are linked to the growth and the saturation time of the KIR
process, respectively. In addition, the diffusion entropy analysis of increments
∆Yt yields curves that resemble a smoother version of the global KIR process.
44
CHAPTER 1
INTRODUCTION
Ballistic deposition [5] can be regarded as the simplest model of sedimentation. In its
simplest form ((1+1)-D, on-lattice) square particles of volume 1 are dropped on an initially
flat line and stick upon first contact. First contact might be established by sticking sideways
to a nearest neighbor particle or by sticking on top of a particle beneath. Since it was first
proposed in 1956, an enormous amount of research has been done on BD and related models,
which can be easily appreciated by looking at the number of papers published on this and
related topics since its invention. In fact, the basic so-called kinetic interface roughening
process that can be investigated using BD turned out to be universal. This macroscopic
roughening process is usually described in terms of universal scaling exponents and resembles
in this sense the critical indices found in the phase-transitions of thermodynamics. Kinetic
interface roughening behavior can be found in a huge number of growth models - not limited
to sedimentation models at all - including such processes of current interest as thin-film
growth[3] and a simple model of growth of bacterial colonies[4].
A BD simulation is a system governed by an extremely simple microscopic rule that,
when iterated, generates macroscopic behavior which is impossible to understand just from
looking at the rule alone. The interface between the simulated rock and air roughens in
time, as the simulated rock grows. This roughening happens in a way which is different from
ordinary diffusion processes. First, the roughness grows as a power law. This growth is not
proportional to√
t, as expected for an ordinary diffusion process. Then, at a certain system-
size-dependent time, the interface roughness saturates at a certain system-size-dependent
value. Quantities describing this global process are the roughness exponent α = 0.5, the
growth exponent β = 1/3 and the dynamic exponent z = α/β.
In this study, I focus on numerical evaluation of single column behavior in BD, as is
expressed in time series. This time series approach is the popular approach of nonlinear
science when analyzing complex systems [27].
45
The question I address is the following: How is the global kinetic interface roughening
process reflected in the evolution of a single column in time? I limit myself to the analysis of
the following quantities. From the single column perspective, I define two types of events,
the so-called sticking and recurrence events. These two types of events are intuitively
deemed important for the emergence of complexity in BD. I build up binary time series
with these events and numerically evaluate the waiting time distributions. The time series
are also analyzed with the method of diffusion entropy (DE) to detect diffusion scaling. For
this purpose, the DE-method is modified from its original definition (single-column DE) to
a version that takes into account many time series, one for each column (multi-column DE).
The multi-column DE succeeds in finding diffusion scaling for both types of events, where
the single-column DE fails. Last, the time series of height fluctuation increments of a single
column is analyzed with the single-column DE method.
Regarding sticking events, ordinary statistics are found, with no sign of departure from
shot-noise dominated behavior. Regarding recurrence events, a power law decaying waiting
time distribution is found with an anomalous exponent µ ≈ 1.74 ≈ 2−β. Regarding height
fluctuation increments, the DE curves obtained from just a single column and a single
realization of growth resemble the global kinetic interface roughening, which in contrast has
to be obtained from all the columns plus an ensemble average over many realizations of
growth.
Results are compared to the trivial case of random deposition (RD), whenever applicable.
In section 2, the BD model, the concept of waiting time distributions, the original
diffusion entropy method and the many-columns diffusion entropy are explained. In section
3, sticking events are analyzed. For these events, the definition, methods and results are
presented, followed by remarks. In section 4, the same is done for recurrence events. In
section 5, the DE analysis of height fluctuation increments is discussed in a similar vein.
Section 6 closes with the conclusions.
46
CHAPTER 2
CONCEPTS
2.1 Ballistic Deposition
Ballistic Deposition, the model under study, was originally proposed in 1956 as a simple
model for sedimentation [5]. It is shown schematically in Fig. 2.1. Imagine an initially flat
line of length L that is partitioned into L sites (columns) of unit length. Starting from
this 1-D substrate, square particles of unit volume are dropped vertically from above into
random columns, one at a time. Each particle falls down and sticks upon first contact with
another particle or the substrate. Thus, sticking might happen sideways or on top of a
particle below.
Figure 2.1: Ballistic deposition (BD): Starting with an initially flat substrate, square par-ticles are dropped from random positions above the substrate one at a time and stick uponfirst contact. A’,B’,C’: sticking positions of particles A,B,C.
During growth of this simulated rock the interface h(x, t) evolves in time. h(x, t) is a
discrete integer-valued function of discrete integer-valued variables. It gives the height h
of the topmost particle in a given column x at a certain time t. The time evolution of the
interface is given by a simple cellular automaton rule:
h(x′, t + 1) = max[h(x′ − 1, t), h(x′, t) + 1, h(x′ + 1, t)
], (2.1)
where h(x, t) is the interface and x′ is a randomly chosen column number (1 ≤ x′ ≤ L),
chosen at time t. In this notation, a time interval of one corresponds to one particle drop.
Cylindrical boundary conditions are adopted: h(L + 1, t) = h(1, t).
47
The time t used to formulate Eq. (2.1) corresponds to the number of particles dropped.
However, in BD, time is usually measured as t = N/L [3], where N is the total number of
particles deposited. In fact, the analysis in this study will be done wholly on this coarse-
grained timescale, where a time increase of one corresponds to the deposition of L particles.
The joint action of randomness and order in BD creates complex behavior that is usually
expressed in terms of the interface width. The interface width is usually defined as the
standard deviation of h(x, t) and is also known as the interface roughness [3, 8]:
w(L, t) =
√√√√ 1L
L∑i=1
(h(x, t)− 〈h〉(t))2. (2.2)
How does w evolve in time? After deposition of about the first ten layers of particles w
increases as a power law w ∼ tβ (growth regime). Then, at a system size dependent time
tsat ∼ Lz (saturation time), the width saturates at the constant value w∞ ∼ Lα (saturation
width)[3]. It is now widely believed that BD is described accurately by the Kardar-Parisi-
Zhang (KPZ) equation [13] in the asymptotic limit of L → ∞ and of a continuous height
profile. The KPZ equation predicts values for the growth exponent β = 1/3, the roughness
exponent α = 0.5 and the dynamic exponent z = α/β = 1.5. Both the quantity w and the
KPZ-equation take into account all of the columns to describe the process. For a further
review of the basic scaling concepts in BD, I refer to my first problems in lieu of thesis work,
which is part 1 of this document, or e.g. Ref. [3]. Here, I limit myself to showing three
illustrative figures and making some additional remarks that should make the general idea
of the macroscopic phenomena clear.
First, the interface roughening of actual numerical simulations of BD is shown in Fig. 2.2
for ensemble averages of w. Ensemble averaging is necessary to reduce noise [3]. Second,
the noisy curve found for a single realization of growth is shown in Fig. 2.3. Third, the
definitions of the initial, growth and saturation regimes are schematically represented in the
”ideal” case of Fig. 2.4.
Why label this system as ”complex”? It is the language of Metzler and Klafter [28]. In
48
0.1
1
10
0.01 0.1 1 10 100 1000 10000 100000
w
t
L=4096L=2048L=1024L=512L=256L=128
Figure 2.2: Anomalous diffusion in BD growth: Interface width w vs. time for an ensembleaverage for BD. Numerical simulation data. Note the double-logarithmic scale, the anoma-lous power law growth with w ∼ t1/3 and the times when saturation sets in tsat ∼ Lz withz = 1.5.
their report on ”The Random Walk’s Guide to Anomalous Diffusion: A Fractional Dynamics
Approach”, they stated that an anomalous diffusion process should be labelled ”complex”.
To understand why this applies to BD, define each single column to be a ”walker”. The
BD process can then be interpreted as a diffusion process generated by the trajectories of
these ”walkers”. The behavior of the standard deviation of this diffusion process is simply
given by the interface roughening shown in Fig. 2.2. Note that this diffusion is anomalous
as the standard deviation grows as t1/3 and not as t0.5 as expected for normal diffusion.
This means that the central limit theorem breaks down for this diffusion process. Now,
according to Metzler and Klafter, a system showing anomalous diffusion should be labelled
”complex”. Note that an ideal anomalous diffusion process such as the ones studied by
Metzler and Klafter in their report would imply a standard deviation that increases forever.
49
0.1
1
10
0.01 0.1 1 10 100 1000 10000 100000
w
t
L=1024
Figure 2.3: Scaling of the interface width w as described in the text, for a single realizationof BD growth. Numerical simulation data.
The phenomenon of saturation complicates things further in BD.
The emergence of complexity from a simple growth rule will be studied in this work,
adopting the single column (single trajectory) perspective.
50
0.1
1
10
0.01 0.1 1 10 100 1000 10000 100000
w
t
Poisson Regime~ t0.5
Growth Regime~tβ
Saturation Regime~Lα
Saturation times
ts~Lz~Lα/β
L=128
L=4096
α
z=α/βSlope=β
Figure 2.4: Ideal scaling of the interface width w as described in the text. Drawn ”byhand”.
51
2.2 Random Deposition
Random deposition (RD) is a deposition model similar to BD, but with no complex behavior
at all. In RD, particles are only allowed to stick on top of others. No sideways sticking
is allowed, and therefore each column grows independently. The interface roughening of
RD can easily be evaluated analytically. Consider a single column of an RD system of size
L. Let us adopt a combinatorial perspective. The probability of a particle arriving on the
column of interest is
p =1L
, (2.3)
otherwise, it will stick elsewhere in the sample. The probability that the column of interest
has height h after deposition of a total of N particles is then just
P (h, N) =(
N
h
)ph (1− p)N−h. (2.4)
The average height is, using q := 1− p,
〈h〉 =N∑
h=0
h P (h, N)
=N∑
h=0
(N
h
)h ph qN−h
= p∂
∂p
∞∑h=0
(N
h
)ph qN−h
= p∂
∂p(p + q)N
= p N (p + q)N−1
= p N (p + 1− p)N−1
= N/L
⇒ 〈h〉 = t,
52
using the definition of integer time t = N/L, where one timestep corresponds to L particles
dropped. Similarly, the second moment of h is
〈h2〉 =N∑
h=0
h2 P (h, N)
= p∂
∂pp
∂
∂p(p + q)N
= p∂
∂pp N (p + q)N−1
= p N (p + q)N−1 + p2 N (N − 1) (p + q)N−2
= p N (p + 1− p)N−1 + p2 N (N − 1)(p + 1− p)N−2
= p N + p2 N2 − p2 N
⇒ 〈h2〉 = t + t2 − p t.
Then, the interface width w becomes
w =√〈(h− 〈h〉)2〉
=√〈h2〉 − 〈h〉2
=√
t + t2 − p t− t2
=√
(1− p) t
⇒ w ∼ t1/2,
yielding the ordinary diffusion behavior with a standard deviation increasing as√
t. Thus,
we have β = 0.5 and no saturation.
This is expected also on the following grounds. In RD each column can be thought of as
an independent walker. Together, those walkers create an ordinary diffusion process, known
as Brownian motion. Remember that the Central Limit Theorem assures us that if random
walk increments are drawn from a distribution with a finite second moment and existing
mean, the resulting diffusion process will always be ordinary with a Gaussian probability
53
distribution. After a few initial steps, the distribution P will thus converge to a Gaussian
with a standard deviation increasing as√
N , where N is the number of random walker steps.
For RD, we have N ∝ t as it is a discrete time process. Thus P (h, N) will converge to a
Gaussian after a few timesteps, and the interface width grows with w ∼ tβ with β = 0.5.
Therefore, it is the Central Limit Theorem that is responsible for the fact that there is no
sign of anomalous behavior in RD. It is simply ordinary diffusion.
2.3 Waiting Time Distribution
The concept of waiting time distributions is important in the context of analyzing time series
consisting of events. Given an unambiguous definition of an event that happens at certain
times during the evolution of a system, and given a discrete time-axis, one can construct
what I will call a ”binary time series of events”. This binary time series will simply consist
of a finite number of zeros and ones:
ηi ηi ε {0, 1} t = i∆t. (2.5)
To construct it, one observes the system and simply puts a ”one” whenever an event occurs,
otherwise one puts a ”zero”. Let us define τ as the waiting time between two consecutive
events. Thus, the possible values of τ are
τ ε { (i− j) ∆t | ηi = 1 ∧ ηj = 1 ∧ (ηk = 0 ∀ k ε ]i, j[)}. (2.6)
Possible τ are simply the number of ”zeros” between two consecutive ”ones” in the time
series (plus one), multiplied with ∆t. The waiting time distribution Ψ(τ) then gives the
probability of finding the waiting time τ somewhere in the sequence.
Two remarks need to be made. First, in the discrete picture used here, the values of
τ are integers and thus a binning of the τ -axis with bin-sizes 6= ∆t is not necessary in
principle. Second, given a time series consisting of a total of N values, Ψ(τ) will suffer from
poor statistics for all large τ . Here, ”large” means all τ that are close to N ∆t in terms
54
of few orders of magnitude. This can make the experimental determination of Ψ(τ) quite
difficult.
2.4 Diffusion Entropy
The method of diffusion entropy was used for the first time in [29] in 2001 to detect memory
in a time series related to teen-birth. The connection between the diffusion scaling detected
by DE and the waiting time distribution in the case of rare events has been further clarified
in Ref. [30]. Since then, DE has been used successfully in a number of cases of complex
system analysis. For example, it has been used to detect the waiting time distribution
of large earthquakes in California [31], to analyze two distinct types of memory in heart
beating [32], to detect bilinear scaling of the Levy walk[33], and recently to analyze weak
chaoticity of vortices in jet exhaustion [34]. In the following I will give a short explanation
of DE, found also e.g. in [29].
Diffusion Entropy (DE) is a method of time series analysis. It uses the idea of converting
a time series into a diffusion process. It can be used as a scaling detector and it is indeed
able to detect scaling without any need for subtracting local bias values from the data
(detrending). It can also detect what is sometimes called ”complexity”. ”Complexity” here
refers to the deviation from total randomness in a time series. Furthermore, concerning
events, DE has been found to be sensitive only to the unpredictable (main) events, even if
they are embedded in a sea of predictable (pseudo) events. This was shown for a special
case of artificially generated time series in Ref. [35]. By converting a binary time series of
events into a diffusion process, DE can be used to detect properties of the waiting time
distribution, even if direct evaluation of the waiting time distribution is not possible due to
poor statistics.
Now, what is Diffusion Entropy? Consider a discrete time series of N values
ai i = 0..N t = i∆t, (2.7)
55
that might be the result of measurements from e.g. a complex physical, sociological or
financial system, or the result of numerical simulation. The DE method converts this time
series into a diffusion process in an auxiliary one-dimensional space. It does so by creating
a set of many Brownian-motion like trajectories from the single time series.
How is this achieved by the DE algorithm? First the time series is covered with overlap-
ping windows of size l, resulting in a total of N − l+1 windows. Next, each of the N − l+1
windows is assigned a ”walker” that exists in an auxiliary space. For l = 0 all of these
walkers start moving from the origin of auxiliary space x = 0 and each one moves ahead
or backwards, according to the sequence of numbers in his assigned window. The numbers
in each window are simply the steplengths of the walker. The window size l is the number
of steps the walker makes. A crucial point now is that l can be interpreted as a time t′,
imagining that each step a walker makes takes a time of ”one”. For window length l, each
walker has then walked for a ”diffusion time” t′ = l. At the start of the diffusion entropy
algorithm, we set l = 1 and have a total of N walkers. All of them have made just one step
and the steplength was determined for each walker by the number in his window. In the
next step of the algorithm, we increase l by one and we get N−1 walkers that started at the
origin and have already made 2 steps, one for each number in their window. This process
is continued until a given maximum time t′ is reached, determined by the total length of
the time series under study. Formally
xj(t′) =t′∑
i=0
aj+i j = 0..(N − t′), (2.8)
where xj(t) is the diffusion trajectory of walker number j. All of the walkers together form
the diffusion process in auxiliary space (see also Fig. 2.5).
The next step consists in finding the approximate probability distribution function (pdf)
of the diffusion process created in this way. For this purpose, the x-axis is divided into bins
of equal size ∆x(t′), where the size might depend on t′. I will denote each of these bins by
its center point xi. By counting the number of walkers per bin and normalizing, the pdf
56
Figure 2.5: Diffusion entropy: A time series of N values is covered with N−l+1 overlappingwindows of size l each. Note that it is not limited to integer numbers. The window size isinterpreted as diffusion time t′, while the sum of numbers in each window give the positionof the corresponding walker at time t′. All the walkers together form a diffusion process.The diffusion entropy is the Shannon entropy of the resulting pdf.
P (x, t) can be easily evaluated,
P (xi, t′) =N−l+1∑
j=0
∫ xi+1/2 ∆x(t′)
xi−1/2 ∆x(t′)δ(xj(t′)− y) dy. (2.9)
P (xi, t′)∆x(t′) gives the probability of finding a walker in the i-th bin, centered at xi. To
obtain good statistics, the bin-size ∆x(t′) has to be large enough to find many walkers
in the majority of bins, but small enough as to give a good approximation of the actual
distribution of walkers. For integer-valued time series usually a constant binsize of one is
fine. For real-valued time series, one can select a constant fraction of the standard deviation
57
of the pdf as binsize.
If there is so-called diffusion scaling, the diffusion pdf should obey the scaling relation
P (x, t′) =1t′δ
F( x
t′δ
), (2.10)
which I formulated here for the ideal case of continuous x, t′ and no binning. The scaling
relation only makes sense in the regime of an approximately continuous t′, which will be
true only for the asymptotic case of large times t′ � 1. Furthermore, in the reality of
numerical calculations, this relation will even then only be satisfied approximately, due to
the discrete binning of the x-axis.
Now, for complex systems, the scaling described by Eq. (2.10) is expected to depart from
the ordinary scaling of Brownian motion. Ordinary scaling is characterized by a scaling
exponent δ = 0.5 and a Gaussian scaling function F. The departure from these ordinary
conditions is seen as ”anomalous scaling”, as a measurement of complexity, or likewise as a
measurement of the ”degree of anomality.”
DE assesses the scaling of Eq. (2.10) in a final step that also lends the method its name.
This final step consists of evaluating the Shannon entropy of P , called the diffusion entropy
S(t′) = −∫ +∞
−∞P (x, t′)ln
[P (x, t′)
]dx, (2.11)
where the integral has to be read as the proper sum, depending on the actual binsize used.
If we disregard the subtleties of numerical approximations and binsizes and assume that
there is perfect diffusion scaling, we can plug Eq. (2.10) into Eq. (2.11), to see the expected
form of S(t′)
S(t′) = −∫ +∞
−∞
1t′δ
F( x
t′δ
)ln
[1t′δ
F( x
t′δ
)]dx
58
A change of variables
y :=x
t′δ
dx = t′δ dy
results in
S(t′) = −∫ +∞
−∞
1t′δ
F (y) ln
[1t′δ
F (y)]
t′δdy
= −∫ +∞
−∞F (y) ln
[1t′δ
F (y)]
dy
= −∫ +∞
−∞F (y) ln
[1t′δ
]dy −
∫ +∞
−∞F (y) ln [F (y)] dy
= δ ln(t′)∫ +∞
−∞F (y) dy −
∫ +∞
−∞F (y) ln [F (y)] dy
= δ ln(t′)−∫ +∞
−∞F (y) ln [F (y)] dy
⇒ S(t′) = A + δ ln(t′).
A depends only the shape of F
A = −∫ +∞
−∞F (y) ln [F (y)] dy, (2.12)
and if we make the assumption that the shape of F remains constant in time, then A is just
a constant. In the second to last step above, the first integral is equal to 1 because of the
normalization condition on P(x,t) which yields, with help from Eq. (2.10),
1 =∫ ∞
−∞P (x, t′) dx
=∫ ∞
−∞
1t′δ
F( x
t′δ
)dx
=∫ ∞
−∞
1t′δ
F (y)t′δ dy
⇒ 1 =∫ ∞
−∞F (y) dy.
59
From the relation
S(t′) = A + δ ln(t′), (2.13)
we can see that the diffusion scaling exponent δ is the slope of the diffusion entropy in a
plot with logarithmic t′-axis. As mentioned above, the scaling exponent δ will depend on
time for short times and the scaling characterized by the constant δ of Eq. (2.10) can be
expected to emerge only in the regime of approximately continuous diffusion times (t′ � 1).
2.5 Diffusion Entropy and Events
In this study, I will use the DE method to analyze sequences of events obtained from
numerical simulation of BD. Consider a binary time series consisting of rare and uncorrelated
events. Namely, this is a sequence of zeros and ones, with more zeros than ones. In this
time series, let a one represents the occurrence of an event. Now, several theorems exist on
the behavior of DE acting on such a time series. The treatment for a certain kind of events
following an asymptotic power law waiting time distribution was done by the authors of
Ref. [30]. Here I focus on using the ”Asymmetric Jump Model” (AJM) of that reference. In
the case of AJM, a binary time series of events is analyzed by DE ”as is”. That means that
the walkers in auxiliary space can only do two different things: jump ahead by a distance
of one when they encounter an event, or stay where they are when there is no event. Thus,
they can never jump backwards. This is why the AJM rule is called asymmetric. Using
AJM the following results have been obtained [30].
If the events are uncorrelated and the waiting time distribution of events has the asymp-
totic power law form
Ψ(τ) ∼ 1τµ
, (2.14)
the diffusion scaling exponent as detected by the DE method yields the asymptotic value
δ = µ− 1 1 < µ < 2
δ = 1(µ−1) 2 < µ < 3
60
δ = 12 3 < µ.
δ depends on the value of µ and can in fact be used to detect µ with great accuracy. Note
that µ is often difficult to detect directly from the waiting time distribution, due to a lack
of statistics.
2.6 Many-columns Diffusion Entropy
Figure 2.6: Many-column diffusion entropy (MCDE): Each column in BD yields one timeseries of events. Each of those time series is associated with just one walker. Starting fromthe bottom, each walker’s position is the sum of numbers in his window, up to a maximumpoint l. l is interpreted as ”diffusion time” t′. Here t′ corresponds to the real time t = N/L.All the walkers together form a diffusion process. The Shannon entropy of the resulting pdfis the MCDE.
The above described method did not yield a well-defined scaling exponent for the events
under study in any computationally accessible interval of time. Thus, a variant of the
DE method was used, which in fact detected scaling in the accessible timeframes. In the
61
following, I will call this variant many-columns DE (MCDE), contrasted to the single-
column DE (SCDE) described above. In MCDE a binary time series of events is created
for each column in a large BD system (L = 105), yielding a total of L distinct time series.
Then each of these time series is treated as single walker in auxiliary space. Call those time
series
bjt t = 1..tmax j = 1..L, (2.15)
as illustrated in Fig. 2.6. The superscript j denotes the column and the subscript t denotes
the time. Starting at height zero, each walkers’ position at time t′ is then
xj(t′) =t′∑
i=1
bji . (2.16)
And with these trajectories, one can proceed as for SCDE by evaluating the Shannon
entropy of the diffusion process. Note that with this method, the diffusion time t′ coincides
exactly with the deposition time t. This method yields a well-defined scaling exponent for
the two types of events considered, and the scaling agrees up to a margin of about 5% with
the waiting time distributions found.
2.7 Two Types of Events
In search for signs of complexity in the single-column evolution in BD, I analyzed two
distinct types of events that are perceived by a single column. For both types, waiting time
distributions were evaluated and many-columns diffusion entropy was applied on binary
time series of those events. In the next two chapters, the two types of events are defined,
the data sets are described and results are presented.
62
CHAPTER 3
STICKING EVENTS
3.1 Definition
”What distinguishes ballistic from random deposition?” The first intuitive answer is of
course ”the sideways sticking of particles”. Consequently, I first consider the sideways
sticking of a particle and call it a sticking event (SE). Whenever a particle falls into the
column of interest, a sticking event is recorded only if the particle sticks sideways to a
neighboring column. When the particle does not stick sideways no sticking event is recorded.
In the respective time series of sticking events for that column, there will be a one for every
SE and a zero for the rest of particles that got stuck on top of others. Note that in this
picture, the time increases by one only whenever a particle is dropped into the column of
interest. As there is on average 1 particle falling into a selected column per dropping of L
particles, this timescale coincides with the usual one only on average.
3.2 Data
To gain good enough statistics for the waiting time distribution of sticking events ΨSE(τ) , a
large number of events needs to be observed. Therefore, waiting times were first calculated
separately for each column in the sample. Then the waiting times were collected together
from all of the columns in the sample to evaluate ΨSE(τ).
As seen before in Fig. 2.4, the BD growth process consists of three regimes, These three
regimes are the initial Poissonian, the growth and the saturation regime. As the actual
regime of analysis might influence ΨSE(τ), I evaluated it three times:
• Growth regime only. I took a system of large size L = 106 where the growth regime
lasts long enough. I then recorded a time series of zeros and ones for each column
63
starting at the beginning of the growth regime, tmin = 10, and subsequently recorded
a total of 108 zeros and ones.
• Saturation regime only. I took a system of small size L=128, where the saturation
happens early enough. I then recorded a time series of zeros and ones, starting inside
the saturation regime, tmin = 234, and subsequently recorded 108 zeros and ones.
• All regimes. I took a system of size L=1024. I then recorded the time series of zeros
and ones, starting at the flat substrate, and subsequently recorded 108 particle drops.
The MCDE was then applied to time series of sticking events. For that purpose, L = 105
and tmax = 105 was chosen. The large L is needed to get an approximately continuous
diffusion pdf.
3.3 Results
As shown in fig 3.1, sticking events obey an ordinary exponential waiting time distribution
ΨSE . Such a waiting time distribution is expected for shot-noise dominated processes (see
next subsection). There is no dependence of Ψ(t) on the regime of analysis or the system
size, stressing further the ordinary statistics of sticking events.
Furthermore, the many-columns DE method gives the result δ = 0.5 corresponding to
what is expected for a process determined purely by randomness. This is seen in Fig. 3.2.
64
1e-09
1e-08
1e-07
1e-06
1e-05
0.0001
0.001
0.01
0.1
1
5 10 15 20 25 30 35 40 45 50
Ψ
τ
L=1,000,000, Growth RegimeL=128, Saturation Regime
L=1024, All regimes0.4 e-0.375 τ
Figure 3.1: Waiting time distribution of sticking events. The distribution is exponential,with no dependence on the regime of analysis or on L. The events are Poissonian.
65
0
1
2
3
4
5
6
7
1 10 100 1000 10000 100000
S(t’
)
t’
Diffusion Entropy of Sticking Events, BDδ=0.5
Figure 3.2: Many-columns diffusion entropy of sticking events. The ordinary Brownianmotion scaling of δ = 0.5 is found. Parameters: L = 105, t = 0..105
66
3.4 Remarks
Why is the occurrence of an exponential decaying waiting time distribution a sign of ran-
domness? Consider the case of RD. Here, I will show that also in RD, an exponential
waiting time distribution occurs. However, this distribution is not related to the one found
above for the sticking events. Rather, the following shall simply serve as an example of why
one can consider such an exponential distribution as a sign of randomness. Of course, it
can only be interpreted as a sign of randomness if the times τ are uncorrelated, which is
clear in the case of RD and is supported in the case of BD by the ordinary diffusion entropy
scaling of δ = 0.5.
Of course, in RD there are no sticking events, so we define as an event simply the arrival
of a particle at a certain column i. Let us also use a different timescale than for the sticking
events: at each timestep, exactly one particle is deposited somewhere in the sample. The
waiting time τ is the time between two consecutive events. That time is also known as
recurrence time. The problem is to find the distribution density of recurrence times. This
problem is known as Bernoulli trials. For RD, we know that the probability of a particle
falling into the column of interest at a certain time is
p = 1/L. (3.1)
The probability of it falling somewhere else is simply (1 − p). Consider a time interval T,
during which T depositions (T trials) take place. The probability of having n events in T
trials is then
pn(1− p)T−n. (3.2)
If the last timepoint in T contains an event, we have
(T − 1n− 1
)=
(T − 1)!(T − n)!(n− 1)!
(3.3)
distinct ways of distributing the remaining n− 1 events over the previous T − 1 timepoints.
67
The total probability to have n events of probability p in T, with the last one happening at
time T is, with Eqs. (3.2) and (3.3),
P (T, n, p) =(T − 1)!
(T − n)!(n− 1)!pn(1− p)T−n. (3.4)
We are interested only in consecutive events, so we set n := 1 and T := τ to obtain the
waiting time distribution
Ψτ = p (1− p)(τ−1)
=p
1− peln(1−p) τ ,
and for small probabilities (large L) this goes to the Poisson distribution
Ψ(τ) = p e−p τ , (3.5)
for random deposition. Assuming no correlations between the τ , one can consider an expo-
nential waiting time distribution as a sign of events governed by randomness. The random-
ness here is just ordinary shot-noise that results in a Poisson distribution. However, this
is not a proof of the results found for the sticking events in BD. In contrast to the waiting
time distribution for RD found here, the waiting time distribution found for sticking events
can not be explained by such a simple argument, as it is related to the nonlinear cellular
automaton rule, Eq. (2.1). In addition, a different time-scale was used for the sticking events
in BD, where ∆t = 1 corresponds on average to L particles dropped, which corresponds
to replacing t in Eq. (3.5) with t/1024. If one adopts this change and plots the resulting
Ψ(τ) for RD, an exponential decay results with exponent −at, where a = 1. In contrast, for
sticking events in BD, a ≈ 0.375 was found and thus, as expected, waiting times between
sticking events in a selected column of BD are longer than waiting times between particle
depositions in a selected column of RD. In other word, the probability of having a sticking
event in a certain column of BD is less than the probability of having a particle dropped in
68
a certain column in RD.
69
CHAPTER 4
RECURRENCE EVENTS
4.1 Definition
After ordinary statistics were found for sticking events, a second set of events was in-
vestigated. For this purpose an intermediate step is to define the time series of height
fluctuations. Let us choose a column of interest, call it column x, and consider the time
series of height fluctuations
Yt = h(x, t)− 〈h〉(t) t = N/L = 0, 1, 2, ..., (4.1)
The notation 〈h〉(t) here stands for the average height of the simulated rock. Yt is just the
distance of column x from the mean height of the system at time t. Here, discrete time
is again measured as N/L, with N being the total number of particles dropped and L the
substrate length. Now, as the simulated rock grows Yt can become positive or negative,
depending on whether the column of interest is higher or lower than the average height.
Now, what I will call a recurrence event (RE) is the occurrence of a change of sign in
the sequence Yt of Fig. 4.2. A binary time series of these events is generated simply in the
following way. At each timepoint, the quantity Yt is inspected for the column of interest
and is compared to its value at the last timepoint, Yt−1. If Yt did change sign with respect
to the last timepoint, an event is recorded. Otherwise, a zero is recorded.
From the time series of RE, the waiting time distribution ΨRE(τ) was evaluated. Similar
to the procedure used for SE, time series were first generated separately for each column in
a given sample. Then the waiting times were collected from all of the time series into one
file to gain better statistics.
The same definition of events and waiting time distribution was used very recently for
a real experiment concerning the evolution of combustion fronts in paper burning [36]. The
70
same authors demonstrated earlier that paper-burning processes indeed belong to the KPZ
universality class [1], the same class that BD belongs to.
4.2 Data
As an example of the structure of the series Yt, the raw values were first plotted for a system
of L = 1000, both for a long and a short interval of time.
ΨRE might depend on the actual regime where analysis takes place, namely the growth
or saturation regime. To address this point next, I generated two ΨRE ’s for a system with
L = 3000:
• Growth regime only. I took a system of size L = 3000. I then recorded a time series
of zeros and ones for each column starting at the beginning of the growth regime,
tmin = 100 (see Fig. 2.2). I subsequently recorded a total of 24 ∗ 106 zeros and ones,
8000 per column.
• Saturation regime only. Using a system of the same size, analysis was started inside
the saturation regime, tmin = 50000 (see Fig. 2.2). Again, I subsequently recorded
24 ∗ 106 zeros and ones.
ΨRE was found to depend on the system size L. To shed light on that dependence, ΨRE
was evaluated for system sizes L = 2n with n = 5..13 starting at tmin = 100 and going up
to tmax = 105 for all sizes.
Furthermore, ΨRE for BD was compared with the one obtained for RD to exhibit the
differences. For this purpose, large systems of L = 105 were simulated up to tmax = 105 for
both RD and BD.
Finally, MCDE was applied to time series of RE with L = tmax = 105.
4.3 Results
The example of a time series Yt is shown in Fig. 4.1 (long term) and Fig. 4.2 (short term).
71
-40
-30
-20
-10
0
10
20
30
40
50
10000 12000 14000 16000 18000 20000
Yt
t
L=1000
Figure 4.1: Time series of height fluctuations Yt. Long term behavior, L=1000.
72
-35
-30
-25
-20
-15
-10
-5
0
5
10
10000 10050 10100 10150 10200 10250 10300 10350 10400 10450 10500
Yt
t
L=1000
Figure 4.2: Time series of height fluctuations Yt. Short term behavior, L=1000.
73
1e-07
1e-06
1e-05
0.0001
0.001
0.01
0.1
1
1 10 100 1000 10000
Ψ(τ
)
τ
L=3000, Growth RegimeL=3000, Saturation Regime
µ=1.74
Figure 4.3: Waiting time distribution of recurrence events. Inverse power law decay withno dependence on regime of analysis. Parameters: L = 3000, t = 100..8000; L = 3000,t = 50000..58000
The results for the waiting time distribution in both growth and saturation regimes for
L = 3000 can be seen in Fig. (4.3). The recurrence events have an inverse power law waiting
time distribution. No dependence of the decay exponent on the regime of analysis could be
found.
The results of analysis concerning L-dependence are shown in Fig. 4.4 for a bin-size of
1 and Fig. 4.5 for logarithmic binning, respectively. From that last figure, it becomes clear
that the power law behavior is actually truncated. Truncation happens at earlier and earlier
values as L is decreased. Let us denote the approximate time of truncation by τtrunc. After
truncation, for τ � τtrunc, the power law becomes an exponential. By comparing truncation
times of Fig. 4.5 with the saturation times of Fig. 2.2, we can see that the truncation time
τtrunc is approximately equal to the saturation time tsat of the interface width. The power
law relation tsat ∼ Lz can then be applied. Recall z = α/β.
74
1e-14
1e-12
1e-10
1e-08
1e-06
0.0001
0.01
1
1 10 100 1000 10000 100000
Ψ(τ
)
τ
L=32, shifted down
L=8192, original height
Figure 4.4: Waiting time distribution of recurrence events for different L. Parameters:tmin = 100..105. Graphs for L < 8192 are shifted down for clarity.
The result of the numerical waiting time analysis can thus be expressed as the truncated
power law
ΨRE(τ) ∼ 1τµ
10� τ � τtrunc(L) (4.2)
where
τtrunc ≈ tsat (4.3)
and an exponential ΨRE for τ � τtrunc.
Thus, the truncation is related to the saturation of the interface width.
Keeping truncation in mind, another test was made against any regime dependence of µ.
This time, systems of different size were compared in the region of τ before truncation. A
system of size L = 105 (growth regime) is compared to a system of size L = 3000 (saturation
regime). The result can be seen in Fig. 4.6, using logarithmic binning.
75
1e-12
1e-10
1e-08
1e-06
0.0001
0.01
1
1 10 100 1000 10000 100000
Ψ(τ
)
τ
µ=1.7L=32L=64
L=128L=256L=512
L=1024L=2048L=4096L=8192
Figure 4.5: Same as Fig. 4.4 with logarithmic binning and no downwards shift of curves.
Again, no dependence of µ on the regime was found. Here, it has to be noted that the
authors of Ref. [37] found a dependence of µ on the regime in their numerical and analytic
study of deposition models described by linear Langevin equations. However, BD is a not a
process that can be described a linear Langevin equation. The KPZ-equation that describes
the BD process in the continuous limit (L → ∞) includes an important non-linear term
that models the sticking of particles.
Now, to find the value of µ, the large system size of L = 105 was used to gain optimum
statistics. For comparison, the same analysis was performed also on a simulation of random
deposition (RD) with the same system parameters. The results are compared in Fig. 4.7,
where logarithmic binning was used. The difference in slope between RD and BD can be
seen. The result for the slopes is
µRD = 1.50± 0.02
76
1e-14
1e-12
1e-10
1e-08
1e-06
0.0001
0.01
1
1 10 100 1000 10000 100000
Ψ(τ
)
τ
L=100000, Growth RegimeL=3000, Saturation Regime
µ=1.74
Figure 4.6: Waiting time distribution of recurrence events. No dependence of µ on theregime of analysis is detectable. Parameters: L = 100000, tmin = 100, tmax = 100000;L = 3000, tmin = 5000, tmax = 1000000; logarithmic binning.
µBD = 1.74± 0.02,
where the error was estimated by eyesight.
The result from MCDE entropy analysis is shown in Fig. 4.8. The diffusion scaling
exponent delta yields
δRD = 0.50± 0.01
δBD = 0.70± 0.01,
close to the expected values δ = µ − 1. Again the error was estimated by eyesight. The
vanishing dependence of δ on diffusion time t′ shows that the series of events transforms well
into a diffusion process, showing scaling over large time intervals. However, this is most
77
1e-08
1e-07
1e-06
1e-05
0.0001
0.001
0.01
0.1
1
1 10 100 1000 10000
Ψ(τ
)
τ
RDµ=1.5
BDµ=1.74"
Figure 4.7: Waiting time distribution of recurrence events. Comparison between randomdeposition and ballistic deposition. Parameters: L=100000, t = 100..105
certainly due to the fact that the many-columns diffusion process is generated by using
each column as a walker instead of creating many walkers from one and the same column.
If the latter method is used, indeed no constant diffusion scaling can be detected in the
computationally accessible timescales.
I conclude that both macroscopic properties of interface roughening in BD are mirrored
in single column behavior. They are mirrored in the waiting time distribution of recurrence
events. First, the anomalous subdiffusional growth of the interface width with a β = 1/3 <
0.5 is represented by µ ≈ 1.74 > 1.5. Second, saturation of the interface width is represented
by the truncation of the waiting time distribution.
In addition to the above results, numerical results suggest that the waiting times are
uncorrelated. This was obtained by numerical evaluation of the correlation function of the
78
-1
0
1
2
3
4
5
6
7
8
9
1 10 100 1000 10000 100000
S(t’
)
t’
Diffusion Entropy of Recurrence Events, BDδ=0.7
Diffusion Entropy of Recurrence Events, RDδ=0.5
Figure 4.8: Many-columns diffusion entropy of recurrence events. Comparison betweenrandom and ballistic deposition. Parameters: L = 100000, t = 0..105.
waiting time sequence τi obtained for a single column1.
1This result was obtained using a program written by Roberto Failla and is mentioned by kind permission
79
4.4 Remarks
There exists a definite link between power law growth of the interface width w and the power
law decay of the waiting time distribution ΨRE . From paper combustion experiments, which
also belong to the KPZ universality class, the authors of Ref. [36] found µ ≈ 1.66 and raised
the conjecture that generally β = 2−µ, which would provide the exact link between growth
of the interface width and ΨRE . The fact that here µ = 1.74 6= 5/3 might be explained
by the small lengths considered. Indeed, for BD and finite L, effective exponents α′ and
β′ do depend strongly on L for small L. They approach the analytical KPZ-values β = 1/3
and α = 0.5 only in the asymptotic limit of L→∞ and likewise h→continuous. This was
shown numerically e.g. by Reis [2] or in my earlier Problem-in-lieu-of thesis work, which is
part I of this document. For example, for L=4096 one gets β′ = 0.2954± 0.0072 < 1/3. For
real paper combustion, on the other hand, the microscopic effects take place on a molecular
scale, so that the process can be expected to be much closer to the ideal continuous h than
the BD simulation.
The conjecture that in the asymptotic limit of large t
β = 2− µ (4.4)
has been derived for free fractional Brownian motion e.g. in Ref. [37]. With free, I mean
the disregard of any saturation effects. In fact, as will be shown here, the conjecture can
be derived analytically for any continuous free subdiffusional processes that obey scaling.
Here, for brevity, I will limit myself to showing this relation only for a certain special form
of ΨRE(t). With this limitation, four assumptions have to be made:
1. There is scaling, which is responsible for the increase of the standard deviation w
during diffusion.
2. There is no possibility that a diffusion trajectory can stay at the origin for any time.
Trajectories can only cross.
80
3. The waiting times of recurrence events of the process are uncorrelated.
4. The waiting time distribution of recurrence events has the form
Ψ(t) = (µ− 1)Tµ−1
(t + T )µ(4.5)
with T = 1 and µ < 2. This form fulfills the asymptotic power law behavior for large
t and has the additional benefit that its Laplace transform was already studied by the
authors of Ref. [38].
Again, the last assumption is certainly not the only valid form of Ψ(t), but it simplifies
the calculations. This is so because the authors of Ref. [38] found the Laplace transform of
Ψ(t) for the asymptotic case t→∞ (u→ 0) to be
Ψ(u) = 1− cuµ−1, (4.6)
with
c = Γ(2− µ). (4.7)
Now, consider a diffusion process with a standard deviation
√〈(y − 〈y〉)2〉(t) ∼ tβ (4.8)
that increases forever. Here, β is not restricted to the ordinary value of 0.5 but can be any
value 0 < β < 0.5. The zero-crossing of one of the diffusion trajectories y(t) is defined as
a recurrence event. As we are only interested in times between recurrence events, let us
focus only on diffusion ”walkers” y(t) that start at y(0) = 0. By assumption 1 the way
the standard deviation w increases is due to scaling of the probability density function, and
thus
p(y, t) =1tβ
F (y
tβ), (4.9)
81
where p(y,t) starts as a Dirac-delta at the origin. The trick is to set y = 0 and obtain
p(0, t) ∼ 1tβ
. (4.10)
p(0, t) is the probability density that the diffusion trajectory h returns to its original value
of exactly h(0) = 0 at exactly the time t. Thus, it is the probability density that an event
happens at t = 0 and another one happens at time t. However, there might be any number
of events in between. Thus, define
Ψn(t) (4.11)
as the probability density function that n events happen inbetween t = 0 to t, the last of
which happens exactly at time t. The probability density to find the trajectory at the origin
after any number of previous jumps is then
p(0, t) =∞∑
n=1
Ψn(t); (4.12)
here, we have to force our trajectories to always make at least one jump, using assumption
2. Therefore
Ψ0(t) = δ(t− t′). (4.13)
Under assumption 1, probabilities Ψ can be multiplied to combine them. The Ψn(t) on the
right hand side of Eq. (4.12) can then be rewritten as a convolution, yielding
p(0, t) =∞∑
n=1
∫ t
0Ψn−1(t′)Ψ1(t− t′) dt′. (4.14)
Under the integral, there is the product of two probabilities: the probability that (n-1)
events occur before time t with the last one occurring at time t′ < t, and the probability
that the last event occurs exactly at time t. Ψ1(t − t′) is the waiting time distribution
of interest here, as it is the probability to find the waiting time τ = t − t′ between two
82
consecutive events. After a Laplace-transformation the convolution becomes a product
p(0, u) =∞∑
n=1
Ψn−1(u)Ψ1(u). (4.15)
The argument that was used above to write Ψn as convolution as can also be applied to
Ψn−1, Ψn−2 and so on, leading to the relation
Ψn−1(u) = Ψn−2(u)Ψ1(u)
= Ψn−3(u)Ψ21(u)
...
= Ψn−n(u)Ψn−11 (u)
= Ψn−11 (u)
⇒ Ψn−1(u) = Ψn−11 (u)
where the fact was used that the Laplace transform of a Dirac-delta function is one and
thus
Ψ0(u) = 1, (4.16)
Let us define
Ψ(t) := Ψ1(t). (4.17)
Eq. (4.15) becomes
p(0, u) =∞∑
n=1
Ψ(u)n
=∞∑
n=0
Ψ(u)n − 1
=Ψ(u)
1− Ψ(u)− 1
=Ψ(u)
1− Ψ(u)
83
⇒ L 1tβ
=Ψ(u)
1− Ψ(u),
where L denotes the Laplace transform. As we are only interested in the asymptotic be-
haviour for large t, we consider the corresponding asymptotic case of
u→ 0. (4.18)
As stated in the beginning of this section, the quantity Ψ(u) on the right hand side is known
in this limit [38]. It is
Ψ(u) ∼ 1− cuµ−1, (4.19)
with
c = Γ(2− µ). (4.20)
Thus for small u, the right hand side of Eq. (4.18) (rhs) becomes
rhs ∼ Ψ(u)1− Ψ(u)
∼ 1− cuµ−1
cuµ−1
∼ 1/cuµ−1.
Note that β = 1/3 < 1 and the Laplace transform on the left hand side of Eq. (4.18)
(lhs) can be obtained from any well equipped handbook of mathematics. It is
lhs = L t−β
=d
u1−β,
for any real number β < 1, with
d = Γ(1− β). (4.21)
Arguing that the asymptotic power law behavior for u→ 0 of rhs and lhs must be the same,
84
we get the relation
1− β = µ− 1 (4.22)
or
β = 2− µ. (4.23)
Thus, for BD we expect µ = 5/3 = 1.6666666.... This is what was found in the aforemen-
tioned paper-burning experiment of Ref. [36]. Also for BD, the value µ = 1.74± 0.02 found
is at least closer to the prediction than it is to the value of RD. I have to refer again to
the statement made at the beginning of the section, namely that effective exponents in BD
are always lower than the KPZ predictions for finite L (for values, see e.g. [2]). Indeed,
according Eq. (4.23), a growth exponent β < 1/3 would result in a µ > 5/3 and this might
explain the value µ ≈ 1.74 > 5/3 found here. Furthermore, while BD simulations result in
lower effective exponents, experimental studies usually obtain exponents larger than KPZ
predictions. This fact has historically been attributed to the effect of correlated noise in
real systems [3].
85
CHAPTER 5
HEIGHT FLUCTUATION INCREMENTS
5.1 Definition
Let us look again at the time series of single-column height fluctuations
Yt = h(x, t)− 〈h〉(t) t = N/L = 0, 1, 2, ..., (5.1)
which was used in the last section to define the recurrence events. If we interpret Yt as a
diffusion process, it makes sense to look at the increments of Yt,
Yt := Yt+1 − Yt (5.2)
and then use the SCDE method on exactly that time series of increments to recreate a new
diffusion process out of a single column time series. Then, we can see if there is resemblance
of this diffusion scaling to the global scaling of the interface width.
5.2 Data
Yt was analyzed by plotting the raw values and by using the SCDE method on the time
series. Systems of sizes L = 2n with n = 7..13 were used with t = 0..106.
5.3 Results
The time series of Yt for BD is shown in Fig. 5.1. The sequence is asymmetric, which
is logical, as columns can only increase in height but never decrease. The mean of the
sequence is zero, as expected. Note that the values for Yt are not distributed uniformly.
Fig. 5.2 shows just the datapoints. They gather around values of 0.86 + n, with integer n.
86
-4
-2
0
2
4
6
8
10
12
14
16
18
10000 10050 10100 10150 10200 10250 10300 10350 10400 10450 10500
Yt+
1-Y
t
t
L=3000
Figure 5.1: Time series of height fluctuation increments for BD. Note the asymmetric nature.L=3000
For comparison, the series for RD is shown in Fig. 5.3.
The result of the SCDE method is shown in Fig. 5.4. For comparison, the result for
RD is shown in Fig. 5.5. The diffusion entropy curves S(t′) resemble the global scaling of
the interface width w(t) (as seen before in Fig. 2.2), if one identifies S ←→ a log(w) and
t′ ←→ t, choosing the right constant a. The early saturation for the two largest L can
be explained by a numerical artifact of the DE method due to a lack of statistics. This
happens if t′ is approaching the length N of the time series under study and is an extra
type of saturation that occurs due to the way the diffusion entropy works. Recall that the
SCDE uses windows over the whole sequence of window length t′. If t′ is within one or two
orders of magnitude of the total length of the sequence under study, the walkers can walk
very large distances and there are not enough walkers to give a good approximation of the
diffusion pdf. This results in a saturation of S(t′) simply because of the finite length of the
time series. In fact, due to this effect even random deposition seems to saturate, as seen in
87
-5
0
5
10
15
20
25
30
10000 20000 30000 40000 50000 60000 70000 80000 90000 100000
Yt+
1-Y
t
t
L=3000
Figure 5.2: Time series of height fluctuation increments as points. Larger time range thanfig. 5.1. Note the non-uniform distribution of values: They are distributed in small stripsaround values 0.86 + n, which is explained in section 5.4.
Fig. 5.5.
The diffusion entropy curves in Fig. 5.4 reproduce to some extent the evolution of the
interface width, Fig. 2.2. The power law growth of the interface width with β = 1/3 is
reproduced quite well in form of a scaling δ = 1/3 for the larger lengths. For the smaller
lengths, the DE yields a lower slope and seems to be more sensitive to small lengths.
Saturation times and values also resemble the interface width.
The smoothness of the SCDE curves is quite amazing, considering that they are the
results of analysis of a single column in a single realization of growth. Recall that the
similar-looking scaling of the interface width (Fig. 2.2) was obtained using all the columns
per realization of growth and - in addition - ensemble averages of many realizations of
growth. Thus Fig. 5.4 should be compared with the noisy scaling of the interface width
obtained for a single realization of growth, which was shown in Fig. 2.3.
88
-2
-1
0
1
2
3
4
5
6
10000 10050 10100 10150 10200 10250 10300 10350 10400 10450 10500
(Yt+
1-Y
t) RD
t
L=1000, RD
Figure 5.3: For comparison with Fig. 5.1: Time series of height fluctuation increments forRD. L=1000
89
0
1
2
3
4
5
1 10 100 1000 10000 100000
S(t’
)
t’
δ=0.333333L=8192L=4096L=2048L=1024L=512L=256L=128
Figure 5.4: Single-column diffusion entropy of the time series Yt for BD. A time series oftmax = 106 numbers was used.
90
1
2
3
4
5
6
7
8
1 10 100 1000 10000 100000
S(t’
)
t’
RD, L=1024δ=0.5
Figure 5.5: Single-column diffusion entropy of the time series Yt for RD. A time series oftmax = 106 numbers was used. System size L = 1024. Note how even random depositionseems to saturate at large t′. This is a numerical artifact due to a loss of statistics (seetext).
91
5.4 Remarks
We have seen that Y takes only values close to 0.86 + n. Why is this so? First, note that
the average height is always a rational number
〈h〉(t) =1L
L∑j=1
h(j, t)
=k
L
with an integer k, as each column always has an integer height.
After the first few layers are deposited, it is well-known that the average height in BD
increases linearly with t [3], and that the slope is approximately equal to 2.14 with small
fluctuations. Thus
〈h〉(t) ≈ 2.14 t. (5.3)
Furthermore, the values for Yt are discrete, as the column height is an integer
Yt = h(x, t)− 〈h〉
≈ n− 2.14 t
where n is an integer. Thus
Yt = Yt+1 − Yt
≈ n′ − 2.14(t + 1)− n + 2.14 t
= ∆n− 2.14,
where n′, n and ∆n are integers. This explains why values of Yt are always close to 0.86+n,
with integer n ≥ −2. They are not exactly equal to 0.86 + n, because the average height
only approximately increases linearly in time; fluctuations are in fact expected due to the
random choice of deposition columns. A similar argument to the one above holds also for
92
RD, but now the average height is simply equal to t, as there are no empty spaces in the
RD-rock. Consequently, Yt is integer-valued for RD.
93
CHAPTER 6
CONCLUSIONS
The idea behind this study was to analyze time series obtained from single columns in
ballistic deposition (BD) and see how the global process of kinetic interface roughening
(KIR), described by the behavior of the interface width w, is represented in the single
column behavior. I limited myself to the analysis of two types of events and the time series
of height fluctuation increments.
First, a sticking event was defined as simply the sideways sticking of a particle. For
sticking events, an ordinary exponentially decaying waiting time distribution was found and
the many-columns diffusion entropy analysis yielded the ordinary diffusion scaling δ = 0.5,
suggesting that they are Poissonian events.
Second, recurrence events were defined as the change of sign of the time series Yt :=
h(x, t) − 〈h〉(t). The integer timescale used has t increasing by one for every L particles
deposited. For this type of events a truncated power law waiting time distribution was
found. The decay exponent found is µ = 1.74± 0.02 for BD which is significantly different
from µ = 1.5 ± 0.02 for random deposition (RD) and close to the theoretically expected
value of µ = β − 2 = 5/3. Furthermore, the truncation of power law behavior sets in at
times close to the saturation time of the KIR process.
Third, the time series of height fluctuation increments Yt = Yt+1−Yt was analyzed with
single-column diffusion entropy (SCDE) for different system sizes L. The curves obtained
from this analysis resemble a smoother version of the curves obtained for the KIR process.
I conclude that the main features of the global KIR process are in fact mirrored in single
column behavior. They are mirrored both in the waiting time distribution of recurrence
events as well as in the diffusion entropy of height fluctuation increments. The subdiffusional
power law growth of w in the growth regime with exponent β ≈ 1/3 6= 0.5 is mirrored by
the anomalous power law index µ ≈ 2 − β of the waiting time distribution of recurrence
events. The saturation of w at times tsat ∼ t1.5 is mirrored by the truncation at times
94
ttrunc ≈ tsat of the waiting time distribution. In addition, the diffusion entropy reproduces
the KIR process quite accurately for large L ≥ 1024, using only the height fluctuations of a
single column. In contrast, the KIR process has to be obtained by making a global analysis
including ensemble averaging.
Further analytical research is necessary to obtain exact analytical expressions, giving
the single column evolution in time without referring to neighboring columns. Such an
analytical expression would explain how the global KIR process is perceived by a single
column as a kind of environmental noise. It would most probably explain the single column
evolution by a basically free anomalous diffusion process that is responsible for the power
law growth, superposed with an additional memory term that results in saturation.
95
APPENDIX A
PROGRAMS USED FOR PROBLEM I
For brevity, only the program ”BDvar.cpp” which generates the interface width (w(t))
curves is reproduced here. Several more programs were written, e.g. to implement the con-
secutive fit method used to define the growth regime and find β′, using a datafile generated
by ”BDvar.cpp”. The random number generator ”ran2()” was taken from [25] and included
without changes as ”random.cpp”. ”ran2()” is a rather slow function. It was chosen in ac-
cordance with a statistical treatment of the coupling of random number generators to BD,
Ref. [26]. In that treatment, the authors concluded that ”ran2()” behaves significantly
better than a standard generator included with c++ in a variety of statistical tests. For a 2
GHz Pentium 4, calculation times for w(t) curves ranged from the order of minutes for the
small L, small B systems to the order of days for the L=4096 B=19 systems (t=1..100000).
Program BDvar.cpp
------------------------------------
//*****************************************************************************
// BDvar.cpp
// Interface width for deposition with growth rules according to "model B"
//
// Input: Command Line arguments
// Start without arguments for list
// Output: Self explanatory ASCII data file
// 3 Column Format
// See header of datafile for explanation
//
// Requires: "random.cpp" the ran2() generator from numerical recipes
//
// Arne Schwettmann
// 2003
//*****************************************************************************
// Include Standard Headers
#include <fstream> #include <string> #include <cmath> #include
<cstdlib> #include <stdlib.h> #include <iostream> #include
<iomanip> #include <sstream> #include <time.h>
// random.cpp is the Random Generator ran2() from "Numerical Recipes"
96
#include "random.cpp"
using namespace std; typedef unsigned long ulong;
// Variables
long idum=0; //random number
generator uses this variable double
Rand_period=1000000000000000000; //random number
generator period!
// Functions
//***********
// max returns maximum of two long
//***********
inline long max(long a,long b) {
if (a<=b)
return(b);
else
return(a);
};
//***********
// min returns minimum of two long
//***********
inline long min(long a,long b) {
if (a<=b)
return(a);
else
return(b);
};
//***********
// max returns maximum of three ulong
//***********
inline long max(ulong a, ulong b, ulong c) {
if (a<=b)
{
if (b<=c)
return(c);
else
return(b);
}
else
{
if (a<=c)
return(c);
else
return(a);
}
};
//***********
// h_avg returns the mean value of long array S, length L
//***********
97
double h_avg(ulong *S,long L) {
double h=0.0;
for (long i=0;i<L;i++) h+=(double) S[i];
return(h/((double) L));
};
//***********
// variance returns the standard deviation
// of _long_ array S, length L from the given mean value h
//***********
double variance(ulong *S,long L, double h) {
double V=0.0;
for (long i=0;i<L;i++) V+=(((double) S[i])-h)*(((double) S[i])-h);
V*=(1.0/(double)L);
return(sqrt(V));
};
//***********
// mean returns the mean value of double array S, length n
//***********
double mean(double *S,int n) {
double temp=0.0;
for (int j=0;j<n;j++) temp+=S[j];
return(temp/((double) n));
};
//***********
// BDdeposit deposits N particles
// on surface S of length L.
//
// Global variable
// long idum
// is needed for random number generator ran2()
//
// Input:
// long N: Number of particles to deposit
// double p[]: Unimplemented
// long range: Sticking distance (model B), range=B+1
// ulong *S: Pointer to height profile array,
// S[i]=Height of Col i, i=0..L-1
// long L: Length of S
// char Periodicbcflag: Use periodic bc, if equal to ’y’
//
// Output:
// Nothing, array S will change according to depositions
// long idum will change due to random number generator
//***********
void BDdeposit(long N,double p[], long range,ulong *S, long L,
char Periodicbcflag) {
long j=0;
// K-neighbour sticking!
ulong h_max=0; // h_max: current maximum height among k neighbors
long k=0; // loop counter
for (long i=1;i<=N;i++) // do N particles total
{
98
do // get a random column: j
{
j=(long) ((float)L*(float)ran2((long *) &idum));
} while (j==L);
h_max=S[j]+1; // initialize h_max to column j
// (stick on top, if no high neighbors)
k=0; // reset counter
// find maximum height of all the neighbour columns and column j
while(k<=range) // check all k nearest neighbors
{ // moving outwards
if (Periodicbcflag==’y’) // New height = Maximum among k neighbors
h_max=max( S[((j-k)<0)?L+((j-k)%L):(j-k)],h_max,S[(j+k)%L]);
else
h_max=max(S[max(j-k,0)],h_max,S[min(j+k,L-1)]); // No periodic bc
k++;
}
S[j]=h_max; // set column j to new height
};
};
//***********
// BDvar will calculate the variance versus mean height and time for a generalized BD-model (model B)
// An ensemble average of the data will be produced and a logarithmic timescale adopted
//
// Program options
//
// L: Length of Surface
// K: Sticking distance (k=B+1)
// Nrsys: Number of Systems in Ensemble
// Tmax: maximum time in units of N/L, where N is the number of particles dropped until time t
// Rand_init: seed for random generator
// Outfilename: File where the data goes
// PeriodicBC?: ’y’ if one wants to adopt periodic boundary conditions (cylindrical wrap around)
//***********
int main(int argc,char *argv[]) {
// First, parse the arguments
if (argc!=8) cout<<"wrong number of arguments:" << endl
<< "L, K, NrSys, t_max, rand_init, outputfilename, PeriodicBC?" << endl << endl
<< "L: Length of Surface (integer)" << endl
<< "K: K-th neighbour sticking (1: B=0) (2: B=1) etc." << endl
<< "NrSys: Number of single systems in ensemble" << endl
<< "t_max: Maximum time in units of L" << endl
<< "rand_init: seed e.g. 123 (integer)" << endl
<< "outputfilename: outp. one file, giving ensemble averages for v, h" << endl
<< "PeriodicBC? : Do you want periodic boundaries? y or n!" << endl;
else
{
char *Arg=argv[1]; //read the program arguments
long L=atol(Arg);
Arg=argv[2];
long k=atol(Arg);
Arg=argv[3];
int Nrsys=atoi(Arg);
Arg=argv[4];
99
int Tmax=atoi(Arg);
Arg=argv[5];
int Rand_init=atol(Arg);
Arg=argv[6];
char *Outfilename=Arg;
Arg=argv[7];
char Periodicbcflag=*Arg;
// Setup
//
//
ofstream Output(Outfilename); // Write ensemble average data here in the end
Output << setprecision(24);
double t=0; // Time is a double
long Tpoints=0; // Get the total number of Timepoints
while(t<Tmax)
{
// Logarithmic Timestep
ulong N=(ulong) (t*(double)L/100.0+1.0);
if (N==1) // But not too small
N=L;
t+=((double)N/(double)L);
cout << t << endl;
Tpoints++;
};
double (* Data)[3][Nrsys];
Data=new double[Tpoints+1][3][Nrsys]; // 2-D array Data contains temporary
// accum. ensemble data, [0=variance,1=h_avg,2=t/L]
for(long i=0;i<=Tpoints;i++)
for(int j=0;j<3;j++)
for(int k=0;k<Nrsys;k++)
Data[i][j][k]=0;
ulong *S; // Declare Surface array
S=new ulong[L];
Output << "# L=" << L // Write header to data file
<< ", Tmax=" << Tmax
<< ", Rand_init=" << Rand_init
<< ", Nr. systems in Ensemble =" << Nrsys
<< ", K=" << k
<< ", Periodic boundary conditions? =" << Periodicbcflag << endl;
// File will contain: w,h_avg,t
Output << "# variance in ensemble, avg. h. in ensemble., time t/L" << endl;
// *** Ensemble Loop, goes through all systems
//
//
for (int j=0;j<Nrsys;j++)
{
cout << "Ens. Nr. " << j+1 << endl; // Which system are we doing?
// initialize Random seed,
// dependent on actual system
idum=-1 * abs((long)L+(long) Rand_init+j);
ran2((long *) &idum);
for (int i=0;i<L;i++) S[i]=0; // init S, the Surface array S[i]=h(x_i)
100
double p=1.0, h=0.0, v=0.0; // init needed variables, p: dummy
long i=0; // init counter for ensemble data array
ulong N=0; // N is the number of particles to be dropped next
t=0; // time starts at zero
// BD Loop *** for single system in the ensemble
//
//
while (t<Tmax)
{
N=(ulong) (t*(double)L/100.0+1.0);// Number of particles to drop
// (logarithmic timescale)
if (N==1)
N=L;
// Deposit N particles on S
BDdeposit(N, &p, k, S, L, Periodicbcflag);
t+=((double)N/(double)L); // advance time (time is in units of L)
h=h_avg(S,L); // compute average height
v=variance(S,L,h); // compute variance, interface width
Data[i][0][j]=v; // put data into temporary storage (ensemble add!)
Data[i][1][j]=h; // later, data will be ensemble averaged
Data[i++][2][0]=t; // time doesn’t need averaging,
// just put in zero array position!
};
};
// Done Depositing
// Now, we have to calculate ensemble averages and output them to file
for (int i=0;i<=Tpoints;i++)
{
double Avgv=mean(Data[i][0],Nrsys); // mean variance
double Avgh=mean(Data[i][1],Nrsys); // mean height
double time=Data[i][2][0]; // time doesn’t need averaging
Output << Avgv /* output ensemble mean Variance */
<< " "
<< Avgh /* output ensemble mean Height */
<< " "
<< Data[i][2][0] /* output ensemble time (t/L) */
<< " "
<< endl;
};
// Clean up
Output.close();
delete []S;
delete []Data;
};
return(0);
};
101
APPENDIX B
PROGRAMS USED FOR PROBLEM II
The program ”BDcol.cpp” reproduced in this appendix is one of the main programs used:
It generates two single column time series: a binary time series of recurrence events and the
sequence of Yt. Yt was generated from the output of that program by using the powerful
Linux shell command ”awk”. More programs were written, e.g. to generate the many-
columns DE (MCDE) of recurrence events. Those are not reproduced here for brevity.
Again, for the pseudo-random number generator (PRNG), the routine ran2() from
Ref. [25] was used without modification. According to the authors of that reference, it
has a period exceeding 1018 that well exceeds any amount of random numbers needed for
the simulations.
Program BDcol.cpp
----------------------
//*****************************************************************************
// BDcol.cpp
// Create time series of recurrence events and of Y=h-h_avg for a single
// column of BD growth
//
// Input: Command Line arguments
// Start without arguments for list
// Output: Self explanatory ASCII data file
// 4 Column Format
// See header of datafile for explanation
//
// Requires: "random.cpp" the ran2() generator from numerical recipes
//
// Arne Schwettmann
// 2003
//*****************************************************************************
// Include Standard Headers
#include <fstream> #include <string> #include <cmath> #include
<cstdlib> // Declare "system()"
#include <stdlib.h> #include <iostream> #include <iomanip>
#include <sstream> #include <time.h> #include "random.cpp"
// random.cpp is the Random Generator ran2() from "Numerical Recipes"
using namespace std;
102
typedef unsigned long ulong;
// Variables
long idum=0; //random number
generator uses this variable double
Rand_period=1000000000000000000; //random number
generator period! int Buflen=1000;
// Length of Writebuffer
// Functions
/*
max returns maximum of two ulong
*/
long max(long a,long b) {
if (a<=b)
return(b);
else
return(a);
};
/*
min returns minimum of two long
*/
long min(long a,long b) {
if (a<=b)
return(a);
else
return(b);
};
/*
max returns maximum of three ulong (OVERLOADED)
*/
ulong max(ulong a, ulong b, ulong c) {
if (a<=b)
{
if (b<=c)
return(c);
else
return(b);
}
else
{
if (a<=c)
return(c);
else
return(a);
}
};
/*
BDdeposit deposits N particles
on surface S of length L.
Global variable
103
long idum
is needed for random number generator ran2()
Input:
long N: Number of particles to deposit
double p[]: Unimplemented, parse any double pointer
long range: Sticking distance (for BD: range=1)
ulong *S: Pointer to height profile array,
S[i]=Height of Col i, i=0..L-1
long L: Length of S
char Periodicbcflag: Use periodic bc, if equal to ’y’
Output:
Nothing, array S will change according to depositions
long idum will change due to random number generator
*/
void BDdeposit(long N,double p[], long range,ulong *S, long L,
char Periodicbcflag) {
long j=0;
// K-neighbour sticking!
ulong h_max=0; // h_max: current maximum height among k neighbors
long k=0; // loop counter
for (long i=1;i<=N;i++) // do N particles total
{
do // get a random column: j
{
j=(long) ((float)L*(float)ran2((long *) &idum));
} while (j==L);
h_max=S[j]+1; // initialize h_max to column j
// (stick on top, eventually)
k=0; // reset counter
// find maximum height of all the neighbour columns and column j
while(k<=range) // check all k nearest neighbors
{ // moving outwards
if (Periodicbcflag==’y’) // New height = Maximum among k neighbors
h_max=max( S[((j-k)<0)?L+((j-k)%L):(j-k)],h_max,S[(j+k)%L]);
else
// No periodic bc
h_max=max(S[max(j-k,0)],h_max,S[min(j+k,L-1)]);
k++;
}
S[j]=h_max; // set column j to new height
};
};
/*
h_rms returns the avg value of array S, length L
*/
double h_avg(ulong *S,long L) {
double h=0.0;
for (long i=0;i<L;i++) h+=(double) S[i];
return(h/((double) L));
};
104
/*
BDcol will calculate the (height-avg.height) vs. time for a single column
and also the binary time series of change of
one column from "bigger than avg." to "smaller than avg." and vice versa,
which is the time series of recurrence events
Program options
L: Length of Surface
ColNr: Column of interest
Tmax: maximum time in units of t/L
Rand_init: seed for random generator
Outfilename: File where the data goes
PeriodicBC?: ’y’ if one wants to adopt periodic boundary conditions (cylindrical wrap around)
time_step: TimeStep in number of particles. Always put "one"
*/
int main(int argc,char *argv[]) {
// First, parse the arguments
if (argc!=9) cout<<"wrong number of arguments" << endl << endl
<< "L, ColNr, k, t_max, rand_init, outputfilename, PeriodicBC?, time_step" << endl << endl
<< "where" << endl
<< "L: Length of Surface (integer)" << endl
<< "ColNr: Number of column of interest (1<=ColNr<=L)" << endl
<< "k: Number nearest STICKY neighbors (BD=1)" << endl
<< "t_max: Maximum timesteps in units of timestep/L particles dropped" << endl
<< "rand_init: seed e.g. 123 (integer)" << endl
<< "outputfilename: one file will be output,
giving a binary timeline of recurrence events,
the column height minus avg. height,
avg height, and time (see header of file)" << endl
<< "PeriodicBC? : Do you want periodic BC? y or n!" << endl
<< "time_step: Time Step value in particle numbers" << endl;
else
{
char *Arg=argv[1];
long L=atol(Arg);
Arg=argv[2];
long Colnr=atol(Arg);
Arg=argv[3];
long k=atol(Arg);
Arg=argv[4];
ulong Tmax=atol(Arg);
Arg=argv[5];
int Rand_init=atol(Arg);
Arg=argv[6];
char *Outfilename=Arg;
Arg=argv[7];
char Periodicbcflag=*Arg;
Arg=argv[8];
long time_step=atol(Arg);
//Init
ofstream Output(Outfilename); // Write data here
Output << setprecision(24);
idum=-1 * abs((long)L+(long) Rand_init); // initialize Random seed
105
ran2((long *) &idum);
ulong *S; // initialize Surface array
S=new ulong[L];
for (int i=0;i<L;i++) S[i]=0;
// Buffered output, init Buffers
int Bbuffer[Buflen]; // binary timeline Buffer
double Hbuffer[Buflen]; // height Buffer
double Hmeanbuffer[Buflen];
double Tbuffer[Buflen]; // time buffer
int Bufpos=0; // buffer position
double p=1.0, h=0.0; // initialize needed variables
ulong t=0; // t is time in units of L
bool Col_bigger=false; // is the column of interest
// bigger than avg. height?
ulong Col=0; // how high is the column of interest
int Col_chg=0; // did the column of interest
// change from bigger to smaller or vice versa?
// Write header to output file
Output << "# L=" << L
<< ", Tmax=" << Tmax
<< ", Rand_init=" << Rand_init
<< ", Col Nr.=" << Colnr
<< ", K=" << k << endl;
Output << "# binary timelime, h minus h avg., h avg, time t/L" << endl;
// Main loop, deposits and outputs
while (t<=Tmax)
{
// Deposit L particles
BDdeposit(time_step, &p, k, S, L, Periodicbcflag);
t++; // Advance time in units of Time_step/L
h=h_rms(S,L); // Calculate mean h
Col=S[Colnr]; // Look at column of interest
if (Col_bigger && ((double) Col < h)) // Did it change from bigger than mean h to smaller?
{ // if YES, col_chg=true
Col_chg=1;
Col_bigger=false;
}
// Did it change from smaller than mean h to bigger?
else if (!Col_bigger && ((double) Col > h))
{ // if YES, col_chg=true
Col_chg=1;
Col_bigger=true;
}
else // else no change
Col_chg=0;
// Buffered OUTPUT to File
Bbuffer[Bufpos]=Col_chg; // Write into Buffer, binary timeline
Hbuffer[Bufpos]=((double) Col)-h; // ... h_i-h_mean
Hmeanbuffer[Bufpos]=h; // h_mean
// time
106
Tbuffer[Bufpos++]=(double) t* (double) time_step/(double) L;
if (Bufpos==Buflen) // Buffer full? >> Output to File
{
for (int i=0;i<Buflen;i++)
Output << Bbuffer[i] << " "
<< Hbuffer[i] << " "
<< Hmeanbuffer[i] << " "
<< Tbuffer[i] << endl;
Bufpos=0;
}
// End of OUTPUT to File
// Reset Column change
Col_chg=0;
// OUTPUT to screen
if ( (long) t % 500 == 0) cout << "t= " << t << endl;
};
// Empty the Buffer, caution: loop leaves Bufpos one ahead
if (Bufpos!=0)
for (int i=0;i<Bufpos-1;i++)
Output << Bbuffer[i] << " "
<< Hbuffer[i] << " "
<< Hmeanbuffer[i] << " "
<< Tbuffer[i] << endl;
// clean up
delete []S;
Output.close();
};
return(0);
};
107
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