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figures Symbolic link
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../data-analysis/figures

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prelude.tex
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\usepackage{todonotes}
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\usepackage{fontspec}
\usepackage{amsmath}
%\setmainfont{New York}
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@ -43,6 +44,23 @@
\renewcommand{\cftsecfont}{\rmfamily\mdseries\upshape}
\renewcommand{\cftsecpagefont}{\rmfamily\mdseries\upshape} % No bold!
\usepackage{authblk}
\usepackage{hyperref}
\newcommand{\singlecolumnabstract}[1]{
\twocolumn[
\begin{@twocolumnfalse}
\maketitle
\begin{abstract}
#1
\bigskip
\end{abstract}
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163
references.bib
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@article{ballDiffusioncontrolledAggregationContinuum1984,
title = {Diffusion-Controlled Aggregation in the Continuum Approximation},
author = {Ball, R. and Nauenberg, M. and Witten, T. A.},
date = {1984-04-01},
journaltitle = {Physical Review A},
shortjournal = {Phys. Rev. A},
volume = {29},
number = {4},
pages = {2017--2020},
issn = {0556-2791},
doi = {10.1103/PhysRevA.29.2017},
url = {https://link.aps.org/doi/10.1103/PhysRevA.29.2017},
urldate = {2023-03-15},
langid = {english},
file = {/Users/joshuacoles/Zotero/storage/DWGMBBIH/Ball et al. - 1984 - Diffusion-controlled aggregation in the continuum .pdf}
}
@article{battyUrbanGrowthForm1989,
title = {Urban {{Growth}} and {{Form}}: {{Scaling}}, {{Fractal Geometry}}, and {{Diffusion-Limited Aggregation}}},
shorttitle = {Urban {{Growth}} and {{Form}}},
author = {Batty, M and Longley, P and Fotheringham, S},
date = {1989-11},
journaltitle = {Environment and Planning A: Economy and Space},
shortjournal = {Environ Plan A},
volume = {21},
number = {11},
pages = {1447--1472},
issn = {0308-518X, 1472-3409},
doi = {10.1068/a211447},
url = {http://journals.sagepub.com/doi/10.1068/a211447},
urldate = {2023-03-15},
abstract = {In this paper, we propose a model of growth and form in which the processes of growth are intimately linked to the resulting geometry of the system. The model, first developed by Witten and Sander and referred to as the diffusion-limited aggregation or DLA model, generates highly ramified tree-like clusters of particles, or populations, with evident self-similarity about a fixed point. The extent to which such clusters fill space is measured by their fractal dimension which is estimated from scaling relationships linking population and density to distances within the cluster. We suggest that this model provides a suitable baseline for the development of models of urban structure and density which manifest similar scaling properties. A typical DLA simulation is presented and a variety of measures of its structure and dynamics are developed. These same measures are then applied to the urban growth and form of Taunton, a small market town in South West England, and important similarities and differences with the DLA simulation are discussed. We suggest there is much potential in extending analogies between DLA and urban form, and we also suggest future research directions involving variants of DLA and better measures of urban density.},
langid = {english},
file = {/Users/joshuacoles/Zotero/storage/F2M3CGET/batty1989.pdf.pdf;/Users/joshuacoles/Zotero/storage/YAGMYPYZ/Batty et al. - 1989 - Urban Growth and Form Scaling, Fractal Geometry, .pdf}
}
@article{botetClusteringClustersProcesses1985,
title = {Clustering of Clusters Processes above Their Upper Critical Dimensionalities},
author = {Botet, R},
@ -75,6 +111,24 @@
file = {/Users/joshuacoles/Zotero/storage/65P85MN4/Lyons - 1983 - A Simple Criterion for Transience of a Reversible .pdf}
}
@article{nicolas-carlockUniversalDimensionalityFunction2019,
title = {A Universal Dimensionality Function for the Fractal Dimensions of {{Laplacian}} Growth},
author = {Nicolás-Carlock, J. R. and Carrillo-Estrada, J. L.},
date = {2019-02-04},
journaltitle = {Scientific Reports},
shortjournal = {Sci Rep},
volume = {9},
number = {1},
pages = {1120},
issn = {2045-2322},
doi = {10.1038/s41598-018-38084-3},
url = {https://www.nature.com/articles/s41598-018-38084-3},
urldate = {2023-03-15},
abstract = {Abstract Laplacian growth, associated to the diffusion-limited aggregation (DLA) model or the more general dielectric-breakdown model (DBM), is a fundamental out-of-equilibrium process that generates structures with characteristic fractal/non-fractal morphologies. However, despite diverse numerical and theoretical attempts, a data-consistent description of the fractal dimensions of the mass-distributions of these structures has been missing. Here, an analytical model of the fractal dimensions of the DBM and DLA is provided by means of a recently introduced dimensionality equation for the scaling of clusters undergoing a continuous morphological transition. Particularly, this equation relies on an effective information-function dependent on the Euclidean dimension of the embedding-space and the control parameter of the system. Numerical and theoretical approaches are used in order to determine this information-function for both DLA and DBM. In the latter, a connection to the Rényi entropies and generalized dimensions of the cluster is made, showing that DLA could be considered as the point of maximum information-entropy production along the DBM transition. The results are in good agreement with previous theoretical and numerical estimates for two- and three-dimensional DBM, and high-dimensional DLA. Notably, the DBM dimensions conform to a universal description independently of the initial cluster-configuration and the embedding-space.},
langid = {english},
file = {/Users/joshuacoles/Zotero/storage/TLE4AFWZ/10.1038@s41598-018-38084-3.pdf.pdf;/Users/joshuacoles/Zotero/storage/ZJDV8CJF/Nicolás-Carlock and Carrillo-Estrada - 2019 - A universal dimensionality function for the fracta.pdf}
}
@article{niemeyerFractalDimensionDielectric1984,
title = {Fractal {{Dimension}} of {{Dielectric Breakdown}}},
author = {Niemeyer, L. and Pietronero, L. and Wiesmann, H. J.},
@ -92,6 +146,23 @@
file = {/Users/joshuacoles/Zotero/storage/LKJWGEMV/niemeyer1984.pdf.pdf}
}
@article{procacciaDimensionDiffusionlimitedAggregates2021,
title = {Dimension of Diffusion-Limited Aggregates Grown on a Line},
author = {Procaccia, Eviatar B. and Procaccia, Itamar},
date = {2021-02-09},
journaltitle = {Physical Review E},
shortjournal = {Phys. Rev. E},
volume = {103},
number = {2},
pages = {L020101},
issn = {2470-0045, 2470-0053},
doi = {10.1103/PhysRevE.103.L020101},
url = {https://link.aps.org/doi/10.1103/PhysRevE.103.L020101},
urldate = {2023-03-15},
langid = {english},
file = {/Users/joshuacoles/Zotero/storage/BA59EJ99/Procaccia and Procaccia - 2021 - Dimension of diffusion-limited aggregates grown on.pdf}
}
@article{sanderDiffusionlimitedAggregationKinetic2000,
title = {Diffusion-Limited Aggregation: {{A}} Kinetic Critical Phenomenon?},
shorttitle = {Diffusion-Limited Aggregation},
@ -110,6 +181,23 @@
file = {/Users/joshuacoles/Zotero/storage/YV9XD9VR/sander2000.pdf.pdf}
}
@article{smithFractalMethodsResults1996a,
title = {Fractal Methods and Results in Cellular Morphology — Dimensions, Lacunarity and Multifractals},
author = {Smith, T.G. and Lange, G.D. and Marks, W.B.},
date = {1996-11},
journaltitle = {Journal of Neuroscience Methods},
shortjournal = {Journal of Neuroscience Methods},
volume = {69},
number = {2},
pages = {123--136},
issn = {01650270},
doi = {10.1016/S0165-0270(96)00080-5},
url = {https://linkinghub.elsevier.com/retrieve/pii/S0165027096000805},
urldate = {2023-03-16},
langid = {english},
file = {/Users/joshuacoles/Zotero/storage/ZWIYRKBT/Smith et al. - 1996 - Fractal methods and results in cellular morphology.pdf}
}
@article{tanInfluenceExternalField2000,
title = {Influence of External Field on Diffusion-Limited Aggregation},
author = {Tan, Zhi-Jie and Zou, Xian-Wu and Zhang, Wen-Bing and Jin, Zhun-Zhi},
@ -122,8 +210,61 @@
issn = {03759601},
doi = {10.1016/S0375-9601(00)00143-2},
url = {https://linkinghub.elsevier.com/retrieve/pii/S0375960100001432},
urldate = {2023-03-13},
langid = {english}
urldate = {2023-03-15},
abstract = {The influence of external electric field on diffusion-limited aggregation ŽDLA. has been investigated by computer simulations. When the parameter l increases from 0 to `, the morphology of aggregates changes from pure DLA to chain-like pattern gradually, where l stands for the relative strength of field-induced dipolar interaction to thermal energy. The structure transition is the transition between a prototype disorder structure and a relative order one in essence. The reason of the transition is the interaction controlling systems changes from thermal force to field-induced dipolar interactions with l rising. q 2000 Elsevier Science B.V. All rights reserved.},
langid = {english},
file = {/Users/joshuacoles/Zotero/storage/34TKRN2A/Tan et al. - 2000 - Influence of external field on diffusion-limited a.pdf}
}
@article{tentiFractalDimensionDiffusionlimited2021,
title = {Fractal Dimension of Diffusion-Limited Aggregation Clusters Grown on Spherical Surfaces},
author = {Tenti, J. M. and Hernández Guiance, S. N. and Irurzun, I. M.},
date = {2021-01-29},
journaltitle = {Physical Review E},
shortjournal = {Phys. Rev. E},
volume = {103},
number = {1},
pages = {012138},
issn = {2470-0045, 2470-0053},
doi = {10.1103/PhysRevE.103.012138},
url = {https://link.aps.org/doi/10.1103/PhysRevE.103.012138},
urldate = {2023-03-15},
langid = {english},
file = {/Users/joshuacoles/Zotero/storage/G4IEQGY6/tenti2021.pdf.pdf}
}
@article{tokuyamaFractalDimensionsDiffusionlimited1984,
title = {Fractal Dimensions for Diffusion-Limited Aggregation},
author = {Tokuyama, M. and Kawasaki, K.},
date = {1984-02},
journaltitle = {Physics Letters A},
shortjournal = {Physics Letters A},
volume = {100},
number = {7},
pages = {337--340},
issn = {03759601},
doi = {10.1016/0375-9601(84)91083-1},
url = {https://linkinghub.elsevier.com/retrieve/pii/0375960184910831},
urldate = {2023-03-15},
langid = {english},
file = {/Users/joshuacoles/Zotero/storage/9XURJU5X/tokuyama1984.pdf.pdf;/Users/joshuacoles/Zotero/storage/JGBE6A2Y/Tokuyama and Kawasaki - 1984 - Fractal dimensions for diffusion-limited aggregati.pdf}
}
@article{turkevichProbabilityScalingDiffusionlimited1986,
title = {Probability Scaling for Diffusion-Limited Aggregation in Higher Dimensions},
author = {Turkevich, Leonid A. and Scher, Harvey},
date = {1986-01-01},
journaltitle = {Physical Review A},
shortjournal = {Phys. Rev. A},
volume = {33},
number = {1},
pages = {786--788},
issn = {0556-2791},
doi = {10.1103/PhysRevA.33.786},
url = {https://link.aps.org/doi/10.1103/PhysRevA.33.786},
urldate = {2023-03-15},
langid = {english},
file = {/Users/joshuacoles/Zotero/storage/FSHPX9RK/turkevich1986.pdf.pdf}
}
@book{vicsekFractalGrowthPhenomena1992,
@ -137,6 +278,7 @@
langid = {english},
pagetotal = {488},
keywords = {Fractals},
note = {Semi-vetted, source Aengus},
file = {/Users/joshuacoles/Zotero/storage/LJP5C4WG/Vicsek - 1992 - Fractal growth phenomena.pdf}
}
@ -154,5 +296,22 @@
url = {https://link.aps.org/doi/10.1103/PhysRevB.27.5686},
urldate = {2023-02-24},
langid = {english},
note = {Unvetted, source google},
file = {/Users/joshuacoles/Zotero/storage/ZP38RVBK/ZP38RVBK.pdf}
}
@article{wuDependenceFractalDimension2013,
title = {Dependence of Fractal Dimension of {{DLCA}} Clusters on Size of Primary Particles},
author = {Wu, Hua and Lattuada, Marco and Morbidelli, Massimo},
date = {2013-07},
journaltitle = {Advances in Colloid and Interface Science},
shortjournal = {Advances in Colloid and Interface Science},
volume = {195--196},
pages = {41--49},
issn = {00018686},
doi = {10.1016/j.cis.2013.04.001},
url = {https://linkinghub.elsevier.com/retrieve/pii/S0001868613000353},
urldate = {2023-03-15},
langid = {english},
file = {/Users/joshuacoles/Zotero/storage/KMWTNPKN/wu2013.pdf.pdf}
}

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report.tex
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\input prelude.tex
\addbibresource{references.bib}
\addbibresource{static.bib}
\newcommand{\singlecolumnabstract}[1]{
\twocolumn[
\begin{@twocolumnfalse}
\maketitle
\begin{abstract}
#1
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\setlength{\marginparwidth}{1.2cm}
% What I wish the title was: Development and Testing of a generalised computational model for efficient diffusion limited aggregation modelling and experimentation.
@ -35,19 +22,27 @@ Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Aenean commodo ligula
\section*{Introduction}
Diffusion-limited aggregation (DLA) models processes where the diffusion of small particles into a larger cluster is the limiting factor in a system's growth. It is applicable to a wide range of systems such as,
Diffusion-limited aggregation (DLA) models processes where the diffusion of small particles into a larger aggregate is the limiting factor in a system's growth. It is applicable to a wide range of systems such as, A, B, and C.
\begin{enumerate}
\item \todo{think of better workds}
\end{enumerate}
This process gives rise to structures which are fractal in nature (for example see Figure \ref{dla-eg}), ie objects which contain detailed structure at arbitrarily small scales. These objects are associated with a fractal dimension, $\mathrm{fd}$ or $df$. This number relates how measures of the object, such as mass, scale when the object itself is scaled. For non fractal this will be its traditional dimension: if you double the scale of a square, you quadruple its area, $2 ^ 2$; if you double the scale of a sphere, you octuple its volume, $2 ^ 3$. For a DLA aggregate in a 2D embedding space, its "traditional" dimension would be 1, it is not by nature 2D, but due to its fractal dimension it has a higher fractal dimension higher than that.
We will model diffusion limited aggregation under a number of novel and unusual conditions to determine the effects of various physical and modelling properties \todo{better word} on the system
% TODO We need to clean up the symbol
% TODO Source the fractal dimension
In this paper we will consider a number of alterations the standard DLA process and the effect they have on the fractal dimension of the resulting aggregate. This data will be generated by a number of computational models derived initially from the code provided \cite{IPC} but altered and optimised as needed for the specific modelling problem.
% Mention MVA I think so I can reference it in the section on spaces alteration.
% TODO Explain Fractal Dimension
\begin{figure}[t]
\includegraphics[width=\columnwidth]{figures/dla-eg}
\caption{A $5000$ particle aggregate on}
\label{dla-eg}
\end{figure}
\section*{Discussion}
In physical systems modelled by the DLA process the growing aggregate is sitting within a medium through which particles diffuse until "sticking" to the aggregate adding to its extent. A high level description of the DLA algorithm is given as follows,
As mentioned the DLA process models the growth of an aggregate (otherwise known as a cluster) within a medium through which smaller free moving particles can diffuse. These particles move freely until they "stick" to the aggregate adding to its extent. A high level description of the DLA algorithm is given as follows,
\begin{enumerate}
\item An initial seed aggregate is placed into the system, without mathematical loss of generality, at the origin. This is normally a single particle.
@ -56,7 +51,7 @@ In physical systems modelled by the DLA process the growing aggregate is sitting
\item At this point the new particle stops moving and becomes part of the aggregate a new particle is released.
\end{enumerate}
An actual implementation of this system will involve a number of computational parameters determining for example the radius that the particles are spawned at, or defining a "kill circle" past which we consider the particle lost and stop simulating it \cite[p.~27]{sanderDiffusionlimitedAggregationKinetic2000} (this is especially important in $d > 2$ dimensional spaces where random walks are not guaranteed to reoccur \cite{lawlerIntersectionsRandomWalks2013} and could instead tend off to infinity).
An actual implementation of this system will involve a number of computational parameters and simplification for computational modelling. For example particles are spawned at a consistent radius from the aggregate, $r_{\mathrm{add}}$, rather than existing uniformly throughout the embedding medium. Further it is traditional to define a "kill circle", $r_{\mathrm{kill}}$ past which we consider the particle lost and stop simulating it \cite[p.~27]{sanderDiffusionlimitedAggregationKinetic2000} (this is especially important in $d > 2$ dimensional spaces where random walks are not guaranteed to reoccur \cite{lawlerIntersectionsRandomWalks2013} and could instead tend off to infinity).
While these are interesting and important to the performant modelling of the system, we aim to choose these such to maximise the fidelity to the original physical system, whilst minimising the computational effort required for simulation. From a modelling perspective however there are a number of interesting orthogonal behaviours within this loose algorithm description which we can vary to potentially provide interesting results.
@ -64,37 +59,110 @@ The first is the seed which is used to start the aggregation process. The tradit
The next behaviour is in the spawning of the active particle. The choice of spawning location is traditionally made in accordance to a uniform distribution, which bar any physical motivation from a particular system being modelled, seems to intuitive choice. However the choice of a single particle is one which is open to more investigation. This is interesting in both the effect varying this will have on the behaviour of the system, but also if it can be done in a way to minimise the aforementioned effects, as a speed up for long running simulations.
Another characteristic behaviour of the algorithm is the choice of diffusion mechanism. Traditionally this is implemented as a
Another characteristic behaviour of the algorithm is the choice of diffusion mechanism. Traditionally this is implemented as a random walk, with each possible neighbour being equally likely. This could be altered for example by the introduction of an external force to the system.
Finally we arrive at the final characteristic we will consider: the space that the DLA process takes place within. Traditionally this is done within a 2D orthogonal gridded space, however other gridded systems, such as hexagonal, can be used to explore any effect the spaces \cite[pp.~210-211]{sanderDiffusionlimitedAggregationKinetic2000}.
%TODO Include a note on long running and exploration simulations in the methodology section?
\subsection*{Method}
To this end we designed a generic system such that that these different alterations of the traditional DLA model could be written and explored quickly, collecting sufficient data for statistical measurements.
%TODO Include a note on long running and exploration simulations in the methodology section?
We first took the initially provided code \cite{IPC} and made minimal alterations such such that the code ran in reasonable time\footnote{When running on macOS systems the rendering code slows down the model by several orders of magnitude making it unsuitable for large scale modelling, hence it is removed, visualisation was handled externally.} and output data for analysis.
To this end we designed a generic system such that these different alterations of the traditional DLA model could be written, explored, and composed quickly, whilst generating sufficient data for statistical measurements.
For large configuration space exploring runs the code was run using \cite{GNUParallel} to allow for substantially improved throughput.
This code was based off the
The code was written such that it is reproducible based on a user provided seed for the random number generator, this provided the needed balance between reproducibility and repeated runs.
The code was written such that it is reproducible based on a user provided seed for the random number generator, this provided the needed balance between reproducibility and repeated runs. Instructions for building the specific models used in the paper can be found in the appendix.
\subsection*{Fractal Dimension Calculation}
There are two methods of determining the fractal dimension. The first is by noting that the number of particles in an aggregate $N_c$ grows with the maximum radius $r_\mathrm{max}$ as
\section*{Alteration 1: Probabilistic Sticking}
\begin{equation*}
N_c(r_{\mathrm{max}}) = (\alpha r_{\mathrm{max}})^{df} + \beta
\end{equation*}
The first alteration of the system is the introduction of a probabilistic component to the sticking behaviour of the DLA system. Here we introduced a probability $p_{stick}$ to the initial grid based sticking behaviour of the particles, with the particle being given this probability to stick at each site (for example, if the particle was adjacent to two cells in the aggregate, then the probabilistic aspect would apply twice).
where $\alpha, \beta$ are two unknown constants. Taking the large $r_\mathrm{max}$ limit we can take $(\alpha r_{\mathrm{max}})^{df} \gg \beta$ and hence,
This was also the case used to ground both the minimally altered code, and our new generic system, to ensure they are functioning correctly. The data for both is presented in Figure \ref{sp-fig}.
\begin{align*}
N_c(r_{\mathrm{max}}) &= (\alpha r_{\mathrm{max}})^{df} + \beta \\
&\approx (\alpha r_{\mathrm{max}})^{df} \\
\log N_c &\approx df \cdot \log\alpha + df \cdot \log r_{\mathrm{max}} \\
\end{align*}
from which we can either perform curve fitting on our data, or by taking $\alpha = 1$ and hence giving us,
\begin{align*}
\log N_c &= df \cdot \log r_{\mathrm{max}} \\
df &= \frac{\log N_c}{\log r_{\mathrm{max}}}
\end{align*}
This gives us a way to determine "instantaneous" fractal dimension.
A second method for determining the fractal dimension is known as box-count \cite{smithFractalMethodsResults1996a}. This involves placing box-grids of various granularities onto the aggregate and observing the number of boxes which have at least one particle within them. The number of these boxes $N$ should grow as,
\begin{equation*}
N \propto w^{-d}
\end{equation*}
where $w$ is the granularity of the box-grid and $d$ is the fractal dimension we wish to find. By a similar process as before we end up with,
\begin{equation*}
\log N = \log N_0 - d \log w
\end{equation*}
where $N_0$ is some proportionality constant. We will expect a plot of $(w, N)$ to exhibit two modes of behaviour,
\begin{enumerate}
\item A linear region from which we can extract fractal dimension data.
\item A saturation region where the box-grid is sufficiently fine such there each box contains either $1$ or none particles.
\end{enumerate}
we will fit on the linear region, dropping some data for accuracy.
\todo{How much of this is actually in the Fractal Dimension section}
\section{Results}
\subsection*{Preliminary Work: Testing Initial Implementation and Fractal Dimension Calculations}
\label{ii-fdc}
\begin{figure}
\includegraphics[width=\columnwidth]{newton.png}
\caption{Sticking probability verThe minimally altered code and new generic system }
\label{sp-fig}
\includegraphics[width=\columnwidth]{figures/rmax-n.png}
\caption{The growth of $N$ vs $r_{\mathrm{max}}$ for $20$ runs of the standard DLA model. When fitting the first $1000$ points are not included when fitting to improve accuracy. The
$50$ data points are not included as the data contains to much noise to be meaningfully displayed. Also included in the figure is the value from theory, $1.71 \pm 0.01$ from \cite[Table 1, $\langle D(d = 2)\rangle$]{nicolas-carlockUniversalDimensionalityFunction2019}.}
\label{rmax-n}
\end{figure}
\begin{figure}[t!]
\includegraphics[width=\columnwidth]{figures/nc-fd-convergence.png}
\caption{The converge of the fractal dimension of $20$ runs of the standard DLA model. This uses the the simple calculation method. The first $50$ data points are not included as the data contains to much noise to be meaningfully displayed. Also included in the figure is the value from theory, $1.71 \pm 0.01$ from \cite[Table 1, $\langle D(d = 2)\rangle$]{nicolas-carlockUniversalDimensionalityFunction2019}.}
\label{nc-fd-convergence}
\end{figure}
To start we do $20$ runs, with seeds $1, 2, \dots, 20$, of the standard DLA model using the minimally altered initially provided code. The fractal dimension is calculated using the simple method \todo{do I want to ref this?} and averaged across the $20$ runs. This is shown in Figure \ref{nc-fd-convergence} along with the result from literature, $d = 1.7 \pm 0.6$ \cite[Table 1, $\langle D(d = 2)\rangle$]{nicolas-carlockUniversalDimensionalityFunction2019}.
\todo{Do I need to find a grid value for this I think this might be continuous}
Taking an average of the trailing $5000$ readings we come to a value of $d = 1.73$ \todo{errors and rounding}, while this values diverges slightly from the value reported in literature, this result provides a reasonable grounding for our model as being roughly correct and provides a useful point of comparison for future work.
This also allows us to say with reasonable confidence that we can halt our model around $N_C = 5000$ as a reasonable trade off between computational time and accuracy. This should be verified for particular model variations however.
\subsection{Probabilistic Sticking}
\begin{figure}[t!]
\includegraphics[width=\columnwidth]{figures/sp-fd}
\caption{The fractal dimension for the DLA system on a 2D grid lattice a sticking probability $p_{stick}$. This value was obtained from $100$ runs with different seeds, by computing the value of the fractal dimension using the simple method, taking a mean across the last $100$ measurements on a $2000$ particle cluster.
% TODO These numbers are way too small given the results of Figure 1.
}
\label{sp-fd}
\end{figure}
As discussed one of the possible alterations of the system is the introduction of a probabilistic component to the sticking behaviour of the DLA system. Here we introduced a probability $p_{stick}$ to the initial grid based sticking behaviour of the particles, with the particle being given this probability to stick at each site (for example, if the particle was adjacent to two cells in the aggregate, then the probabilistic aspect would apply twice).
This was also the case used to ground both the minimally altered code, and our new generic system, to ensure they are functioning correctly. The data for both is presented in Figure \ref{sp-fig}. As we would expect we see a g
\section*{Hexagonal}
% TODO Formatting of these (for one its in american date formats ughhh)