Pre final spell check and code assembly
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@ -27,8 +27,6 @@ Once rough accordance with literature was obtained (see Figure \ref{nc-fd-conver
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This then provided sufficient data for us to transition to our new generic framework, verifying that it agreed with this dataset to ensure correctness.
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%TODO Should we reference git commits here? Or keep them all in one repo. Maybe a combo and have them as submodules in a report branch allowing for a linear history and also concurrent presentation for a report.
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\subsection*{Auxiliary Programs}
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@ -1,31 +1,25 @@
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\singlecolumnabstract{
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Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Aenean commodo ligula eget dolor. Aenean massa. Cum sociis natoque penatibus et magnis dis parturient montes, nascetur ridiculus mus. Donec quam felis, ultricies nec, pellentesque eu, pretium quis, sem. Nulla consequat massa quis enim. Donec pede justo, fringilla vel, aliquet nec, vulputate eget, arcu. In enim justo, rhoncus ut, imperdiet a, venenatis vitae, justo. Nullam dictum felis eu pede mollis pretium. Integer tincidunt.
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Diffusion-limited aggregation is a well known model simulating the growth of complex bodies across a range of disciplines. Modelling the process under a variety of conditions is useful in exploring its behaviours in novel applications. Here we discuss possible altered conditions for the DLA model and discuss the development of a framework to test this behaviour. Of these conditions we determine a fractal dimension for the standard DLA model in 2D as $\mathrm{fd} = 1.735 \pm 0.020$ and in 3D as $\mathrm{fd} = 2.03 \pm 0.06$, as well as exploring the change in the fractal dimension in these two settings, when introducing probabilistic sticking behaviour
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}
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% TODO Write abstract
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\medskip
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% TODO Do I want a TOC?
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%\tableofcontents
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\section*{Introduction}
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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.
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% TODO Provide examples
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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 fields and systems such as:
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the growth of dust particles, modelling dielectric breakdown, and in urban growth.
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\begin{figure}[htb]
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\includegraphics[width=\columnwidth]{figures/dla-eg}
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\caption{A $5000$ particle aggregate on a 2D square grid.}
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\caption{A $5000$ particle aggregate on a 2D square grid, the lighter colours being placed later in the process.}
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\label{dla-eg}
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\end{figure}
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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, $\mathrm{fd}$, (occasionally written as $df$ or $d$). 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 \enquote{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. Fractals are often associated with a scale invariance, ie they have the same observables at various scales. This can be observed for DLA aggregates in Figure \ref{scale-comparison} where we have two aggregates of different sizes, scaled as too fill the same physical space.
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This process gives rise to structures which are fractal in nature (for example see Figure \ref{dla-eg}), i.e. objects which contain detailed structure at arbitrarily small scales. These objects are associated with a fractal dimension, $\mathrm{fd}$, (occasionally written as $\mathrm{fd}$ or $d$). 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 \enquote{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. Fractals are often associated with a scale invariance, i.e. they have the same observables at various scales. This can be observed for DLA aggregates in Figure \ref{scale-comparison} where we have two aggregates of different sizes, scaled as to fill the same physical space.
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% TODO We need to clean up the symbol
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% TODO Source the fractal dimension
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In this paper we will consider a number of alterations the standard DLA process and the affect 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.
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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.
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\begin{figure}[htb]
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\includegraphics[width=\columnwidth]{figures/scale-comparison.png}
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@ -33,9 +27,6 @@ In this paper we will consider a number of alterations the standard DLA process
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\label{scale-comparison}
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\end{figure}
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% TODO Do I want to show something akin to the comparison image with a 2x2 grid of different sizes?
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% TODO Extension, can do we do something akin to renormalisation with that scaling property?
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\section*{Discussion}
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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 \enquote{stick} to the aggregate adding to its extent. A high level description of the DLA algorithm is given as follows,
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@ -53,54 +44,47 @@ While these are interesting and important to the performant modelling of the sys
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The first is the seed which is used to start the aggregation process. The traditional choice of a single seed models the spontaneous growth of a cluster, but the system could be easily extended to diffusion onto a plate under influence of an external force field \cite{tanInfluenceExternalField2000}, or cluster-cluster aggregation where there are multiple aggregate clusters, which are capable of moving themselves \cite[pp.~210-211]{sanderDiffusionlimitedAggregationKinetic2000}.
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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.
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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.
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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.
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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}.
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Finally, we arrive at the last 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}.
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We will explore a number of these alterations in the report that follows.
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\section*{Method}
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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. This involved separating the various orthogonal behaviours of the DLA algorithm into components which could be combined in a variety of ways enabling a number of distinct models to be exist concurrently within the same codebase.
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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. This involved separating the various orthogonal behaviours of the DLA algorithm into components which could be combined in a variety of ways enabling a number of distinct models to be coexist within the same codebase.
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This code was based off the initially provided code, altered to allow for data extraction and optimised for performance. For large configuration space exploring runs the code was run using GNU Parallel \nocite{GNUParallel} to allow for substantially improved throughput (this is opposed to long running, high $N$ simulations where they were simply left to run).
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This code was based off the initially provided code (IPC), altered to allow for data extraction and optimised for performance. For large configuration space exploring runs the code was run using GNU Parallel \nocite{GNUParallel} to allow for substantially improved throughput (this is opposed to long-running, high $N$ simulations where they were simply left to run).
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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.
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% TODO Verify stats for said statistical measurements!!!
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%\subsection*{Statistical Considerations}
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% TODO Is this something we need to talk about? Or should it be in the appendix?
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\subsection*{Fractal Dimension Calculation}
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We will use two methods of determining the fractal dimension of our aggregates. The first is the mass method and the second box-count\cite{smithFractalMethodsResults1996a}.
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% TOOD Replace simple method with mass method
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We will use two methods of determining the fractal dimension of our aggregates. The first is the mass method and the second box-count \cite{smithFractalMethodsResults1996a}.
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For the mass method we note that the number of particles in an aggregate $N_c$ grows with the maximum radius $r_\mathrm{max}$ as
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\begin{equation*}
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N_c(r_{\mathrm{max}}) = (\alpha r_{\mathrm{max}})^{df} + \beta
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N_c(r_{\mathrm{max}}) = (\alpha r_{\mathrm{max}})^{\mathrm{fd}} + \beta,
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\end{equation*}
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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,
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where $\alpha, \beta$ are two unknown constants. Taking the large $r_\mathrm{max}$ limit we can take $(\alpha r_{\mathrm{max}})^{\mathrm{fd}} \gg \beta$ and hence,
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\begin{align*}
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N_c(r_{\mathrm{max}}) &= (\alpha r_{\mathrm{max}})^{df} + \beta \\
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&\approx (\alpha r_{\mathrm{max}})^{df} \\
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\log N_c &\approx df \cdot \log\alpha + df \cdot \log r_{\mathrm{max}} \\
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N_c(r_{\mathrm{max}}) &= (\alpha r_{\mathrm{max}})^{\mathrm{fd}} + \beta \\
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&\approx (\alpha r_{\mathrm{max}})^{\mathrm{fd}} \\
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\log N_c &\approx \mathrm{fd} \cdot \log\alpha + \mathrm{fd} \cdot \log r_{\mathrm{max}} \\
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\end{align*}
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from which we can either perform curve fitting on our data.
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In addition if we take $\alpha = 1$ as this is an entirely computational model and we can set our length scales without loss of generality we obtain,
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In addition if we take $\alpha = 1$, as this is an entirely computational model and we can set our length scales without loss of generality we obtain,
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\begin{align*}
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\log N_c &= df \cdot \log r_{\mathrm{max}} \\
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df &= \frac{\log N_c}{\log r_{\mathrm{max}}}
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\log N_c &= \mathrm{fd} \cdot \log r_{\mathrm{max}} \\
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\mathrm{fd} &= \frac{\log N_c}{\log r_{\mathrm{max}}}
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\end{align*}
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giving us a way to determine \enquote{instantaneous} fractal dimension at any particular point the modelling process.
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@ -126,6 +110,4 @@ where $N_0$ is some proportionality constant. We will expect a plot of $(w, N)$
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\item A saturation region where the box-grid is sufficiently fine such there each box contains either $1$ or none particles.
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\end{enumerate}
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we will fit on the linear region, dropping some data for accuracy.
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\todo{How much of this is actually in the Fractal Dimension section}
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We will fit on the linear region, dropping some data for accuracy.
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@ -10,7 +10,7 @@
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%\usepackage[none]{hyphenat}
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\usepackage{geometry} % to change the page dimensions
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\geometry{a4paper, left=17.5mm, right=17.5mm, textwidth=85mm,columnsep=5mm, top=32mm}
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\geometry{a4paper, left=17.5mm, right=17.5mm, textwidth=85mm,columnsep=5mm, top=30mm}
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\setlength{\parskip}{6pt}
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%\setlength{\belowdisplayskip}{0pt} \setlength{\belowdisplayshortskip}{0pt}
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%\setlength{\abovedisplayskip}{0pt} \setlength{\abovedisplayshortskip}{0pt}
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@ -6,8 +6,6 @@
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\addbibresource{static.bib}
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\setlength{\marginparwidth}{1.2cm}
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% What I wish the title was: Development and Testing of a generalised computational model for efficient diffusion limited aggregation modelling and experimentation.
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\title{\textbf{Modelling Diffusion Limited Aggregation under a Variety of Conditions}}
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\author{Candidate Number: 24829}
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\affil{Department of Physics, University of Bath}
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@ -17,7 +15,6 @@
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\input{introduction-dicussion-method.tex}
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\input{results.tex}
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% TODO Formatting of these (for one its in american date formats ughhh)
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\printbibliography
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\input appendix
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64
results.tex
64
results.tex
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\begin{figure}[t]
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\includegraphics[width=\columnwidth]{figures/rmax-n.png}
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\caption{The growth of $N$ vs $r_{\mathrm{max}}$ for $20$ runs of the standard DLA model. Also included is a line of best fit for the data, less the first $50$ which are removed to improve accuracy.
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% TODO Check all of my captions are correct.
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% TODO Add information for this
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\caption{The growth of $N$ vs $r_{\mathrm{max}}$ for $20$ runs of the standard DLA model to a maximum value of $N_C = 10000$. Also included is a line of best fit for the data, less the first $50$ which are removed to improve accuracy, with form $\log N_C = a_0 + \mathrm{fd} \cdot \log r_{\mathrm{max}}$ and coefficients $\mathrm{fd} = 1.7685 \pm 0.0004$, $a_0 = -0.1815 \pm 0.002$. % TODO Verify rounding
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}
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\label{rmax-n}
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\end{figure}
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@ -14,16 +12,16 @@
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\begin{figure}[hbt]
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\includegraphics[width=\columnwidth]{figures/nc-fd-convergence.png}
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\caption{The converge of the fractal dimension of $20$ runs of the standard DLA model. This uses the mass 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 literature, $1.71 \pm 0.01$ from \cite[Table 1, $\langle D(d = 2)\rangle$]{nicolas-carlockUniversalDimensionalityFunction2019}.}
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\caption{The converge of the fractal dimension of $20$ runs of the standard DLA model. This uses the mass 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 literature, $\mathrm{fd} = 1.71 \pm 0.01$ from \cite[Table 1, $\langle D(d = 2)\rangle$]{nicolas-carlockUniversalDimensionalityFunction2019}.}
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\label{nc-fd-convergence}
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\end{figure}
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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 mass method 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}.
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To start we do $20$ runs, with seeds $1, 2, \dots, 20$, of the standard DLA model using the minimally altered IPC. We use both the instantaneous and line-fitting mass method, as shown in Figure \ref{nc-fd-convergence} and Figure \ref{rmax-n} respectively. For the instantaneous case, the fractal dimension is calculated using the mass method and averaged across the $20$ runs. This is shown in Figure \ref{nc-fd-convergence} along with the result from literature, $\mathrm{fd} = 1.7 \pm 0.6$ \cite[Table 1, $\langle D(d = 2)\rangle$]{nicolas-carlockUniversalDimensionalityFunction2019}.
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% TODO Errors
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Taking an average of the trailing $5000$ readings we come to a value of $fd = 1.73$. As can be seen on the figure this is divergence from the literature (we suspect due to the gridded nature of the embedding space) the result is reasonable and consistent across runs. We consider this, along with the sourcing of the initially provided code, to be sufficient grounding the start of our trust chain.
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Taking an average of the trailing $5000$ readings we come to a value of $\mathrm{fd} = 1.735 \pm 0.020$. As can be seen on the figure this is divergence from the literature (we suspect due to the gridded nature of the embedding space) the result is reasonable and consistent across runs. We consider this, along with the sourcing of the IPC, to be sufficient grounding the start of our trust chain.
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This also allows us to say with reasonable confidence that we can halt our model around $N_C = 5000$ as a trade off between computational time and accuracy. This should be verified for particular model variations however.
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This also allows us to say with reasonable confidence that we can halt our model around $N_C = 5000$ as a trade-off between computational time and accuracy. However care must be taken to verify this is appropriate for any particular model variation.
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\subsection*{Probabilistic Sticking}
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@ -42,49 +40,43 @@ This also allows us to say with reasonable confidence that we can halt our model
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The first alteration we shall make to the DLA model is the introduction of a probabilistic component to the sticking behaviour. We parametrise this behaviour by a sticking probability $p_{stick} \in (0, 1]$, 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).
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Comparing first the clusters for different values of $p_{stick}$ we can see in Figure \ref{sp-dla-comparison} a clear thickening of the arms with lower values of $p_{stick}$. This aligns with data for the fractal dimension, as seen in Figure \ref{sp-fd}, with thicker arms bringing the cluster closer to a non-fractal two dimensional object.
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Comparing first the clusters for different values of $p_{stick}$ we can see in Figure \ref{sp-dla-comparison} a clear thickening of the arms with lower values of $p_{stick}$. This aligns with data for the fractal dimension, as seen in Figure \ref{sp-fd}, with thicker arms bringing the cluster closer to a non-fractal two-dimensional object.
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In the low $p_{stick}$ domain we record values of $\mathrm{fd} > 2$ which is the embedding domain. This is unexpected and points towards a possible failure of our mass-method fractal dimension calculation. More work and analysis is required to verify these results.
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% TODO Conclusion point
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In the low $p_{stick}$ domain we record values of $\mathrm{fd} > 2$, greater that the 2D space they are embedded within. This is unexpected and points towards a possible failure of our mass-method fractal dimension calculation. More work and analysis is required to verify these results.
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As discussed in the Appendix, \nameref{generic-dla}, this also provides the next chain of grounding between the initially provided code, and the new generic framework. Further details can be found in the aforementioned appendix.
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%As discussed in the Appendix, \nameref{generic-dla}, this also provides the next chain of grounding between the initially provided code, and the new generic framework (see the aforementioned appendix for more).
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\subsection*{Higher Dimensions}
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The next alteration to explore is changing the embedding space to be higher dimensional. Here we use a k-dimensional tree structure to store the aggregate as opposed to an array based grid allowing us to greatly reduce memory consumption ($O(\text{grid\_size}^D) \to O(n)$ where $n$ is the number of particles in the aggregate) whilst retaining a strong access and search time complexity of $O(n \log n)$\cite{bentleyMultidimensionalBinarySearch1975}.
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\begin{figure}[hbt]
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\includegraphics[width=\columnwidth]{figures/3d-eg}
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\caption{A 3D DLA aggregate on a 3D orthogonal grid, with $N_C = 5000$, coloured by deposition time. Note that the view of the particles as spheres is an artifact of the rendering process, they are in fact cubes. Here we can observe the expected tendril structure of a DLA aggregate.}
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\label{3d-eg}
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\end{figure}
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To start we model two styles of random walk: direct, where only those cells which are directly adjacent to its current location current location are accessible; off-axis, where all the full $3 \times 3 \times 3$ cubic (bar the centre position) are available. The ($N_c, fd$) correspondence is shown in Figure \ref{3d-nc-fd-convergence} where we can see that both walk methods, as expected produce identical results, varying only slightly from a naive implementation in the initially provided codebase included to ensure correct behaviour. These off axis walks do however offer a speed boost as the larger range of motion leads to faster movement within the space.
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Modelling the system across the range of $p_{stick}$ we obtain results as shown in Figure \ref{sp-fd-2d-3d}. These show a similar pattern as was seen in the 2D case of Figure \ref{sp-fd}. We note that whilst these lines are similar, they are not parallel showing distinct behaviour.
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At this moment we do not have an analytical form for this relation but further work may provide such.
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% TODO Conclusion point
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\begin{figure}
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\begin{figure}[hbt]
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\includegraphics[width=\columnwidth]{figures/sp-fd-2d-3d}
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\caption{A comparison of the fractal dimension of DLA aggregates in 2- and 3-dimensional embedding space. The datasets were obtained by averages of $100$ and $200$ for 2D and 3D respectively, both with data recorded at a increment of $\Delta p_{stick} \approx 0.1$, and an aggregate size of $2000$.}
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\label{sp-fd-2d-3d}
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\end{figure}
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\begin{figure}
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\includegraphics[width=\columnwidth]{figures/3d-nc-fd-convergence}
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\caption{A comparison of direct and off-axis walks in 3 dimensions, using both the new framework (NF) and the initial provided code (IPC). Note a slight divergence between the NF and IPC lines but a complete agreement between the direct and off-axis walks for the NF. Errors are not displayed as they are to small to be visible on this graph due to the large sample size.} % TODO Verify literature line here and rename it.
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\label{3d-nc-fd-convergence}
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\end{figure}
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%\begin{figure}[hbt]
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%\includegraphics[width=\columnwidth]{figures/3d-nc-fd-convergence}
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%\caption{A comparison of direct and off-axis walks in 3 dimensions, using both the new framework (NF) and the initial provided code (IPC). Note a slight divergence between the NF and IPC lines but a complete agreement between the direct and off-axis walks for the NF. Errors are not displayed as they are to small to be visible on this graph due to the large sample size. Also included is the result from literature, $\mathrm{fd} = 2.51 \pm 0.01$\cite[Table 1, $\langle D(d = 3)\rangle$]{nicolas-carlockUniversalDimensionalityFunction2019}.}
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%\label{3d-nc-fd-convergence}
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%\end{figure}
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% TODO Do I want to do higher dimensions still 4d?
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% TODO how am I going to cope with the fact we don't agree with theory?
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% TOOD Look at theory to see if I can find a curve for these sp-fd graphs or at the very least note similarities and differences between them. "Given the erroneous behaviour for low sp we are uncertain as to the correctness). Maybe take another crack at boxcount since you've mentioned it and it might be interesting.
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The next alteration to explore is changing the embedding space to be higher dimensional, an example of which can be seen in Figure \ref{3d-eg}. Here we use a k-dimensional tree structure to store the aggregate as opposed to an array based grid allowing us to greatly reduce memory consumption ($O(\text{grid\_size}^D) \to O(n)$ where $n$ is the number of particles in the aggregate) whilst retaining a strong access and search time complexity of $O(n \log n)$\cite{bentleyMultidimensionalBinarySearch1975}.
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% MORE EXTENSIONS
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To start we model two forms of random walk: direct, where the particle can only access directly adjacent cells and off-axis, where all the full $3 \times 3 \times 3$ cubic (bar the centre position) are available. These behave identically to each other, varying only slightly from a naive implementation in the IPC included to provide assurance of the correct behaviour. These off axis walks do however offer a speed boost as the larger range of motion leads to faster movement within the space.
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|
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%\subsection*{Continuous Space}
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%
|
||||
%\begin{enumerate}
|
||||
% \item We get a divergence from theory, what happens if we use continuous
|
||||
%\end{enumerate}
|
||||
Modelling the system across the range of $p_{stick}$ we obtain results as shown in Figure \ref{sp-fd-2d-3d}. These show a similar pattern as was seen in the 2D case of Figure \ref{sp-fd}. We note that whilst these lines are similar, they are not parallel showing distinct behaviour.
|
||||
|
||||
Note that the divergence from the value expected from the literature is greater here than in the 2D case, with a fractal dimension reported of $\mathrm{fd} = 2.03 \pm 0.06$. This along with our inability to find a satisfactory analytic form for this behaviour suggests further analysis on different grids is required.
|
||||
|
||||
% Extensions Do I want to do higher dimensions still 4d?
|
||||
% Look at theory to see if I can find a curve for these sp-fd graphs or at the very least note similarities and differences between them. "Given the erroneous behaviour for low sp we are uncertain as to the correctness). Maybe take another crack at boxcount since you've mentioned it and it might be interesting.
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\section*{Conclusion}
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||||
|
||||
In conclusion, we have validated that the new framework provides consistent behaviour aligned with previous models, whilst allowing for a wider range of behaviours to be easily tested. Future work is required to determine analytic or physical explanations for the data presented. However there are a number of pointers to places of immediate interest and avenues waiting to be explored.
|
||||
|
||||
In this report we have presented findings for the fractal dimension of DLA aggregates in 2D and 3D on orthogonal grids, as well as qualitative assessments of the variation of the fractal dimension across a range of sticking probabilities. In addition, we have validated the framework used to be consistent with previous models allowing for quick iteration. Future work is required to determine analytic or physical explanations for the data presented, specifically the $(p_{stick}, \mathrm{fd})$ relation, in addition to identifying the cause of the divergence between reported results and previous models, with literature, possibly through different geometries.
|
||||
|
||||
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Reference in New Issue
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