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% TODO Do I want a TOC?
%\tableofcontents

\section*{Introduction}

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. 

% TODO Provide examples

\begin{figure}[htb]
\includegraphics[width=\columnwidth]{figures/dla-eg}
\caption{A $5000$ particle aggregate on a 2D square grid.}
\label{dla-eg}
\end{figure}

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.

% 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 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.

\begin{figure}[htb]
\includegraphics[width=\columnwidth]{figures/scale-comparison.png}
\caption{A $5000$ and $10000$ particle aggregate scaled to fill the same physical space. Note the similar structure and pattern between the two objects.}
\label{scale-comparison}
\end{figure}

% TODO Do I want to show something akin to the comparison image with a 2x2 grid of different sizes?
% TODO Extension, can do we do something akin to renormalisation with that scaling property?

\section*{Discussion}

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,

\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.
	\item A new particle is then released at some sufficient distance from the seeded aggregate.
	\item This particle is allowed to then diffuse until it sticks to the aggregate.
	\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 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 \enquote{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.

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}.

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 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}.

We will explore a number of these alterations in the report that follows.

\section*{Method}

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.


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).

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.

% TODO Verify stats for said statistical measurements!!!
%\subsection*{Statistical Considerations}
% TODO Is this something we need to talk about? Or should it be in the appendix?

\subsection*{Fractal Dimension Calculation}

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}.

% TOOD Replace simple method with mass method

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

\begin{equation*}
N_c(r_{\mathrm{max}}) = (\alpha r_{\mathrm{max}})^{df} + \beta
\end{equation*}

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,

\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. 

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,

\begin{align*}
\log N_c &= df \cdot \log r_{\mathrm{max}} \\
df       &= \frac{\log N_c}{\log r_{\mathrm{max}}}
\end{align*}

giving us a way to determine \enquote{instantaneous} fractal dimension at any particular point the modelling process. 

% TODO If we don't end up using this, bin this section it is just going to be 

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}
