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Joshua Coles
e9b8aca97f Stash 2023-05-28 14:37:09 +01:00
Joshua Coles
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Joshua Coles
5dc97497b5 Some writing 2023-05-02 16:46:44 +01:00
Joshua Coles
9e0c932109 Stash 2023-05-02 15:10:54 +01:00
Joshua Coles
17a624cc84 Pre final spell check and code assembly 2023-03-20 15:02:42 +00:00
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Ising Model Report.bib Normal file
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@book{yeomansStatisticalMechanicsPhase1992,
title = {Statistical Mechanics of Phase Transitions},
author = {Yeomans, J. M.},
date = {1992},
series = {Oxford Science Publications},
publisher = {{Clarendon Press ; Oxford University Press}},
location = {{Oxford [England] : New York}},
isbn = {978-0-19-851730-6 978-0-19-851729-0},
langid = {english},
pagetotal = {153},
keywords = {Phase transformations (Statistical physics)},
file = {/Users/joshuacoles/Zotero/storage/HZ5HD4LV/HZ5HD4LV.pdf;/Users/joshuacoles/Zotero/storage/TNHQFTLU/(Oxford Science Publications) J. M. Yeomans - Statistical Mechanics of Phase Transitions-Oxford University Press, USA (1992).pdf}
}

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\section*{Appendix}
\subsection*{Generic DLA Model}
\label{generic-dla}
The main tool used to generate the data in this report was the generic DLA framework written to support the report. Here we will briefly discuss the process of creating and verifying this framework.
An intrinsic problem of developing computational models for exploratory work is the question of correctness: is some novel result you find a bug in your model, or exactly the interesting new behaviour you set out to explore.
To mitigate this issue the model for this system was created iteratively, with each step being checked against the last where their domains overlap (naturally the newer model is likely to cover a superset of the domain of the old model so there will be some areas where they do not) and unit testing of specific behaviours and verifying expectations\fnmark{unit-test-egs}.
\fntext{unit-test-egs}{Examples that came up in development include: ensuring our uniform random walks are indeed uniform, that they visit all the desired neighbours, etc}
This creates a chain of grounding between one model and the next, where our trust model $N+1$ is grounded in our trust of model $N$ and our unit testing however this trust chain, depends on our trust of $N = 0$, the initially provided code. For this we rely on both the extensive history of the code, and (rough) agreement with literature (see the results section for this comparison).
To this end, starting with the initially provided code we made the minimal alterations necessary such that it would run in reasonable time\fnmark{macos-speed} and output the data required for later analysis. This was done explicitly with the goal of perturbing the initial code's behaviour as little as possible, including not performing relatively obvious performance improvement that might introduce bugs (the previously mentioned performance improvements were predominantly code removal as opposed to code change). This allowed us to collect the data we needed and ground the initial model in theory.
\fntext{macos-speed}{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 and replaced with image generation mitigation as discussed later.}
Once rough accordance with literature was obtained (see Figure \ref{nc-fd-convergence}), and most importantly, consistency between runs (verifying against a ill behaved system is a fruitless and painful endeavour), we added the sticking probability alteration as the simplest alteration the DLA algorithm, verifying agreement between the traditional and probabilistic sticking models at $p_{stick} = 1$. See Figure \ref{sp-fd-rust-vs-c} for this comparison.
\begin{figure}[t]
\includegraphics[width=\columnwidth]{figures/sp-fd-rust-vs-c.png}
\caption{A comparison of the reported fractal dimension the probabilistic sticking extension of the Initially Provided Code (IPC + PS) in blue, and the New Framework with probabilistic sticking enabled (NF) in red. We can clearly see a high degree of agreement grounding our new framework and the basic functions of the model.}
\label{sp-fd-rust-vs-c}
\end{figure}
This then provided sufficient data for us to transition to our new generic framework, verifying that it agreed with this dataset to ensure correctness.
%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.
\subsection*{Auxiliary Programs}
A number of auxiliary programs were also developed to assist in the running and visualisation of the model. Most notably was the image generation tool which allowed for the model to focus on one thing: modelling DLA, separating out generating visualisations of the system. This was used to generate images such as that shown in Figure \ref{dla-eg} which are both useful for presentation, and visual qualitative assessment of model correctness. Additional tools can be found under the tools executable of the rust-codebase.

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\singlecolumnabstract{
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.
{\lipsum[1]}
}
% TODO Write abstract
\medskip
% 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}
\begin{figure}[hbt]
\label{eg-phase-diagram}
\includegraphics[width=\columnwidth]{Phase-diag2.png}
\caption{An example phase diagram of water showing different phase transitions and phases separating them. The critical point a the end of the liquid-gaseous phase is clearly shown, beyond this point a supercritical fluid is observed and the thermodynamic variables can be varied without a corresponding discontinuity in a state function. Source \parencite{wiki:phase-digram-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.
In statistical mechanics phase transitions occur when there is a discontinuity singularity in the free energy or one of its derivatives, such as specific heat capacity, density, or other thermodynamic observables. Examples include the liquid-gas transition, the appearance super conductors, or ferromagnetic materials at the Curie Temperature $T_C$. Phase diagrams such as the one shown in Figure \ref{eg-phase-diagram} are used to represent these transitions, with the lines representing where discontinuities are present.
% TODO We need to clean up the symbol
% TODO Source the fractal dimension
As the magnitude of the discontinuity varies across the extent of the phase transition, it will often reach a point such that it becomes $0$ and thus the phase transition line abruptly ends. Beyond this point one can continuously transition between different phases without crossing the now terminated phase transition line.
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.
Of these transitions one of the easiest to model is that of ferromagnets at their Curie Temperature. This involves heating a ferromagnetic material up, possibly in the presence of an external magnetic field $H$, and observing the magnetisation $M$ of the material itself, the order parameter of the system. All phase transitions occur when $H = 0$ due to the symmetry of the system to sign reversal of the external magnetic field, making this a useful model for exploring critical points themselves.
\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}
This has birthed a number of models for ferromagnetic materials around their critical point including: the Ising Model which models the magnetic as a lattice of interacting magnetic spin sites\fnmark{ising-model-generalisations}; Mean Field Theory which models interactions of spins with a hypothetical mean field of all other spins in the material; and Renormalisation Group Theory which is a more general mathematical framework for understanding systems which can exhibit scale invariance.
% 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?
\fntext{ising-model-generalisations}{
There are a number of generalisations of the Ising Model including the Heisenberg Model, XY model, Potts Model, among others which increase the possible states of spins. These however are not discussed in this paper.
}
% https://en.wikipedia.org/wiki/Classical_Heisenberg_model
% https://en.wikipedia.org/wiki/Classical_XY_model
% https://en.wikipedia.org/wiki/Potts_model
%7. **Renormalization Group Theory**: The Renormalization Group (RG) theory is a powerful mathematical framework for studying the critical behavior of ferromagnetic materials. It involves a systematic procedure for studying the behavior of systems at different length scales, enabling the determination of critical exponents, fixed points, and the universality of phase transitions. The RG theory has significantly advanced our understanding of critical phenomena in ferromagnetic systems and beyond.
Of these we will focus most on the Ising Model, which we will use to model ferromagnetic materials under various conditions, comparing it to Mean Field Theory. (\TKK will we mention renormalisation), with the aim of extracting quantitative data (\TKK we are not mentioning critical) about the behaviour of systems at their critical points (\TKK Do we want to mention universality classes? Is that a thing which we can?).
In the discussion we will explore the theory behind the models discussed, as well as phase transitions and critical points themselves. This will be followed by a brief discussion of the Monte-Carlo (\TKK do I need to mention this before) method employed in the simulation and relevant statistical methods. Finally we will present data from these simulations, comparing it to the result from theory and concluding on the effectiveness of our model in desired exploration.
%All transitions occur at zero magnetic field, H = 0, because of the symmetry of a ferromagnet to reversals in the field. The additional symmetry means that it is often easier to work in magnetic language and we shall do so throughout most of this book.
%Of these te
%These phase transitions are occur at phase transition lines on phase diagrams, which termi
%A phase transition occurs when there is a singularity in the free energy or one of its derivatives. What is often visible is a sharp change in the properties of a substance. The transitions from liquid to gas, from a normal conductor to a superconductor, or from paramagnet to ferromagnet are common examples. The phase diagram of a typical fluid is shown in Fig. 1.1.
%As the temperature and pressure are varied water can exist as a solid, a liquid, or a gas. Well-defined phase boundaries separate the regions in which each state is stable. Crossing the phase boundaries there is a jump in the density and a latent heat, signatures of a first-order transition. Consider moving along the line of liquid-gas coexistence. As the temperature increases the difference in density between the liquid and the gas decreases continuously to zero as shown in Fig. 1.2. It becomes zero at the critical point beyond which it is possible to move continuously from a liquid-like to a gas-like fluid. The difference in densities, which becomes non-zero below the critical temperature, is called the order parameter of the liquid- gas transition. Seen on the phase diagram of water the critical point looks insignificant. However, there are clues that this might not be the case. Fig. 1.3 shows the specific heat of argon measured along the critical isochore, p = p-. There is a striking signature of criticality: the specific heat diverges and is infinite at the critical temperature itself. Analogous behaviour is seen in magnetic phase transitions. The phase diagram of a simple ferromagnet is shown in Fig. 1.4. Just as in the case of liquid—gas coexistence there is a line of first-order transitions ending in a critical point. All transitions occur at zero magnetic field, H = 0, because of the symmetry of a ferromagnet to reversals in the field. The additional symmetry means that it is often easier to work in magnetic language and we shall do so throughout most of this book.
%In thermodynamics, a critical point (or critical state) is the end point of a phase equilibrium curve. One example is the liquid–vapor critical point, the end point of the pressure–temperature curve that designates conditions under which a liquid and its vapor can coexist. At higher temperatures, the gas cannot be liquefied by pressure alone. At the critical point, defined by a critical temperature Tc and a critical pressure pc, phase boundaries vanish. Other examples include the liquid–liquid critical points in mixtures, and the ferromagnet–paramagnet transition (Curie temperature) in the absence of an external magnetic field.[2]
%
%Ferromagnetic systems are a useful and widely used model for phase transitions
%
%Phase diagrams, critical points
%
%There are a number of models for ferromagnetic materials critical,
\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,
Here we will discuss the theory behind the different models mentioned as well as phase transitions themselves.
\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}
\subsection*{Thermodynamics}
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).
\begin{figure}[hbt]
\label{thermodynamic-variables}
\includegraphics[width=\columnwidth]{thermodynamic-variables.png}
\caption{The relationship between the partition function and various thermodynamic observables. From \parencite{yeomansStatisticalMechanicsPhase1992}}
\end{figure}
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.
Whichever model we apply standard statistical mechanics to give us relations between the partition function, itself derived from the energy of the system, and other thermodynamic observables. These relationships can be seen in Figure \ref{thermodynamic-variables}, from \cite[p. 17]{yeomansStatisticalMechanicsPhase1992}.
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}.
We obtain the magnetisation and energy from our chosen model, deriving the specific heat capacity, and magnetic susceptibility (labeled as \enquote{Isothermal susceptibility} in the diagram) from these base quantities by the equations,
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.
\begin{align}
c &= \frac{\p}{\p T} \left\langle\frac{E}{N}\right\rangle \\
\chi &= \frac{\p}{\p h} \left\langle\frac{M}{N}\right\rangle.
\end{align}
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.
where $E$ is the total energy of the system, $M$ is total magnetisation of the system and then $N$ is the number of cells in the Ising Model. For convenience we write $m = \frac MN$.
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 can then apply the Fluctuation-dissipation theorem \cite{???} to transform the derivatives in these equations into expressions about the thermodynamic variance of the property at equilibrium. These are much more useful for the purposes of simulation and are given by,
\begin{align}
c &= \frac{1}{N k_B T^2} \Var(E) \\
\chi &= \frac{N}{k_B T} \Var(m).
\end{align}
Which in terms of our dimensionless energy gives us \TKK.
\subsection*{The Ising Model}
%The Ising model was invented by the physicist Wilhelm Lenz (1920), who gave it as a problem to his student Ernst Ising. The one-dimensional Ising model was solved by Ising (1925) alone in his 1924 thesis;[2] it has no phase transition. The two-dimensional square-lattice Ising model is much harder and was only given an analytic description much later, by Lars Onsager (1944). It is usually solved by a transfer-matrix method, although there exist different approaches, more related to quantum field theory.
The Ising Model will be the main model used to explore the behaviour of our system. It was developed by \TKK in \TKK for \TKK.
It is comprised of a grid of spin-sites which can take a value of either $+1$ or $-1$, representing up or down spins respectively. These spins interact in a local manner with their nearest neighbours in the grid. We write the interaction energy as,
\begin{equation}
E = -J\sum_{\ip{ij}} s_i s_j
\end{equation}
where $J > 0$ is the strength of the interaction, $\sum_{\ip{ij}}$ represents a sum over all pairs of nearest neighbours, with the neighbours being written as sites $i, j$\fnmark{sites}. We can see trivially that this energy expression preferences same spin neighbours having $(\pm 1)^2 = 1 \implies E_i < 0$.
\fntext{sites}{When performing a calculation a \emph{site} is mapped to two integers $(x, y) \in \N^2$ representing the location of this site in the overall grid, however in mathematics we abstract this to a single index for conceptual ease.}
In the fully general model this adds to energy from the the externally imposed magnetic field $H$ giving a total energy of,
\begin{equation*}
E = -J\sum_{\ip{ij}} s_i s_j - H \sum_{i} s_i.
\end{equation*}
From these expressions we can obtain the partition function, in the standard manner as,
\begin{equation}
Z(T) = \sum_{\omega \in \Omega} \exp\left(\frac{E(\omega)}{k_B T}\right)
\end{equation}
where $\omega \in \Omega$ is a particular micro-state, corresponding to a particular set of choices for $s_i \in \set{\pm 1}$. This has a total phase space of $\Omega = \set{1, -1}^N$ where $N$ is the number of cells in the grid.
\subsection*{Mean Field Theory}
Whereas the Ising Model restrict interactions of spins to nearest-neighbours, Mean Field Theory, also known as Curie-Weiss theory (\TKK consistent capitalisation of terms), instead chooses to average the effect that each other site on the cell has on a chosen site. In this sense a site interacts with a \enquote{mean field}, representing all other sites in the lattice.
Taking this spin to be $s_0 = \pm 1$ we can determine its energy as
\begin{equation}
E(s_0) = -s_0\left(J\sum_{\ip{0 j}} s_j + H \right)
\end{equation}
where this time one site, $0$, is fixed in our nearest neighbour sum. On a square 2D grid this sum has a constant number of terms, $q = 4$ which allows us to re-write this expression as,
\begin{equation}
E(s_0) = -s_0(qJm + H) - Js_0 \sum_{\ip{0 j}}(s_j - m)
\end{equation}
where $s_j - m$ is the effect of the variation of site $s_j$ from the overall mean on the energy of $s_0$. If we take this variation to be $0$ for all $s_j$ then we obtain,
\begin{equation}
E(s_0) = -s_0(qJm + H)
\end{equation}
which is an expression containing no cross terms and hence allows for treatment as a non-interacting thermodynamic system with
\begin{equation}
\begin{split}
Z_1(s_0) &= \sum_{s \in \set{\pm 1}} \exp(-\beta E(s)) \\
&= 2\cosh({(\beta(qJm + H))}) \\
&= 2\cosh({\beta_0(qm + H)})
\end{split}
\end{equation}
where $\beta_0$ is the dimensionless $\beta$ discussed in \TKK.
\begin{equation}
Z = (Z_1(s_0))^N
\end{equation}
where $N$ is the number of sites in the system. Applying the identities seen in Figure \ref{thermodynamic-variables} we obtain an expression for the magnetisation as,
\begin{equation}
m = \tanh(\beta_0(qJm + H))
\end{equation}
which must be solved to find a self consistent value of $m$ for a given $H, \beta_0$.
\subsection*{Phase transitions and Critical Points}
%
%In dimensions greater than four, the phase transition of the Ising model is described by mean-field theory. The Ising model for greater dimensions was also explored with respect to various tree topologies in the late 1970’s, culminating in an exact solution of the zero-field, time-independent Barth (1981) model for closed Cayley trees of arbitrary branching ratio, and thereby, arbitrarily large dimensionality within tree branches. The solution to this model exhibited a new, unusual phase transition behavior, along with non-vanishing long-range and nearest-neighbor spin-spin correlations, deemed relevant to large neural networks as one of its possible applications.
%
%The Ising problem without an external field can be equivalently formulated as a graph maximum cut (Max-Cut) problem that can be solved via combinatorial optimization.
%A phase transition is defined as a defined by a singularity in thermodynamic potential or its derivatives.
%
%When crossing a critical point we use an order parameter, in this case $m$ the mean magnetisation, to characterise which phase we in. For the ordered phase this parameter has a non-zero value, in the ordered phase it has a zero value, up to thermodynamic variance.
%
%A phase transition occurs when there is a singularity in the free energy or one of its derivatives. What is often visible is a sharp change in the properties of a substance. The transitions from liquid to gas, from a normal conductor to a superconductor, or from paramagnet to ferromagnet are common examples. The phase diagram of a typical fluid
%
%
%
%In chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states of matter: solid, liquid, and gas, and in rare cases, plasma. A phase of a thermodynamic system and the states of matter have uniform physical properties. During a phase transition of a given medium, certain properties of the medium change as a result of the change of external conditions, such as temperature or pressure. This can be a discontinuous change; for example, a liquid may become gas upon heating to its boiling point, resulting in an abrupt change in volume. The identification of the external conditions at which a transformation occurs defines the phase transition point.
%
%
%A wide variety of physical systems undergo rearrangements of their internal constituents in response to the thermodynamic conditions to which they are subject. Two classic examples of systems displaying such phase transitions are the ferromagnet and fluid systems. As the temperature of a ferromagnet is increased, its magnetic moment is observed to decrease smoothly, until at a certain temperature known as the critical temperature, it vanishes altogether.
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.
\item Monte-Carlo
\item relevant statistical methods
\item Anything about how we fit shit I suppose.
\item Breaking symmetry with the initial $s_i = -1$ state.
\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}

407
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\usepackage[utf8]{inputenc}
\usepackage{fontspec}
\usepackage{amsmath}
\usepackage{amsfonts}
%\setmainfont{New York}
%\setsansfont{New York}
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%\usepackage[none]{hyphenat}
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%\setlength{\abovedisplayskip}{0pt} \setlength{\abovedisplayshortskip}{0pt}
@ -46,6 +47,7 @@
\renewcommand{\cftsecfont}{\rmfamily\mdseries\upshape}
\renewcommand{\cftsecpagefont}{\rmfamily\mdseries\upshape} % No bold!
\usepackage{authblk}
\usepackage{lipsum}
\usepackage{hyperref}
@ -63,6 +65,7 @@
\setlength{\marginparwidth}{1.2cm}
\usepackage{csquotes}
\usepackage{url}
\usepackage{refcount}% http://ctan.org/pkg/refcount
\newcounter{fncntr}
\newcommand{\fnmark}[1]{\refstepcounter{fncntr}\label{#1}\footnotemark[\getrefnumber{#1}]}
@ -71,4 +74,21 @@
%%% END Article customizations
\include{shortcuts}
\newcommand{\nab}{\nabla}
\newcommand{\divrg}{\nab \cdot}
\newcommand{\curl}{\nab \cp}
\newcommand{\lap}{\Delta}
\newcommand{\p}{\partial}
\renewcommand{\d}{\mathrm{d}}
\newcommand{\rd}{~\mathrm{d}}
\newcommand{\ip}[1]{\left\langle#1\right\rangle}
\newcommand{\N}{\mathbb{N}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\TKK}{\textbf{TKK} }
\newcommand{\set}[1]{\left\{#1\right\}}
\newcommand{\abs}[1]{\left|#1\right|}
\DeclareMathOperator{\Var}{Var}
%%% The "real" document content comes below...

342
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@ -1,335 +1,13 @@
@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{bentleyMultidimensionalBinarySearch1975,
title = {Multidimensional Binary Search Trees Used for Associative Searching},
author = {Bentley, Jon Louis},
date = {1975-09},
journaltitle = {Communications of the ACM},
shortjournal = {Commun. ACM},
volume = {18},
number = {9},
pages = {509--517},
issn = {0001-0782, 1557-7317},
doi = {10.1145/361002.361007},
url = {https://dl.acm.org/doi/10.1145/361002.361007},
urldate = {2023-03-18},
abstract = {This paper develops the multidimensional binary search tree (or k -d tree, where k is the dimensionality of the search space) as a data structure for storage of information to be retrieved by associative searches. The k -d tree is defined and examples are given. It is shown to be quite efficient in its storage requirements. A significant advantage of this structure is that a single data structure can handle many types of queries very efficiently. Various utility algorithms are developed; their proven average running times in an n record file are: insertion, O (log n ); deletion of the root, O ( n ( k -1)/ k ); deletion of a random node, O (log n ); and optimization (guarantees logarithmic performance of searches), O ( n log n ). Search algorithms are given for partial match queries with t keys specified [proven maximum running time of O ( n ( k - t )/ k )] and for nearest neighbor queries [empirically observed average running time of O (log n ).] These performances far surpass the best currently known algorithms for these tasks. An algorithm is presented to handle any general intersection query. The main focus of this paper is theoretical. It is felt, however, that k -d trees could be quite useful in many applications, and examples of potential uses are given.},
langid = {english},
file = {/Users/joshuacoles/Zotero/storage/EZJWE76J/Bentley - 1975 - Multidimensional binary search trees used for asso.pdf}
}
@article{botetClusteringClustersProcesses1985,
title = {Clustering of Clusters Processes above Their Upper Critical Dimensionalities},
author = {Botet, R},
date = {1985-04-01},
journaltitle = {Journal of Physics A: Mathematical and General},
shortjournal = {J. Phys. A: Math. Gen.},
volume = {18},
number = {5},
pages = {847--855},
issn = {0305-4470, 1361-6447},
doi = {10.1088/0305-4470/18/5/017},
url = {https://iopscience.iop.org/article/10.1088/0305-4470/18/5/017},
urldate = {2023-02-24},
file = {/Users/joshuacoles/Zotero/storage/CDI267Y4/botet1985.pdf.pdf}
}
@article{botetSizeDistributionClusters1984,
title = {Size Distribution of Clusters in Irreversible Kinetic Aggregation},
author = {Botet, R and Jullien, R},
date = {1984-08-21},
journaltitle = {Journal of Physics A: Mathematical and General},
shortjournal = {J. Phys. A: Math. Gen.},
volume = {17},
number = {12},
pages = {2517--2530},
issn = {0305-4470, 1361-6447},
doi = {10.1088/0305-4470/17/12/022},
url = {https://iopscience.iop.org/article/10.1088/0305-4470/17/12/022},
urldate = {2023-02-24},
file = {/Users/joshuacoles/Zotero/storage/CV6NW42A/botet1984.pdf.pdf}
}
@book{lawlerIntersectionsRandomWalks2013,
title = {Intersections of Random Walks},
author = {Lawler, Gregory F.},
date = {2013},
series = {Modern {{Birkhäuser}} Classics},
publisher = {{Birkhäuser}},
location = {{New York}},
isbn = {978-1-4614-5971-2 978-1-4614-5972-9},
pagetotal = {223},
keywords = {Random walks (Mathematics)},
annotation = {OCLC: ocn812067146},
file = {/Users/joshuacoles/Zotero/storage/C3TSALXD/Lawler - 2013 - Intersections of random walks.pdf}
}
@article{liStorageAddressingScheme2013,
title = {Storage and Addressing Scheme for Practical Hexagonal Image Processing},
author = {Li, Xiangguo},
date = {2013-01-31},
journaltitle = {Journal of Electronic Imaging},
shortjournal = {J. Electron. Imaging},
volume = {22},
number = {1},
pages = {010502},
issn = {1017-9909},
doi = {10.1117/1.JEI.22.1.010502},
url = {http://electronicimaging.spiedigitallibrary.org/article.aspx?doi=10.1117/1.JEI.22.1.010502},
urldate = {2023-03-04},
langid = {english}
}
@article{lyonsSimpleCriterionTransience1983,
title = {A {{Simple Criterion}} for {{Transience}} of a {{Reversible Markov Chain}}},
author = {Lyons, Terry},
date = {1983-05-01},
journaltitle = {The Annals of Probability},
shortjournal = {Ann. Probab.},
volume = {11},
number = {2},
issn = {0091-1798},
doi = {10.1214/aop/1176993604},
url = {https://projecteuclid.org/journals/annals-of-probability/volume-11/issue-2/A-Simple-Criterion-for-Transience-of-a-Reversible-Markov-Chain/10.1214/aop/1176993604.full},
urldate = {2023-03-13},
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.},
date = {1984-03-19},
journaltitle = {Physical Review Letters},
shortjournal = {Phys. Rev. Lett.},
volume = {52},
number = {12},
pages = {1033--1036},
issn = {0031-9007},
doi = {10.1103/PhysRevLett.52.1033},
url = {https://link.aps.org/doi/10.1103/PhysRevLett.52.1033},
urldate = {2023-03-02},
langid = {english},
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},
author = {Sander, Leonard M.},
date = {2000-07},
journaltitle = {Contemporary Physics},
shortjournal = {Contemporary Physics},
volume = {41},
number = {4},
pages = {203--218},
issn = {0010-7514, 1366-5812},
doi = {10.1080/001075100409698},
url = {http://www.tandfonline.com/doi/abs/10.1080/001075100409698},
urldate = {2023-03-02},
langid = {english},
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},
date = {2000-04},
journaltitle = {Physics Letters A},
shortjournal = {Physics Letters A},
volume = {268},
number = {1-2},
pages = {112--116},
issn = {03759601},
doi = {10.1016/S0375-9601(00)00143-2},
url = {https://linkinghub.elsevier.com/retrieve/pii/S0375960100001432},
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,
title = {Fractal Growth Phenomena},
author = {Vicsek, Tamás},
@book{yeomansStatisticalMechanicsPhase1992,
title = {Statistical Mechanics of Phase Transitions},
author = {Yeomans, J. M.},
date = {1992},
edition = {2nd ed},
publisher = {{World Scientific}},
location = {{Singapore ; New Jersey}},
isbn = {978-981-02-0668-0 978-981-02-0669-7},
series = {Oxford Science Publications},
publisher = {{Clarendon Press ; Oxford University Press}},
location = {{Oxford [England] : New York}},
isbn = {978-0-19-851730-6 978-0-19-851729-0},
langid = {english},
pagetotal = {488},
keywords = {Fractals},
note = {Semi-vetted, source Aengus},
file = {/Users/joshuacoles/Zotero/storage/LJP5C4WG/Vicsek - 1992 - Fractal growth phenomena.pdf}
}
@article{wittenDiffusionlimitedAggregation1983,
title = {Diffusion-Limited Aggregation},
author = {Witten, T. A. and Sander, L. M.},
date = {1983-05-01},
journaltitle = {Physical Review B},
shortjournal = {Phys. Rev. B},
volume = {27},
number = {9},
pages = {5686--5697},
issn = {0163-1829},
doi = {10.1103/PhysRevB.27.5686},
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}
pagetotal = {153},
keywords = {Phase transformations (Statistical physics)},
file = {/Users/joshuacoles/Zotero/storage/HZ5HD4LV/HZ5HD4LV.pdf;/Users/joshuacoles/Zotero/storage/TNHQFTLU/(Oxford Science Publications) J. M. Yeomans - Statistical Mechanics of Phase Transitions-Oxford University Press, USA (1992).pdf}
}

9
report.tex
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@ -6,18 +6,17 @@
\addbibresource{static.bib}
\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.
\title{\textbf{Modelling Diffusion Limited Aggregation under a Variety of Conditions}}
\title{\textbf{Comparison of Models for Ferromagnetic Systems near the Critical Point}}
\author{Candidate Number: 24829}
\affil{Department of Physics, University of Bath}
\date{March 21, 2023} % Due Date
\date{May 12, 2023} % Due Date
\begin{document}
\input{introduction-dicussion-method.tex}
\input{results.tex}
% TODO Formatting of these (for one its in american date formats ughhh)
\printbibliography
\input appendix

85
results.tex
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@ -1,90 +1,39 @@
\section*{Results}
\begin{figure}[t]
\includegraphics[width=\columnwidth]{figures/rmax-n.png}
\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.
% TODO Check all of my captions are correct.
% TODO Add information for this
}
\label{rmax-n}
\end{figure}
\subsection*{Relaxation Time (Time to Equilibrium)}
\subsection*{Preliminary Work: Testing Initial Implementation and Fractal Dimension Calculations}
\label{ii-fdc}
The first question of modelling was to determine when the system had reached equilibrium. As discussed in Section \ref{system-convergence} \TKK, the method chosen to determine the number of sweeps after which this had occurred, $N_0$ (also known as the transient boundary) was autocorrelation. The data processed was the 1\textdegree thermodynamic variables, the mean dimensionless energy $\widetilde{E}$ and mean magnetisation $m$.
To start we can show that the system does indeed attain equilibrium by observing the behaviour at a singular value of $\beta_0 = 0.25$ as shown in Figure \ref{single-beta-convergence}. Here we can see the system attains a stable state at $N_s \approx 25$.
\begin{figure}[hbt]
\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 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}.}
\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 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}.
% TODO Errors
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.
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.
\subsection*{Probabilistic Sticking}
\begin{figure}[hbt]
\includegraphics[width=\columnwidth]{figures/eg-across-sp/sp-range.png}
\caption{Here we see the result of three different DLA simulations with $p_{stick} = 0.1,0.5,1.0$ from left to right. Note the thickening of the arms at low probabilities.}
\label{sp-dla-comparison}
\label{single-beta-convergence}
\includegraphics[width=\columnwidth]{figures/single-beta-convergence.png}
\caption{The mean magnetisation $m$ and mean dimensionless energy $\widetilde{E}$ as they evolve over sweeps of the Monte-Carlo simulation. Both these quantities are normalised relative to their maximum modulus value, so that they can be shown on the same value and compared. This results in a scaling to the range of $[-1, 1]$ such that the sign of the value is preserved.}
\end{figure}
\begin{figure}[hbt]
\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 data was obtained in two batches: in the $p_{stick} \in [0.1, 1]$ range $100$ samples were taken with different seeds with $N_C= 2000$, the fractal dimension being computed by the mass method; for the $p_{stick} \in (0.001, 0.1)$ range a $100$ samples of $N_C = 5000$ clusters were used.
}
\label{sp-fd}
\label{beta-relaxation-time}
\includegraphics[width=\columnwidth]{figures/transient-boundary-vs-beta.png}
\caption{The number of sweeps required for equilibrium to be reached (aka the transient boundary) compared to the $\beta_0$, the dimensionless inverse temperature. Also shown is the a line showing the position of $\beta_c$. This data is comprised of a mean of value of $10$ runs with different seeds. Here we focus on the region surrounding the critical temperature $\beta_c$. A $\beta_0$ increment of $0.001$ was used, with $N_\mathrm{max}$, the maximum number of sweeps being $500$ far from the critical temperature, and $10,000$ close to it. TKK define all these terms}
\end{figure}
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).
Next the question is show this varies across the range of $\beta_0$. This is shown in Figure \ref{beta-relaxation-time} where we can see the relaxation time when computed in terms of the $\widetilde E$ and $m$ both. Here we can see expected that far from the critical temperature (relatively speaking) $N_0$ decays quickly, spiking just before the critical temperature and decaying after it.
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.
This data allows us to determine the portion of each run which is at equilibrium and hence to which we can apply the relevant statistical and statistical mechanical methods for analysis (most notably the Ergodic hypothesis TKK in thermodynamics section.)
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.
% TODO Conclusion point
\subsection*{Thermodynamic Observables}
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.
Using this data to extract steady state information we are able to determine values of the other thermodynamic observables we desire including, the modulus magnetisation $\abs{m}$, the magnetic susceptibility $\chi$, and heat-capacity $c$.
\subsection*{Higher Dimensions}
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}.
\subsection*{Comparison of Models}
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.
\subsection*{Correlation Length}
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.
\subsection*{Critical Exponents}
At this moment we do not have an analytical form for this relation but further work may provide such.
% TODO Conclusion point
\begin{figure}
\includegraphics[width=\columnwidth]{figures/sp-fd-2d-3d}
\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 $2000$.}
\label{sp-fd-2d-3d}
\end{figure}
\begin{figure}
\includegraphics[width=\columnwidth]{figures/3d-nc-fd-convergence}
\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.
\label{3d-nc-fd-convergence}
\end{figure}
% TODO Do I want to do higher dimensions still 4d?
% TODO how am I going to cope with the fact we don't agree with theory?
% 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.
% MORE EXTENSIONS
%\subsection*{Continuous Space}
%
%\begin{enumerate}
% \item We get a divergence from theory, what happens if we use continuous
%\end{enumerate}
\section*{Conclusion}
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.

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@article{GNUParallel,
title = {GNU Parallel - The Command-Line Power Tool},
author = {O. Tange},
address = {Frederiksberg, Denmark},
journal = {;login: The USENIX Magazine},
month = {Feb},
number = {1},
volume = {36},
url = {http://www.gnu.org/s/parallel},
year = {2011},
pages = {42-47}
}
@online{IPC,
author = {Various Physics Lecturers},
title = {{Initially Provided DLA Code Model}},
url = {https://moodle.bath.ac.uk/course/view.php?id=1876&section=3},
urldate = {2023-03-14}
}
@misc{wiki:phase-digram-eg,
author = "Wikimedia Commons",
title = "File:Phase-diag2.svg --- Wikimedia Commons{,} the free media repository",
year = "2022",
url = "\url{https://commons.wikimedia.org/w/index.php?title=File:Phase-diag2.svg&oldid=667966844}",
note = "[Online; accessed 28-May-2023]"
}

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