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13
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}
}

137
introduction-dicussion-method.tex
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@ -30,8 +30,6 @@ Of these we will focus most on the Ising Model, which we will use to model ferro
In the discussion we will explore the theory behind the models discussed, as well as phase transitions (\TKK is this a hyphenated word or not?) 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.
---
@ -55,68 +53,135 @@ In the discussion we will explore the theory behind the models discussed, as wel
\section*{Discussion}
Here we will discuss the theory behind
Here we will discuss the theory behind the different models mentioned as well as phase transitions themselves.
\subsection*{Thermodynamics}
\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}
Whichever model we use standard Statistical Mechanics gives us relations between the partition function, itself derived from the energy of the system, and other thermodynamic observables. This relationship can be seen in Figure \ref{thermodynamic-variables}, from \cite[p. 17]{yeomansStatisticalMechanicsPhase1992}.
We obtain the magnetisation and energy from our chosen model, relying on statistical mechanics for the specific heat capacity, and magnetic susceptibility (labeled as \enquote{Isothermal susceptibility} in the diagram), given by
\begin{align}
c &= \frac{\p}{\p T} \left\langle\frac{E}{N}\right\rangle \\
\chi &= \frac{\p}{\p h} \left\langle M \right\rangle.
\end{align}
\TKK define E and M
To these we can apply the Fluctuation-dissipation theorem \cite{???} \TKK to transform the derivatives in these equations to 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}
\subsection*{The Ising Model}
The Ising Model is a simplified model of Ferromagnetic Materials to explore behaviour across the magnetised—non-magnetised phase transition. 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, having an interaction energy equal to,
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, having an interaction energy equal to,
\subsection*{Mean Field Theory}
$$
E_i = -J\sum_{\ip{ij}}
$$
\begin{equation}
E = -J\sum_{\ip{ij}} s_i s_j
\end{equation}
where $J > 0$ is the strength of the \todo{what are the units?} interaction, and $i, j$ are specific sites in the grid\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 $(x, y) \in \N$ 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,
A phase transition is defined as a defined by a singularity in thermodynamic potential or its derivatives.
\begin{equation*}
E = -J\sum_{\ip{ij}} s_i s_j - H \sum_{i} s_i.
\end{equation*}
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.
From these expressions we can obtain the partition function, in the standard manner as,
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
\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 microstate, 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}
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.
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.
\TKK
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.
\begin{equation}
E(s_0) = -s_0\left(J\sum_{\ip{s_0 s_j}} s_j + H \right)
\end{equation}
where $\ip{s_0 s_j}$ represents the nearest-neighbours to some fixed $s_0$. By counting noting that the number of neighbours to any given site, $q$, is constant with a value of $q = 4$ we can obtain,
\begin{enumerate}
\item What does the Ising Model model?
\item Why do we care
\item What is the Ising Model
\end{enumerate}
\begin{equation}
E(s_0) = -s_0(qJm + H) - Js_0 \sum_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 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 and hence,
The Ising model (German pronunciation: [ˈiːzɪŋ]) (or Lenz-Ising model or Ising-Lenz model), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that represent magnetic dipole moments of atomic "spins" that can be in one of two states (+1 or −1). The spins are arranged in a graph, usually a lattice (where the local structure repeats periodically in all directions), allowing each spin to interact with its neighbors. Neighboring spins that agree have a lower energy than those that disagree; the system tends to the lowest energy but heat disturbs this tendency, thus creating the possibility of different structural phases. The model allows the identification of phase transitions as a simplified model of reality. The two-dimensional square-lattice Ising model is one of the simplest statistical models to show a phase transition.[1]
\begin{align}
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{align}
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.
where $\beta_0$ is the dimensionless $\beta$ discussed in \TKK.
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.
\begin{equation}
Z = (Z_1(s_0))^N
\end{equation}
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.
where $N$ is the number of sites in the system. Applying the identities seen in Figure \ref{thermodynamic-identities-graph} we obtain an expression for the magnetisation as,
\begin{equation}
m = \tanh(\beta(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}
%The Ising model (German pronunciation: [ˈiːzɪŋ]) (or Lenz-Ising model or Ising-Lenz model), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that represent magnetic dipole moments of atomic "spins" that can be in one of two states (+1 or −1). The spins are arranged in a graph, usually a lattice (where the local structure repeats periodically in all directions), allowing each spin to interact with its neighbors. Neighboring spins that agree have a lower energy than those that disagree; the system tends to the lowest energy but heat disturbs this tendency, thus creating the possibility of different structural phases. The model allows the identification of phase transitions as a simplified model of reality. The two-dimensional square-lattice Ising model is one of the simplest statistical models to show a phase transition.[1]
%
%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.
%
%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.
\section*{Discussion}
\section*{Method}
The Ising Model was implemented on a square cell grid with periodic boundary conditions.
\subsection*{Convergence}
\begin{figure}[hbt]
\includegraphics[width=\columnwidth]{./figures/convergence-rate-varying-beta.png}
\caption{Her}
\end{figure}
\begin{enumerate}
\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}

6
prelude.tex
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@ -4,6 +4,7 @@
\usepackage[utf8]{inputenc}
\usepackage{fontspec}
\usepackage{amsmath}
\usepackage{amsfonts}
%\setmainfont{New York}
%\setsansfont{New York}
%\setmonofont{New York}
@ -72,7 +73,7 @@
%%% END Article customizations
\include{shortcuts.tex}
\include{shortcuts}
\newcommand{\nab}{\nabla}
\newcommand{\divrg}{\nab \cdot}
@ -85,5 +86,8 @@
\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
references.bib
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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},
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\input prelude.tex
\addbibresource{references.bib}
\addbibresource{static.bib}
\setlength{\marginparwidth}{1.2cm}
\title{\textbf{Comparison of Models for Ferromagnetic Systems near the Critical Point}}

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\section*{Results}
\subsection*{Relaxation Time (Time to Equilibrium)}
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]
\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]
\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}
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.
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.)
\subsection*{Thermodynamic Observables}
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*{Comparison of Models}
\subsection*{Correlation Length}
\subsection*{Critical Exponents}
\section*{Conclusion}

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