Add to method and conclusion.

latex/references.bib

@@ -1,4 +1,10 @@
 # Ising model related literature
+@misc{hj:2015:comp_phys,
+  author       = {Morten Hjorth-Jensen},
+  howpublished = {\url{https://raw.githubusercontent.com/CompPhysics/ComputationalPhysics/master/doc/Lectures/lectures2015.pdf}},
+  title        = {Computational Physics, Lecture Notes Fall 2015},
+  year         = {2015}
+}
 @misc{britannica:2023:ferromagnetism,
   author  = {Britannica, The Editors of Encyclopaedia},
   title   = {ferromagnetism},

latex/sections/conclusion.tex

@@ -2,5 +2,18 @@
 
 \begin{document}
 \section{Conclusion}\label{sec:conclusion}
-
+% Draft based on abstract
+We have used the Ising model to study the behavior in ferromagnets, when undergoing
+a phase transition near a critical temperature. We generated samples using the 
+Markov chain Monte Carlo method, while utilizing methods of parallelization. 
+Finding the burn-in time to be approx. 3000 Monte Carlo cycles. For temperature 
+$T = 1.0 J / k_{B}$ we found a propability distrobution with an expected mean 
+value of $\mu \approx -1.9969$ and variation $\sigma^{2} = 0.0001$. Whereas the 
+pdf close to the critical temperature is $\mu \approx -1.2370$ and variation 
+$\sigma^{2} = 0.0203$. We estimated the expected energy and magnetization per spin, 
+in addition to the heat capacity and susceptibility. Using the values from 
+finite sized lattices, we approximated the critical temperature of an infinite 
+sized lattice. Using linear regression, we numerically estimated $T_{c}$ $T_{C}(L = \infty) \approx 2.2695$ 
+which is close to the analytical solution $T_{C}(L = \infty) \approx 2.269 J/k_{B}$ 
+found by Lars Onsager. 
 \end{document}

latex/sections/methods.tex

@@ -80,7 +80,8 @@ have two spins oriented up the total energy have two possible values, as shown i
     conditions.}
     \label{tab:lattice_config}
 \end{table}
-We use the analytical values, found in Table for both for lattices where $L = 2$ and $L > 2$. 
+We use the analytical values, found in Table for both for lattices where $L = 2$ 
+and $L > 2$. 
 
 However, to compare the quantities for lattices where $L > 2$, we find energy 
 per spin given by 
@@ -94,6 +95,7 @@ and magnetization per spin given by
     \label{eq:magnetization_spin}
 \end{equation}
 
+
 \subsection{Statistical mechanics}\label{subsec:statistical_mechanics}
 When we study ferromagnetism, we have to consider the probability for a microstate 
 $\mathbf{s}$ at a fixed temperature $T$. The probability distribution function 
@@ -103,13 +105,19 @@ $\mathbf{s}$ at a fixed temperature $T$. The probability distribution function
     \label{eq:boltzmann_distribution}
 \end{equation}
 known as the Boltzmann distribution. This is an exponential distribution, where 
-$\beta$ and $Z$ are given by
-\begin{align*}
-    \beta =& \frac{1}{k_{B}} \ , &
-    Z &= \sum_{\text{all possible } \mathbf{s}} e^{-\beta E(\mathbf{s})} \ , \\
-\end{align*}
-and $k_{B}$ is the Boltzmann constant. $Z$ is a normalizing factor of the pdf, 
-known as the partition function, which we derive in Appendix \ref{sec:partition_function}
+$\beta$ is given by 
+\begin{equation}
+    \beta = \frac{1}{k_{B}} \ ,
+    \label{eq:beta}
+\end{equation}
+where and $k_{B}$ is the Boltzmann constant. $Z$ is a normalizing factor of the 
+pdf, given by
+\begin{equation}
+    Z = \sum_{\text{all possible } \mathbf{s}} e^{-\beta E(\mathbf{s})} \ , 
+    \label{eq:partition}
+\end{equation}
+and is known as the partition function. We derive $Z$ in Appendix \ref{sec:partition_function},
+which gives us
 \begin{equation*}
     Z = 4 \cosh (8 \beta J) + 12 \ .
 \end{equation*}
@@ -117,27 +125,28 @@ Using the partition function and Eq. \eqref{eq:boltzmann_distribution}, the pdf
 of a microstate at a fixed temperature is given by 
 \begin{equation}
     p(\mathbf{s} \ | \ T) = \frac{1}{4 \cosh (8 \beta J) + 12} e^{-\beta E(\mathbf{s})} \ .
+    \label{eq:pdf}
 \end{equation}
 % Add something about why we use the expectation values?
 We derive the analytical expressions for expectation values in Appendix. 
 \ref{sec:expectation_values}. We find the expected total energy
-\begin{equation*} %\label{eq:energy_total_first}
+\begin{equation}\label{eq:energy_total_result}
 	\langle E \rangle = -\frac{8J \sinh(8 \beta J)}{\cosh(8 \beta J) + 3} \ ,
-\end{equation*}
+\end{equation}
 and the expected energy per spin
-\begin{equation*} %\label{eq:energy_spin_first}
+\begin{equation}\label{eq:energy_spin_result}
 	\langle \epsilon \rangle = \frac{-2J \sinh(8 \beta J)}{ \cosh(8 \beta J) + 3} \ .
-\end{equation*}
+\end{equation}
 We find the expected absolute total magnetization
-\begin{equation*} %\label{eq:magnetization_total_first}
+\begin{equation}\label{eq:magnetization_total_result}
 	\langle |M| \rangle = \frac{2(e^{8 \beta J} + 2)}{\cosh(8 \beta J) + 3} \ ,
-\end{equation*}
+\end{equation}
 and the expected magnetization per spin
-\begin{equation*} %\label{eq:magnetization_spin_first}
+\begin{equation}\label{eq:magnetization_spin_result}
 	\langle |m| \rangle = \frac{e^{8 \beta J} + 1}{2( \cosh(8 \beta J) + 3)} \ .
-\end{equation*}
+\end{equation}
 
-We will also determine the heat capacity 
+We also need to determine the heat capacity 
 \begin{equation}
     C_{V} = \frac{1}{k_{B} T^{2}} (\mathbb{E}(E^{2}) - [\mathbb{E}(E)]^{2}) \ ,
     \label{eq:heat_capacity}
@@ -180,6 +189,12 @@ Boltzmann constant we derive the remaining units, which can be found in Table
 
 \subsection{Phase transition and critical temperature}\label{subsec:phase_critical}
 % P9 critical temperature
+When a ferromagnetic material is heated, it will change at a macroscopic level. 
+Based on a $2 \times 2$ lattice, we can show that the total energy is equal to the 
+energy where all spins have the orientation up \cite[p. 426]{hj:2015:comp_phys}.
+Increasing the temperature of the external field, the Ising model move from an 
+ordered to an unordered phase. At the critical temperature the heat capacity $C_{V}$, 
+and the magnetic susceptibility $\chi$ diverge \cite[p. 431]{hj:2015:comp_phys}.
 
 \subsection{The Markov chain Monte Carlo method}\label{subsec:mcmc_method}
 Markov chains consist of a sequence of samples, where the probability of the next