Need to rewrite burn-in time subsection, include speed-up, and fix appendices.

latex/ising_model.tex

@@ -90,15 +90,16 @@
 % Method
 % Problem 1: OK
 % Problem 2: OK
-% Problem 3: OK - missing theory of phase transition
+% Problem 3: OK
-% Problem 4: validate numerical results using the analytical results
+% Problem 4: OK
-% Problem 7: division of workload mpi and openmp
+% Problem 9: OK
-% Problem 9: rewrite theoretical background of phase transition and critical temperature 
 
 % Results
 % Problem 5: burn-in time
-% Problem 6: estimate probability distribution based on histogram
+% Problem 6: OK
+% Problem 7: OK
 % Problem 8: investigate phase transitions, and estimate critical temperature based on plot
 % Problem 9: use critical temperature to estimate the ctritical temperature of infinite lattice
-%
+
-%
+% Appendices
+% Problem 0: move in front of references, and change back to double column

latex/sections/appendices.tex

@@ -180,34 +180,34 @@ $\Delta E = E_{\text{after}} - E_{\text{before}}$. We find the $3^{2}$ values as
 where the five distinct values are $\Delta E = \{-16J, -8J, 0, 8J, 16J\}$.
 
 
-\section{Extra}\label{sec:extra_results}
+\section{Additional results}\label{sec:additional_results}
-We increased number of MC cycles to 10 million
+Results of 1 million MC cycles.
 \begin{figure}
     \centering
     \includegraphics[width=\linewidth]{../images/phase_transition/fox/wide/10M/energy.pdf}
-    \caption{$\langle \epsilon \rangle$ for $T \in [2.1, 2.4]$, 10000000 MC cycles.}
+    \caption{$\langle \epsilon \rangle$ for $T \in [2.1, 2.4]$, $10^{6}$ MC cycles.}
-    \label{fig:phase_energy_10M}
+    \label{fig:phase_energy_1M}
 \end{figure}
 
 \begin{figure}
     \centering
     \includegraphics[width=\linewidth]{../images/phase_transition/fox/wide/10M/magnetization.pdf}
-    \caption{$\langle |m| \rangle$ for $T \in [2.1, 2.4]$, 10000000 MC cycles.}
+    \caption{$\langle |m| \rangle$ for $T \in [2.1, 2.4]$, $10^{6}$ MC cycles.}
-    \label{fig:phase_magnetization_10M}
+    \label{fig:phase_magnetization_1M}
 \end{figure}
 
 \begin{figure}
     \centering
     \includegraphics[width=\linewidth]{../images/phase_transition/fox/wide/10M/heat_capacity.pdf}
-    \caption{$C_{V}$ for $T \in [2.1, 2.4]$, 10000000 MC cycles.}
+    \caption{$C_{V}$ for $T \in [2.1, 2.4]$, $10^{6}$ MC cycles.}
-    \label{fig:phase_heat_10M}
+    \label{fig:phase_heat_1M}
 \end{figure}
 
 \begin{figure}
     \centering
     \includegraphics[width=\linewidth]{../images/phase_transition/fox/wide/10M/susceptibility.pdf}
-    \caption{$\chi$ for $T \in [2.1, 2.4]$, 10000000 MC cycles.}
+    \caption{$\chi$ for $T \in [2.1, 2.4]$, $10^{6}$ MC cycles.}
-    \label{fig:phase_susceptibility_10M}
+    \label{fig:phase_susceptibility_1M}
 \end{figure}
 
 \end{document}

latex/sections/methods.tex

@@ -212,11 +212,13 @@ correlation length is proportional with the lattice size, resulting in the criti
 temperature scaling relation
 \begin{equation}
     T_{c}(L) - T_{c}(L = \infty) = aL^{-1} \ ,
+    \label{eq:critical_infinite}
 \end{equation}
 where $a$ is a constant. For the Ising model in two dimensions, with an lattice of 
-infinite size, the critical temperature is 
+infinite size, the analytical solution of the critical temperature is 
 \begin{equation}
     T_{c}(L = \infty) = \frac{2}{\ln (1 + \sqrt{2})} J/k_{B} \approx 2.269 J/k_{B}
+    \label{eq:critical_solution}
 \end{equation}
 This result was found analytically by Lars Onsager in 1944. We can estimate the 
 critical temperature of an infinite lattice, using finite lattices critical temperature 
@@ -288,7 +290,7 @@ distribution of the system.
 
 \subsection{Implementation and testing}\label{subsec:implementation_test}
 We implemented a test suite, and compared the numerical estimates to the analytical 
-results from \ref{subsec:statistical_mechanics}. In addition, we set a tolerance to 
+results from Section \ref{subsec:statistical_mechanics}. In addition, we set a tolerance to 
 verify convergence, and a maximum number of Monte Carlo cycles to avoid longer 
 runtimes during implementation.
 

latex/sections/results.tex

@@ -2,22 +2,8 @@
 
 \begin{document}
 \section{Results}\label{sec:results}
-
+\subsection{Burn-in time}\label{subsec:burnin_time}
-% 2.1-2-4 divided into 40 steps, which gives us a step size of 0.0075.
+$\boldsymbol{Draft}$
-
-% 10 MPI processes
-% - 10 threads per process
-% = 100 threads total
-% Not a lot of downtime for the threads
-
-However, when the temperature is close to the critical point, we observe an increase 
-in expected energy and a decrease in magnetization. Suggesting a higher energy and 
-a loss of magnetization close to the critical temperature.
-
-% We did not set the seed for the random number generator, which resulted in 
-% different numerical estimates each time we ran the model. However, all expectation 
-% values are calculated using the same data. The burn-in time varied each time. 
-% We see a burn-in time t = 5000-10000 MC cycles. However, this changed between runs. 
 We decided with a burn-in time parallelization trade-off. That is, we set the 
 burn-in time lower in favor of sampling. To take advantage of the parallelization 
 and not to waste computational resources. The argument to discard samples generated 
@@ -26,25 +12,6 @@ parallelize using MPI. We generated
 samples for the temperature range $T \in [2.1, 2.4]$. Using Fox we generated both 
 1 million samples and 10 million samples.
 
-It is worth mentioning that the time (number of MC cycles) necessary to get a 
-good numerical estimate, compared to the analytical result, foreshadowing the 
-burn-in time.
-
-Markov chain starting point can differ, resulting in different simulation. By 
-discarding the first samples, the ones generated before system equilibrium we can 
-get an estimate closer to the real solution. Since we want to estimate expectation 
-values at a given temperature, the samples should represent the system at that 
-temperature.
-
-Depending on number of samples used in numerical estimates, using the samples 
-generated during burn-in can in high bias and high variance if the ratio is skewed. 
-However, if most samples are generated after burn-in the effect is not as visible. 
-Can't remove randomness by starting around equilibrium, since samples are generated 
-using several ising models we need to sample using the same conditions that is 
-system state equilibrium.
-
-\subsection{Burn-in time}\label{subsec:burnin_time}
-$\boldsymbol{Draft}$
 We start with a lattice where $L = 20$, to study the burn-in time, that is the 
 number of Monte Carlo cycles necessary for the system to reach an equilibrium. 
 We consider two different temperatures $T_{1} = 1.0 J/k_{B}$ and $T_{2} = 2.4 J/k_{B}$, 
@@ -52,7 +19,7 @@ where $T_{2}$ is close to the critical temperature. We can use the correlation
 time $\tau \approx L^{d + z}$ to determine time, where $d$ is the dimensionality 
 of the system and $z = 2.1665 \pm 0.0012$ \footnote{This value was determined by 
 Nightingale and Blöte for the Metropolis algorithm.}
-% Need to include a section of Onsager's analytical result.
+% 
 We show the numerical estimates for temperature $T_{1}$ of $\langle \epsilon \rangle$ 
 in Figure \ref{fig:burn_in_energy_1_0} and $\langle |m| \rangle$ in Figure 
 \ref{fig:burn_in_magnetization_1_0}. For temperature $T_{2}$, the numercal estimate 
@@ -97,85 +64,125 @@ pdf using $T = 1.0$ result in
 
 
 \subsection{Probability distribution}\label{subsec:probability_distribution}
-$\boldsymbol{Draft}$
 % Histogram figures
-We use the estimated burn-in time to set starting time for sampling, then generate 
+We used the estimated burn-in time as starting time for sampling, and generated 
-samples to plot in a histogram for $T_{1}$ in Figure \ref{fig:histogram_1_0} and 
+samples. To visualize the distribution of energy per spin $\epsilon$, we used histograms 
-$T_{2}$ in Figure \ref{fig:histogram_2_4}. For $T_{1}$ we can see that most samples 
+with a bin size (...). In Figure \ref{fig:histogram_1_0} we show the distribution 
-have the expected value $-2$, we have a distribution with low variance.
+for $T_{1}$, where the majority of the energy per spin is $-2$. The resulting expectation 
+value of energy per spin is $\langle \epsilon \rangle = -1.9969$, with a low variance 
+of Var$(\epsilon) = 0.0001$. %
 \begin{figure}[H]
     \centering
     \includegraphics[width=\linewidth]{../images/pd_estimate_1_0.pdf}
     \caption{Histogram $T = 1.0 J / k_{B}$}
     \label{fig:histogram_1_0}
-\end{figure}
+\end{figure} %
+In Figure \ref{fig:histogram_2_4}, for $T_{2}$, the samples of energy per spin is 
+centered around the expectation value $\langle \epsilon \rangle = -1.2370$. %
 \begin{figure}[H]
     \centering
     \includegraphics[width=\linewidth]{../images/pd_estimate_2_4.pdf}
     \caption{Histogram $T = 2.4 J / k_{B}$}
     \label{fig:histogram_2_4}
-\end{figure}
+\end{figure} % 
+However, we observe a higher variance of Var$(\epsilon) = 0.0203$. When the temperature 
+increase, the system moves from an ordered to an onordered state. The change in 
+system state, or phase transition, indicates the temperature is close to a critical 
+point. 
 
 
 \subsection{Phase transition}\label{subsec:phase_transition}
 $\boldsymbol{Draft}$
 % Phase transition figures
+We continue investigating the behavior of the system around the critical temperature.
+First, we generated $10$ million samples of spin configurations for lattices of 
+size $L \in \{20, 40, 60, 80, 100\}$, and temperatures $T \in [2.1, 2.4]$. We divided the 
+temperature range into $40$ steps, with an equal step size of $0.0075$. The samples 
+were generated in parallel, where we allocated $4$ sequential temperatures to $10$ 
+MPI processes. Each process was set to spawn $10$ thread, resulting in a total of 
+$100$ threads working in parallel. We include results for $1$ million MC cycles 
+in Appendix \ref{sec:additional_results}
+
+$\boldsymbol{Rewrite}$ We ran a profiler to make sure the program was fully optimized which found that the 
+workload was balanced, the threads was not left idle to long/not a lot of downtime.
+
+In Figure \ref{fig:phase_energy_10M}, for the larger lattices we observe a sharper 
+increase in $\langle \epsilon \rangle$ in the temperature range $T \in [2.25, 2.35]$. %]
 \begin{figure}
     \centering
-    \includegraphics[width=\linewidth]{../images/phase_transition/fox/wide/1M/energy.pdf}
+    \includegraphics[width=\linewidth]{../images/phase_transition/fox/wide/10M/energy.pdf}
-    \caption{$\langle \epsilon \rangle$ for $T \in [2.1, 2.4]$, 1000000 MC cycles.}
+    \caption{$\langle \epsilon \rangle$ for $T \in [2.1, 2.4]$, $10^7$ MC cycles.}
-    \label{fig:phase_energy}
+    \label{fig:phase_energy_10M}
-\end{figure}
+\end{figure} % 
-
+We observe a deacrese in $\langle |m| \rangle$ for the same temperature range in 
+Figure \ref{fig:phase_magnetization_10M}, suggesting the system moves from an ordered 
+magnetized state to a state of no net magnetization. The system energy increase, 
+however, there is a loss of magnetization close to the critical temperature.
 \begin{figure}[H]
     \centering
-    \includegraphics[width=\linewidth]{../images/phase_transition/fox/wide/1M/magnetization.pdf}
+    \includegraphics[width=\linewidth]{../images/phase_transition/fox/wide/10M/magnetization.pdf}
-    \caption{$\langle |m| \rangle$ for $T \in [2.1, 2.4]$, 1000000 MC cycles.}
+    \caption{$\langle |m| \rangle$ for $T \in [2.1, 2.4]$, $10^7$ MC cycles.}
-    \label{fig:phase_magnetization}
+    \label{fig:phase_magnetization_10M}
-\end{figure}
+\end{figure} %
-
+In Figure \ref{fig:phase_heat_10M}, we observe an increase in heat capacity in the 
+temperature range $T \in [2.25, 2.35]$. In addition, we observed a sharper peak 
+value of heat capacity the lattice size increase. 
 \begin{figure}[H]
     \centering
-    \includegraphics[width=\linewidth]{../images/phase_transition/fox/wide/1M/heat_capacity.pdf}
+    \includegraphics[width=\linewidth]{../images/phase_transition/fox/wide/10M/heat_capacity.pdf}
-    \caption{$C_{V}$ for $T \in [2.1, 2.4]$, 1000000 MC cycles.}
+    \caption{$C_{V}$ for $T \in [2.1, 2.4]$, $10^7$ MC cycles.}
-    \label{fig:phase_heat}
+    \label{fig:phase_heat_10M}
-\end{figure}
+\end{figure} % 
-
+The magnetic susceptibility in Figure \ref{fig:phase_susceptibility_10M}, showed 
+the sharp peak in the same temperature range as that of the heat capacity. Since 
+shape of the curve for both heat capacity and the magnetic susceptibility become 
+sharper when we increase lattice size, we are moving closer to the critical temperature.
 \begin{figure}[H]
     \centering
-    \includegraphics[width=\linewidth]{../images/phase_transition/fox/wide/1M/susceptibility.pdf}
+    \includegraphics[width=\linewidth]{../images/phase_transition/fox/wide/10M/susceptibility.pdf}
-    \caption{$\chi$ for $T \in [2.1, 2.4]$, 1000000 MC cycles.}
+    \caption{$\chi$ for $T \in [2.1, 2.4]$, $10^7$ MC cycles.}
-    \label{fig:phase_susceptibility}
+    \label{fig:phase_susceptibility_10M}
-\end{figure}
+\end{figure} %
-We include results for 10 million MC cycles in Appendix \ref{sec:extra_results}
+% Add something about the correlation length?
 
 
 \subsection{Critical temperature}\label{subsec:critical_temperature}
-$\boldsymbol{Draft}$
+Based on the heat capacity (Figure \ref{fig:phase_heat_10M}) and susceptibility 
-We use the critical temperatures found in previous section, in addition to the 
+(Figure \ref{fig:phase_susceptibility_10M}), we estimated the critical temperatures 
-scaling relation in Equation \eqref{eq:critical_intinite}
+of lattices of size $L \in \{20, 40, 60, 80, 100\}$ found in Table \ref{tab:critical_temperatures}.
-\begin{equation}
+\begin{table}[H]
-    T_{c} - T_{c}(L = \infty) = \alpha L^{-1}
+    \centering
-    \label{eq:critical_intinite}
+    \begin{tabular}{cc} % @{\extracolsep{\fill}}
-\end{equation}
+        \hline
-to estimate the critical temperature for a lattize of infinte size. We also 
+        $L$ & $T_{c}(L)$ \\
-compared the estimate with the analytical solution 
+        \hline 
-\begin{equation}
+        $20$ & $J$ \\
-    T_{c}(L = \infty) = \frac{2}{\ln (1 + \sqrt{2})} J/k_{B} \approx 2.269 J/k_{B}
+        $40$ & $1$ \\
-\end{equation}
+        $60$ & $J / k_{B}$ \\
-using linear regression. In Figure \ref{fig:linreg} we find the critical 
+        $80$ & $k_{B}$ \\
-temperatures as function of the inverse lattice size. When the lattice size increase 
+        $100$ & $1 / J$ \\
-toward infinity, $1/L$ goes toward zero, we find the intercept which gives us an 
+        \hline
-estimated value of the critical temperature for a lattice of infinite size.
+    \end{tabular}
-
+    \caption{Estimated critical temperatures for lattices $L \times L$.}
-% Critical temp regression figure 
+    \label{tab:critical_temperatures}
+\end{table}
+We used the critical temperatures of finite lattices and the scaling relation in 
+Equation \eqref{eq:critical_infinite}, Section \ref{subsec:phase_critical}, to estimate 
+the critical temperature of a lattice of infinite size. In Figure \ref{fig:linreg_10M}, 
+we plot the critical temperatures $T_{c}(L)$ of the inverse lattice size $1/L$. When 
+the lattice size increase toward infinity, $1/L$ approaches zero. %
 \begin{figure}[H]
     \centering
-    \includegraphics[width=\linewidth]{../images/phase_transition/fox/wide/1M/linreg.pdf}
+    \includegraphics[width=\linewidth]{../images/phase_transition/fox/wide/10M/linreg.pdf}
-    \caption{Linear regression, where $\beta_{0}$ is the intercept $T_{c}(L = \infty)$ and $\beta_{1}$ is the slope.}
+    \caption{Linear regression, where $\beta_{0}$ is the intercept approximating $T_{c}(L = \infty)$, and $\beta_{1}$ is the slope.}
-    \label{fig:linreg}
+    \label{fig:linreg_10M}
 \end{figure}
-
+Using linear regression, we find the intercept which gives us an estimated value 
-
+of the critical temperature for a lattice of infinite size. We find the critical 
+temperature to be $T_{c \text{num}} \approx 2.2695 J/k_{B}$. We also compared the 
+estimate with the analytical solution, the relative error of our estimate is
+\begin{equation*}
+    \text{Relative error} =  \frac{T_{c \text{ numerical}} - T_{c \text{ analytical}}}{T_{c \text{ analytical}}} \approx 0.001 J/k_{B}
+\end{equation*}
 
 \end{document}