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

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Janita Willumsen 2023-12-03 15:47:53 +01:00
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15
latex/ising_model.tex
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@ -90,15 +90,16 @@
% Method
% Problem 1: OK
% Problem 2: OK
% Problem 3: OK - missing theory of phase transition
% Problem 4: validate numerical results using the analytical results
% Problem 7: division of workload mpi and openmp
% Problem 9: rewrite theoretical background of phase transition and critical temperature
% Problem 3: OK
% Problem 4: OK
% Problem 9: OK
% 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

20
latex/sections/appendices.tex
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@ -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}
We increased number of MC cycles to 10 million
\section{Additional results}\label{sec:additional_results}
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.}
\label{fig:phase_energy_10M}
\caption{$\langle \epsilon \rangle$ for $T \in [2.1, 2.4]$, $10^{6}$ MC cycles.}
\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.}
\label{fig:phase_magnetization_10M}
\caption{$\langle |m| \rangle$ for $T \in [2.1, 2.4]$, $10^{6}$ MC cycles.}
\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.}
\label{fig:phase_heat_10M}
\caption{$C_{V}$ for $T \in [2.1, 2.4]$, $10^{6}$ MC cycles.}
\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.}
\label{fig:phase_susceptibility_10M}
\caption{$\chi$ for $T \in [2.1, 2.4]$, $10^{6}$ MC cycles.}
\label{fig:phase_susceptibility_1M}
\end{figure}
\end{document}

6
latex/sections/methods.tex
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@ -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.

179
latex/sections/results.tex
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@ -2,22 +2,8 @@
\begin{document}
\section{Results}\label{sec:results}
% 2.1-2-4 divided into 40 steps, which gives us a step size of 0.0075.
% 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.
\subsection{Burn-in time}\label{subsec:burnin_time}
$\boldsymbol{Draft}$
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
samples to plot in a histogram for $T_{1}$ in Figure \ref{fig:histogram_1_0} and
$T_{2}$ in Figure \ref{fig:histogram_2_4}. For $T_{1}$ we can see that most samples
have the expected value $-2$, we have a distribution with low variance.
We used the estimated burn-in time as starting time for sampling, and generated
samples. To visualize the distribution of energy per spin $\epsilon$, we used histograms
with a bin size (...). In Figure \ref{fig:histogram_1_0} we show the distribution
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}
\caption{$\langle \epsilon \rangle$ for $T \in [2.1, 2.4]$, 1000000 MC cycles.}
\label{fig:phase_energy}
\end{figure}
\includegraphics[width=\linewidth]{../images/phase_transition/fox/wide/10M/energy.pdf}
\caption{$\langle \epsilon \rangle$ for $T \in [2.1, 2.4]$, $10^7$ MC cycles.}
\label{fig:phase_energy_10M}
\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}
\caption{$\langle |m| \rangle$ for $T \in [2.1, 2.4]$, 1000000 MC cycles.}
\label{fig:phase_magnetization}
\end{figure}
\includegraphics[width=\linewidth]{../images/phase_transition/fox/wide/10M/magnetization.pdf}
\caption{$\langle |m| \rangle$ for $T \in [2.1, 2.4]$, $10^7$ MC cycles.}
\label{fig:phase_magnetization_10M}
\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}
\caption{$C_{V}$ for $T \in [2.1, 2.4]$, 1000000 MC cycles.}
\label{fig:phase_heat}
\end{figure}
\includegraphics[width=\linewidth]{../images/phase_transition/fox/wide/10M/heat_capacity.pdf}
\caption{$C_{V}$ for $T \in [2.1, 2.4]$, $10^7$ MC cycles.}
\label{fig:phase_heat_10M}
\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}
\caption{$\chi$ for $T \in [2.1, 2.4]$, 1000000 MC cycles.}
\label{fig:phase_susceptibility}
\end{figure}
We include results for 10 million MC cycles in Appendix \ref{sec:extra_results}
\includegraphics[width=\linewidth]{../images/phase_transition/fox/wide/10M/susceptibility.pdf}
\caption{$\chi$ for $T \in [2.1, 2.4]$, $10^7$ MC cycles.}
\label{fig:phase_susceptibility_10M}
\end{figure} %
% Add something about the correlation length?
\subsection{Critical temperature}\label{subsec:critical_temperature}
$\boldsymbol{Draft}$
We use the critical temperatures found in previous section, in addition to the
scaling relation in Equation \eqref{eq:critical_intinite}
\begin{equation}
T_{c} - T_{c}(L = \infty) = \alpha L^{-1}
\label{eq:critical_intinite}
\end{equation}
to estimate the critical temperature for a lattize of infinte size. We also
compared the estimate with the analytical solution
\begin{equation}
T_{c}(L = \infty) = \frac{2}{\ln (1 + \sqrt{2})} J/k_{B} \approx 2.269 J/k_{B}
\end{equation}
using linear regression. In Figure \ref{fig:linreg} we find the critical
temperatures as function of the inverse lattice size. When the lattice size increase
toward infinity, $1/L$ goes toward zero, we find the intercept which gives us an
estimated value of the critical temperature for a lattice of infinite size.
% Critical temp regression figure
Based on the heat capacity (Figure \ref{fig:phase_heat_10M}) and susceptibility
(Figure \ref{fig:phase_susceptibility_10M}), we estimated the critical temperatures
of lattices of size $L \in \{20, 40, 60, 80, 100\}$ found in Table \ref{tab:critical_temperatures}.
\begin{table}[H]
\centering
\begin{tabular}{cc} % @{\extracolsep{\fill}}
\hline
$L$ & $T_{c}(L)$ \\
\hline
$20$ & $J$ \\
$40$ & $1$ \\
$60$ & $J / k_{B}$ \\
$80$ & $k_{B}$ \\
$100$ & $1 / J$ \\
\hline
\end{tabular}
\caption{Estimated critical temperatures for lattices $L \times L$.}
\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}
\caption{Linear regression, where $\beta_{0}$ is the intercept $T_{c}(L = \infty)$ and $\beta_{1}$ is the slope.}
\label{fig:linreg}
\includegraphics[width=\linewidth]{../images/phase_transition/fox/wide/10M/linreg.pdf}
\caption{Linear regression, where $\beta_{0}$ is the intercept approximating $T_{c}(L = \infty)$, and $\beta_{1}$ is the slope.}
\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}