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Missing estimate of speed-up and critical temperatures in table.

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Janita Willumsen
2023-12-04 10:08:43 +01:00
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latex/sections/results.tex
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@@ -3,72 +3,72 @@
\begin{document}
\section{Results}\label{sec:results}
\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
during the burn-in time is ... Increasing number of samples outweigh the ...
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.
We started with a lattice size $L = 20$ and considered the temperatures $T_{1} = 1.0 J/k_{B}$,
and $T_{2} = 2.4 J/k_{B}$, where $T_{2}$ is close to the analytical critical
temperature given by Equation \eqref{eq:critical_solution}.
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 $T_{2} = 2.4 J/k_{B}$,
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.}
%
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
of $\langle \epsilon \rangle$ is shown in Figure \ref{fig:burn_in_energy_2_4} and
$\langle |m| \rangle$ in Figure \ref{fig:burn_in_magnetization_2_4}. The lattice
is initialized in both an ordered and an unordered state. We observe that for
$T_{1}$ there is no change in either expectation value with increasing number of
Monte Carlo cycles, when we start with an ordered state. As for the unordered
lattice, we observe a change for the first 5000 MC cycles, where it stabilizes.
The approximated expected energy is $-2$ and expected magnetization is $1.0$,
which is to be expected for temperature 0f $1.0$. T is below the critical and the
pdf using $T = 1.0$ result in
\begin{align*}
p(s|T=1.0) &= \frac{1}{e^{-\beta \sum E(s)}} e^{-\beta E(s)} \\
&= \frac{1}{e^{-(1/k_{B}) \sum E(s)}} e^{-(1/k_{B}) E(s)} \ .
\end{align*}
To determine the burn-in time, we used the numerical estimates of energy
per spin for $T_{1}$ in Figure \ref{fig:burn_in_energy_1_0}, and $T_{2}$ in Figure
\ref{fig:burn_in_energy_2_4}. We also considered the estimates of magnetization
per spin for $T_{1}$ in Figure \ref{fig:burn_in_magnetization_1_0}, and for $T_{2}$
in Figure \ref{fig:burn_in_energy_2_4}.
% Not sure about the relevance of the paragraph below
We used random numbers to set the initial state and the index of spin to flip, and
observed different graphs ... However, when we set the seed of the random number
engine, we could reproduce the results at every run. It is possible to determine
the burn-in time analytically, using the correlation time given by $\tau \approx L^{d + z}$.
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.}.
However, when we increased number of Monte Carlo cycles the sampled generated during
the burn-in time did not affect the mean value (...). % Should add result showing this
The lattice was initialized in an ordered and an unordered state, for both temperatures. We observed
no change in expectation value of energy or magnetization for $T_{1}$, when we
initialized the lattice in an ordered state. As for the unordered initialized
lattice, we first observed a change in expectation values, and a stabilization around
$5000$ Monte Carlo cycles. The expected energy per spin is $\langle \epsilon \rangle = -2$
and the expected magnetization per spin is $\langle |m| \rangle = 1.0$. % add something about what is expected for $T_{1}$ ?
For $T_{2}$ we observed a change in expectation values for both the ordered and the
unordered lattice.
% \begin{align*}
% p(s|T=1.0) &= \frac{1}{e^{-\beta \sum E(s)}} e^{-\beta E(s)} \\
% &= \frac{1}{e^{-(1/k_{B}) \sum E(s)}} e^{-(1/k_{B}) E(s)} \ .
% \end{align*}
For $T_{2}$ we observe an increase in expected energy per spin $\langle \epsilon \rangle \approx -1.23$,
and a decrease in expected magnetization per spin $\langle |m| \rangle \approx 0.46$.
% Burn-in figures
\begin{figure}[H]
\centering
\includegraphics[width=\linewidth]{../images/burn_in_time_magnetization_1_0.pdf}
\caption{$\langle |m| \rangle$ as a function of time, for $T = 1.0 J / k_{B}$}
\caption{Magnetization per spin $\langle |m| \rangle$ as a function of time $t$ given by Monte Carlo cycles, for $T = 1.0 J / k_{B}$}
\label{fig:burn_in_magnetization_1_0}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[width=\linewidth]{../images/burn_in_time_energy_1_0.pdf}
\caption{$\langle \epsilon \rangle$ as a function of time, for $T = 1.0 J / k_{B}$}
\caption{Energy per spin $\langle \epsilon \rangle$ as a function of time $t$ given by Monte Carlo cycles, for $T = 1.0 J / k_{B}$}
\label{fig:burn_in_energy_1_0}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[width=\linewidth]{../images/burn_in_time_magnetization_2_4.pdf}
\caption{$\langle |m| \rangle$ as a function of time, for $T = 2.4 J / k_{B}$}
\caption{Magnetization per spin $\langle |m| \rangle$ as a function of time $t$ given by Monte Carlo cycles, for $T = 2.4 J / k_{B}$}
\label{fig:burn_in_magnetization_2_4}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[width=\linewidth]{../images/burn_in_time_energy_2_4.pdf}
\caption{$\langle \epsilon \rangle$ as a function of time, for $T = 2.4 J / k_{B}$}
\caption{Energy per spin $\langle \epsilon \rangle$ as a function of time $t$ given by Monte Carlo cycles, for $T = 2.4 J / k_{B}$}
\label{fig:burn_in_energy_2_4}
\end{figure}
\subsection{Probability distribution}\label{subsec:probability_distribution}
% Histogram figures
We used the estimated burn-in time as starting time for sampling, and generated
We used the estimated burn-in time of $5000$ Monte Carlo cycles as starting time, 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
with a bin size $0.02$. In Figure \ref{fig:histogram_1_0} we show the distribution
for $T_{1}$. Where 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]
@@ -85,16 +85,15 @@ centered around the expectation value $\langle \epsilon \rangle = -1.2370$. %
\caption{Histogram $T = 2.4 J / k_{B}$}
\label{fig:histogram_2_4}
\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
However, we observed a higher variance of Var$(\epsilon) = 0.0203$. When the temperature
increased, the system moved 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.
We continued 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
@@ -133,7 +132,7 @@ value of heat capacity the lattice size increase.
\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 magnetic susceptibility in Figure \ref{fig:phase_susceptibility_10M}, show
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.
@@ -156,14 +155,14 @@ of lattices of size $L \in \{20, 40, 60, 80, 100\}$ found in Table \ref{tab:crit
\hline
$L$ & $T_{c}(L)$ \\
\hline
$20$ & $J$ \\
$40$ & $1$ \\
$60$ & $J / k_{B}$ \\
$80$ & $k_{B}$ \\
$100$ & $1 / J$ \\
$20$ & $ J / k_{B}$ \\
$40$ & $ J / k_{B}$ \\
$60$ & $ J / k_{B}$ \\
$80$ & $ J / k_{B}$ \\
$100$ & $ J / k_{B}$ \\
\hline
\end{tabular}
\caption{Estimated critical temperatures for lattices $L \times L$.}
\caption{Estimated critical temperatures for lattices $L \times L$, where $L$ denote the lattice size.}
\label{tab:critical_temperatures}
\end{table}
We used the critical temperatures of finite lattices and the scaling relation in