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Fix expectation values

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Janita Willumsen
2023-12-05 19:17:54 +01:00
parent e4baf0bbca
commit b99abbc291
4 changed files with 14 additions and 14 deletions
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14
latex/sections/results.tex
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@@ -26,16 +26,16 @@ The lattice was initialized in an ordered and an unordered state, for both tempe
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 $T_{1}$ ?
$5000$ Monte Carlo cycles. The expected energy per spin is $\langle \epsilon \rangle \approx -2$
and the expected magnetization per spin is $\langle |m| \rangle \approx 1$. % 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 $\langle |m| \rangle \approx 0.46$.
For $T_{2}$ we observe an increase in expected energy per spin $\langle \epsilon \rangle \approx -1.25$,
and a decrease in expected magnetization per spin $\langle |m| \rangle \approx 0.47$.
% Burn-in figures
\begin{figure}[H]
\centering
@@ -69,7 +69,7 @@ We used the estimated burn-in time of $5000$ Monte Carlo cycles as starting time
samples. To visualize the distribution of energy per spin $\epsilon$, we used histograms
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
value of energy per spin is $\langle \epsilon \rangle = -1.9972$, with a low variance
of Var$(\epsilon) = 0.0001$. %
\begin{figure}[H]
\centering
@@ -78,14 +78,14 @@ of Var$(\epsilon) = 0.0001$. %
\label{fig:histogram_1_0}
\end{figure} %
In Figure \ref{fig:histogram_2_4}, for $T_{2}$, the samples of energy per spin is
centered around the expectation value $\langle \epsilon \rangle = -1.2370$. %
centered around the expectation value $\langle \epsilon \rangle = -1.2367$. %
\begin{figure}[H]
\centering
\includegraphics[width=\linewidth]{../images/pd_estimate_2_4.pdf}
\caption{Distribution of values of energy per spin, when temperature is $T = 2.4 J / k_{B}$}
\label{fig:histogram_2_4}
\end{figure} %
However, we observed a higher variance of Var$(\epsilon) = 0.0203$. When the temperature
However, we observed a higher variance of Var$(\epsilon) = 0.0202$. When the temperature
increased, the system moved from an ordered to an unordered state. The change in
system state, or phase transition, indicates the temperature is close to a critical
point.