Fix expectation values

latex/sections/abstract.tex

@@ -8,10 +8,10 @@
     Carlo sampling method. We determined the time of equilibrium to be approximately 
     $5000$ Monte Carlo cycles, and used the following samples to find the probability 
     distribution at temperature $T_{1} = 1.0 J / k_{B}$, and $T_{2} = 2.4 J / k_{B}$.
-    For $T_{1}$ the mean energy per spin is $\langle \epsilon \rangle \approx -1.9969 J$, 
+    For $T_{1}$ the mean energy per spin is $\langle \epsilon \rangle \approx -1.9972 J$, 
     with a variance $\text{Var} (\epsilon) = 0.0001$. And for $T_{2}$, close to the critical 
-    temperature, the mean energy per spin is $\langle \epsilon \rangle \approx -1.2370 J$, 
-    with a variance $\text{Var} (\epsilon) = 0.0203$. In addition, we estimated 
+    temperature, the mean energy per spin is $\langle \epsilon \rangle \approx -1.2367 J$, 
+    with a variance $\text{Var} (\epsilon) = 0.0202$. In addition, we estimated 
     the expected energy and magnetization per spin, the heat capacity and magnetic 
     susceptibility. We have estimated the critical temperatures of finite lattice sizes, 
     and used these values to approximate the critical temperature of a lattice of 

latex/sections/appendices.tex

@@ -216,7 +216,7 @@ the magnetic susceptibility.
     \label{fig:phase_susceptibility_1M}
 \end{figure}
 
-Result of profiling using Score-P in Figure \ref{fig:scorep_assessment}.
+Assessment of profiling using Score-P in Figure \ref{fig:scorep_assessment}.
 \begin{figure}[H]
     \centering
     \includegraphics[width=\linewidth]{../images/profiling.pdf}

latex/sections/conclusion.tex

@@ -21,9 +21,9 @@ Continuing, we used the generated samples to compute energy per spin $\langle \e
 magnetization per spin $\langle |m| \rangle$, heat capacity $C_{V}$, and $\chi$. 
 In addition, we estimated the probability distribution for temperatures $T_{1} = 1.0 J / k_{B}$,
 and $T_{2} = 2.4 J / k_{B}$. We found that for $T_{1}$ the expected mean energy 
-per spin is $\langle \epsilon \rangle \approx -1.9969 J$, with a variance $\text{Var} (\epsilon) = 0.0001$. 
-And for $T_{2}$, the mean energy per spin is $\langle \epsilon \rangle \approx -1.2370 J$, 
-with a variance $\text{Var} (\epsilon) = 0.0203$. 
+per spin is $\langle \epsilon \rangle \approx -1.9972 J$, with a variance $\text{Var} (\epsilon) = 0.0001$. 
+And for $T_{2}$, the mean energy per spin is $\langle \epsilon \rangle \approx -1.2367 J$, 
+with a variance $\text{Var} (\epsilon) = 0.0202$. 
 
 We estimated the expected energy and magnetization per spin, in addition to the 
 heat capacity and susceptibility for lattices of size $L = {20, 40, 60, 80, 100}$. 

latex/sections/results.tex

@@ -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.