Missing estimate of speed-up and critical temperatures in table.
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Janita Willumsen
@@ -40,12 +40,15 @@ the spin up as visualized in Figure \ref{fig:tikz_neighbor}.
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Using the values estimated for the $2 \times 2$ case, found in \ref{tab:lattice_config},
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we find the partition function
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\begin{align*}
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Z &= 1 \cdot e^{-\beta (-8J)} + 4 \cdot e^{-\beta (0)} + 4 \cdot e^{-\beta (0)} + 2 \cdot e^{-\beta (8J)} \\
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& \quad + 4 \cdot e^{-\beta (0)} 1 \cdot e^{-\beta (-8J)} \\
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Z &= 1 \cdot e^{-\beta (-8J)} + 4 \cdot e^{-\beta (0)} + 4 \cdot e^{-\beta (0)} \\
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& \quad + 2 \cdot e^{-\beta (8J)} + 4 \cdot e^{-\beta (0)} 1 \cdot e^{-\beta (-8J)} \\
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&= 2e^{8 \beta J} + 2e^{-8 \beta J} + 12.
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\end{align*}
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We rewrite the expression using the identity $\cosh(8 \beta J) = 1/2 \big( e^{8 \beta J} + e^{-8 \beta J})$,
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and get
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We rewrite the expression using the identity
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\begin{align*}
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\cosh(8 \beta J) &= 1/2 \big( e^{8 \beta J} + e^{-8 \beta J})
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\end{align*}
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and find
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\begin{align*}
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z &= 4 \cosh (8 \beta J) + 12 \ .
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\end{align*}
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@@ -55,12 +58,10 @@ and get
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For a linear function of a stochastic random variable $X$, with a known probability
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distribution, the expected value of $x$ is given by
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\begin{align*}
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\mathbb{E}(aX + b) &= a \cdot \mathbb{E}(X) + b & \text{\cite[p. 131]{springer:2012:modernstat}}
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\langle aX + b \rangle &= a \cdot \langle X \rangle + b & \text{\cite[p. 131]{springer:2012:modernstat}}
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\end{align*}
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In our case the discrete random variable is the spin configuration, and we want
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to find the expected value of the function $E(\mathbf{s})$. Continuing, we will
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use the notation $\langle E \rangle$ for the expectation value of a given variable,
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in this case $E$.
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to find the expected value of the function $E(\mathbf{s})$.
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Both energy per spin and magnetization per spin are functions of $\mathbf{s}$.
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In addition, the number of spins is given as a constant for each lattice. We can
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@@ -75,9 +76,10 @@ The same applies to magnetization per spin
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\begin{align*}
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\langle |m| \rangle = \frac{1}{N} \sum_{i=1}^{N} |M(s_{i})| p(s_{i} \ | \ T) \ .
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\end{align*}
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Continuing with the expectation values for a $2 \times 2$ lattice, excluding the terms which give zero we get
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Continuing with the expectation values for a $2 \times 2$ lattice, excluding the terms which result in zero, we get
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\begin{align*}
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\langle E \rangle &= (-8J) \cdot \frac{1}{Z} e^{8 \beta J} + 2 \cdot (8J) \cdot \frac{1}{Z} e^{-8 \beta J} + (-8J) \cdot \frac{1}{Z} e^{8 \beta J} \\
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\langle E \rangle &= (-8J) \cdot \frac{1}{Z} e^{8 \beta J} + 2 \cdot (8J) \cdot \frac{1}{Z} e^{-8 \beta J} \\
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& \quad + (-8J) \cdot \frac{1}{Z} e^{8 \beta J} \\
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&= \frac{16J}{Z} \big(e^{-8 \beta J} - e^{8 \beta J}) \\
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&= -\frac{32J \sinh(8 \beta J)}{4(\cosh(8 \beta J) + 3)} \\
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&= -\frac{8J \sinh(8 \beta J)}{\cosh(8 \beta J) + 3} \ ,
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@@ -90,9 +92,10 @@ and
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&= \frac{4 (2e^{8 \beta J} + 4)}{4(\cosh(8 \beta J) + 3)} \\
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&= \frac{2(e^{8 \beta J} + 2)}{\cosh(8 \beta J) + 3} \ .
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\end{align*}
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The squared function
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The squared energy function
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\begin{align*}
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\langle E^{2} \rangle &= (-8J)^{2} \cdot \frac{1}{Z} e^{8 \beta J} + 2 \cdot (8J)^{2} \cdot \frac{1}{Z} e^{-8 \beta J} + (-8J)^{2} \cdot \frac{1}{Z} e^{8 \beta J} \\
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\langle E^{2} \rangle &= (-8J)^{2} \cdot \frac{1}{Z} e^{8 \beta J} + 2 \cdot (8J)^{2} \cdot \frac{1}{Z} e^{-8 \beta J} \\
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& \quad + (-8J)^{2} \cdot \frac{1}{Z} e^{8 \beta J} \\
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&= \frac{128J^{2}}{Z} \big(e^{8 \beta J} + e^{-8 \beta J} \big) \\
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&= \frac{128J^{2} \cosh(8 \beta J)}{4(\cosh(8 \beta J) + 3)} \\
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&= \frac{64J^{2} \cosh(8 \beta J)}{\cosh(8 \beta J) + 3} \ ,
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@@ -116,6 +119,7 @@ and
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&= \frac{4(e^{8 \beta J} + 2)^{2}}{(\cosh(8 \beta J) + 3)^{2}} \ .
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\end{align*}
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\section{Heat capacity and magnetic susceptibility}\label{sec:heat_susceptibility}
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To find the heat capacity in Eq. \ref{eq:heat_capacity}, we normalize to heat
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capacity per spin
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@@ -149,8 +153,9 @@ and the variance of the total magnetization is given by
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\mathbb{V}(M) &= \mathbb{E}(M^{2}) - [\mathbb{E}(|M|)]^{2} \\
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&= \frac{8e^{8 \beta J} + 8}{\cosh(8 \beta J) + 3} - \frac{4(e^{8 \beta J} + 2)^{2}}{(\cosh(8 \beta J) + 3)^{2}} \\
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&= \frac{(8(e^{8 \beta J} + 1)) \cdot (\cosh(8 \beta J) + 3) - 4(e^{8 \beta J} + 2)^{2}}{(\cosh(8 \beta J) + 3)^{2}} \\
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&= \frac{4(e^{8 \beta J} + 1) \cdot (e^{8 \beta J} + e^{-8 \beta J}) + 24(e^{8 \beta J} + 1) - 4(e^{8 \beta J} + 1)^{2}}{(\cosh(8 \beta J) + 3)^{2}} \\
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&= \frac{4e^{2(8 \beta J)} + 4e^{8 \beta J} 4e^{0} + 4e^{-8 \beta J} 24e^{8 \beta J} + 24 - 4e^{2(8 \beta J)} - 16e^{8 \beta J} - 16}{(\cosh(8 \beta J) + 3)^{2}} \\
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&= \frac{4(e^{8 \beta J} + 1) \cdot (e^{8 \beta J} + e^{-8 \beta J})}{(\cosh(8 \beta J) + 3)^{2}} \\
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& \quad + \frac{24(e^{8 \beta J} + 1) - 4(e^{8 \beta J} + 1)^{2}}{(\cosh(8 \beta J) + 3)^{2}}\\
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% &= \frac{4e^{2(8 \beta J)} + 4e^{8 \beta J} 4e^{0} + 4e^{-8 \beta J} 24e^{8 \beta J} + 24 - 4e^{2(8 \beta J)} - 16e^{8 \beta J} - 16}{(\cosh(8 \beta J) + 3)^{2}} \\
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&= \frac{4(3e^{8 \beta J} + e^{-8 \beta J} + 3)}{(\cosh(8 \beta J) + 3)^{2}} \ .
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\end{align*}
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We find the heat capacity
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@@ -181,29 +186,30 @@ where the five distinct values are $\Delta E = \{-16J, -8J, 0, 8J, 16J\}$.
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\section{Additional results}\label{sec:additional_results}
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Results of 1 million MC cycles.
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\begin{figure}
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We also did the phase transition experiment using 1 million MC cycles. In Figure \ref{fig:phase_energy_1M}
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we show expected energy per spin, and in Figure \ref{fig:phase_magnetization_1M}
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expected magnetization per spin.
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\begin{figure}[H]
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\centering
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\includegraphics[width=\linewidth]{../images/phase_transition/fox/wide/10M/energy.pdf}
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\caption{$\langle \epsilon \rangle$ for $T \in [2.1, 2.4]$, $10^{6}$ MC cycles.}
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\label{fig:phase_energy_1M}
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\end{figure}
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\begin{figure}
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\end{figure} %
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\begin{figure}[H]
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\centering
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\includegraphics[width=\linewidth]{../images/phase_transition/fox/wide/10M/magnetization.pdf}
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\caption{$\langle |m| \rangle$ for $T \in [2.1, 2.4]$, $10^{6}$ MC cycles.}
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\label{fig:phase_magnetization_1M}
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\end{figure}
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\begin{figure}
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\end{figure} %
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In Figure \ref{fig:phase_heat_1M} we show heat capacity, and in Figure \ref{fig:phase_susceptibility_1M}
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the magnetic susceptibility.
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\begin{figure}[H]
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\centering
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\includegraphics[width=\linewidth]{../images/phase_transition/fox/wide/10M/heat_capacity.pdf}
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\caption{$C_{V}$ for $T \in [2.1, 2.4]$, $10^{6}$ MC cycles.}
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\label{fig:phase_heat_1M}
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\end{figure}
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\begin{figure}
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\end{figure} %
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\begin{figure}[H]
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\centering
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\includegraphics[width=\linewidth]{../images/phase_transition/fox/wide/10M/susceptibility.pdf}
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\caption{$\chi$ for $T \in [2.1, 2.4]$, $10^{6}$ MC cycles.}
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