Equilibrium and Variance

I implemented the Swedson-Wang algorithm for my Ising model. I used a lattice of size 256 by 256. In order to determine a good cut-off for the number of steps required to reach equilibrium, I performed a series of measurements of the magnetism of the lattice at various temperatures after a variety of time steps. From examining Figures 6 and 17 I decided to use 128 iterations of the Swendson-Wang algorithm as my benchmark for equilibrium. I provide Figures 16 and 16 in order to further demonstrate the similarities between the distributions of magnetization samples after 64 and 128 time steps.

After settling on 128 time steps for my "time for equilibrium" I wanted to answer two questions. First, I was curious about the variance between the samples I had taken. Figures 27 and 31 display the variance measurements between the samples from Figures 7 and 17 respectively. Note that the plots are semilog and that the variances peak near the temperature

as expected.

**Figure 1:** Lattice magnetization after 2 time steps. Temperature range: 0.0 - 5.0. Temperature delta: 0.01. Number of trials per temperature: 256.
$\begin{figure}\begin{center} \mbox{\epsfig{figure=swendwang/out2.eps}} \end{center} \end{figure}$

**Figure 2:** Lattice magnetization after 4 time steps. Temperature range: 0.0 - 5.0. Temperature delta: 0.01. Number of trials per temperature: 256.
$\begin{figure}\begin{center} \mbox{\epsfig{figure=swendwang/out4.eps}} \end{center} \end{figure}$

**Figure 3:** Lattice magnetization after 8 time steps. Temperature range: 0.0 - 5.0. Temperature delta: 0.01. Number of trials per temperature: 256.
$\begin{figure}\begin{center} \mbox{\epsfig{figure=swendwang/out8.eps}} \end{center} \end{figure}$

**Figure 4:** Lattice magnetization after 16 time steps. Temperature range: 0.0 - 5.0. Temperature delta: 0.01. Number of trials per temperature: 256.
$\begin{figure}\begin{center} \mbox{\epsfig{figure=swendwang/out16.eps}} \end{center} \end{figure}$

**Figure 5:** Lattice magnetization after 32 time steps. Temperature range: 0.0 - 5.0. Temperature delta: 0.01. Number of trials per temperature: 256.
$\begin{figure}\begin{center} \mbox{\epsfig{figure=swendwang/out32.eps}} \end{center} \end{figure}$

**Figure 6:** Lattice magnetization after 64 time steps. Temperature range: 0.0 - 5.0. Temperature delta: 0.01. Number of trials per temperature: 256.
$\begin{figure}\begin{center} \mbox{\epsfig{figure=swendwang/out64.eps}} \end{center} \end{figure}$

**Figure 7:** Lattice magnetization after 128 time steps. Temperature range: 0.0 - 5.0. Temperature delta: 0.01. Number of trials per temperature: 256.
$\begin{figure}\begin{center} \mbox{\epsfig{figure=swendwang/out128.eps}} \end{center} \end{figure}$

**Figure 8:** Lattice magnetization after 256 time steps. Temperature range: 0.0 - 5.0. Temperature delta: 0.01. Number of trials per temperature: 256.
$\begin{figure}\begin{center} \mbox{\epsfig{figure=swendwang/out256.eps}} \end{center} \end{figure}$

**Figure 9:** Lattice magnetization after 512 time steps. Temperature range: 0.0 - 5.0. Temperature delta: 0.01. Number of trials per temperature: 256.
$\begin{figure}\begin{center} \mbox{\epsfig{figure=swendwang/out512.eps}} \end{center} \end{figure}$

**Figure 10:** Lattice magnetization after 1024 time steps. Temperature range: 0.0 - 5.0. Temperature delta: 0.01. Number of trials per temperature: 256.
$\begin{figure}\begin{center} \mbox{\epsfig{figure=swendwang/out1024.eps}} \end{center} \end{figure}$

**Figure 11:** Lattice magnetization after 2 time steps. Temperature range: 2.0 - 3.0. Temperature delta: 0.001. Number of trials per temperature: 256.
$\begin{figure}\begin{center} \mbox{\epsfig{figure=swendwangtight/out2.eps}} \end{center} \end{figure}$

**Figure 12:** Lattice magnetization after 4 time steps. Temperature range: 2.0 - 3.0. Temperature delta: 0.001. Number of trials per temperature: 256.
$\begin{figure}\begin{center} \mbox{\epsfig{figure=swendwangtight/out4.eps}} \end{center} \end{figure}$

**Figure 13:** Lattice magnetization after 8 time steps. Temperature range: 2.0 - 3.0. Temperature delta: 0.001. Number of trials per temperature: 256.
$\begin{figure}\begin{center} \mbox{\epsfig{figure=swendwangtight/out8.eps}} \end{center} \end{figure}$

**Figure 14:** Lattice magnetization after 16 time steps. Temperature range: 2.0 - 3.0. Temperature delta: 0.001. Number of trials per temperature: 256.
$\begin{figure}\begin{center} \mbox{\epsfig{figure=swendwangtight/out16.eps}} \end{center} \end{figure}$

**Figure 15:** Lattice magnetization after 32 time steps. Temperature range: 2.0 - 3.0. Temperature delta: 0.001. Number of trials per temperature: 256.
$\begin{figure}\begin{center} \mbox{\epsfig{figure=swendwangtight/out32.eps}} \end{center} \end{figure}$

**Figure 16:** Lattice magnetization after 64 time steps. Temperature range: 2.0 - 3.0. Temperature delta: 0.001. Number of trials per temperature: 256.
$\begin{figure}\begin{center} \mbox{\epsfig{figure=swendwangtight/out64.eps}} \end{center} \end{figure}$

**Figure 17:** Lattice magnetization after 128 time steps. Temperature range: 2.0 - 3.0. Temperature delta: 0.001. Number of trials per temperature: 256.
$\begin{figure}\begin{center} \mbox{\epsfig{figure=swendwangtight/out128.eps}} \end{center} \end{figure}$

**Figure 18:** Lattice magnetization after 256 time steps. Temperature range: 2.0 - 3.0. Temperature delta: 0.001. Number of trials per temperature: 256.
$\begin{figure}\begin{center} \mbox{\epsfig{figure=swendwangtight/out256.eps}} \end{center} \end{figure}$

**Figure 19:** Lattice magnetization after 512 time steps. Temperature range: 2.0 - 3.0. Temperature delta: 0.001. Number of trials per temperature: 256.
$\begin{figure}\begin{center} \mbox{\epsfig{figure=swendwangtight/out512.eps}} \end{center} \end{figure}$

**Figure 20:** Lattice magnetization after 1024 time steps. Temperature range: 2.0 - 3.0. Temperature delta: 0.001. Number of trials per temperature: 256.
$\begin{figure}\begin{center} \mbox{\epsfig{figure=swendwangtight/out1024.eps}} \end{center} \end{figure}$

**Figure 21:** Variance among samples from Figure 1 (after two time steps).
$\begin{figure}\begin{center} \mbox{\epsfig{figure=swendwang/out2var.eps}} \end{center} \end{figure}$

**Figure 22:** Variance among samples from Figure 2 (after four time steps).
$\begin{figure}\begin{center} \mbox{\epsfig{figure=swendwang/out4var.eps}} \end{center} \end{figure}$

**Figure 23:** Variance among samples from Figure 3 (after eight time steps).
$\begin{figure}\begin{center} \mbox{\epsfig{figure=swendwang/out8var.eps}} \end{center} \end{figure}$

**Figure 24:** Variance among samples from Figure 4 (after sixteen time steps).
$\begin{figure}\begin{center} \mbox{\epsfig{figure=swendwang/out16var.eps}} \end{center} \end{figure}$

**Figure 25:** Variance among samples from Figure 5 (after thirty-two time steps).
$\begin{figure}\begin{center} \mbox{\epsfig{figure=swendwang/out32var.eps}} \end{center} \end{figure}$

**Figure 26:** Variance among samples from Figure 6 (after sixty-four time steps).
$\begin{figure}\begin{center} \mbox{\epsfig{figure=swendwang/out64var.eps}} \end{center} \end{figure}$

**Figure 27:** Variance among samples from Figure 7 (after 128 time steps).
$\begin{figure}\begin{center} \mbox{\epsfig{figure=swendwang/out128var.eps}} \end{center} \end{figure}$

**Figure 28:** Variance among samples from Figure 8 (after 256 time steps).
$\begin{figure}\begin{center} \mbox{\epsfig{figure=swendwang/out256var.eps}} \end{center} \end{figure}$

**Figure 29:** Variance among samples from Figure 9 (after 512 time steps).
$\begin{figure}\begin{center} \mbox{\epsfig{figure=swendwang/out512var.eps}} \end{center} \end{figure}$

**Figure 30:** Variance among samples from Figure 10 (after 1024 time steps).
$\begin{figure}\begin{center} \mbox{\epsfig{figure=swendwang/out1024var.eps}} \end{center} \end{figure}$

**Figure 31:** Variance among samples from Figure 17 (after 128 time steps).
$\begin{figure}\begin{center} \mbox{\epsfig{figure=var2.eps}} \end{center} \end{figure}$