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## Thermo-dynamic wave equation.

• It is a good idea to first check if the LCEE engine is online, click here and make sure you get a response with a build date and version number, if there is no response or you see an error, please be patient as the site may be down while undergoing maintenance.
• Assuming LCEE is live, open a bash terminal, navigate to a clean working directory and type the following into the terminal.

which will download an octave m file, now fire up octave by typing the following into the terminal.

 # octave

When the Octave terminal is ready to go, simply execute the m-file by typing

 www_theta

This will begin a single frame of 10000 iterations (65,536 decimal) which will take just a couple of seconds, after which you will see two graphs pop up on the screen. Figure 1 is the Total torque of the ant as it traverses the lattice.
Figure 2 is the accompanying surface plot.

If you zoom into Figure 1 above and capture the graph between 0 and 13,000 iteration steps, you will get the following plot below. Click here to see the same graph produced on another website Now to get a longer run, edit the www_theta.m file and change the number of frames from '0001' to '00ff'
 system ("wget 'http://buzwordsalad.com/lcee/theta.sh?-d 2 -xy 8050 -f 00ff -t' -O output_t.bin");

Now run the mfile again to get the following plots. Figure 1 is the Total torque of the ant as it traverses the lattice.
Figure 2 is the accompanying surface plot. Due to limitations on bandwidth, file sizes are limitted, so iteration frame counts are capped between 0001 and 00ff

By generating a further two 'theta' plots, combining them with the data in figure 1 above and ensuring that all three binary lattice sizes are different, we can plot the tripplet 'theta data' as a parametric plot on the x,y and z axis as a function of 'n' (itertation count), this gives us the following combined spatial and density graph, note that the density data is inferred by how many times the same points are plotted over each other as the data points are not monotonic) Looking at the rotated image clearly demonstrates that the system has structure, within this structure there are hidden random walks, these can be found in my paper on langtons ant.

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