%;*******************************************************************************
%;*       Last edited date: 1st June 2016 Ver 1.0
%;*       langtons_ant_3.m 
%;*       m-file for langtons ant (closed form)
%;*
%;*       Written by: Graham Medland
%;*       gmail.com: graham.medland@
%;*******************************************************************************
clear
E=100              % Size of lattice.
N=11200         % Number of iterations.
z(1:1:E^2)=-1;
k(1:1:E^2)=0;
k(1)=(E^2+E)/2;
for t=1:1:N;
k(t+1)=mod(k(t)+real(round((0.5*(E+1)*exp(i*pi/4*(trace(diag(z))-E^2)))-(0.5*(E-1)*exp(-i*pi/4*(trace(diag(z))-E^2)))))+imag(round((0.5*(E+1)*exp(i*pi/4*(trace(diag(z))-E^2)))-(0.5*(E-1)*exp(-i*pi/4*(trace(diag(z))-E^2))))),E^2);
z(k(t+1)+1)=real(exp(2*i*pi/4*(1+z(k(t+1)+1))));
endfor;

imagesc(reshape(z,E,E))
disp("\n      Langtons Ant characteristic equation Delta_k \n      By Graham Medland )")

Copy and paste into octave-online to see Langtons ant.

Mapping the ant world to a one dimensional flat torus or manifold using Eulers identity,

${\displaystyle {f:{\textbf {k}}\rightarrow \mathbb {C} }}$

${\displaystyle \left(n\cos \left[{\frac {\pi }{2}}\left(\theta +{\textbf {z}}_{{\textbf {k}}_{t}}\right)\right]+i\sin \left[{\frac {\pi }{2}}\left(\theta +{\textbf {z}}_{{\textbf {k}}_{t}}\right)\right]:\theta \in [-\pi ,+\pi ]\right)\rightarrow \left[(\Delta x,i\Delta y):\in \mathbb {C} \right]}$

This can now be recast using partial fractions into the form.

${\displaystyle {\frac {1}{2}}\left(n+1\right)e^{i\theta }+{\frac {1}{2}}\left(n-1\right)e^{-i\theta }=n\cos(\theta )+i\sin(\theta )}$

With variables.

• ${\displaystyle K={\frac {\pi }{4}}}$ this represents the orthogonal changes in direction (Electric / Magnetic field lines).
• ${\displaystyle \psi _{\vartriangle }^{\star }}$ and ${\displaystyle \psi _{\theta }^{\star }}$ be the complex conjugates of ${\displaystyle \psi _{\vartriangle }}$ and ${\displaystyle \psi _{\theta }}$
• ${\displaystyle A={\frac {1}{2}}\left(E+1\right)}$ and ${\displaystyle B={\frac {1}{2}}\left(E-1\right)}$ where ${\displaystyle E=}$ the length of one side of the lattice.
• ${\displaystyle i={\sqrt {-1}}}$ and ${\displaystyle {\tilde {z}}}$ is the basis vector for the z axis, the same principle applies to the other two axis.

We create the complex oscillator (which gives rise to the discrete rotations) where.

${\displaystyle {\textbf {z}}_{{\textbf {k}}_{t}}=\psi _{\vartriangle _{t}}}$ and. ${\displaystyle \psi _{\vartriangle _{t\vert t+1}}=e^{-i2K\psi _{\vartriangle _{t\vert t}}}}$

Now we create the rotate equation, using the sum over all paths integral, see Feynman chessboard since the ant visits all of the squares over time.

${\displaystyle \psi _{\theta _{t}}=e^{iK\sum \limits _{n=1}^{E^{2}}\left(\psi _{n\vert t}-1\right)}}$

Noting that

${\displaystyle \lim _{t\to \infty }K\sum \limits _{n=1}^{E^{2}}\left(\psi _{n\vert {t}}-1\right)\to {\frac {1}{2}}E^{2}}$

And since the continuum is elliptical we need two of these functions

${\displaystyle \Delta k_{t}=A\psi _{\theta _{t}}+B\psi _{\theta _{t}}^{\star }}$

Now we make the Quantum rotor equations

${\displaystyle \psi _{L}=\psi _{\vartriangle }\left(A\psi _{\theta }+B\psi _{\theta }^{\star }\right)}$

${\displaystyle \psi _{R}=\psi _{\vartriangle }^{\star }\left(A\psi _{\theta }+B\psi _{\theta }^{\star }\right)}$

These rotors (torsion based spinor's) when combined expand to form,

${\displaystyle \psi _{L}=e^{-i2K\psi _{\vartriangle _{t}}}\left({\frac {1}{2}}\left(E+1\right)e^{iK\sum \limits _{n=1}^{E^{2}}\left(\psi _{n}-1\right)}+{\frac {1}{2}}\left(E-1\right)e^{-iK\sum \limits _{n=1}^{E^{2}}\left(\psi _{n}-1\right)}\right)}$

${\displaystyle \psi _{R}=e^{i2K\psi _{\vartriangle _{t}}}\left({\frac {1}{2}}\left(E+1\right)e^{iK\sum \limits _{n=1}^{E^{2}}\left(\psi _{n}-1\right)}+{\frac {1}{2}}\left(E-1\right)e^{-iK\sum \limits _{n=1}^{E^{2}}\left(\psi _{n}-1\right)}\right)}$

This then becomes a superposition of both rotors where in the limit as t tends to infinity we get an equal spread of both functions. At the start of the iteration process, after four iterations of ${\displaystyle \psi _{L}}$ we get ${\displaystyle \psi _{R}}$ which represents a step change in direction of phase, after another four iterations of ${\displaystyle \psi _{R}}$ we get back to ${\displaystyle \psi _{L}}$, these 8 steps are represented by a simple sum,

${\displaystyle \sum _{t=1}^{8}\Delta \Psi _{(t)}=\sum _{t=1}^{4}\psi _{L_{(t)}}+\sum _{t=5}^{8}\psi _{R_{(t)}}}$

Thus in its native state as a 2x2 SU2 Special Unitary Matrix, we have

${\displaystyle \psi _{L}=\psi _{\theta _{t}}+\psi _{\vartriangle _{t}}\quad }$ and ${\displaystyle \psi _{R}=\psi _{\theta _{t}}-\psi _{\vartriangle _{t}}\quad }$

The Langtons ant Binary matrix ${\displaystyle {\textbf {z}}}$ becomes an oscillating spinor (a model for a single subatomic particle), with the matrix increased in size we start to see a complex chaotic system which now models an ensemble of particles (a model for the Higgs field interaction).

Starting with a quadratic equation with ${\displaystyle B\psi _{\vartriangle _{t}}}$ and ${\displaystyle A\psi _{\vartriangle _{t}}}$ forming the eigen wave functions (roots of the quadratic), expanding, we get

${\displaystyle \left(\psi _{\theta _{t}}+B\psi _{\vartriangle _{t}}\right)\left(\psi _{\theta _{t}}^{\star }+A\psi _{\vartriangle _{t}}\right)=\psi _{\theta _{t}}.\psi _{\theta _{t}}^{\star }+\psi _{\theta _{t}}.A\psi _{\vartriangle _{t}}+B\psi _{\vartriangle _{t}}.\psi _{\theta _{t}}^{\star }+B\psi _{\vartriangle _{t}}.A\psi _{\vartriangle _{t}}}$

Ths simplifies to

${\displaystyle 1+\psi _{\theta _{t}}.A\psi _{\vartriangle _{t}}+B\psi _{\vartriangle _{t}}.\psi _{\theta _{t}}^{\star }+AB=1+\psi _{\vartriangle }\left(A\psi _{\theta }+B\psi _{\theta }^{\star }\right)+AB}$

Therefore the Langtons ant wave equation can be expressed as a quadratic equation.

${\displaystyle \psi _{L}=\left(\psi _{\theta _{t}}+B\psi _{\vartriangle _{t}}\right)\left(\psi _{\theta _{t}}^{\star }+A\psi _{\vartriangle _{t}}\right)-(1+AB)}$

As an SHM equation we can remove ${\displaystyle \psi _{L}}$, rearange to form the 'langtons ant wave equation'. ${\displaystyle \left(\psi _{\theta _{t}}+B\psi _{\vartriangle _{t}}\right)\left(\psi _{\theta _{t}}^{\star }+A\psi _{\vartriangle _{t}}\right)=1+AB}$

The super attractor of the bounded wave equation (Langtons ant) produces a Gaussian distributed system, where at the start of the algorithm there is a low value of entropy (since all squares are the same colour) after several million iterations half of the squares are now black and the other half are white and the ant reaches a steady state.

Category:Thermodynamics Category:Chaos theory Category:Equations Category:Spin models