Kramers-Wannier Duality at the Heart of Traffic

Abstract

To model high-risk traffic densities a cellular automaton model is constructed, exhibiting Ising model-like properties. The attraction-repulsion forces between vehicles are evaluated as coupling, and the coupling function $p(1-p)=g/8$, where p is the initial distribution of cells with state 1, and g is the Moore neighbor count, which has roots $\cos^2(\pi/8)$ and $\sin^2(\pi/8)$ for $g=1$. This is achieved without the use of any trigonometric functions in the code. From the roots, the tangent polynomial $\tan^2(x) + \tan(x)$ emerges. Kramers-Wannier duality is recovered and it is conjectured that the $1/\sqrt{2}$ difference between the roots serves as a fixed point for a projection mechanism from the 2-dimensional Ising model onto a 1-dimensional Ising chain through the Gudermannian function. The sigmoid evaluated at the proposed fixed point is substituted into the derivative of the logistic coupling function, $\sigma(1/\sqrt{2})(1-\sigma(1/\sqrt{2}))$, yielding a numerical approximation to the three-dimensional Ising inverse critical coupling. Finally, the results are linked to risk densities in traffic and vehicle types, accounting for the amplification of fatal accidents.