交通核心的Kramers-Wannier对偶性
Kramers-Wannier Duality at the Heart of Traffic
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中文总结 AI 辅助
构建类似伊辛模型的元胞自动机模型研究交通高风险密度，通过耦合函数得出特定根，恢复Kramers-Wannier对偶性，推测根差异为投影固定点，代入函数得三维伊辛逆临界耦合近似值，并关联交通风险与车辆类型。
中文摘要 AI 辅助
为了对高风险交通密度进行建模，构建了一个具有类似伊辛模型性质的元胞自动机模型。将车辆间的吸引-排斥力评估为耦合，耦合函数为\(p(1 - p)=g/8\)，其中\(p\)是状态为1的单元格的初始分布，\(g\)是摩尔邻居数，\(g = 1\)时其根为\(\cos^2(\pi/8)\)和\(\sin^(\pi/8)\)。代码中未使用任何三角函数得出此结果，进而出现正切多项式\(\tan^2(x)+\tan(x)\)。恢复了Kramers-Wannier对偶性，并推测根之间的\(1/\sqrt{2}\)差异是通过古德曼函数从二维伊辛模型到一维伊辛链的投影机制的固定点。将在提议的固定点处评估的Sigmoid代入逻辑耦合函数的导数\(\sigma(1/\sqrt{2})(1-\sigma(1/\sqrt{2}))\)，得到三维伊辛逆临界耦合的数值近似。最后，将结果与交通中的风险密度和车辆类型相关联，解释致命事故的放大情况。
英文摘要
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 $\cos^2(π/8)$ and $\sin^2(π/8)$ 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, $σ(1/\sqrt{2})(1-σ(1/\sqrt{2}))$, 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.
