[论文解读] Finding the Edge of Chaos in a Ferromagnet: Quantifying the "Complexity" of 2D Ising Phase Transitions with Image Compression

Cooper Jacobus|arXiv (Cornell University)|Feb 16, 2026

Theoretical and Computational Physics被引用 0

一句话总结

该论文提出一种基于压缩的、模型无关的二维伊辛配置结构复杂度度量，在临界温度 Tc 处显示出尖峰，表明临界性时复杂度达到最大。

ABSTRACT

The data-driven characterization of the ``complexity'' present in dynamical systems remains an open problem with broad applications across the physical sciences. We investigate the ``structural complexity'' of the 2D ferromagnetic Ising model, a paradigmatic system exhibiting a second-order phase transition at a certain critical temperature which is often cited as a canonical example of complex morphology. We define a quantitative metric for this structural complexity, $\mathcal C_s$, through the lens of algorithmic information theory by approximating the Kolmogorov complexity of lattice configurations via standard lossless image compression algorithms. We regularize our proposed metric, $\mathcal C_s$, by comparing the compressibility of a configuration to that of its pixel-wise sorted and randomly shuffled counterparts. We arrive at a definition of $\mathcal C_s$ as a product of two components representing the systems departure from perfect order and disorder respectively which we then plot as a function of temperature. Our numerical simulations reveal a distinct peak in $\mathcal C_s$ at the known critical temperature $T_c$. This result demonstrates that such information-theoretic measures can act as sensitive, model-agnostic indicators of criticality, directly quantifying the emergence of complex structure at the boundary between order and chaos, opening the door to data-driven applications in domains where analytic solutions are unavailable.

研究动机与目标

• 为图像结构数据提供一个实用的、模型无关的复杂度度量。
• 使用近似于无损压缩的算法信息理论来量化相变中复杂结构的出现。
• 将该度量应用于二维伊辛模型以测试对临界性的敏感性。
• 通过与保持组成但破坏结构的基线进行比较来正则化复杂度。

提出的方法

• 将二维伊辛晶格配置转换为灰度图像，并使用 PNG（LZ77/Huffman）测量其无损压缩大小。
• 定义可压缩比 rho = 压缩大小 / 未压缩大小，作为算法复杂度的代理。
• 通过基线归一化：rho_shuffle[S]（具有相同组成的随机化）和 rho_sort[S]（排序到最小无序）。
• 定义结构有序 Os[S] = rho_shuffle[S] - rho[S]。
• 定义结构无序 Ds[S] = rho[S] - rho_sort[S]。
• 定义结构复杂度 Cs[S] = sqrt( Os[S] × Ds[S] )。
• 使用带 Wolff 群更新和 Metropolis 步骤的蒙特卡罗方法，在 Tc 附近的温度范围内生成伊辛配置。

Figure 1: Example images of “structurally complex” systems from various domains over many orders of magnitude in scale. Image Credits: (a) Dartmouth College, (b) Nancy Kedersha, (c) Emmanuel Coupé, (d), ESA, (e,f,g) NASA, (h) Volker Springel.

实验结果

研究问题

• RQ1 基于压缩的、模型无关的复杂度度量是否在已知的二维伊辛临界温度 Tc 处出现峰值？
• RQ2 该度量能否仅基于空间结构在有序、临界和无序区域之间进行区分？
• RQ3 对无损压缩算法的选择和有限尺寸效应，该峰是否鲁棒？
• RQ4 该度量是否能有效地区分复杂的、类似分形的结构与简单的周期序列和随机性？

主要发现

• 在已知的二维伊辛模型临界温度 Tc 附近出现结构复杂度 Cs 的明显峰值。
• 基于压缩的可压缩性度量 Os 与 Ds 一起分离出非平凡的空间结构并在临界时达到最大。
• Cs 保持有界并对系统对称性不变，对有限尺寸效应敏感（ larger Ns 导致峰值更尖锐 ）。
• 在所报告的测试中，Cs 对不同的无损压缩算法（如 PNG）表现出鲁棒性。
• 该框架提供了一种数据驱动的临界性指示，量化在有序与混沌边界处复杂结构的出现。

Figure 2: Visualization of thermalized 2D Ising lattices generated by our simulation method from initially random conditions with width $N=256$ cells. Samples illustrate the ordered, critical, and disordered regimes relative to the critical temperature.