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

TL;DR

The paper presents a compression-based framework to quantify structural complexity in image-like data and demonstrates its effectiveness on the canonical 2D Ising model. By defining $\mathcal{C}_s = \sqrt{\mathcal{O}_s \mathcal{D}_s}$ with baseline-driven components $\mathcal{O}_s$ and $\mathcal{D}_s$, the authors show that $\mathcal{C}_s$ attains a pronounced peak at the critical temperature $T_c$, signaling maximal emergent structure at the order–disorder boundary. The approach remains robust across compression algorithms and lattice sizes, with the peak sharpening as system size grows in a manner consistent with a second-order phase transition. This model-agnostic, data-driven indicator of criticality holds promise for broad applications where analytic solutions are unavailable and complex morphology is of interest.

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.

Figures (21)

• 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.
• 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.
• Figure 3: Visualizations of the Metropolis and Wolff lattice configuration update rules, each showing one simulation step.
• Figure 4: Behavior of traditional known standard quantities vs temperature for the 2D Ising model: (a) The average absolute magnetization, $\langle|M|\rangle$. (b) The entropy per site, $S/k_B$.
• Figure 5: Visual schematic of the compression ratio, $\rho$.