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

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.