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.