A Statistical Physics View of the S&P 500: Pairwise Interactions and Time-Varying Dynamics

We analyze a fixed panel of S\&P 500 stocks from 1996 to 2026 using complementary static and kinetic Ising models applied to daily binary open-to-close movements. The static pairwise model provides a long-run maximum-entropy summary of low-order dependence and reveals a sectorally organized interaction network with modest small-world structure and within-sector couplings about 2.8 times stronger than between-sector couplings, with especially coherent real estate and energy sectors. The kinetic model incorporates smooth time-varying external fields, self-memory, and directed lagged couplings to describe next-day dynamics. It reveals slow field-regime shifts around three major market-wide perturbations -- the dot-com bust, the global financial crisis, and the COVID-19 episode. Self-memory is generally weak, and the directed coupling structure is much less sector-concentrated and more asymmetric than the static network, while still reproducing the broad evolution of aggregate market movement. Taken together, the two complementary models characterize both persistent market organization and short-horizon cross-stock dynamics, providing a compact statistical physics view of interaction structure and time-varying behavior in the S\&P 500.