The Geodesic Dispersion Phenomenon in Random Field Dynamics: Probing the Shape of Time

Abstract Gaussian random fields (GRFs) provide a fundamental framework for modeling stochastic spatial phenomena with broad applications in physics and related fields. Using the tools of information geometry, we investigate the geometric structure of the statistical manifold associated with a three-parameter isotropic GRF model. By deriving the Fisher information metric and computing its Christoffel symbols, we formulate the geodesic equations governing the dynamics in parameter space. A hybrid computational approach, combining Markov Chain Monte Carlo (MCMC) estimation of statistical quantities with Runge-Kutta integration, enables us to numerically explore these geodesic flows. Our simulations reveal a phenomenon we term geodesic dispersion, where forward and backward geodesic trajectories deviate in regions of strong curvature gradients. While this effect is not claimed as direct evidence of thermodynamic irreversibility, it provides a geometric marker of sensitivity in the statistical manifold that may underlie asymmetric behavior in driven random field dynamics. Additionally, singularities in the Fisher metric are observed to coincide with parameter regimes reminiscent of phase transitions, suggesting that geometric properties can serve as indicators of critical phenomena.Our study establishes two key insights: 1) time asymmetry in Gaussian random fields correlates with local manifold curvature variations during forward and backward simulations; 2) this asymmetry, arising from sharply curvature variations, underscores a fundamental insight: curvature in the statistical manifold acts as a geometric source of asymmetry in stochastic dynamics. These findings establish geodesic dispersion as a useful geometric lens for studying complex stochastic systems, and point to future connections with nonequilibrium thermodynamics, entropy production, and phase transition detection.