The geodesic dispersion phenomenon in random fields dynamics

Random fields are ubiquitous mathematical structures in physics, with applications ranging from thermodynamics and statistical physics to quantum field theory and cosmology. Recent works on information geometry of Gaussian random fields proposed mathematical expressions for the components of the metric tensor of the underlying parametric space, allowing the computation of the Gaussian curvature in each point of the manifold that represents the space of all possible parameter values that define such mathematical model. A key result in the dynamics of these random fields concerns the curvature effect, a series of variations in the curvature that happens in the parametric space when there are significant increase/decrease in the inverse temperature parameter. In this paper, we propose a numerical algorithm for the computation of geodesic curves in the Gaussian random fields manifold by deriving the 27 Christoffel symbols of the metric required for the definition of the Euler-Lagrange equations. The fourth-order Runge-Kutta method is applied to solve the Euler-Lagrange equations using an iterative approach based in Markov Chain Monte Carlo simulation. Our results reveal that, when the system undergoes phase trasitions, the geodesic dispersion phenomenon emerges: the geodesic curve obtained by reversing the system of differential equations in time, diverges from the original geodesic curve, as we move from zero curvature configurations (Euclidean geometry) to negative curvature configurations (hyperbolic-like geometry), and vice-versa. This phenomenon suggest that, time irreversibility in random field dynamics can be a direct consequence of the geometry of the underlying parametric space.