  Random fields are mathematical structures used to model the spatial
interaction of random variables along time, with applications ranging from
statistical physics and thermodynamics to system's biology and the simulation
of complex systems. Despite being studied since the 19th century, little is
known about how the dynamics of random fields are related to the geometric
properties of their parametric spaces. For example, how can we quantify the
similarity between two random fields operating in different regimes using an
intrinsic measure? In this paper, we propose a numerical method for the
computation of geodesic distances in Gaussian random field manifolds. First, we
derive the metric tensor of the underlying parametric space (the 3 x 3
first-order Fisher information matrix), then we derive the 27 Christoffel
symbols required in the definition of the system of non-linear differential
equations whose solution is a geodesic curve starting at the initial
conditions. The fourth-order Runge-Kutta method is applied to numerically solve
the non-linear system through an iterative approach. The obtained results show
that the proposed method can estimate the geodesic distances for several
different initial conditions. Besides, the results reveal an interesting
pattern: in several cases, the geodesic curve obtained by reversing the system
of differential equations in time does not match the original curve, suggesting
the existence of irreversible geometric deformations in the trajectory of a
moving reference traveling along a geodesic curve.
