Information Geometry in Gaussian Random Fields: Searching for an Arrow of Time in Complex Systems

Random fields are characterized by intricate non-linear relationships between their elements over time. However, what is a reasonable intrinsic definition for time in such complex systems? Here, we discuss the problem of characterizing the notion of time in isotropic pairwise Gaussian random fields. In particular, we are interested in studying the behavior of these fields when temperature deviates from infinity. Our investigations are focused in the relation between entropy and Fisher information, by the definition of the Fisher curve. The results suggest the emergence of an arrow of time as a consequence of asymmetrical geometric deformations in the random field model's metric tensor. In terms of information geometry, the process of taking a random field from a lower entropy state A to a higher entropy state B and then bringing it back to A, induces a natural intrinsic one-way direction of evolution. In practical terms, there are different trajectories in the information space, suggesting that the deformations induced by the metric tensor into the parametric space (manifold) are not reversible for positive and negative displacements in the inverse temperature parameter direction. In other words, there is only one possible orientation to move through different entropic states along the Fisher curve.