  Random fields are useful mathematical objects in the characterization of
non-deterministic complex systems. A fundamental issue in the evolution of
dynamical systems is how intrinsic properties of such structures change in
time. In this paper, we propose to quantify how changes in the spatial
dependence structure affect the Riemannian metric tensor that equips the
model's parametric space. Defining Fisher curves, we measure the variations in
each component of the metric tensor when visiting different entropic states of
the system. Simulations show that the geometric deformations induced by the
metric tensor in case of a decrease in the inverse temperature are not
reversible for an increase of the same amount, provided there is significant
variation in the system entropy: the process of taking a system 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 this context,
Fisher curves resemble mathematical models of hysteresis in which the natural
orientation is pointed by an arrow of time.
