Glimpsing God playing dice over water and climate

Dice throw has been a popular metaphor for the notion of randomness as reflected in many famous aphorisms. Indeed, the outcome of a die throw is not predictable. However, the die motion is fully describable in deterministic terms as it obeys Newton’s laws of motion. This makes its motion predictable for short time horizons. Even without using the equations of motion, by monitoring the die trajectory and analysing video frames it is possible to make predictions for short lead times (smaller enough than 1 s) using simple data-driven techniques (the method of analogues) or simple stochastic techniques. Thus, experimenting with dice using visualization techniques we can gain some understanding of how physical systems behave. Actually, dice behave like any other common physical system: predictable for short horizons, unpredictable for long horizons. The difference of dice from other common physical systems is that they enable unpredictability very quickly. Another experiment, this time using a mathematical model of a hydrological system deliberately made extraordinarily simple and fully deterministic, leads to the same conclusion: that the system trajectory is predictable for short horizons and becomes unpredictable for longer horizons. The two experiments show that the traditional notion of randomness and uncertainty, according to which natural phenomena are separated into two mutually exclusive components, random (or stochastic) and deterministic, is incorrect. A more correct view is that uncertainty is an intrinsic property of nature, that causality implies dependence of natural processes in time, thus suggesting predictability, but even the tiniest uncertainty (e.g., in initial conditions or in external perturbation) may result in unpredictability after a certain time horizon. On these premises it is possible to shape a consistent stochastic representation of natural processes, in which predictability (suggested by deterministic laws) and unpredictability (randomness) coexist and are not separable or additive components. Deciding which of the two dominates is simply a matter of specifying the time horizon and scale of the prediction. Long horizons of prediction are inevitably associated with high uncertainty, whose quantification relies on the long-term stochastic properties of the processes. While outcomes of different dice throws are typically independent to each other and comply with classical statistics, the motion in a specific die throw reveals strong dependence. Likewise, trajectories of physical systems are characterized by a strong dependence structure whose convergence to zero is slow. Essentially, this behaviour manifests that long-term changes are much more frequent and intense than commonly perceived and modelled through classical statistics. This makes classical statistics inappropriate for the study of physical systems and suggests the necessity of an advanced model, the so called Hurst-Kolmogorov stochastics. According to the latter, the future states are much more uncertain and unpredictable on long time horizons than implied by standard approaches. The omnipresence of this behaviour in Nature, as well as its implications, are illustrated using examples related to water and climate.