Randomness, determinateness, and predictability

The basic conventions regarding randomness employed in mathematics (set-theoretical approach, algorithmic approach) and in physics (decaying correlations, continuous spectrum, hyperbolicity, fractal nature, uncontrollability, nonrepeatability, nonreproducibility, nonpredictability, etc.) are analyzed. It is pointed out that phenomena that are random from one viewpoint may be determinate from another viewpoint. The concept of partially determinate processes, i.e., processes that admit prediction over bounded time intervals, is discussed. The theory of partially determinate processes is based on identifying randomness with unpredictability and establishes the interrelation between the real physical process $x(t)$, the observed process $y(t)$, and the model (predictive, hypothetical) process $t(t)$. In this theory the degree of determinateness, which is denned as the correlation coefficient between the observed process and prediction, is employed as a measure of the quality of predictability. Diverse theoretical, experimental, and numerical measures of partially determinate processes as well as examples of partially determinate fields are presented. It is emphasized that the time of determinate (i.e., predictable) behavior $\tau_{\det}$ of an observed process $y(t)$ can be much longer than the correlation time $\tau_c$, and the degree of coherence is the worst estimate of the degree of determinateness. From the viewpoint expounded determinate chaos stands out as a completely determinate process over short time intervals ($\tau\ll\tau_{\det}$), as a completely random process over long intervals ($\tau\gg\tau_{\det}$), and as a partially determinate process over intermediate time intervals $\tau\sim\tau_{\det}$. It is significant that in the interval between $\tau_c$ and $\tau_{\det}$ chaotic and turbulent fields admit both a determinate and statistical (kinetic) description.