Random musings on stochastics (Lorenz Lecture)

Abstract:

In 1960 Lorenz identified the chaotic nature of atmospheric dynamics, thus highlighting the importance of the discovery of chaos by Poincare, 70 years earlier, in the motion of three bodies. Chaos in the macroscopic world offered a natural way to explain unpredictability, that is, randomness. Concurrently with Poincare’s discovery, Boltzmann introduced statistical physics, while soon after Borel and Lebesgue laid the foundation of measure theory, later (in 1930s) used by Kolmogorov as the formal foundation of probability theory. Subsequently, Kolmogorov and Khinchin introduced the concepts of stochastic processes and stationarity, and advanced the concept of ergodicity. All these areas are now collectively described by the term “stochastics”, which includes probability theory, stochastic processes and statistics.

As paradoxical as it may seem, stochastics offers the tools to deal with chaos, even if it results from deterministic dynamics. As chaos entails uncertainty, it is more informative and effective to replace the study of exact system trajectories with that of probability densities. Also, as the exact laws of complex systems can hardly be deduced by synthesis of the detailed interactions of system components, these laws should inevitably be inferred by induction, based on observational data and using statistics.

The arithmetic of stochastics is quite different from that of regular numbers. Accordingly, it needs the development of intuition and interpretations which differ from those built upon deterministic considerations. Using stochastic tools in a deterministic context may result in mistaken conclusions. In an attempt to contribute to a more correct interpretation and use of stochastic concepts in typical tasks of nonlinear systems, several examples are studied, which aim (a) to clarify the difference in the meaning of linearity in deterministic and stochastic context; (b) to contribute to a more attentive use of stochastic concepts (entropy, statistical moments, autocorrelation, power spectrum), in model identification and parameter estimation from data; and (c) to provide interpretations to scaling laws based on maximization of entropy or entropy production, or else natural amplification of uncertainty, which are alternative to more common ones, like self-organization.