The prediction of future from the past: an old problem from a modern perspective

TL;DR

This paper revisits the classic problem of predicting future states from past data using modern ergodic theory, revealing that the effective degrees of freedom, not chaos, limit predictability.

Contribution

It applies ergodic theory, specifically Kac's lemma, to establish fundamental limits on predictability based on system degrees of freedom, supported by numerical models.

Findings

Predictability is limited by the effective degrees of freedom, not chaos.

Large degrees of freedom make accurate prediction practically impossible.

Ergodic theory provides intrinsic bounds on forecasting future states.

Abstract

The idea of predicting the future from the knowledge of the past is quite natural when dealing with systems whose equations of motion are not known. Such a long-standing issue is revisited in the light of modern ergodic theory of dynamical systems and becomes particularly interesting from a pedagogical perspective due to its close link with Poincar\'e's recurrence. Using such a connection, a very general result of ergodic theory - Kac's lemma - can be used to establish the intrinsic limitations to the possibility of predicting the future from the past. In spite of a naive expectation, predictability results to be hindered rather by the effective number of degrees of freedom of a system than by the presence of chaos. If the effective number of degrees of freedom becomes large enough, regardless the regular or chaotic nature of the system, predictions turn out to be practically impossible. The discussion of these issues is illustrated with the help of the numerical study of simple models.