Predicting the future from the past: An old problem from a modern perspective

Abstract

The idea of predicting the future from the knowledge of the past is quite natural, even when dealing with systems whose equations of motion are not known. This 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 Poincaré’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 is hindered more 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, whether the system is chaotic or not, predictions turn out to be practically impossible. The discussion of these issues is illustrated with the help of the numerical study of simple models.