4 - Recurrence

Summary

The sum total of the energy in the universe is determinate, it is not infinite. Consequently the number of positions, changes, combinations of this energy, although tremendously large and practically “innnumerable”, is nevertheless determinate and not infinite. But time, in which the universe exercises its energy, is infinite … Consequently the development at this moment must be a repetition, so too that which it produces and that from which it arises, and so forwards and backwards … the total arrangement of all forms of energy ever recurs.

Friedrich Nietzsche (1844–1900)

Introduction: random systems and recurrence

Imagine an experiment where an unbiased coin is tossed at random over and over again. Then, recurrence in this process is illustrated by the fact that, sooner or later, we expect to get the outcome “heads”. What is more, if we continue tossing, sooner or later we expect to get the result “heads” again—the outcome “heads” will then have recurred.

More generally, imagine a process developing over time. One state of the process may be observed and then, at a later time, the same state, or one close to it, may be observed again. We might say that the original state of the process has recurred or has recurred in some approximate sense. Such recurrence phenomena are common in nature. For example, the seasons recur at fixed times of the year, the sun rises once every twenty four hours and many astronomical phenomena recur in a like manner.