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I know that the Poincaré recurrence tells us that every given physical arrangement in a finite physical space will eventually recur given enough time. However, as I was thinking about it, even if my current position and physical state are closely repeated at some point in the future, the chaotic nature of macroscopic systems means that a future near identical state might not be followed by the same set of events. How can we calculate the Poincaré recurrence time for a given set of sequential physical states (e.g. not just me right near exactly how I am now but also me going on to do whatever it is I’m doing today in the same order)? Or is the universe deterministic enough that if we reach the same physical state at some point, it will also be followed by the same set of events?

Toasted Uranium
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The simplest way to do this generalization might be to extend the state space to include time: this way, your recurrence region will define not only the desired phase distance, but also the desired length of time during which this (possibly time-dependent) distance is respected.

Or is the universe deterministic enough that if we reach the same physical state at some point, it will also be followed by the same set of events?

This is a very different question, and a quite philosophical one to which it can be argued we don't know the answer. But, in the context of the recurrence theorem we're not necessarily talking about the whole of the universe, but of a specific volume-preserving system modeled by a given set of equations and then, yes, if the model is deterministic, then trivially (by definition) its phase space trajectory is completely determined by its starting point. But here we should remember that the theorem doesn't guarantee a continuous system will come back to exactly the same state, only arbitrarily close to it.

stafusa
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