3 ms·

Not necessarily. I suspect the Poincare recurrence Theorem provides us with enormous durations, like 10^(number of dynamical variables), with the unit of time o

by cjfd 3y ago

Not necessarily. I suspect the Poincare recurrence Theorem provides us with enormous durations, like 10^(number of dynamical variables), with the unit of time omitted because of utter irrelevance, or something even more horrific. It may be the case that the 2nd law of thermodynamics does not hold on such time scales but how could we ever know?

mamonster 3y ago
It doesn't really provide you with durations. The handwavy proof goes as follows(The proof of this theorem is extremely simple in measure theory language, it's actually a quite common exam question in bachelor's level measure theory courses): You take the set of items that you think never return to their original position.Call this set X. Now take the infinite sequence of pre-images of this set (i.e look at the items that map onto some part of X after one "evolution" of the volume preserving flow, then look at the items that map onto some part of X after 2 flows, etc), for natural numbers. Note that a key part of the measure theoretic version is that you can assign a finite "volume" to the overall system and some subsets of the system. This sequence (of pre-images under the volume preserving flow) has items that are pairwise disjoint (i.e an item that flows onto X after 1 evolution cannot also be an item that flows onto X after n evolutions). This is obvious from definition of X(as otherwise that item would recur). Now here is the tricky part(but basic result in measure theory): The sum of the "volumes" of these sets (of pre-images) is equal to the "volume" of set Y, where Y is this infinite union of pre-image sets together(with union). Because this big set Y is some subset of your system, clearly it has a finite "volume" This means the "volume" of your non-recurrent points has to be 0. Why? You assume the flow is volume preserving, which means that all the terms in your pre-image sequence have the same volume. Thus each volume has to be 0 (as you have an infinite sequence summing to a finite number).
xmcqdpt2 3y ago
In practice, for quantum mechanical systems, you get astronomical durations once the systems get even slightly complicated. It's possible to estimate and bound the recurrence time using the distribution of energy levels. Even the simplest chemical systems, like a single 10 atom molecule in vacuum, can have recurrences of years. A glass of water would eclipse the age of the universe, etc.

by cjfd 3y ago

mamonster 3y ago

It doesn't really provide you with durations. The handwavy proof goes as follows(The proof of this theorem is extremely simple in measure theory language, it's actually a quite common exam question in bachelor's level measure theory courses): You take the set of items that you think never return to their original position.Call this set X. Now take the infinite sequence of pre-images of this set (i.e look at the items that map onto some part of X after one "evolution" of the volume preserving flow, then look at the items that map onto some part of X after 2 flows, etc), for natural numbers. Note that a key part of the measure theoretic version is that you can assign a finite "volume" to the overall system and some subsets of the system. This sequence (of pre-images under the volume preserving flow) has items that are pairwise disjoint (i.e an item that flows onto X after 1 evolution cannot also be an item that flows onto X after n evolutions). This is obvious from definition of X(as otherwise that item would recur). Now here is the tricky part(but basic result in measure theory): The sum of the "volumes" of these sets (of pre-images) is equal to the "volume" of set Y, where Y is this infinite union of pre-image sets together(with union). Because this big set Y is some subset of your system, clearly it has a finite "volume" This means the "volume" of your non-recurrent points has to be 0. Why? You assume the flow is volume preserving, which means that all the terms in your pre-image sequence have the same volume. Thus each volume has to be 0 (as you have an infinite sequence summing to a finite number).

xmcqdpt2 3y ago

In practice, for quantum mechanical systems, you get astronomical durations once the systems get even slightly complicated. It's possible to estimate and bound the recurrence time using the distribution of energy levels. Even the simplest chemical systems, like a single 10 atom molecule in vacuum, can have recurrences of years. A glass of water would eclipse the age of the universe, etc.