# Poincare Recurrence and the Infinite N Limit

1. ### NickJ

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(One version of) Poincare's Recurrence Theorem states that for any conservative system whose possible states S form a compact set in phase space, that system will "almost always" return arbitrarily close to its initial state, provided we wait long enough. ('Almost always' means 'all but a set of Lebesgue measure zero'.)

Carlo Cercignani, in a discussion of the Boltzmann equation, has this to say: the set of possible states for a conservative system is not compact in phase space for the limit where the number of particles in the system goes to infinity (N --> infty). (See The Mathematical Theory of Dilute Gases, Springer-Verlag: 1994, page 56 first paragraph.)

Why?

He also says that, in the N --> infty limit, the recurrence time for the system is expected to go to infinity (albeit at a much faster rate than N).

Why?

Can someone please explain? Does the non-compactness of S in the N --> infty limit have to do with S becoming unbounded? or not closed? Which one? And is there some intuitive reasoning to explain this? Or maybe a rigorous explanation too? I've been through lots of Cercignani's work; I can't find an explanation.

The answer is not this: the recurrence time is really really long, and that is why the Boltzmann equation is consistent with the recurrence theorem. Cercignani thinks that his claim *exempts* the Boltzmann equation from the strictures of the recurrence theorem, since he proves the validity of the Boltzmann equation for the N --> infty limit.