# Proof of Poincare Recurrence Theorem

1. Aug 31, 2006

### NickJ

Does anyone know of an accessible reference that sketches a proof of Poincare's recurrence theorem? (This is not a homework question.)

I'm coming up short in my searches -- either the proof is too sketchy, or it is inaccessible to me (little background in maths, but enough to talk about phase points, their trajectories).

If possible, I'd like the proof to provide a reductio of the following assumptions:

1. A is a set of phase points in some region of Gamma-space, such that each point in A represents a system with fixed and finite energy E and finite spatial extension.

2. B is a non-empty subset of A consisting of those points on trajectories that never return to A having once left A.

3. The Lebesgue measure of B is both finite and non-zero.

I know these three assumptions are jointly inconsistent -- but I can't figure out why.

Thanks!

2. Aug 31, 2006

### Sir.Aaron

Did you try Wikipedia, and google?

3. Sep 1, 2006

### NickJ

yup. no luck.

4. Sep 1, 2006

### Careful

Ohw, the proof is quite easy to understand. First you must know that dynamics deals with a Hamiltonian, that is give me 2N numbers which we call position (first N) and momentum (second N) and I can tell you how the system evolves. Now, suppose you don't know precisely what the initial momenta and positions are and you take some volume in 2N space, then the Liouville theorem says that this volume is preserved if you drag it along the flow. Now assume that the points in your original neighborhood all belong to different trajectories, then the snake moves in a finite volume and cannot self intersect herself (because different fluid trajectories cannot intersect each other). This clearly leads to a contradiction.

Careful

Last edited: Sep 1, 2006