the Poincaré recurrence theorem
/ pwan-kah-RAY /
The Poincaré recurrence theorem is a startling promise about bounded, conservative motion: wait long enough and the system will return arbitrarily close to where it started. Every configuration of positions and momenta it ever visits will, sooner or later, be almost exactly reproduced. It is a purely geometric consequence of Liouville's theorem, and it sits in famous tension with the thermodynamic arrow of time.
The theorem states: for a Hamiltonian system whose motion is confined to a bounded region of phase space, almost every initial state (all but a set of zero volume) returns arbitrarily close to itself after some finite time. The proof is a pigeonhole argument on phase-space volume. Because Liouville's theorem makes the flow volume-preserving, and the accessible region has finite total volume, the images of any small neighborhood under repeated evolution cannot all be disjoint forever -- they would fill more than the available volume. Two of them must overlap, and running time backward shows the neighborhood must revisit itself. The catch is entirely in the word 'eventually'.
This theorem is why the 'recurrence paradox' haunted the founders of statistical mechanics. If an isolated gas must eventually return to its initial low-entropy state, how can the second law say entropy only increases? The resolution is scale: the recurrence time for a macroscopic system is astronomically long -- for a mole of gas it vastly exceeds the age of the universe, roughly exp(10^23) times any ordinary timescale. Entropy increase is overwhelmingly probable on human timescales; recurrence is certain only on timescales no experiment could ever probe.
For a handful of gas molecules in a box, recurrence is quick and observable. For one mole (about 6 x 10^23 molecules), the estimated recurrence time is of order exp(N) with N ~ 10^23 -- a number so vast that whether you measure it in seconds or in ages of the universe makes no visible difference. The theorem is true and utterly irrelevant to everyday thermodynamics.
Poincaré recurrence is certain in principle but, for a macroscopic gas, occurs on timescales dwarfing the age of the universe.
Recurrence requires bounded phase space and a measure-preserving (Hamiltonian) flow. It fails for dissipative systems (which contract onto attractors) and for unbounded motion (a particle escaping to infinity never returns). It also does not contradict the second law -- the two describe utterly different timescales.