Is the Poincaré theorem valid for our universe?

Question

Poincaré tell us (roughly speaking) that any hamiltonian system come up arbitrarily close to the initial condition if you wait enough time. For example, this theorem is valid for gases, and in general is one of the many key theorems that sustain Statistical Physics. And if this theorem is valid to cosmological level , what that implies?
I mean, at the end, the universe is just evolving along the most probable line which could be one posible option of many more.

Answer 1

Poincare's recurrence theorem contradicts the second law of thermodynamics,which states that the entropy of an isolated system is non decreasing. The theorem suggests that a bounded dynamical system satisfying certain constraints, may return arbitrarily close to its initial state within some finite time. However, it is not necesary for all parts of a system to reach this arbitrarily close state at the same time. Physically this implies that various parts of the universe may revert back to the stages of the Big Bang after a long enough time, although the second law forbids such a situation.

For further reference,give the wikipedia article a read here