Poincaré recurrence theorem
In mathematics and physics, the Poincaré recurrence theorem states that certain dynamical systems will, after a sufficiently long but finite time, return to a state arbitrarily close to (for continuous state systems), or exactly the same as (for discrete state systems), their initial state. The Poincaré recurrence time is the length of time elapsed until the recurrence. This time may vary greatly depending on the exact initial state and required degree of closeness. The result applies to isolated mechanical systems subject to some constraints, e.g., all particles must be bound to a finite volume. The theorem is  commonly discussed in the context of ergodic theory, dynamical systems and statistical mechanics. Systems to which the Poincaré recurrence theorem applies are called conservative systems. The theorem is named after Henri Poincaré, who discussed it in 1890 and proved by Constantin Carathéodory using measure theory in 1919. Precise formulation. Any dynamical system defined by an ordinary differential equation determines a flow map "f" mapping phase space on itself. The system is said to be volume-preserving if the volume of a set in phase space is invariant under the flow. For instance, all Hamiltonian systems are volume-preserving because of  Liouville's theorem. The theorem is then: If a flow preserves volume and has only bounded orbits, then, for each open set, any orbit that intersects this open set intersects it infinitely often. Discussion of proof. The proof, speaking qualitatively, hinges on two premises: Imagine any finite starting volume formula_1 of the phase space and to follow its path under the dynamics of the system. The volume evolves through a "phase tube" in the phase space, keeping its size constant. Assuming a finite phase space, after some number of steps formula_2 the phase tube must intersect itself. This means that  at least a finite fraction formula_3 of the starting volume is recurring. Now, consider the size of the non-returning portion formula_4 of the starting phase volume – that portion that never returns to the starting volume. Using the principle just discussed in the last paragraph, we know that if the non-returning portion is finite, then a finite part formula_5 of it must return after formula_6 steps. But that would be a contradiction, since in a number formula_7lcmformula_8 of step, both formula_3 and formula_5 would be returning, against the hypothesis that only formula_3 was. Thus, the non-returning portion of  the starting volume cannot be the empty set, i.e. all formula_1 is recurring after some number of steps. The theorem does not comment on certain aspects of recurrence which this proof cannot guarantee: Formal statement. Let be a finite measure space and let be a measure-preserving transformation. Below are two alternative statements of the theorem. Theorem 1. For any formula_15, the set of those points formula_16 of formula_17 for which there exists formula_18 such that formula_19 for all formula_20 has zero measure. In other words, almost every point of formula_17 returns to formula_17. In fact, almost every  point returns infinitely often; "i.e." For a proof, see the cited reference. Theorem 2. The following is a topological version of this theorem: If formula_24 is a second-countable Hausdorff space and formula_25 contains the Borel sigma-algebra, then the set of recurrent points of formula_26 has full measure. That is, almost every point is recurrent. For a proof, see the cited reference. More generally, the theorem applies to conservative systems, and not just to measure-preserving dynamical systems. Roughly speaking, one can say that conservative systems are precisely those to which the recurrence theorem applies. Quantum mechanical version. For  time-independent quantum mechanical systems with discrete energy eigenstates, a similar theorem holds. For every formula_27 and formula_28 there exists a time "T" larger than formula_29, such that formula_30, where formula_31 denotes the state vector of the system at time "t". The essential elements of the proof are as follows. The system evolves in time according to: where the formula_33 are the energy eigenvalues (we use natural units, so formula_34 ), and the formula_35 are the energy eigenstates. The squared norm of the difference of the state vector at time "formula_36" and time zero, can be written as: We can truncate  the summation at some "n" = "N" independent of "T", because formula_38 which can be made arbitrarily small by increasing "N", as the summation formula_39, being the squared norm of the initial state, converges to 1. The finite sum can be made arbitrarily small for specific choices of the time "T", according to the following construction. Choose an arbitrary formula_41, and then choose "T" such that there are integers formula_42 that satisfies for all numbers formula_44. For this specific choice of "T", As such, we have: The state vector formula_47 thus returns arbitrarily close to the initial state formula_48. Poincaré