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. A proof was presented by Constantin Carathéodory using measure theory in 1919.

Precise formulation

Any dynamical system defined by an ordinary differential equation determines a flow map ft 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 D_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 k_1 the phase tube must intersect itself. This means that at least a finite fraction R_1 of the starting volume is recurring. Now, consider the size of the non-returning portion D_2 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 R_2 of it must return after k_2 steps. But that would be a contradiction, since in a number k_3=lcm(k_1, k_2) of step, both R_1 and R_2 would be returning, against the hypothesis that only R_1 was. Thus, the non-returning portion of the starting volume cannot be the empty set, i.e. all D_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 E\in \Sigma, the set of those points x of E for which there exists such that for all n>N has zero measure. In other words, almost every point of E returns to E. In fact, almost every point returns infinitely often; i.e.

Theorem 2

The following is a topological version of this theorem: If X is a second-countable Hausdorff space and \Sigma contains the Borel sigma-algebra, then the set of recurrent points of f has full measure. That is, almost every point is recurrent. 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 and T_0>0 there exists a time T larger than T_0, such that, where 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 E_n are the energy eigenvalues (we use natural units, so \hbar = 1 ), and the are the energy eigenstates. The squared norm of the difference of the state vector at time T and time zero, can be written as: We can truncate the summation at some n = N independent of T, because which can be made arbitrarily small by increasing N, as the summation, 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 \delta>0, and then choose T such that there are integers k_n that satisfies for all numbers. For this specific choice of T, As such, we have: The state vector thus returns arbitrarily close to the initial state.