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.

Any dynamical system defined by an ordinary differential equation determines a flow map f ^t 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.

The proof, speaking qualitatively, hinges on two premises:

Imagine any finite starting volume ${\displaystyle D_{1}}$ 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 ${\displaystyle k_{1}}$ the phase tube must intersect itself. This means that at least a finite fraction ${\displaystyle R_{1}}$ of the starting volume is recurring. Now, consider the size of the non-returning portion ${\displaystyle D_{2}}$ 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 ${\displaystyle R_{2}}$ of it must return after ${\displaystyle k_{2}}$ steps. But that would be a contradiction, since in a number ${\displaystyle k_{3}=}$lcm${\displaystyle (k_{1},k_{2})}$ of step, both ${\displaystyle R_{1}}$ and ${\displaystyle R_{2}}$ would be returning, against the hypothesis that only ${\displaystyle R_{1}}$ was. Thus, the non-returning portion of the starting volume cannot be the empty set, i.e. all ${\displaystyle D_{1}}$ is recurring after some number of steps.

The theorem does not comment on certain aspects of recurrence which this proof cannot guarantee:

Let