# đŸ”— PoincarĂ© Recurrence Theorem

đŸ”— Physics đŸ”— Systems đŸ”— Systems/Dynamical systems

In mathematics and physics, the PoincarĂ© 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 PoincarĂ© 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 PoincarĂ© recurrence theorem applies are called conservative systems.

The theorem is named after Henri PoincarĂ©, who discussed it in 1890 and proved by Constantin CarathĂ©odory using measure theory in 1919.