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.
The theorem is named after Henri Poincaré, who discussed it in 1890.
[1][2] A proof was presented by Constantin Carathéodory using measure theory in 1919.
[3][4] 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.
[5] The proof, speaking qualitatively, hinges on two premises:[6] Imagine any finite starting volume
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
Thus, the non-returning portion of the starting volume cannot be the empty set, i.e. all
The theorem does not comment on certain aspects of recurrence which this proof cannot guarantee: Let be a finite measure space and let be a measure-preserving transformation.
In fact, almost every point returns infinitely often; i.e.
The following is a topological version of this theorem: If
contains the Borel sigma-algebra, then the set of recurrent points of
Roughly speaking, one can say that conservative systems are precisely those to which the recurrence theorem applies.
For time-independent quantum mechanical systems with discrete energy eigenstates, a similar theorem holds.
denotes the state vector of the system at time t.[7][8][9] The essential elements of the proof are as follows.
are the energy eigenvalues (we use natural units, so
The squared norm of the difference of the state vector at time
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.
For this specific choice of T, As such, we have: The state vector
thus returns arbitrarily close to the initial state
This article incorporates material from Poincaré recurrence theorem on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.