In mathematics and physics, the Poincaré Recurrence Theorem states that if a system has a finite volume (or total probability measure) and preserves that volume over time, then almost every initial state will eventually return arbitrarily close to its starting position, and it will do so infinitely many times.
In simple terms, it is a foundational result in both dynamical systems and statistical mechanics. The theorem is named after Henri Poincaré, who discussed it in 1890. A proof was presented by Constantin Carathéodory using measure theory in 1919.
Contents
Definitions and Poincaré's formulation
Any dynamical system defined by an ordinary differential equation determines a flow map f t mapping phase space to itself. Each point of the phase space describes the entire state of the system, typically the positions and velocities of all involved particles. The flow map tells us how the system evolves: if it starts in state x, then after time t the system will be in state f t(x). If we follow this point for all times t, we obtain the orbit of x. The system is said to be volume-preserving if the volume of each set in phase space is invariant under the flow. Poincaré's formulation of the theorem then states:
If a flow preserves volume and has only bounded orbits, then, for each open set, there is an orbit that intersects this open set infinitely often.
A consequence of the theorem is that conservative dynamical systems will, after a sufficiently long but finite time, almost certainly return to a state arbitrarily close to their initial state. The result applies to isolated mechanical systems subject to some constraints, e.g., all particles must be bound to a finite volume. Systems to which the Poincaré recurrence theorem applies are called conservative systems.
The Poincaré recurrence time estimated by Kac's lemma, 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.
Discussion, example and applications
We may think of the open set as tiny; the claim is that there will be orbits that return to this tiny set. (In fact, according to more modern formulations of the theorem given below, almost all orbits that start in the tiny set will return infinitely often.)
The assumption that all orbits be bounded is crucial: otherwise the orbits could "go off to infinity", never to return.
The other important assumption is that the system is volume-preserving. This is a common feature of physical systems when there is no friction or air resistance, no energy is pumped in and no energy is ever lost to heat. Technically, all Hamiltonian systems are volume-preserving because of Liouville's theorem. (Systems that don't preserve phase space volume and lose energy over time could start in a tiny open set and then slowly come to a standstill somewhere else, never to return.)
As an example of the theorem, consider a billiard table (without pockets) and a set of billiard balls, moving on the table without friction and undergoing completely elastic collisions with the cushions and the other balls. This ensures that no energy is ever lost and the flow preserves phase space volume. We can even allow for hills and valleys on the table. A point in phase space specifies the positions and velocities of all balls. If we impose a certain maximum total energy on the balls, then the system has bounded orbits and Poincaré's theorem applies. It states: when specifying the initial positions and velocities of all balls with arbitrary (but finite) precision, there will be orbits that start according to these specifications and that will return infinitely often to conditions that also accord with those specifications. Note however that not all orbits will necessarily return: it's possible to find orbits so finely tuned that some balls will slowly come to rest at the top of hills, never to return.
The Poincaré recurrence theorem has applications in statistical mechanics, dynamical systems, and ergodic theory. In statistical mechanics, it is used to predict the long-term behavior of isolated systems, such as gases confined to a finite volume, where the system is expected to return arbitrarily close to its initial state after sufficient length.
Proof idea and further discussion
The qualitative proof hinges on two premises:
A finite upper bound can be set on the total potentially accessible phase space volume. For a mechanical system, this bound can be provided by requiring that the system is contained in a bounded physical region of space (so that it cannot, for example, eject particles that never return)–combined with the conservation of energy, this locks the system into a finite region in phase space.
The phase volume of a finite element under dynamics is conserved.
Imagine any finite starting volume
D
{\displaystyle D}
of the phase space and follow its path under the dynamics of the system. ("Finite" here means that the volume is positive.) The volume evolves through a "phase tube" in the phase space, keeping its size constant. Since the phase space is finite, after some number of steps
k
1
{\displaystyle k_{1}}
the phase tube must intersect itself in some positive volume. Translating this intersection backwards in time, we find that at least a finite fraction of the starting volume is recurring.
Now, consider the size of the non-returning portion
D
1
{\displaystyle D_{1}}
of the starting phase volume–that portion that never returns to the starting volume
Formal statement
The physical paradox
In physics, this theorem creates a philosophical paradox (the Zermelo paradox) against the second law of thermodynamics. If you open a bottle of perfume in a sealed, finite room, thermodynamics says that the gas molecules will disperse and never spontaneously go back into the bottle.
However, because classical mechanics in a closed box preserves phase space volume (by Liouville's theorem), Poincaré's theorem guarantees that after enough time, the molecules will spontaneously reassemble back inside the bottle. Though, the "Poincaré recurrence time" for macroscopic systems is staggeringly long—vastly longer than the current age of the universe.
Extensions and generalizations
Over the years, Poincaré's theorem has been extended significantly to apply to more complex mathematical frameworks:
Topological recurrence: If
X
{\displaystyle X}
is a compact metric space and
T
:
X
→
X
{\displaystyle T:X\to X}
is a continuous map, a point
x
{\displaystyle x}
is topologically recurrent if
lim inf
n
→
∞
d
(
T
n
(
x
)
,
x
)
=
0
Quantum mechanical version
For time-independent quantum mechanical systems with discrete energy eigenstates, a similar theorem holds. For every
ε
>
0
{\displaystyle \varepsilon >0}
and
T
0
>
0
{\displaystyle T_{0}>0}
there exists a time T larger than
T
0
{\displaystyle T_{0}}
, such that
|
|
ψ
(
T
)
⟩
−
|
ψ
(
0
)
⟩
|
<
ε