In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
and
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
, 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 and proved by
Constantin Carathéodory using
measure theory in 1919.
Precise formulation
Any
dynamical system
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water i ...
defined by an
ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...
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
Flow may refer to:
Science and technology
* Fluid flow, the motion of a gas or liquid
* Flow (geomorphology), a type of mass wasting or slope movement in geomorphology
* Flow (mathematics), a group action of the real numbers on a set
* Flow (psyc ...
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:
# 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 (for a mechanical system, this is ensured by
Liouville's theorem).
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
the phase tube must intersect itself. This means that at least a finite fraction
of the starting volume is recurring.
Now, consider the size of the non-returning portion
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
of it must return after
steps. But that would be a contradiction, since in a number
lcm of step, both
and
would be returning, against the hypothesis that only
was. Thus, the non-returning portion of the starting volume cannot be the empty set, i.e. all
is recurring after some number of steps.
The theorem does not comment on certain aspects of recurrence which this proof cannot guarantee:
* There may be some special phases that never return to the starting phase volume, or that only return to the starting volume a finite number of times then never return again. These however are extremely "rare", making up an infinitesimal part of any starting volume.
* Not all parts of the phase volume need to return at the same time. Some will "miss" the starting volume on the first pass, only to make their return at a later time.
* Nothing prevents the phase tube from returning completely to its starting volume before all the possible phase volume is exhausted. A trivial example of this is the
harmonic oscillator
In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force ''F'' proportional to the displacement ''x'':
\vec F = -k \vec x,
where ''k'' is a positive const ...
. Systems that do cover all accessible phase volume are called
ergodic (this of course depends on the definition of "accessible volume").
* What ''can'' be said is that for "almost any" starting phase, a system will eventually return arbitrarily close to that starting phase. The recurrence time depends on the required degree of closeness (the size of the phase volume). To achieve greater accuracy of recurrence, we need to take smaller initial volume, which means longer recurrence time.
* For a given phase in a volume, the recurrence is not necessarily a periodic recurrence. The second recurrence time does not need to be double the first recurrence time.
Formal statement
Let
:
be a finite
measure space and let
:
be a
measure-preserving transformation
In mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of dynamical systems, and ergodic theory in particular. Measure-preserving systems obey the Poincaré recurrence theorem, and are a special ca ...
. Below are two alternative statements of the theorem.
Theorem 1
For any
, the set of those points
of
for which there exists
such that
for all
has zero measure.
In other words, almost every point of
returns to
. In fact, almost every point returns infinitely often; ''i.e.''
:
For a proof, see the cited reference.
Theorem 2
The following is a topological version of this theorem:
If
is a
second-countable Hausdorff space
In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the ma ...
and
contains the
Borel sigma-algebra, then the set of
recurrent points of
has full measure. That is, almost every point is recurrent.
For a proof, see the cited reference.
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
there exists a time ''T'' larger than
, 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
are the energy eigenvalues (we use natural units, so
), and the
are the energy
eigenstates. The squared norm of the difference of the state vector at time ''
'' and time zero, can be written as:
: