Poincaré Recurrence Theorem
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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 r ...
, the Poincaré recurrence theorem states that certain
dynamical systems 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 in a p ...
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 Ergodic theory (Greek: ' "work", ' "way") is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, statistical properties means properties which are expres ...
,
dynamical systems 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 in a p ...
and
statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...
. Systems to which the Poincaré recurrence theorem applies are called
conservative system In mathematics, a conservative system is a dynamical system which stands in contrast to a dissipative system. Roughly speaking, such systems have no friction or other mechanism to dissipate the dynamics, and thus, their phase space does not shrink ...
s. The theorem is named after
Henri Poincaré Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The ...
, who discussed it in 1890 and proved by
Constantin Carathéodory Constantin Carathéodory ( el, Κωνσταντίνος Καραθεοδωρή, Konstantinos Karatheodori; 13 September 1873 – 2 February 1950) was a Greek mathematician who spent most of his professional career in Germany. He made significant ...
using
measure theory In mathematics, the concept of a measure is a generalization and formalization of geometrical measures ( length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simil ...
in 1919.


Precise formulation

Any
dynamical system In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...
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 w ...
determines a
flow map A flow map is a type of thematic map that uses linear symbols to represent movement. It may thus be considered a hybrid of a map and a flow diagram. The movement being mapped may be that of anything, including people, highway traffic, trade goods ...
''f'' ''t'' mapping
phase space In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usually ...
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 system A Hamiltonian system is a dynamical system governed by Hamilton's equations. In physics, this dynamical system describes the evolution of a physical system such as a planetary system or an electron in an electromagnetic field. These systems can b ...
s 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.


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 In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usually ...
. # 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 D_1 of the
phase space In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usually ...
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 k_1 the phase tube must intersect itself. This means that at least a finite fraction R_1 of the starting volume is recurring. Now, consider the size of the non-returning portion 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 R_2 of it must return after k_2 steps. But that would be a contradiction, since in a number k_3= lcm(k_1, k_2) of step, both R_1 and R_2 would be returning, against the hypothesis that only R_1 was. Thus, the non-returning portion of the starting volume cannot be the empty set, i.e. all D_1 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 Mechanical equilibrium, equilibrium position, experiences a restoring force ''F'' Proportionality (mathematics), proportional to the displacement ''x'': \v ...
. Systems that do cover all accessible phase volume are called
ergodic In mathematics, ergodicity expresses the idea that a point of a moving system, either a dynamical system or a stochastic process, will eventually visit all parts of the space that the system moves in, in a uniform and random sense. This implies tha ...
(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 :(X,\Sigma,\mu) be a finite
measure space A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that i ...
and let :f\colon X\to X 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 E\in \Sigma, the set of those points x of E for which there exists N\in\mathbb such that f^n(x)\notin E for all n>N has zero measure. In other words, almost every point of E returns to E. In fact, almost every point returns infinitely often; ''i.e.'' :\mu\left(\\right)=0. For a proof, see the cited reference.


Theorem 2

The following is a topological version of this theorem: If X is a
second-countable In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly, a topological space T is second-countable if there exists some countable collection \mat ...
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 many ...
and \Sigma contains the
Borel sigma-algebra In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are named a ...
, then the set of recurrent points of f has full measure. That is, almost every point is recurrent. For a proof, see the cited reference. More generally, the theorem applies to
conservative system In mathematics, a conservative system is a dynamical system which stands in contrast to a dissipative system. Roughly speaking, such systems have no friction or other mechanism to dissipate the dynamics, and thus, their phase space does not shrink ...
s, 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 \varepsilon >0 and T_0>0 there exists a time ''T'' larger than T_0, such that , , \psi(T) \rangle - , \psi(0)\rangle, < \varepsilon, where , \psi(t)\rangle 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: :, \psi(t)\rangle = \sum_^\infty c_n \exp(-i E_n t), \phi_n\rangle where the E_n are the energy eigenvalues (we use natural units, so \hbar = 1 ), and the , \phi_n \rangle are the energy
eigenstates In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution i ...
. The squared norm of the difference of the state vector at time ''T'' and time zero, can be written as: :, , \psi(T) \rangle - , \psi(0)\rangle, ^2 = 2\sum_^\infty , c_n, ^2 -\cos(E_n T)/math> We can truncate the summation at some ''n'' = ''N'' independent of ''T'', because \sum_^\infty , c_n, ^2 -\cos(E_n T)\leq 2\sum_^\infty , c_n, ^2 which can be made arbitrarily small by increasing ''N'', as the summation \sum_^\infty , c_n, ^2, being the squared norm of the initial state, converges to 1. The finite sum :\sum_^N , c_n, ^2 -\cos(E_n T)/math> can be made arbitrarily small for specific choices of the time ''T'', according to the following construction. Choose an arbitrary \delta>0, and then choose ''T'' such that there are integers k_n that satisfies :, E_n T -2\pi k_n, <\delta, for all numbers 0 \leq n \leq N. For this specific choice of ''T'', :1-\cos(E_n T)<\frac. As such, we have: :2\sum_^N , c_n, ^2 -\cos(E_n T)< \delta^2 \sum_^N , c_n, ^2<\delta^2. The state vector , \psi(T)\rangle thus returns arbitrarily close to the initial state , \psi(0)\rangle.


See also

* Arnold's cat map *
Boltzmann brain The Boltzmann brain thought experiment suggests that it might be more likely for a single brain to spontaneously form in a void (complete with a memory of having existed in our universe) rather than for the entire universe to come about in the ma ...
*
Ergodic hypothesis In physics and thermodynamics, the ergodic hypothesis says that, over long periods of time, the time spent by a system in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e., th ...
*
Recurrence period density entropy Recurrence period density entropy (RPDE) is a method, in the fields of dynamical systems, stochastic processes, and time series analysis, for determining the periodicity, or repetitiveness of a signal. Overview Recurrence period density entropy i ...
*
Recurrence plot In descriptive statistics and chaos theory, a recurrence plot (RP) is a plot showing, for each moment i in time, the times at which the state of a dynamical system returns to the previous state at i, i.e., when the phase space trajectory visits rou ...
*
Wandering set In dynamical systems and ergodic theory, the concept of a wandering set formalizes a certain idea of movement and mixing. When a dynamical system has a wandering set of non-zero measure, then the system is a dissipative system. This is the opposit ...


References


Further reading

*


External links

* * {{DEFAULTSORT:Poincare recurrence theorem Theorems in dynamical systems Ergodic theory Statistical mechanics Recurrence theorem