Recurrence sentence

The Poincaré recurrence theorem is a mathematical theorem on dynamical systems . It says that in autonomous Hamiltonian systems , the phase space of which has a finite volume, there are states in every open set in the phase space, the trajectories of which return as often as desired . In particular, Poincaré's return theorem is a proposition of ergodic theory and can also be seen as the first result of chaos theory . ${\displaystyle U}$ ${\displaystyle U}$

origin

Poincaré's recurrence theorem was first published in 1890 in the Swedish journal Acta Mathematica in a paper by Henri Poincaré on the three-body problem . The first formulation of the recurrence clause can be found on page 69:

Théorème I. Supposons que le point reste à distance finie, et que le volume soit un invariant intégral; si l'on considère une région quelconque, quelque petite que soit cette région, il y aura des trajectoires qui la traversent une infinité de fois. ${\displaystyle P}$ ${\displaystyle \int dx_{1}dx_{2}dx_{3}}$ ${\displaystyle r_{0}}$

(Proposition I. Let us assume that the point remains at a finite distance, and that the volume is an invariant integral; if we now consider any area, however small it may be, there will always be orbits that traverse it infinitely often .) ${\displaystyle P}$ ${\displaystyle \int dx_{1}dx_{2}dx_{3}}$

Poincaré proves this proposition on the next two pages of his work; it is clear from his evidence that the dimension of volume does not matter. In fact, Poincaré formulates it on page 72f. this rate also for any dimension . Poincaré's context is the Hamilton formalism of classical mechanics, where the point describes the state of the mechanical system that changes over time and the Hamilton function is autonomous, i.e. not explicitly dependent on time. For example, the three-body problem has a total of 18 components, namely three (generalized) position and three (generalized) momentum coordinates for each body; in this case the phase space is 18-dimensional. In the case of autonomous Hamiltonian systems, Liouville's theorem shows that the volume in phase space is preserved under motion. ${\displaystyle n>3}$ ${\displaystyle P}$ ${\displaystyle P}$

mathematics

Taking into account the original context, the following formulation of Poincaré's recurrence theorem results:

Let be an autonomous Hamilton function on a phase space with finite volume. Then for every open set there is a trajectory of the associated Hamiltonian system, which runs through infinitely often. ${\displaystyle H}$ ${\displaystyle \Omega }$ ${\displaystyle U\subset \Omega }$ ${\displaystyle U}$

Essential ideas of evidence

The most important steps of Poincaré's proof are (in today's notation):

1. The vector field that defines the Hamiltonian system arises from partial derivatives of the Hamilton function. Because this is autonomous according to the assumption, the vector field is free of divergence .
2. It follows from Liouville's volume formula that the flow generated by the Hamiltonian system is volume-preserving. This means in formulas: The flow defines a bijective mapping for each . If it is measurable , it is also measurable, and it applies . ${\displaystyle \Phi }$ ${\displaystyle t}$ ${\displaystyle \Phi _{t}\colon \Omega \to \Omega }$ ${\displaystyle U\subset \Omega }$ ${\displaystyle \Phi _{t}^{-1}(U)=\Phi _{-t}(U)}$ ${\displaystyle vol(U)=vol(\Phi _{t}(U))}$
3. One now concentrates on integer points in time ; all quantities have the same volume . Because the phase space has finite volume, the sets cannot be pairwise disjoint. So there is such that . This also applies . ${\displaystyle t=n}$ ${\displaystyle \Phi _{n}(U)\subset \Omega }$ ${\displaystyle vol(U)>0}$ ${\displaystyle \Omega }$ ${\displaystyle \Phi _{n}(U)}$ ${\displaystyle j>k\geq 0}$ ${\displaystyle \Phi _{j}(U)\cap \Phi _{k}(U)\neq \emptyset }$ ${\displaystyle \emptyset \neq U\cap \Phi _{j-k}(U)=\Phi _{-k}(\Phi _{j}(U)\cap \Phi _{k}(U))}$
4. If it is found in such a way that , then by the same argument the sets cannot be pairwise disjoint. So there is with . For true order . ${\displaystyle n_{0}}$ ${\displaystyle U\cap \Phi _{n_{0}}(U)\neq \emptyset }$ ${\displaystyle \Phi _{n\cdot n_{0}}(U)}$ ${\displaystyle j>k\geq 0}$ ${\displaystyle \Phi _{j\cdot n_{0}}(U)\cap \Phi _{k\cdot n_{0}}(U)\neq \emptyset }$ ${\displaystyle n_{1}:=(j-k)n_{0}}$ ${\displaystyle U\cap \Phi _{n_{1}}(U)=\Phi _{-kn_{0}}(\Phi _{jn_{0}}(U)\cap \Phi _{kn_{0}}(U))\neq \emptyset }$

Steps 1 and 2 of this line of argument were well known before Poincaré. The remaining ideas for proof are found for the first time in Poincaré's work.

Dimension theory formulation and tightening

In Poincaré's proof, the concept of volume plays an important role. With the help of measure theory and the associated terms, the proof can be structured more clearly. You start with a measurement space and name a measurable image ${\displaystyle (\Omega ,\Sigma ,\mu )}$

${\displaystyle g\colon \Omega \to \Omega }$

maßerhaltend if for any measurable amount , the equation is true, so when the measure and its size under the same. Furthermore, one has to assume the finiteness of the dimensional space, that is . This leads to the measure theoretical variant , where the -fold iteration of denotes: ${\displaystyle U\subset \Omega }$ ${\displaystyle \mu (g^{-1}(U))=\mu (U)}$ ${\displaystyle \mu }$ ${\displaystyle g}$ ${\displaystyle \mu (\Omega )<\infty }$ ${\displaystyle g^{n}}$ ${\displaystyle n}$ ${\displaystyle g}$

Let be a finite measure space, a measure-preserving mapping and a measurable set with . Then there are points with the property that for an unlimited ascending sequence . ${\displaystyle (\Omega ,\Sigma ,\mu )}$ ${\displaystyle g\colon \Omega \to \Omega }$ ${\displaystyle U\subset \Omega }$ ${\displaystyle \mu (U)>0}$ ${\displaystyle x\in U}$ ${\displaystyle g^{n_{k}}(x)\in U}$ ${\displaystyle (n_{k})}$

A precise analysis of Poincaré's proof with the help of the measure theory leads to the following tightening of the measure theory :

Let be a finite measure space, a measure-preserving mapping and a measurable set with . Then the points , the iterates of which do not return to arbitrarily often , form a zero set. ${\displaystyle (\Omega ,\Sigma ,\mu )}$ ${\displaystyle g\colon \Omega \to \Omega }$ ${\displaystyle U\subset \Omega }$ ${\displaystyle \mu (U)>0}$ ${\displaystyle x\in U}$ ${\displaystyle g^{n_{k}}(x)}$ ${\displaystyle U}$ ${\displaystyle \mu }$

Discrete dynamic systems

The theoretical variants can easily be applied to discrete dynamic systems, but they do not bring anything new: The measure here is simply the counting measure . The requirement then means that the underlying set is finite. Thus, measure preserving becomes synonymous with bijective , and the statement of Poincaré's recurrence theorem becomes the simple fact that every permutation of a finite set breaks down into cycles . ${\displaystyle \mu (\Omega )<\infty }$

physics

Physically, Poincaré's law of recurrence means that a mechanical system whose orbits remain restricted (e.g. the solar system) has the property that there are system states in every environment of the initial state, the orbits of which return as often as desired in the said environment of the initial state. This gives the following result: If you connect two containers that contain different gases , they first mix. After the recurrence principle, however, there is an arbitrarily small change in the initial state with the consequence that the gases separate and are segregated at a later point in time. The segregation contradicts a deterministic formulation of the second law of thermodynamics , which excludes a decrease of the entropy . A dispute arose over this between Ernst Zermelo and Ludwig Boltzmann , in the course of which Boltzmann wrote several articles about the connections between Poincaré's recurrence theorem and the second law of thermodynamics. After that, the contradiction disappears if the second law is interpreted statistically:

“Already Clausius , Maxwell u. a. have repeatedly pointed out that the theorems of gas theory have the character of statistical truths. I have emphasized especially often and as clearly as I was able to say that Maxwell's law of the velocity distribution among gas molecules can by no means be proven from the equations of motion alone like a theorem of ordinary mechanics, that one can rather only prove that it has by far the greatest probability and in the case of a large number of molecules all other states are so improbable compared with them that they cannot be considered in practice. At the same point I also emphasized that the second law from the molecular theoretical standpoint is a mere probability theorem. "

Accordingly, a decrease in entropy is not impossible in principle, but very unlikely within a “short” period of time. However, if one considers the behavior of a Hamiltonian system with a restricted phase space for arbitrarily large times, the return is almost certain - as follows from the weight-theoretical tightening of Poincaré's return theorem. In the appendix to the cited paper, Boltzmann gives an estimate of the return time for the molecules of air of normal density in a vessel with a volume of one cm³. After about one page of combinatorial considerations, he comes to a number (where is an estimate for the number of combinations of discretized particle impulses and describes the number of gas particle collisions per second) that still has to be “multiplied by a second of a similar order of magnitude”, and from the he writes: ${\displaystyle N/b}$ ${\displaystyle N}$ ${\displaystyle b}$

“How big the number is, you get an idea of ​​that when you consider that it has many trillion digits. If, on the other hand, as many planets revolved around each fixed star visible with the best telescope as around the sun, if there were as many people on each of these planets as there were on earth, and if each of these people lived a trillion years, then the number of seconds would have what experience all together, nowhere near fifty places. " ${\displaystyle N/b}$

literature

• Konrad Jacobs (Ed.): Selecta Mathematica. IV (=  Heidelberg Pocket Books . Volume 98 ). Springer-Verlag , Berlin, Heidelberg, New York 1972, ISBN 3-540-05782-X .
• Ricardo Mañé : Ergodic Theory and Differentiable Dynamics . Translated from the Portuguese by Silvio Levy (=  results of mathematics and their border areas . Volume 3 ). Springer-Verlag , Berlin, Heidelberg, New York, London, Paris, Tokyo 1987, ISBN 3-540-15278-4 ( MR0889254 ).
• JC Oxtoby : Measure and Category . Translated from English by K. Schürger (= university text  ). Springer-Verlag, Berlin, Heidelberg, New York 1971, ISBN 3-7643-0839-7 ( [1] ).
• Mark Pollicott, Michiko Yuri: Dynamical Systems and Ergodic Theory . Transferred to digital printing 2008 (=  London Mathematical Society Student Texts . Volume 40 ). Cambridge University Press , Cambridge 1998, ISBN 978-0-521-57294-1 ( MR1627681 ).
• Xiong Ping Dai: From the first Borel-Cantelli lemma to Poincaré's recurrence theorem . In: American Mathematical Monthly . tape 122 , 2015, p. 173-174 ( MR3324694 ).

Individual evidence

1. ^ Henri Poincaré : Sur leproblemème des trois corps et les equations de la dynamique , Acta Math. 13 (1890), 1-270. Poincaré had originally submitted a work to a tender by the Swedish King Oskar II , and thus won the prize. The work published in Volume 13 of Acta Mathematica is a revision of it, in which a serious error has been eliminated, but which no longer contains the supposed main result of the price work.
2. ↑ Konrad Jacobs : Selecta Mathematica IV. Some basic concepts of topological dynamics. Poincaré's recurring phrase. Springer-Verlag 1972.
3. ↑ Ludwig Boltzmann : Reply to the heat theoretical considerations of Mr. E. Zermelo , Ann. Phys. 293 [= re. Ann. 57], pp. 773-784 (1896). In: Scientific papers by Ludwig Boltzmann , ed. by Fritz Hasenöhrl, III. Band, New York 1968