Poincaré recurrence theorem

The Poincaré recurrence theorem is a mathematical theorem about dynamic systems. He says that, in every open set in the phase space are states in autonomous Hamiltonian systems whose phase space has a finite volume, often return their trajectories any back. In particular, the Poincaré recurrence theorem is a set of ergodic theory, and can also be viewed as the first result of chaos theory.

Origin

The Poincaré recurrence theorem was published in 1890 in the Swedish journal Acta Mathematica in a work of Henri Poincaré on the three-body problem for the first time. The first formulation of the recurrence rate can be found in it on page 69:

Poincaré proves this proposition on the next two pages of his work; from his evidence is clear that the dimension of the volume does not matter. In fact, Poincaré formulated on page 72f. this theorem for arbitrary dimension. The context at the Poincaré Hamiltonian formalism of classical mechanics, with the point describing the time-variable state of the mechanical system and the Hamiltonian function autonomously, that are not explicitly dependent on time, is. For example, the three-body problem has chapters and three (generalized ) momentum coordinates total of 18 components, namely for every body three (generalized ); in this case the phase space is thus 18 -dimensional. For autonomous Hamiltonian systems results from the set of Liouville that the volume remains in the phase space of the movement.

Mathematics

When adding in the original context leads to the following formulation of the Poincaré recurrence theorem:

Main ideas of the proof

The main steps of the proof are Poincaré ( in modern notation):

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

Measure-theoretical formulation and tightening

In Poincaré's proof of concept volume plays an important role. With the help of measure theory and related concepts can be the proof structure clearer. It begins with a measure space and identifies a measurable map

Maßerhaltend, if the equation is true for any measurable quantity, so if the measure and its size under match. Furthermore, one has to assume the finiteness of Maßraums so. Next comes the measure theoretic variant, with the times of iteration may refer to:

A detailed analysis of the Poincaré proof using the measure theory leads to the following measure theoretic tightening:

Discrete dynamical systems

The measure theoretic variants can be easily applied to discrete dynamical systems bring there but nothing new: As a measure here you just take the counting measure. The demand then means that the underlying set is finite. This is maßerhaltend synonymous with bijective, and the statement, the Poincaré recurrence theorem is that any permutation of a finite set decomposes into the simple fact in Cycles.

Physics

Physically, the Poincaré recurrence theorem, that a mechanical system whose orbits remain limited ( thus eg the solar system ), has the property that there are system states in each neighborhood of the initial state, often return their tracks as desired in said vicinity of the initial state. It follows approximately the following result: If you connect two containers, which contain different gases, then this first mix. After the recurrence rate but there is an arbitrarily small change of the initial state with the consequence that separate the gases at a later time by themselves and are segregated. The segregation contradicts a deterministic formulation of the second law of thermodynamics, which excludes a decrease in entropy. In ensued a dispute between Ernst Zermelo and Ludwig Boltzmann, in the course of Boltzmann wrote some articles about the relationship between the Poincaré recurrence theorem and the second law of thermodynamics. After that, the contradiction disappears when statistically interpreted the second law:

Accordingly, a decrease in entropy principle is not impossible, but very unlikely within a "short" period of time. However, considering the behavior of a Hamiltonian system with limited phase space for arbitrarily large times, the recurrence is almost certain - as follows from the measure theoretic intensification of the Poincaré recurrence theorem. In the appendix of the cited treatise Boltzmann gives an estimate of the recurrence time for the molecules of ordinary air density in a vessel of a cc volume. After about a page of combinatorial considerations, he comes to a number ( with an estimate for the number of combinations is discretized particle momenta and the number of Gasteilchenkollisionen per second describes ), the " be multiplied by a second of similar magnitude " still needed, and the he writes: