User:Vilnius/Kac's lemma

In ergodic theory, Kac's lemma, demonstrated by mathematician Mark Kac in1947, states that in a measure space the orbit of almost all the points contained in a set $$\langle T_R \rangle = \tau/P(A)$$of such space, whose measure is $$\mu(A)$$, return to $$A$$ within an average time inversely proportional to $$\mu(A)$$.

The lemma extends what is stated by Poincarè recurrence theorem, in which it is shown that the points return in $$A$$ infinite times.

Since the phase space of a dynamical system with $$n$$ variables and bounded, i.e. with the $$n$$ variables all having a minimum and a maximum, is, for the Liouville theorem, a measure space, the lemma implies that given a configuration of the system (point of space) the average return period close to this configuration (in the neighbourhood of the point) is inversely proportional to the considered size of volume surrounding the configuration.

Normalizing the measure space to 1, it becomes a probability space and the measure $$P(A)$$ of its set $$A$$ represents the probability of finding the system in the states represented by the points of that set. In this case the lemma implies that the smaller is the probability to be in a certain state (or close to it), the longer is the time of return near that state.

In formulas, if $$A$$ is the region close to the starting point and $$T_R$$ is the return period, its average value is:

$$\langle T_R \rangle = \tau/P(A)$$

Where $$\tau$$ is a characteristic time of the system in question.

Note that since the volume of $$A$$, therefore $$P(A)$$, depends exponentially on the number of variables in the system ($$A = \epsilon ^n$$, with $$\epsilon$$ infinitesimal side, therefore less than 1, of the volume in $$n$$ dimensions), $$P(A)$$ decreases very rapidely as the variables of the system increase and consequently the return period increases exponentially.

In practice, as the variables needed to describe the system increase, the return period increases rapidly.

Intuitively it is quite obvious to understand that if the possible configurations are limited sooner or later they will repeat themselves. At the same time is quite obvious that the most likely configurations will repeat more frequently