Kac's lemma

In ergodic theory, Kac's lemma, demonstrated by mathematician Mark Kac in 1947,^[1] is a lemma stating that in a measure space the orbit of almost all the points contained in a set ${\displaystyle A}$ of such space, whose measure is ${\displaystyle \mu (A)}$, return to ${\displaystyle A}$ within an average time inversely proportional to ${\displaystyle \mu (A)}$.^[2]

The lemma extends what is stated by Poincaré recurrence theorem, in which it is shown that the points return in ${\displaystyle A}$ infinite times.^[3]

Application

In physics, a dynamical system evolving in time may be described in a phase space, that is by the evolution in time of some variables. If this variables are bounded, that is having a minimum and a maximum, for a theorem due to Liouville, a measure can be defined in the space, having a measure space where the lemma applies. As a consequence, given a configuration of the system (a point in the phase 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 ${\displaystyle P(A)}$ of its set ${\displaystyle A}$ 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.^[4]

In formulas, if ${\displaystyle A}$ is the region close to the starting point and ${\displaystyle T_{R}}$ is the return period, its average value is:

${\displaystyle \langle T_{R}\rangle =\tau /P(A)}$

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

Note that since the volume of ${\displaystyle A}$, therefore ${\displaystyle P(A)}$, depends exponentially on the ${\displaystyle n}$ variables in the system (${\displaystyle A=\epsilon ^{n}}$, with ${\displaystyle \epsilon }$ infinitesimal side, therefore less than 1, of the volume in ${\displaystyle n}$ dimensions),^[5] ${\displaystyle P(A)}$ decreases very rapidly as the variables of the system increase and consequently the return period increases exponentially.^[6]

In practice, as the variables needed to describe the system increase, the return period increases rapidly.^[7]

References

1. Kac, Mark (1947). "On the notion of recurrence in discrete stochastic processes" (PDF). Bulletin of the American Mathematical Society. 53 (10): 1002–1010. doi:10.1090/S0002-9904-1947-08927-8.
2. Hochman, Michael (2013-01-27). "Notes on ergodic theory" (PDF). p. 20.
3. Walkden, Charles. "MAGIC: 10 lectures course on ergodic theory – Lecture 5".
4. Pereira, Tiago. "Lecture Notes - Introduction to Ergodic Theory" (PDF). Imperial College London. Department of Mathematics. p. 12.
5. ${\displaystyle \lim _{n\to \infty }\epsilon ^{n}=0\quad {\text{for}}\quad 0<\epsilon <1}$. See List of limits.
6. Gammaitoni, Luca; Vulpiani, Angelo (2019). Perché è difficile prevedere il futuro (in Italian). Bari: Edizioni Dedalo. p. 91. ISBN 978-88-220-6882-8.
7. Petersen, Karl E. (1983). Ergodic Theory. Cambridge: Cambridge University Press. p. 37. ISBN 0521236320.

Further reading

• Kac, Mark (1947). "On the notion of recurrence in discrete stochastic processes" (PDF). Bulletin of the American Mathematical Society. 53 (10): 1002–1010. doi:10.1090/S0002-9904-1947-08927-8.
• Walkden, Charles. "MAGIC: 10 lectures course on ergodic theory – Lecture 5".
• Petersen, Karl E. (1983). Ergodic Theory. Cambridge: Cambridge University Press. p. 46. ISBN 0521236320.
• Hochman, Michael (2013-01-27). "Notes on ergodic theory" (PDF).