Related Experiment Video
Updated: Jun 5, 2026

A Simple Flight Mill for the Study of Tethered Flight in Insects
Published on: December 10, 2015
Generalization of the Khinchin theorem to Lévy flights
Aleksander Weron1, Marcin Magdziarz
1Hugo Steinhaus Center, Institute of Mathematics and Computer Science, Wroclaw University of Technology, Wroclaw, Poland. aleksander.weron@pwr.wroc.pl
Abstract:
One of the most fundamental theorems in statistical mechanics is the Khinchin ergodic theorem, which links the ergodicity of a physical system with the irreversibility of the corresponding autocorrelation function. However, the Khinchin theorem cannot be successfully applied to processes with infinite second moment, in particular, to the relevant class of Lévy flights. Here, we solve this challenging problem. Namely, using the recently developed measure of dependence called Lévy correlation cascade, we derive a version of the Khinchin theorem for Lévy flights. This result allows us to verify the Boltzmann hypothesis for systems displaying Lévy-flight-type dynamics.
Related Concept Videos
Fundamental Theorem of Calculus II
Fundamental Theorem of Calculus I
Thevinin's Theorem
Fundamental Theorem of Algebra
Indeterminate Forms and L’Hôpital’s Rule
Central Limit Theorem
The sample size, n, that...