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Updated: May 24, 2026

An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids
Published on: December 4, 2017
Infinite invariant density determines statistics of time averages for weak chaos
1Department of Physics, Institute of Nanotechnology and Advanced Materials, Bar Ilan University, Ramat-Gan 52900, Israel.
Abstract:
Weakly chaotic nonlinear maps with marginal fixed points have an infinite invariant measure. Time averages of integrable and nonintegrable observables remain random even in the long time limit. Temporal averages of integrable observables are described by the Aaronson-Darling-Kac theorem. We find the distribution of time averages of nonintegrable observables, for example, the time average position of the particle, x[over ¯]. We show how this distribution is related to the infinite invariant density. We establish four identities between amplitude ratios controlling the statistics of the problem.
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