Leaking Chaotic Systems

TL;DR

This paper provides a comprehensive framework for understanding leaking chaotic systems, covering classical and quantum applications, and introduces new methods for modeling partial leaks with broad physical implications.

Contribution

It offers a unified treatment of leaking systems using transient chaos theory and extends the theory to partial leaks, with applications across various physical disciplines.

Findings

Transient chaos theory is essential for modeling real leaks.

True-time maps are necessary for accurate billiard dynamics.

Generalized Perron-Frobenius operators describe partial leaks effectively.

Abstract

There are numerous physical situations in which a hole or leak is introduced in an otherwise closed chaotic system. The leak can have a natural origin, it can mimic measurement devices, and it can also be used to reveal dynamical properties of the closed system. In this paper we provide an unified treatment of leaking systems and we review applications to different physical problems, both in the classical and quantum pictures. Our treatment is based on the transient chaos theory of open systems, which is essential because real leaks have finite size and therefore estimations based on the closed system differ essentially from observations. The field of applications reviewed is very broad, ranging from planetary astronomy and hydrodynamical flows, to plasma physics and quantum fidelity. The theory is expanded and adapted to the case of partial leaks (partial absorption/transmission) with applications to room acoustics and optical microcavities in mind. Simulations in the lima .con family of billiards illustrate the main text. Regarding billiard dynamics, we emphasize that a correct discrete time representation can only be given in terms of the so- called true-time maps, while traditional Poincar \'e maps lead to erroneous results. We generalize Perron-Frobenius-type operators so that they describe true-time maps with partial leaks.