Follow the fugitive: an application of the method of images to open dynamical systems

TL;DR

This paper introduces a theoretical framework using the method of images to analyze escape rates in open dynamical systems, providing explicit formulas and relations that simplify both analytical and numerical studies.

Contribution

It extends the method of images to open dynamical systems, deriving exact escape rates and relations between transfer operators for Markov systems, applicable to various fields.

Findings

Explicit size- and position-dependent escape rates for Markov systems.

Derived relations between transfer operators of closed and open systems.

Computed small hole asymptotic behavior with logarithmic corrections.

Abstract

Borrowing and extending the method of images we introduce a theoretical framework that greatly simplifies analytical and numerical investigations of the escape rate in open dynamical systems. As an example, we explicitly derive the exact size- and position-dependent escape rate in a Markov case for holes of finite size. Moreover, a general relation between the transfer operators of closed and corresponding open systems, together with the generating function of the probability of return to the hole is derived. This relation is then used to compute the small hole asymptotic behavior, in terms of readily calculable quantities. As an example we derive logarithmic corrections in the second order term. Being valid for Markov systems, our framework can find application in information theory, network theory, quantum Weyl law and via Ulam's method can be used as an approximation method in more general dynamical systems.