Ergodic decay laws in Newtonian and relativistic chaotic scattering

TL;DR

This paper challenges the assumption of ergodicity in chaotic scattering systems, demonstrating that the chaotic saddle prevents ergodic decay laws from accurately describing escape dynamics, and proposes a new decay law in the relativistic regime.

Contribution

It reveals the non-ergodic nature of escape dynamics in Hamiltonian systems due to the chaotic saddle and introduces a new decay law applicable in relativistic contexts.

Findings

Chaotic saddle prevents ergodic decay assumptions.

Escape dynamics are non-ergodic regardless of KAM tori presence.

Derived a new decay law for relativistic Hamiltonian systems.

Abstract

In open Hamiltonian systems, the escape from a bounded region of phase space according to an exponential decay law is frequently associated with the existence of hyperbolic dynamics in such a region. Furthermore, exponential decay laws based on the ergodic hypothesis are used to describe escapes in these systems. However, we uncover that the presence of the set that governs the hyperbolic dynamics, commonly known as the chaotic saddle, invalidates the assumption of ergodicity. For the paradigmatic H\'enon-Heiles system, we use both theoretical and numerical arguments to show that the escaping dynamics is non-ergodic independently of the existence of KAM tori, since the chaotic saddle, in whose vicinity trajectories are more likely to spend a finite amount of time evolving before escaping forever, is not utterly spread over the energy shell. Taking this into consideration, we provide a clarifying discussion about ergodicity in open Hamiltonian systems and explore the limitations of ergodic decay laws when describing escapes in this kind of systems. Finally, we generalize our claims by deriving a new decay law in the relativistic regime for an inertial and a non-inertial reference frames under the assumption of ergodicity, and suggest another approach to the description of escape laws in open Hamiltonian systems.