Chaos and Lévy Flights in the Three-Body Problem

Abstract

We study chaos and Lévy flights in the general gravitational three-body problem. We introduce new metrics to characterize the time evolution and final lifetime distributions, namely Scramble Density $\mathcal{S}$ and the LF index $\mathcal{L}$, that are derived from the Agekyan-Anosova maps and homology radius $R_{\mathcal{H}}$. Based on these metrics, we develop detailed procedures to isolate the ergodic interactions and Lévy flight interactions. This enables us to study the three-body lifetime distribution in more detail by decomposing it into the individual distributions from the different kinds of interactions. We observe that ergodic interactions follow an exponential decay distribution similar to that of radioactive decay. Meanwhile, Lévy flight interactions follow a power-law distribution. Lévy flights in fact dominate the tail of the general three-body lifetime distribution, providing conclusive evidence for the speculated connection between power-law tails and Lévy flight interactions. We propose a new physically-motivated model for the lifetime distribution of three-body systems and discuss how it can be used to extract information about the underlying ergodic and Lévy flight interactions. We discuss mass ejection probabilities in three-body systems in the ergodic limit and compare it to previous ergodic formalisms. We introduce a novel mechanism for a three-body relaxation process and discuss its relevance in general three-body systems.

arXiv comment20 pages, 14 figures

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