Abstract
Since Poincaré, the three-body problem is known to be chaotic and is believed to lack a general deterministic solution. Instead, decades ago a statistical solution was marked as a goal. Yet, despite considerable progress, all extant approaches display two flaws. First, probability was equated with phase space volume, thereby ignoring the fact that significant regions of phase space describe regular motion, including post-decay motion. Secondly and relatedly, an adjustable parameter, the strong interaction region, which is a sort of cutoff, was a central ingredient of the theory. This paper introduces remedies and presents for the first time a statistical prediction of decay rates, in addition to outcomes. Based on an analogy with a particle moving within a leaky container, the statistical distribution is presented in an exactly factorized form. One factor is the flux of phase-space volume, rather than the volume itself, and it is given in a cutoff-independent closed form. The other factors are the chaotic absorptivity and the regularized phase space volume. The situation is analogous to Kirchhoff’s law of thermal radiation, also known as greybody radiation. In addition, an equation system for the time evolution of the statistical distribution is introduced; it describes the decay rate statistics while accounting for sub-escape excursions. Early numerical tests indicate a leap in accuracy.
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Notes
We note that Valtonen and Karttunen (2006) includes also the empirical expression \(P_s =\frac{1}{m_s^q}\) with \(q=3/(1+2 L^2/L_{\max }^2),\, L_{\max }=5/2 [(m_1 m_2 + m_2 m_3 + m_3 m_1)/3]^{5/4}/\sqrt{E}\), see equations (7.30, 7.33, 2.67, 7.28) there. We find it appropriate to compare our derived expression with a derived expression, rather than an empirical one. In addition, a comparison to the empirical expression can be found in table 2 of [38], which leads to similar conclusions.
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Acknowledgements
It is a pleasure to thank L. Lederer, N. Leigh, V. Manwadkar, S. Mazumdar, A. Ori, A. Schiller, U. Smilansky, N. Stone, R. Shir and A. Trani for discussions. I thank Sara Kol for linguistic editing help. This work is based in part on ideas on statistical predictions for the double pendulum, a chaotic mechanical system, developed by the author in January 2016, initiated by teaching a course on Analytical Mechanics, and tested against simulations in an unpublished internal report with A. Marmor “Predicting chaos statistically: the double pendulum” (August 2016).
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Kol, B. Flux-based statistical prediction of three-body outcomes. Celest Mech Dyn Astr 133, 17 (2021). https://doi.org/10.1007/s10569-021-10015-x
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DOI: https://doi.org/10.1007/s10569-021-10015-x