Measurement of three-body chaotic absorptivity predicts chaotic outcome distribution

Abstract

The flux-based statistical theory of the non-hierarchical three-body system predicts that the chaotic outcome distribution reduces to the chaotic emissivity function times a known function, the asymptotic flux. Here, we measure the chaotic emissivity function (or equivalently, the absorptivity) through simulations. More precisely, we follow millions of scattering events only up to the point when it can be decided whether the scattering is regular or chaotic. In this way, we measure a tri-variate absorptivity function. Using it, we determine the flux-based prediction for the chaotic outcome distribution over both binary binding energy and angular momentum, and we find good agreement with the measured distribution. This constitutes a detailed confirmation of the flux-based theory, and demonstrates a considerable reduction in computation to determine the chaotic outcome distribution.

Figures (8)

• Figure 1: The figure shows the general outline of this work. (Left Panel) Description of our method to measure chaotic absorptivity from binary-single scattering interactions in different configurations. (Center Panel) Description of procedure to predict the outcome distribution of chaotic three-body interactions that uses the chaotic absorptivity measurement from left panel. (Right Panel) Description of procedure to independently measure the chaotic three-body outcome distribution by means of three-body interaction simulations. The goal of the paper is to make a comparison between this measured outcome distribution and the predicted outcome distribution to test the validity of the flux-based theory presented in Kol2020.
• Figure 2: A 3D schematic representation of the outcome distribution of three-body interactions. Each semi-circular slice in this figure corresponds to a slice with constant binary energy $|\epsilon_{\rm B}|$. These slices are equivalent to the slices in Figure \ref{['fig:measured_calE']}.
• Figure 3: The grid in $l_{\rm B,x} - l_{\rm B,y}$ space (at a fixed $\epsilon_{\rm B}$) we use for measuring chaotic absorptivity. At each of these grid points, we run an ensemble of binary-single scattering experiments to measure the chaotic absorptivity at that given grid point. We use two super-imposed grids for better interpolation across grid. The Chebyshev grid is denoted by black circles (and with grid lines for reference). The smaller, uniform grid is denoted by grey circles. Refer to Section \ref{['sssec:sims_absorb']} for details on the grid.
• Figure 4: The 2D histogram distribution of interactions with a chaotic escape in $\epsilon_{\rm B} - l_{\rm B}$ space. Refer to Section \ref{['sssec:outcome_meas']} for details on the cuts used to consider interactions with chaotic escapes.
• Figure 5: Chaotic absorptivity $\mathcal{E}$ measured as function of $l_{\rm B,x},\, l_{\rm B,y}$ and $\epsilon_{\rm B}$. Note that $l_{\rm B,x},\, l_{\rm B,y}$ are the components of the binary angular momentum, and $\epsilon_{\rm B}$ is the binary energy. $\mathcal{E} \left( l_{\rm B,x},\, l_{\rm B,y},\, \epsilon_{\rm B} \right)$ is presented as a sequence of colored contour plots, with each 2D slice corresponding to a fixed value of $\epsilon_{\rm B}$, shown on the top-left corner of each panel. To be able to resolve the absorptivity structures in each panel, the color scaling for each panel is scaled relative to its maximum absorptivity value, $\mathcal{E}_{\rm max}$, shown in top-right corner of each panel. To better see how these absorptivity structures evolve as a function of $\epsilon_{\rm B}$, we prepared a video out of the sequence of such slices. See YouTube or the following http://phys.huji.ac.il/ barak_kol/resrch_supp/3body/3d_calE.mp4.$^1$