Memory and chance in the three-body problem: a planar test of the statistical theories

We test, in the planar three-body problem with zero total angular momentum, whether the system forgets its initial conditions and with what measure it occupies the final states. In a map of 34,115 free falls with equal masses, the dependence of the outcome on the initial configuration disappears, within the resolution of the test, after five phases of democratic interaction. The energy distribution of the final pair agrees with the phase-volume theory, but the eccentricity distribution does not: the theory predicts 49% of pairs with eccentricity above 0.9, and the simulations give 35%. Since the deviation does not depend on the history, it is not memory but a failure of uniform occupation. We directly measure the absorptivity of Kol's flux theory in 210,000 time-reversed scatterings. The flux multiplied by it reproduces, with no fitted parameter, the distributions with equal masses and reduces the error in the ejection probabilities with unequal masses to about one percentage point, against 4.2 points for the volume theory. Unlike the three-dimensional case, the absorptivity is indispensable for predicting which body is ejected. The residual withstands a finer grid, a larger launch distance, and another chaos criterion, and it is smaller than the variation of the measured fractions themselves with the lifetime of the system, which indicates the limit of a time-independent statistical description.