Statistics – Computation
Scientific paper
Sep 1999
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1999dda....31.0602m&link_type=abstract
American Astronomical Society, DDA meeting #31, #06.02
Statistics
Computation
Scientific paper
The gravitational n-body problem is chaotic. Consider two very similar n-body systems, each represented by a point in the same 6n-dimensional phase space. The two points start very near each other, but they separate rapidly as the systems develop, resulting in phase trajectories that also separate rapidly. The rate looks exponential over long times. Such systems are extremely sensitive to initial conditions. At any instant, phase points separated in certain directions move apart (unstable directions), while those separated in other directions stay at about the same distance (stable directions). Unstable directions lie along eigenvectors of the matrix of force gradients that correspond to positive eigenvalues. The number of positive eigenvalues of that matrix gives the dimensionality of unstable subspace. This number changes extremely rapidly as a system moves through configuration space. On average, there are about 1.2n unstable directions out of 3n. Numerical effects augment the sensitivity, bringing their reliability into question. The issue whether numerical computations yield reliable estimates hinges on whether numerical systems populate the phase space in the same way as physical systems. ``Shadowing'' appears to hold some promise of assuring reliability, but it does not address the question how computed trajectories populate the phase space. In fact, it seems to say more about the character of chaotic trajectories: they are wilder than we'd imagined. For almost any computed trajectory, a segment of a physical trajectory can be found along which the physical phase point remains near the computed system's phase point for a surprisingly long time. Estimates of the rate of trajectory separation require that the force gradient matrix be averaged somehow over realistic phase trajectories. Attempts to do this require lots of additional assumptions. This problem will be discussed, as will the question whether practitioners of the n-body art can safely be reassured by shadowing.
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