Physical limit of prediction for chaotic motion of three-body problem

Abstract

A half century ago, Lorenz found the "butterfly effect" of chaotic dynamic systems and made his famous claim that long-term prediction of chaos is impossible. However, the meaning of the "long-term" in his claim is not very clear. In this article, a new concept, i.e. the physical limit of prediction time, denoted by $T_p^{max}$, is put forwarded to provide us a time-scale for at most how long mathematically reliable (numerical) simulations of trajectories of a chaotic dynamic system are physically correct. A special case of three-body problem is used as an example to illustrate that, due to the inherent, physical uncertainty of initial positions in the (dimensionless) micro-level of $10^{-60}$, the chaotic trajectories are essentially uncertain in physics after $t>T_p^{max}$, where $T_p^{max}\approx 810$ for this special case of the three body problem. Thus, physically, it has no sense to talk about the "accurate, deterministic prediction" of chaotic trajectories of the three body problem after $t>T_p^{max}$. In addition, our mathematically reliable simulations of the chaotic trajectories of the three bodies suggest that, due to the butterfly effect of chaotic dynamic systems, the micro-level physical uncertainty of initial conditions might transfer into macroscopic uncertainty. This suggests that micro-level uncertainty might be an origin of some macroscopic uncertainty. Besides, it might provide us a theoretical explanation about the origin of uncertainty (or randomness) of many macroscopic phenomena such as turbulent flows, the random distribution of stars in the universe, and so on.

24 pages, 17 figures. Accepted by Communications in Nonlinear Science and Numerical Simulations, DOI: 10.1016/j.cnsns.2013.07.008. arXiv admin note: substantial text overlap with arXiv:1304.2089

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