Chaos: a bridge from microscopic uncertainty to macroscopic randomness

TL;DR

This paper demonstrates that microscopic uncertainties, amplified by sensitive dependence on initial conditions, can lead to macroscopic randomness in chaotic systems, challenging traditional views on predictability.

Contribution

It introduces a high-precision numerical method to show how micro-level uncertainties propagate into macroscopic chaos, linking microscopic fluctuations to observable randomness.

Findings

Micro-level initial uncertainties transfer into macroscopic chaos.

Sensitive dependence on initial conditions amplifies microscopic fluctuations.

Long-term prediction of chaotic systems is fundamentally limited.

Abstract

It is traditionally believed that the macroscopic randomness has nothing to do with the micro-level uncertainty. Besides, the sensitive dependence on initial condition (SDIC) of Lorenz chaos has never been considered together with the so-called continuum-assumption of fluid (on which Lorenz equations are based), from physical and statistic viewpoints. A very fine numerical technique (Liao, 2009) with negligible truncation and round-off errors, called here the "clean numerical simulation" (CNS), is applied to investigate the propagation of the micro-level unavoidable uncertain fluctuation (caused by the continuum-assumption of fluid) of initial conditions for Lorenz equation with chaotic solutions. Our statistic analysis based on CNS computation of 10,000 samples shows that, due to the SDIC, the uncertainty of the micro-level statistic fluctuation of initial conditions transfers into the macroscopic randomness of chaos. This suggests that chaos might be a bridge from micro-level uncertainty to macroscopic randomness, and thus would be an origin of macroscopic randomness. We reveal in this article that, due to the SDIC of chaos and the inherent uncertainty of initial data, accurate long-term prediction of chaotic solution is not only impossible in mathematics but also has no physical meanings. This might provide us a new, different viewpoint to deepen and enrich our understandings about the SDIC of chaos.