Crossing the Abyss: From Rigorous Limits to Constructive Multiscale Equations—A Deep Critique of Two Approaches to Hilbert's Sixth Problem

In the grand narrative of mathematical physics, a profound gap separates microscopic particle dynam- ics and macroscopic continuum mechanics. Hilbert’s sixth problem aims to construct the axiomatic foundation of macroscopic physical laws starting from atomism; this is not only a pursuit of mathemat- ical rigor but also an ultimate inquiry into the essential laws of the physical world. This paper provides an in-depth analysis of two milestone works in this field: the rigorous derivation of fluid equations by Deng, Hani, and Ma [1] and the constructive decomposition of kinetic equations by Chang Liu and Kun Xu [2]. The former establishes the inevitability of macroscopic laws by rigorously proving long-time convergence from hard-sphere systems to fluid equations through the introduction of a "Molecular Cutting Algorithm." The latter constructs a unified equation system traversing the entire Knudsen spectrum by introducing physical constraints under finite parameters via a "Local Dynamical Horizon." In particular, the work by Liu and Xu provides intermediate equation sets from the BGK model to the Navier-Stokes equations that vary continuously with scale, creatively solving the immense challenge of "how to establish control equations at designated scales" in multi-scale engineering calculations. This paper systematically compares these two distinct but highly complementary research paradigms from four dimensions: the height of mathematical analysis, the depth of physical modeling, the breadth of computational methods, and the width of application prospects.