A Re-Proof of Hilbert's Sixth Problem via Unified Complex Systems Theory

Abstract

David Hilbert’s Sixth Problem put forward in 1900 assigns two core missions for mathematical physics: first, to realize the axiomatization of the entire physical system on the basis of strict logical self-consistency, and second, to establish a rigorous derivation relationship between microscopic particle deterministic mechanics and macroscopic continuum fluid mechanics. For more than a century, classical solutions represented by Lanford’s short-time proof of the Boltzmann equation and Deng et al.’s long-time extension have been trapped in the idealized hypothesis of dilute hard-sphere systems and perfectly elastic collisions. They cannot describe dense gas systems, gas-liquid phase transitions and real inelastic collision processes, and lack a complete causal mechanism to explain the sustained movement of molecules under energy dissipation conditions, so they cannot truly solve Hilbert’s Sixth Problem. This paper takes the Unified Complex Systems Theory (UCST) as the core framework, and relies on the mind-aether dual ontology and the active-passive force model to complete the re-certification of Hilbert’s Sixth Problem. UCST takes aether as the unobservable elementary substrate for energy transmission, defines gas molecules as ontological living particles with an endogenous energy compensation mechanism, and introduces an active force based on the causal principle of mind to compensate for the energy loss caused by inelastic collisions. Starting from the discrete particle dynamics dominated by the UCST active-passive force model, this paper strictly derives the arbitrary-density gas state equation that is applicable to all density systems and can degenerate into the ideal gas law and Boltzmann equation in the dilute limit, obtains the accurate gas-liquid phase transition conditions based on the critical point of the state equation, and further derives the macroscopic continuum fluid equations including the continuity equation, Euler equation, Navier-Stokes equation and energy conservation equation through the improved Liouville equation and moment expansion method. This research has fully realized the two core requirements of Hilbert’s Sixth Problem. It overcomes all the inherent defects of classical kinetic theory and continuum mechanics, and constructs a set of axiomatic physical theories that are fully applicable to real thermodynamic systems with inelastic collisions, arbitrary densities and phase transitions. The UCST framework maintains complete consistency with the verified conclusions of classical and modern physics in its applicable scope, and realizes a revolutionary breakthrough in the unification of physical theories.