Unified Continuous State-Space Thermodynamics Framework for Kinetic Fluctuations: Closures, Regularization, and Numerical Paradigms

This work presents a comprehensive, mathematically rigorous framework for modeling non-equilibrium fluid dynamics by lifting the internal phase-space from a traditional macroscopic description to an invariant continuous state coordinate $s \in [0, \infty)$, which can represent localized peculiar energy spectra, species distributions, or internal molecular states. By transforming cumulative kinetic balances into pure, locally closed 5D partial differential equations, the classical closure problem for non-linear convective transport is analytically eliminated. We establish consistent rheological and thermal closures based on continuous state density distributions and smeared-flux methodologies to handle sharp gradients. To solve the resulting coupled hyper-dimensional system, we introduce a mathematical background tracer matrix regularization technique and propose three distinct numerical paradigms: a deterministic 4D hyper-mesh solver, an Eulerian Stochastic Fields method driven by Wiener processes, and a data-driven Scientific Machine Learning (SciML) surrogate framework.