A Unified Non-Equilibrium Framework Overcoming the Fickian Limit: From Rough Thermodynamics to Fractional Cahn-Hilliard Dynamics

The fundamental limits of Fick’s laws of diffusion are exposed when dealing with high-concentration, non-ideal multi-component systems, structural stress, and anomalous transport phenomena. This paper establishes a mathematically rigorous, superior framework that subsumes Fickian diffusion as a low-order approximation. By integrating irreversible thermodynamics, Onsager reciprocal relations, Universal Rough Operator Algebra (UROA) [3], and Time-Fractional Continuous-Time Random Walk under Seonggil Theory of Complex Torsion (STCT) [1], we derive a Universal Torsional Transport Equation. Furthermore, we provide a computational implementation via a Semi-Implicit Fourier Spectral Method in Python to simulate non-commutative phase-separation dynamics where classical diffusion theory completely collapses.