Anomalous Diffusion as Structural Memory: An Extended Structural Dynamics Approach

Abstract

Sub-diffusion in biological systems is conventionally treated as anomalous, requiring fractional derivatives, heavy-tailed waiting times, or fitted memory kernels. We argue that this anomaly is an artifact of an incomplete phase space. Standard frameworks model diffusing particles as points. Biological molecules are not points. They are three-dimensional deformable entities whose position, orientation, and internal structure are irreducible physical properties, not modeling conveniences appended to a point mass. Within the Extended Structural Dynamics (ESD) framework, each particle is a primitive structured entity with translational, orientational, and deformational degrees of freedom. When dynamics on this full phase space are projected onto the translational subspace alone, a memory kernel emerges from the projection without phenomenological postulate. The subdiffusion exponent is determined by the internal mode spectrum, independently measurable from B-factors, NMR order parameters, or molecular dynamics simulations, without fitting to transport data. Four falsifiable predictions follow: subdiffusion strength correlates with molecular flexibility; temperature drives crossover to normal diffusion at a characteristic energy scale set by internal mode frequencies; a non-zero rotation-translation cross-correlation spectrum encodes internal dynamics, identically zero in point-particle models; and memory timescales scale as the square of particle size. Quantitative consistency with experimental observations for proteins in crowded media is demonstrated using independently estimated structural parameters. What appears anomalous from the point-particle perspective is the expected behavior of structured matter projected onto an impoverished description. The anomaly is not in the physics. It is in the phase space.