The Flow of Learning Accessibility: From Stochastic Trajectories to Hydrodynamic Models

Abstract

Describing learning as flow is scientifically useful only if the mathematics is derived rather than borrowed as metaphor. This paper develops a conditional micro-to-macro program for the Dynamic Map of Learning Accessibility (MDAA), which conceptualizes learning accessibility not as a stable property of the learner but as a situated and time-dependent relation among subject, learning object, and conditions. The starting point is not a fluid equation but the trajectory model class that survives the identification protocol developed in companion work. A deterministic Markov state model leads to Liouville transport of an ensemble density; a Markov diffusion leads to a Fokker–Planck equation; a hybrid Markov process leads to coupled regime-specific forward equations; and a history-dependent process requires state augmentation or a generalized, possibly fractional, probability description. This branching is necessary because educational and adjacent research already contains direct precedents for test-score flow and diffusion, kinetic learning and competence models, hydrodynamic reductions of social variables, and Navier–Stokes-like equations in abstract social spaces. The candidate contribution is therefore not the importation of fluid mechanics into learning. It is a memory-aware, geometry-aware, conservation-audited derivation-and-rejection protocol with seven admissibility gates: whether a continuum density is estimable over a coherent ensemble, whether a defensible state-space geometry exists, whether coarse graining preserves effective Markovianity, whether candidate collective variables are local and self-predictive, whether any vector field behaves as a slow variable with its own balance law, whether the moment hierarchy admits a stable closure, and whether a Navier–Stokes-like closure improves out-of-sample explanation beyond Fokker–Planck, generalized-memory, or lower-order alternatives. Navier–Stokes remains a live hypothesis, but it is an endpoint that must be earned. Preprint. Theoretical manuscript, not peer reviewed. No empirical results are claimed.