DOI: 10.5281/zenodo.18036386
title: On the Incompleteness of Fourier-Navier-Stokes Heat Transport in Structured Geometries

description: On the Incompleteness of Fourier–Navier–Stokes Heat Transport in Structured Geometries

BY ANDREW S. ELLIOTT and JENNIFER M. BULYAKI

This paper measures, compares, and in some regimes outperforms Fourier–Navier–Stokes heat transport on real experimental data—achieving 5×–20× reductions in mean-squared error—across the Sullivan–Thompson–Williamson rod experiments, Weber packed-bed transients, and Ilmenau turbulent wall-flux measurements. We introduce a minimal entropy–geometry closure that captures geometry-dependent, non-Markovian structure while preserving the classical self-adjoint diffusion framework, suggesting that in these systems randomness is not fundamental but emergent from unresolved geometry.

At its core, the paper challenges a 200-year assumption underlying classical heat and scalar transport: that diffusivity is a constant material parameter and that deviations from Fourier’s law can be treated as noise, turbulence, or effective stochastic corrections. By systematically confronting this assumption with high-resolution experimental data, we show that constant-coefficient closures fail in structured geometries in reproducible, non-random ways—misplacing peaks, distorting decay rates, and producing heavy-tailed, clustered residuals that cannot be reconciled with Gaussian or Poisson models.

Rather than proposing ad hoc corrections, we generalize the diffusion operator itself. The transport law is written in the classical self-adjoint form, but with diffusivity promoted to a geometry-dependent field, D(S), where S encodes local entropy or curvature of the thermal field. This single modification preserves conservation, semigroup evolution, and the Fourier limit in flat geometries, while allowing the operator to respond dynamically to structure. In this sense, Fourier diffusion emerges as a special case of a broader entropy–geometry transport law.

Empirically, this minimal closure collapses systematic residual structure across all three testbeds. In the near-ideal one-dimensional copper rod experiments, it corrects the characteristic fast-then-slow relaxation tails that Fourier theory cannot reproduce. In the Weber packed-bed transients, a highly heterogeneous and convective regime, it achieves order-of-magnitude improvements in error without introducing regime-specific tuning. In turbulent Rayleigh–Bénard wall-flux data, it explains heavy-tailed amplitudes and strongly clustered burst statistics that violate the foundational assumptions of stochastic Fourier closures.

A key finding is that transport energy and curvature are not homogeneously distributed but localize into coherent geometric structures—most notably thermal plumes in turbulent convection—which carry a disproportionate share of the heat flux. Classical models smear this structure into an average diffusivity and then interpret the remaining intermittency as randomness. Our results show instead that once geometry is resolved in the operator, much of this apparent randomness disappears, revealing deterministic organization beneath what had been treated as noise.

Conceptually, the work reframes heat transport as operator evolution on an entropy-curved geometry, continuing the modern progression from constant-coefficient laws toward structure-aware dynamics seen throughout mathematical physics. The philosophy is not to replace classical theory, but to extend it in the same spirit that curvature extended flat geometry and spectral theory extended pointwise PDEs: by adding a single, natural degree of freedom where the data demands it.

The broader implication is that in many structured transport systems, stochasticity may not be fundamental but a modeling artifact of geometry-blind closures. The entropy–geometry framework provides a falsifiable, minimal path beyond Fourier–Navier–Stokes that is immediately testable on existing datasets and compatible with classical limits. In doing so, this work opens a route toward a unified, operator-theoretic model of transport in complex media, with consequences for turbulence, porous flows, energy systems, and the foundations of non-equilibrium thermodynamics.

This perspective continues a long historical arc in the theory of heat and motion. From Newton’s early laws of flux and cooling, through Fourier’s formulation of heat as a linear partial differential equation, transport has progressively shifted from phenomenological rules toward deeper structural descriptions. Einstein’s analysis of Brownian motion reframed diffusion as the macroscopic shadow of microscopic dynamics, while Planck’s resolution of blackbody radiation revealed that thermal laws ultimately encode geometric and spectral structure at a fundamental level. In each case, what appeared as empirical constants were gradually reinterpreted as emergent from deeper organizing principles.

The twentieth century completed this transition by casting physical evolution in operator form. Schrödinger placed dynamics under self-adjoint operators whose spectra encode measurable quantities; Carathéodory gave thermodynamics an axiomatic geometric foundation, showing that entropy is not merely statistical but structurally constrained; and Lyapunov theory formalized stability and irreversible flow as properties of underlying dynamical geometry. Together, these developments established that heat, motion, and irreversibility are most naturally understood through operator evolution on structured state spaces, rather than through fixed coefficients acting on flat backgrounds.

Our work aligns with this lineage and extends it into the modern geometric era shaped by Perelman’s resolution of the Poincaré conjecture via Ricci flow. Just as Perelman showed that apparent topological complexity dissolves when geometry is allowed to evolve under curvature-driven flow, we show that apparent randomness in heat transport dissolves when diffusion is allowed to evolve under entropy geometry. In this sense, the entropy–geometry operator introduced here is not a departure from classical theory but a continuation of the historical progression: from constants to operators, from flat laws to curved flows, and from phenomenological closure to geometry-driven dynamics.