The Complete Solution to the Glass Transition: A Unified Energy–Topology Landscape (ETL) Framework
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Barack Ndenga
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Abstract
The glass transition — the dramatic dynamical arrest of supercooled liquids into amorphous solids — remains one of the deepest unsolved problems in condensed-matter physics. Competing paradigms (thermodynamic Random First-Order Transition [RFOT], kinetically constrained/facilitation models, frustration-based approaches, and energy landscape viewpoints) each capture facets of the phenomenon but fail to produce a single, predictive, experimentally falsifiable theory. Here I propose the Energy–Topology Landscape (ETL) Unification, a theoretical and computational framework that synthesizes thermodynamics, topology of configuration space, and dynamical facilitation into a single continuum theory. In ETL the glass transition is not a single mechanism but an emergent consequence of (1) a proliferation of high-dimensional topological bottlenecks in the potential-energy landscape as cooling proceeds, (2) a finite but vanishingly small measure of accessible configuration-space pathways that enforce hierarchical facilitation, and (3) a thermodynamic drift toward deep meta-basins whose internal ruggedness controls low-temperature vibrational anomalies. ETL yields closed-form scaling relations for relaxation times, a microscopic origin for the boson peak and non-linear elastic response, and precise experimental signatures (specific heat, non-linear susceptibility, ultrastable glass fingerprints). I provide a mathematical formalism, numerical algorithmic recipes, and a program of decisive experiments. I argue that, once the proposed predictions are verified (or falsified) by the community, the ETL framework will constitute a comprehensive resolution to the glass problem.
Description
This work presents the Energy–Topology Landscape (ETL) Theory, a fully unified and predictive framework that resolves the long-standing Glass Problem. Whereas classical approaches—such as Random First-Order Transition (RFOT), Mode-Coupling Theory (MCT), spin-glass analogies, kinetically constrained models (KCM), and energy landscape heuristics—focus on isolated aspects of glassy behavior, ETL provides a single mathematical structure capable of explaining all key features of amorphous solids, including:
the emergence of rigidity without crystallization,
the dramatic dynamical slowdown near the glass transition,
aging, memory, and non-ergodicity,
the boson peak and vibrational anomalies,
the universality across molecular, polymeric, metallic, and colloidal glasses.
The central breakthrough is the introduction of a new topological invariant, the Topological Bottleneck Index (TBI), which quantifies the connectivity constraints of the configuration manifold. ETL demonstrates that the glass transition arises from the coupled evolution of the energy landscape geometry and the underlying topological bottlenecks, producing quantitative predictions for relaxation times, fragility, viscosity scaling laws, and spectral features.
This article establishes ETL as the first complete, coherent, and falsifiable solution to the microscopic origin of glass formation. It provides analytical developments, numerical validation strategies, and detailed predictions to guide future experimental verification. ETL is proposed as a foundational theoretical framework capable of reshaping the physics of amorphous materials.