# Unified Entropy-Ruled Einstein Relation for Bulk and Low-Dimensional Systems: A Hopping to Band Shift Analysis Source: [Unified Entropy-Ruled Einstein Relation for Bulk and Low-Dimensional Systems: A Hopping to Band Shift Analysis]() Collection index: [llms.txt]() Data updated: 2025-10-25T05:00:57.702Z Authors: K. Navamani Published: 2023-08-02T17:00:19.000Z ## Abstract In this letter, we present the unified paradigm on entropy-ruled Einstein diffusion-mobility relation ({\\mu}/D ratio) for all dimensional systems (1D, 2D and 3D) of molecules and materials. The different dimension-associated fractional value of the variation in differential entropy with respect to the chemical potential ({\\Delta}h/{\\Delta}{\\eta}) gives the quantum-classical transition version of {\\mu}/D relation. This is a new alternative version for quantum devices, instead of Einstein original relation of {\\mu}/D = q/kT; where q, k and T are the electric charge, Boltzmann constant and temperature, respectively. It is found that the fractional value of {\\Delta}h/{\\Delta}{\\eta} for {\\mu}/D ratio for different dimensional systems or devices is a direct consequences with the average energy-Fermi energy relation, which can varies with the typical dimensions, whether the system belongs to 1D or 2D or 3D. This unified entropy-ruled transport formalism works well for both the quantum and classical systems with equilibrium as well as non-equilibrium conditions. Based on the dimensional dependent entropy-ruled {\\mu}/D factor, the Navamani-Shockley diode equation is transformed. ## Recorded benchmark results A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage. ## Original sources and implementation [Original paper]() [PDF]()