Non-Statistical Analytic Entropy-Multiplicative Entropy Formula Answers Planck’s Question

This paper introduces a novel entropy formulation — multiplicative entropy — defined as the product of energy values across all units in a quantized homogeneous invariant network. Unlike traditional statistical entropy, this approach explicitly tracks irreversible energy redistribution pathways, offering an analytic and path-dependent description of entropy growth. Logarithmic transformation recovers classical entropy forms, while preserving temporal directionality and quantum-scale resolution. The model enables precise simulation of thermodynamic processes, supporting the development of Analytic Quantum Thermodynamics as a new framework for understanding entropy-driven dynamics. Crucially, the analytical multiplicative entropy formula proposed here responds to Planck’s long-sought vision of an entropy expression in analytic form, enabling the resolution of entropy to match the high resolution of quantum processes.