Unified Structural Equation of Entropy

This paper reexamines the fragmented landscape of entropy concepts that have independently developed in thermodynamics, statistical mechanics, information theory, quantum information, and dynamical systems. Despite their diverse origins, these entropies share a common structure characterized by the disappearance of bias, the loss of distinguishability, and the irreversible dissipation of information. Based on this shared structure, we introduce a unified structural equation of entropy, expressed through four minimal parameters: the initial value, dispersion rate, dispersion resistance, and a promotion term. This equation provides a universal description of entropy evolution across physical, informational, quantum, and social systems. The unified equation naturally incorporates thermodynamic entropy as a special case, thereby repositioning the second law of thermodynamics as one instance of a more general information-loss process. The inclusion of a promotion term allows the framework to describe accelerated entropy production observed in complex systems such as biological, urban, and social networks. Parameter assignments for major entropy definitions (Clausius, Boltzmann--Gibbs, Shannon, von Neumann, and KS entropy) demonstrate that each can be understood as a specific instantiation of the unified structure. This work establishes entropy as a universal principle of irreversibility and provides a foundation for future extensions to nonlinear, hierarchical, and emergent systems.\end{abstract}