A Brief Discussion on the Definition of Liu Entropy

There has long been an academic chaos of theoretical separation and abnormal information measurement standards at the quantum-classical scale in the field of mesoscopic physics. Classical Shannon entropy and quantum von Neumann entropy have inherent scale limitations, which cannot characterize the residual structured information generated by incomplete decoherence of mesoscopic systems. Furthermore, the solidified disciplinary paradigm barriers have led to the long-term lack of a unified theoretical system for the quantitative research of mesoscopic information. Aiming at this problem, this paper discusses and standardizes the scientific definition, mathematical framework and physical connotation of Liu Entropy (mesoscopic modal residual structural entropy) based on modal information theory, S4 modal logic, M₃ non-distributive lattice theory, modal-probability bridging axiom, and Busch generalized Gleason strong-weak dual bridging axiom. By defining the core order parameter, critical criterion and complete analytical formula, this paper clarifies the measurement category, unit system and applicable boundary of Liu Entropy, and deeply compares its core mechanism differences with classical entropy and quantum entropy. The research shows that Liu Entropy effectively fills the theoretical gap of the traditional information entropy system in the mesoscopic transition interval, breaks the paradigm barrier between classical and quantum information disciplines, constructs a globally self-consistent information measurement system covering classical, mesoscopic and quantum scales, and can provide brand-new theoretical support for the research of mesoscopic quantum decoherence, structural phase transition and cross-scale information evolution.