A Unified Theory of Generalized Entropy in Open Dissipative Systems: Energy-Information Coupling and Continuous-Discrete Dual Axiomatic Framework

The unification of thermodynamic entropy, statistical entropy, and information entropy represents a long-standing foundational problem in non-equilibrium statistical mechanics, with profound implications for both fundamental physics and cross-disciplinary applications. Built upon three experimentally validated independent axioms—the total entropy production principle, Landauer’s energy-information equivalence, and the Markov Blanket boundary condition—this work reconstructs irreversible entropy production from first principles via the Crooks fluctuation theorem, and rigorously defines the conditional mutual information flux across system boundaries. We derive a universal Generalized Entropy Equation (GEE) that formally unifies energy dissipation and information processing in open systems. To eliminate the non-physical artificial dissipation introduced by discretizing continuous field theories onto discrete networks, we construct a continuous-discrete dual mathematical structure based on Discrete Exterior Calculus (DEC) over oriented simplicial complexes. We strictly prove that the GEE preserves identical algebraic form, flux conservation laws, and non-negativity of entropy production on both continuous Riemannian manifolds and discrete simplicial complexes, resolving a long-standing numerical artifact in discrete thermodynamic modeling. We further demonstrate that Clausius thermal entropy, Boltzmann–Gibbs statistical entropy, Shannon information entropy, and the Landauer erasure limit are all exact degenerate special cases of the GEE, settling the long debate over the physical nature of information entropy. By decomposing energy-information coupling efficiency into a substrate-topology dual tensor struc ture, we decouple intrinsic material dissipation from dynamical path excess loss. Extending Prigogine’s minimum entropy production principle to energy-information coupled systems, we prove that open dissipative systems spontaneously evolve toward non-equilibrium steady states along geodesics of the Fisher information manifold. This axiomatic framework establishes a first-principles thermodynamic foundation for cross-disciplinary dissipative systems ranging from semiconductor computing devices and biomolecular networks to deep learning systems and complex ecological structures.