Energy Distribution: Reconstructing the First Principles Framework of Quantum Theory Universal Formula for Internal Dissipation Power in Many Body Systems and Rigorous Derivation of the Mesoscopic Diffusion Equation

Traditional quantum theory typically attributes energy dissipation to random perturbations from the external environment. Based on the "Interfacial Projected Realism" framework, this work re-examines the nature of energy dissipation from a novel perspective, treating the qubit, the interfacial defect network, and the bulk dissipation channel as a self-contained total system.

We first rigorously derive the universal formula for internal dissipation power and the energy distribution ratio \(f\) in single-body quantum systems, and then extend these concepts to many-body systems. Starting from a microscopic Hamiltonian coupling multiple qubits to a common quantized field and employing the Nakajima-Zwanzig projection operator technique, we obtain a universal expression for the energy dissipated from a many-body system to the environment. Applying this result to the specific model of multiple two-level systems coupled to a common bosonic field yields a collective master equation where the dissipation coefficient matrix \(\Gamma_{ij}\) encodes all spatial and coupling information.

Through systematic coarse-graining, gradient expansion, and semiclassical approximation, we derive for the first time from an energy-distribution perspective a closed set of macroscopic equations for the local system energy density \(u(\mathbf{r},t)\) and the environmental field energy density \(U(\mathbf{r},t)\):

\[
\frac{\partial u}{\partial t} = R_{\mathrm{total}}(\mathbf{r},t) - \gamma_0(U) u + \nabla \cdot \bigl( \gamma_2(U) \nabla u \bigr)
\]
\[
\frac{\partial U}{\partial t} = D \nabla^2 U - \Gamma_B U + \rho(\mathbf{r}) \gamma_0(U) u
\]

Within this framework, the energy distribution ratio \(f(\mathbf{r},t) = \gamma_0 u / R_{\mathrm{total}}\) emerges naturally as a macroscopic quantity, its physical essence being jointly determined by the system’s microscopic coupling structure (embodied by \(\Gamma_{ij}\)) and quantum statistical state.

The theory quantitatively explains several long-unexplained anomalous phenomena in cutting-edge experiments, demonstrates strong testability, and lays a solid foundation for exploring deeper connections between energy distribution and macroscopic physics.

For the exact mathematical formulation and detailed derivations, please refer to the PDF file.