Thermodynamics Rebuilt by Complex-Number Entropy

Classical thermodynamics is built on the assumptions of equilibrium, near-equilibrium,diffusion-dominated evolution, and monotonic entropy increase. As a result, it cannot describe essential structures observed in real flows, such as vortices, waves, boundary reflections, local entropy decrease, or nonequilibrium oscillations. To overcome these structural limitations, this study introduces the \emph{complex-number entropy general equation} \[S(x,t) = S_{\mathrm{Re}}(x,t) + i\,S_{\mathrm{Im}}(x,t),\] in which the real part represents diffusive spreading, while the imaginary part represents ordered structures such as vortices, phases, pull-back effects, and boundary reflections. Using a unified structural equation characterized by four parameters---initial value, diffusion rate, internal resistance, and external promotion---we reinterpret the secondlaw, heat, free energy, and nonequilibrium steady states. Classical thermodynamics emerges naturally as the projection of this general equation onto its real component,revealing that traditional thermodynamics is a special case within a broader structural framework. The complex-number entropy general equation provides a unified description of thermal, fluid, informational, and nonequilibrium phenomena. It shows that the interplay between entropy increase (real part) and entropy decrease (imaginary part) generates velocity differences, vortices, and nonequilibrium structures. This establishes a new structural thermodynamics based on the antagonistic interaction between diffusion and order. 既存の熱力学は、平衡または準平衡・拡散・単調増加を前提とするため、渦・波・境界反射・局所的エントロピー低下・非平衡振動といった現実の流れの主要構造を扱うことができない。本研究では、エントロピーを複素数として拡張した複素エントロピー一般式\[S(x,t)=S_{\mathrm{Re}}(x,t)+iS_{\mathrm{Im}}(x,t)\]を導入し、実部が拡散構造を、虚部が渦・位相・引き戻し・境界反射などの秩序構造を担うことを示す。さらに、エントロピー進化を初期値・拡散率・内部抵抗・外部促進の四つの構造パラメータで表す統一方程式を用いて、第二法則・熱・自由エネルギー・非平衡定常状態を再解釈する。これまでの熱力学は本一般式の実部への射影として得られる特殊ケースであり、複素エントロピー一般式は、熱・流体・情報・非平衡現象を統一的に記述する新しい構造熱力学の基礎方程式として機能する。