Ergodicity, Fluid Dynamics, and the Complex Entropy General Equation

Classical fluid dynamics and statistical mechanics are usually treated as separate domains: the former describes velocity fields and vortices, while the latter introduces the ergodic hypothesis to justify entropy and ensemble averages. This paper proposes a unified structural view based on the Complex Entropy General Equation. In this framework, the real part of entropy corresponds to diffusive relaxation and ergodic mixing, whereas the imaginary part captures vortical, phase-coherent, and non-ergodic structures. We show that the core equations of fluid dynamics can be reinterpreted as the dynamics of the imaginary component of a complex entropy field, while the ergodic hypothesis appears as the limiting regime in which the real part dominates and the imaginary part becomes negligible. This perspective clarifies why turbulence, coherent vortices, and long-lived structures escape classical ergodic assumptions, and suggests that low-speed physical systems share a common complex-entropy-based mechanism underlying both diffusion and vortex dynamics.