Universality of Quantum Mechanics and a Scale Invariant Statistical Theory of Turbulence

Application of scale-invaraint Boltzmann statistical mechanics to cosmology (10³⁵), astrophysics (10¹⁸), hydrodynamics (10⁰), electrodynamics (10^-18), and chromodynamics (10^-35), leads to a universal statistical theory of turbulence [1]. According to the classical theories, turbulence results either from onset of fluid instabilities as Reynolds number exceeds certain crirical value or onset of chaos following perceptions of Poincaré. In the present study, following Heisenberg analogy between problems of turbulence and kinetic theory of gas, a modified statistical theory of turbulence is presented with spectral energy of eddies ε_iβ = hν_iβ following invariant Planck energy distribution function in accordance with observations of Van Atta and Chen [1]. Therefore, the gap between quantum mechanics and turbulence is closed through derivation of invariant Schrödinger equation from invariant Bernoulli equations for incompressible potential flow [1]. Analytic solutions of modified equation of motion for the problems of turbulent flow over flat plate at eddy-, cluster-, molecular-, and atomic-dynamics (EED, ECD, EMD, EAD) scales covering spatial range of 10⁸ are compared with experimental data available in the literature. Thus, the invariant Reynolds number Re_β = (λ_xβv_xβ)/(λ_xβ-1v_xβ-1) governs transitions from highly-dissipative (turbulent or normal) to weakly-dissipative (laminar or super) flow at hydro-, molecular-, electro-, and chromo-dynamics, … scales in harmony with pioneering observations of superconductivity by Onnes [1]. Finally, invariant Dirac relativistic wave equation is derived from modified invariant Navier-Stokes equation of motion in harmony with quantum gravity as a dissipative deterministic dynamic system [2].

[1] Sohrab, S. H., Examples of Applications of an Invariant Statistical Theory of Field to Cosmology, Astrophysics, Hydrodynamics, Electrodynamics, and Photonics, C.H. Skiadas and Y.Dimotikalis (eds.), 15th Chaotic Modeling and Simulation International Conference, Springer Proceedings in Complexity, 2023.



[2] ‘t Hooft, G., Quantum Gravity as a Dissipative deterministic system, Class. Quantum Grav. 16, 3263 (1999).