Bridging the Gap Between Stationary Homogeneous Isotropic Turbulence and Quantum Mechanics

A statistical theory of stationary isotropic turbulence $^{1}$ is presented with eddies possessing Gaussian velocity distribution, Maxwell-Boltzmann speed distribution in harmony with perceptions of Heisenberg $^{2}$, and Planck energy distribution in harmony with perceptions of Chandrasekhar$^{3}$ and in agreement with experimental observations of Van Atta and Chen (\textit{J. Fluid Mech. }34 (3) \quad 497-515, 1968). Defining the action $S=-m\Phi $ in terms of velocity potential of atomic motion, scale-invariant Schr\"{o}dinger equation is derived$^{1\, }$from invariant Bernoulli equation. Thus, the gap between the problems of turbulence and quantum mechanics is closed through connections between Cauchy-Euler-Bernoulli equations of hydrodynamics, Hamilton-Jacobi equation of classical mechanics, and finally Schr\"{o}dinger equation of quantum mechanics. Transitions of particle (molecular cluster c$_{ji})$ from a small rapidly-oscillating eddy e$_{j}$ (high-energy level-j) to a large slowly-oscillating eddy e$_{i}$ (low energy-level-i) leads to emission of a sub-particle (molecule m$_{ji})$ that carries away the excess energy $\varepsilon_{ji} =h(\nu_{j} -\nu_{i} )$ in harmony with Bohr theory of atomic spectra. $\backslash \backslash $ $^{1}$ Sohrab, S. H.,\textit{ Chaotic Modeling and Simulation} (CMSIM) \textbf{3}, 231-245\textbf{ }(2016). $^{2}$ Heisenberg, W., \textit{Proc. Roy. Soc. }A \quad \textbf{159}, 402-406 (1948). $^{3}$ Chandrasekhar, S., \textit{Stochastic, Statistical, and Hydrodynamic Problems in Physics and Astronomy}, Selected Papers, vol.3, University of Chicago Press, Chicago, 515-528, 1989.