The invariant Planck energy distribution law and its connection to the Maxwell-Boltzmann distribution function

A scale-invariant model of statistical mechanics is described and applied to derive the invariant Planck law of energy distribution from the invariant Boltzmann distribution function. Also, the invariant Schrödinger equation is derived from the invariant Bernoulli equation for potential incompressible flow. It is shown that a homogeneous isotropic turbulent fluid is composed of a spectrum of eddies (energy levels) each composed of a spectrum of molecular clusters with energy spectra governed by the Planck distribution law and harmonious with the Kolmogorov κ^-5/3 law. The stability of clusters, de Broglie wave packets, is shown to be due to a potential that acts as Poincaré stress. Atomic transitions between different size clusters (energy levels) are shown to result in emission/ absorption of energy in harmony with Bohr's theory of atomic spectra.