# Some Implications of a Scale Invariant Model of Statistical Mechanics, Kinetic Theory of Ideal Gas, and Riemann Hypothesis

A scale invariant statistical mechanics model is used to construct invariant versions of conservation equations. The authors provide a modified form of the Cauchy stress tensor for fluid, which leads to a modified Stokes assumption and hence a finite bulk viscosity coefficient. Brownian motion is defined as the state of equilibrium between suspended particles and Brownian-moving molecule clusters. The Casimir vacuum, also known as Dirac’s “stochastic ether” or de Broglie’s “hidden thermostat,” is a tachyonic fluid that is compressible according to Planck’s compressible ether. The Planck h and Boltzmann k constants’ stochastic definitions are demonstrated to be related to the spatial and temporal features of vacuum fluctuations, respectively. As a result, a new thermodynamic temperature definition is provided, resulting in anticipated sound velocity that matches observations. To obtain invariant forms of Planck energy and Maxwell-Boltzmann speed distribution functions, the Boltzmann combinatoric approach was used. The universal gas constant is also identified as a modified value of the Joule-Mayer mechanical equivalent of heat, known as De Pretto number 8338, which appears in his mass-energy equivalence equation. Boltzmann’s combinatoric methods are used to determine invariant versions of Boltzmann, Planck, and Maxwell-Boltzmann distribution functions for equilibrium statistical fields, including those of isotropic stationary turbulence. As a result, (electron, photon, and neutrino) are defined as the most likely equilibrium sizes of (photon, neutrino, and tachyon) clusters, respectively. The physical basis for the normalised spacing between zeros of the Riemann zeta function and the normalised Maxwell-Boltzmann distribution is studied, as well as its ties to the Riemann hypothesis. The zeros of the Riemann zeta function are connected to the zeros of particle velocities or “stationary states” using Euler’s golden key, offering a physical explanation for the crucial line’s location. Because the Schrödinger equation of quantum mechanics will determine the energy spectrum of the Casimir vacuum, it is proposed that physical space be defined by noncommutative spectral geometry of Connes in light of Heisenberg matrix mechanics. The hierarchies of vacua and absolute zero temperatures, as well as invariant forms of transport coefficients assuming finite values of gravitational viscosity, are discussed. The results are examined in terms of their implications for the problem of thermodynamic irreversibility and the Poincaré recurrence theorem. The first law of thermodynamics is invariantly modified, as is a modified definition of entropy, which bridges the gap between radiation and gas theory. Finally, new paradigms for the hydrodynamic foundations of both Schrödinger and Dirac wave equations, as well as transitions between Bohr stationary states, are investigated in quantum mechanics.**Author (S) DetailsSiavash H. Sohrab**Department of Mechanical Engineering, Robert McCormick School of Engineering and Applied Science, Northwestern University, Evanston, 60208-3111, Illinois, U.S.A.

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