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

To derive invariant forms of conservation equations, a scale invariant statistical mechanics model is used. A modified form of the Cauchy stress tensor for fluid is presented, which leads to a modified Stokes assumption and thus a finite bulk viscosity coefficient. Brownian motion is defined as the state of equilibrium between suspended particles and molecular clusters that also have Brownian motion. Physical space, also known as the Casimir vacuum, is a tachyonic fluid that is Dirac's "stochastic ether" or de Broglie's "hidden thermostat," and it is compressible according to Planck's compressible ether. The stochastic definitions of the Planck h and Boltzmann k constants are shown to be related to the spatial and temporal aspects of vacuum fluctuations, respectively. As a result, a modified definition of thermodynamic temperature is introduced, resulting in predicted sound velocity that agrees with observations. Boltzmann combinatoric method was employed to derive invariant forms of Planck energy and Maxwell-Boltzmann speed distribution functions. In addition, the universal gas constant is identified as a modified value of the Joule-Mayer mechanical equivalent of heat known as De Pretto number 8338, which appeared in his mass-energy equivalence equation. Invariant versions of Boltzmann, Planck, and Maxwell-Boltzmann distribution functions for equilibrium statistical fields, including those of isotropic stationary turbulence, are determined using Boltzmann's combinatoric methods. The latter leads to the definitions of (electron, photon, neutrino) as the most likely equilibrium sizes of (photon, neutrino, tachyon) clusters, respectively. The physical basis for the coincidence of the Riemann zeta function's normalized spacing between zeros and the normalized Maxwell-Boltzmann distribution, as well as its connections to the Riemann hypothesis are investigated. Through Euler's golden key, the zeros of the Riemann zeta function are related to the zeros of particle velocities or "stationary states," providing a physical explanation for the location of the critical line. It is proposed that, because the energy spectrum of the Casimir vacuum will be determined by the Schrödinger equation of quantum mechanics, physical space should be characterised by noncommutative spectral geometry of Connes in light of Heisenberg matrix mechanics. Invariant forms of transport coefficients implying finite values of gravitational viscosity, as well as hierarchies of vacua and absolute zero temperatures, are described. Some of the implications of the results for the problem of thermodynamic irreversibility and the Poincaré recurrence theorem are discussed. An invariant modified form of the first law of thermodynamics is obtained, as well as a modified definition of entropy, which closes the gap between radiation and gas theory. Finally, in quantum mechanics, new paradigms for the hydrodynamic foundations of both Schrödinger and Dirac wave equations, as well as transitions between Bohr stationary states, are examined.