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

摘要

A scale invariant model of statistical mechanics is applied to derive invariant forms of conservation equations. A modified form of Cauchy stress tensor for fluid is presented that leads to modified Stokes assumption thus a finite coefficient of bulk viscosity. The phenomenon of Brownian motion is described as the state of equilibrium between suspended particles and molecular clusters that themselves possess Brownian motion. Physical space or Casimir vacuum is identified as a tachyonic fluid that is "stochastic ether" of Dirac or "hidden thermostat" of de Broglie, and is compressible in accordance with Planck's compressible ether. The stochastic definitions of Planck h and Boltzmann k constants are shown to respectively relate to the spatial and the temporal aspects of vacuum fluctuations. Hence, a modified definition of thermodynamic temperature is introduced that leads to predicted velocity of sound in agreement with observations. Also, a modified value of Joule-Mayer mechanical equivalent of heat is identified as the universal gas constant and is called De Pretto number 8338 which occurred in his mass-energy equivalence equation. Applying Boltzmann 's combinatoric methods, invariant forms of Boltzmann, Planck, and Maxwell-Boltzmann distribution functions for equilibrium statistical fields including that of isotropic stationary turbulence are derived. The latter is shown to lead to the definitions of (electron, photon, neutrino) as the most-probable equilibrium sizes of (photon, neutrino, tachyon) clusters, respectively. The physical basis for the coincidence of normalized spacings between zeros of Riemann zeta function and the normalized Maxwell-Boltzmann distribution and its connections to Riemann Hypothesis are examined. The zeros of Riemann zeta function are related to the zeros of particle velocities or "stationary states" through Euler's golden key thus providing a physical explanation for the location of the critical line. It is argued that because the energy spectrum of Casimir vacuum will be governed by Schroedinger equation of quantum mechanics, in view of Heisenberg matrix mechanics physical space should be described by noncommutative spectral geometry of Connes. Invariant forms of transport coefficients suggesting finite values of gravitational viscosity as well as hierarchies of vacua and absolute zero temperatures are described. Some of the implications of the results to the problem of thermodynamic irreversibility and Poincare recurrence theorem are addressed. Invariant modified form of the first law of thermodynamics is derived and a modified definition of entropy is introduced that closes the gap between radiation and gas theory. Finally, new paradigms for hydrodynamic foundations of both Schrodinger as well as Dirac wave equations and transitions between Bohr stationary states in quantum mechanics are discussed.