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

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On a Scale Invariant Model of Statistical Mechanics, Kinetic Theory of Ideal Gas, and Riemann Hypothesis International Journal of Modern Communication Technologies & Research (IJMCTR) ISSN: 2321-0850, Volume-3, Issue-6, June 2015 On a Scale Invariant Model of Statistical Mechanics, Kinetic Theory of Ideal Gas, and Riemann Hypothesis Siavash H. Sohrab Abstract— A scale invariant model of statistical mechanics is I. INTRODUCTION applied to derive invariant forms of conservation equations. A It is well known that the methods of statistical mechanics modified form of Cauchy stress tensor for fluid is presented that leads to modified Stokes assumption thus a finite coefficient of can be applied to describe physical phenomena over a broad bulk viscosity. The phenomenon of Brownian motion is range of scales of space and time from the exceedingly large described as the state of equilibrium between suspended scale of cosmology to the minute scale of quantum optics as particles and molecular clusters that themselves possess schematically shown in Fig. 1. All that is needed is that the Brownian motion. Physical space or Casimir vacuum is system should contain a large number of weakly coupled identified as a tachyonic fluid that is “stochastic ether” of Dirac particles. The similarities between stochastic quantum fields or “hidden thermostat” of de Broglie, and is compressible in [1-17] and classical hydrodynamic fields [18-29] resulted in accordance with Planck’s compressible ether. The stochastic recent introduction of a scale-invariant model of statistical definitions of Planck h and Boltzmann k constants are shown to mechanics [30] and its application to thermodynamics [31, respectively relate to the spatial and the temporal aspects of vacuum fluctuations. Hence, a modified definition of 32], fluid mechanics [33], and quantum mechanics [34]. thermodynamic temperature is introduced that leads to predicted velocity of sound in agreement with observations. UNIVERSE GALACTIC CLUSTER LGD (J + 11/2) Also, a modified value of Joule-Mayer mechanical equivalent of EGD (J +5) GALACTIC CLUSTER heat is identified as the universal gas constant and is called De Lg Pretto number 8338 which occurred in his mass-energy GALAXY Lp LPD (J + 9/2) EPD (J+4 ) GALAXY equivalence equation. Applying Boltzmann’s combinatoric Lp PLANET LHD (J + 7/2) methods, invariant forms of Boltzmann, Planck, and Lh EHD (J +3) PLANET Maxwell-Boltzmann distribution functions for equilibrium Lh HYDRO-SYSTEM LFD (J + 5/2) statistical fields including that of isotropic stationary turbulence Lf EFD (J +2) are derived. The latter is shown to lead to the definitions of Lf HYDRO-SYSTEM FLUID ELEMENT LED (J + 3/2) (electron, photon, neutrino) as the most-probable equilibrium Le EED (J+1) sizes of (photon, neutrino, tachyon) clusters, respectively. The Le FLUID ELEMENT physical basis for the coincidence of normalized spacing between EDDY Lc LCD (J + 1/2) ECD (J) zeros of Riemann zeta function and the normalized Lc EDDY Maxwell-Boltzmann distribution and its connections to Riemann CLUSTER Lm LMD (J - 1/2) Hypothesis are examined. The zeros of Riemann zeta function EMD (J -1) Lm CLUSTER are related to the zeros of particle velocities or “stationary MOLECULE La LAD (J - 3/2) states” through Euler’s golden key thus providing a physical EAD (J -2) La MOLECULE explanation for the location of the critical line. It is argued that ATOM Ls LSD (J - 5/2) because the energy spectrum of Casimir vacuum will be ESD (J -3) Ls ATOM governed by Schrödinger equation of quantum mechanics, in SUB-PARTICLE Lk LKD (J - 7/2) EKD (J-4 ) L view of Heisenberg matrix mechanics physical space should be k SUB-PARTICLE PHOTON Lt LTD (J - 9/2) described by noncommutative spectral geometry of Connes. ETD (J -5) Lt Invariant forms of transport coefficients suggesting finite values TACHYON PHOTON 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 Fig. 1 Scale-invariant model of statistical mechanics. irreversibility and Poincaré recurrence theorem are addressed. Equilibrium--Dynamics on the left-hand-side and Invariant modified form of the first law of thermodynamics is non-equilibrium Laminar--Dynamics on the derived and a modified definition of entropy is introduced that right-hand-side for scales = g, p, h, f, e, c, m, a, s, k, and closes the gap between radiation and gas theory. Finally, new paradigms for hydrodynamic foundations of both Schrödinger t as defined in Section 2. Characteristic lengths of (system, as well as Dirac wave equations and transitions between Bohr element, “atom”) are (L ) and is the stationary states in quantum mechanics are discussed. β mean-free-path [32]. Index Terms— Kinetic theory of ideal gas; Thermodynamics; Statistical mechanics; Riemann hypothesis; TOE. In the present study [35] the invariant model of statistical mechanics and its implications to the physical foundations of thermodynamics, kinetic theory of ideal gas [36-42], and Manuscript received June 22, 2015. quantum mechanics are further examined. Whereas the Siavash H. Sohrab, Department of Mechanical Engineering, Robert McCormick School of Engineering and Applied Science, Northwestern outline of the main ideas are described in this introduction, University, Evanston, Illinois 60208-3111 ([email protected]). 7 www.erpublication.org On a Scale Invariant Model of Statistical Mechanics, Kinetic Theory of Ideal Gas, and Riemann Hypothesis references to most of the specific literature will be presented method to the foundation of quantum mechanics is discussed. in the corresponding Sections. It is suggested that at a given temperature the After a brief introduction of a scale invariant model of Maxwell-Boltzmann distribution function could be viewed as statistical mechanics the invariant definitions of density, spectrum of stochastically stationary sizes of particle clusters. ―atomic‖, element and system velocities are presented in Sec. Since according to the scale invariant model of statistical 2. The invariant forms of conservation equations for mass, mechanics the ―atom‖ of statistical field at scale is energy, linear and angular momentum based on linearization identified as the most-probable cluster size of the lower scale of Boltzmann equation and in harmony with Enskog [43] (Fig. 13), the definitions of (electron, photon, neutrino) methods are described in Sec. 3. Because by definition fluids are introduced as the most-probable equilibrium sizes of can only support normal stresses, following Cauchy a (photon, neutrino, tachyon) clusters. Also, definitions of both modified form of the stress tensor for fluids is introduced that dark energy (electromagnetic mass) and dark matter leads to modified Stokes assumption thus a finite value of (gravitational mass) are introduced. bulk viscosity such that in the limit of vanishing interatomic Next Maxwell-Boltzmann speed distribution is directly spacing all tangential stresses in the fluid vanish in derived from invariant Planck energy distribution in Sec. 7. accordance with the perceptions of Cauchy and Poisson. In Hence, at thermodynamic equilibrium particles of statistical addition, the concept of absolute enthalpy and iso-spin are fields of all scales (Fig. 1) will have Gaussian velocity introduced and incorporated in the derivation of scale distribution, Planck energy distribution, and invariant forms of energy and angular momentum Maxwell-Boltzmann speed distribution. conservation equations following the classical method of Montgomery-Odlyzko law of correspondence between summational invariants [33]. The nature of the equation of distribution of normalized spacing of zeros of Riemann zeta motion for (a) equilibrium flow in absence of iso-spin (b) function and those of eigenvalues of Gaussian unitary laminar potential flow (c) viscous flow in presence of spin are ensemble (GUE) is shown to extend to normalized identified. Maxwell-Boltzmann distribution function in Sec. 8. Thus, a In Sec. 4 hierarchies of embedded statistical fields from connection is established between analytic number theory on Planck to cosmic scales are described. It is shown that the the one hand and the kinetic theory of ideal gas on the other scale factor of 1017 appears to separate the equilibrium hand. The spacing between energy levels are then related to statistical fields of chromodynamics (Planck scale), frequency spacing through Planck formula for quantum of electrodynamics, hydrodynamics, planetary-dynamics energy = h. Next, the frequencies of Heisenberg-Kramers (astrophysics), and galactic-dynamics (cosmology). The virtual oscillators are taken as powers of prime numbers and phenomenon of Brownian motion is described in terms of the expressed in terms of Gauss’s clock calculators or Hensel’s statistical field of equilibrium cluster-dynamics ECD. The p-adic numbers. Finally, the spacing between zeros of stochasticity of cascade of statistical fields is found to particle velocities are related to zeros of Riemann zeta continue to Casimir vacuum that is identified as a tachyonic function through Euler’s golden key. In addition, it is argued fluid that is Dirac stochastic ether or de Broglie hidden that since physical space or Casimir vacuum is identified as a thermostat and is considered to be compressible in tachyonic quantum fluid with energy spectra given by accordance with compressible ether of Planck. Stochastic Schrödinger equation and hence Heisenberg matrix definitions of Planck h and Boltzmann k constants are mechanics, it should be described by noncommutative
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