Implications of a scale invariant model of statistical mechanics to nonstandard analysis and the wave equation (original) (raw)
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Some implications of a scale invariant model of statistical mechanics to turbulent combustion
2009
Some implications of a scale invariant model of statistical mechanics to transport phenomena in general and the iso-spin diffusion in connection to the Onsager's reciprocity principle and the modified Helmholtz vorticity equation are discussed. The invariant forms of mass, energy, linear momentum, and angular momentum conservation equations are derived from an invariant Enskog equation of change. A scale invariant definition of Reynolds number is introduced as Re β = (L xβ w xβ)/(λ xβ−1 v xβ−1) and applied to describe the phenomena of super-fluidity, BEC, superconductivity, and super-luminosity (laser action) as transitions from turbulent (strongly dissipative) to laminar (weakly dissipative) flows at molecular-, atomic-, electro-, and chromo-dynamic scales, respectively. Also, a scale-invariant model of "particle" wave amplification by simulated emission of particles parallel to laser phenomenon is introduced. Finally, a model of Faraday line of force as strings of electrons and positrons is described that is in harmony with the perceptions of Dirac.
Some implications of a scale invariant model of statistical mechanics to transport phenomena
International Conference on Systems, 2009
Some implications of a scale invariant model of statistical mechanics to transport phenomena in general and the iso-spin diffusion in connection to the Onsager's reciprocity principle and the modified Helmholtz vorticity equation are discussed. The invariant forms of mass, energy, linear momentum, and angular momentum conservation equations are derived from an invariant Enskog equation of change. A scale invariant definition of Reynolds number is introduced as Re β = (L xβ w xβ)/(λ xβ−1 v xβ−1) and applied to describe the phenomena of super-fluidity, BEC, superconductivity, and super-luminosity (laser action) as transitions from turbulent (strongly dissipative) to laminar (weakly dissipative) flows at molecular-, atomic-, electro-, and chromo-dynamic scales, respectively. Also, a scale-invariant model of "particle" wave amplification by simulated emission of particles parallel to laser phenomenon is introduced. Finally, a model of Faraday line of force as strings of electrons and positrons is described that is in harmony with the perceptions of Dirac.
International Journal of Thermodynamics, 2014
Some implications of a scale invariant model of statistical mechanics to the mechanical theory of heat of Helmholtz and Clausius are described. Modified invariant definitions of heat and entropy are presented closing the gap between radiation and gas theory. Modified relativistic transformations of pressure, Boltzmann constant, entropy, and density are introduced leading to transformation of ideal gas law. Following Helmholtz, the total thermal energy of thermodynamic system is decomposed into free heat U and latent heat p V and identified as modified form of the first law of thermodynamics Q = H = U + p V. Subjective versus objective aspects of Boltzmann thermodynamic entropy versus Shannon information entropy are discussed. Also, modified thermodynamic properties of ideal gas are presented. The relativistic thermodynamics being described is in accordance with Poincaré-Lorentz dynamic theory of relativity as opposed to Einstein kinematic theory of relativity since the former theory that is based on compressible ether of Planck is causal as was emphasized by Pauli.
International Journal of Thermal Sciences, 1999
A scale-invariant statistical theory of fields is presented that leads to invariant definition of density, velocity, temperature, and pressure. The definition of Boltzmann constant is introduced as kk-k = rakV'k C = 1.381"10-23 J-K-1, suggesting that the Kelvin absolute temperature scale is equivalent to a length scale. Two new state variables called the reversible heat Qrev = TS and the reversible work Wrev-PV are introduced. The modified forms of the first and second law of thermodynamics are presented. The microscopic definition of heat (work) is presented as the kinetic energy due to the random (peculiar) translational, rotational, and pulsational motions. The Gibbs free energy of an element at scale/3 is identified as the total system energy at scale (/3-1), thus leading to an invariant form of the first law of thermodynamics Uf~ = Q~-W~-I-Ne~ U~-I. ~)1999 [~ditions scientifiques et mfidicales Elsevier SAS. fundamental thermodynamics / first and second laws / invariant form / statistical thermodynamics / thermophysical characteristics
Universality of a scale invariant model of turbulence and its quantum mechanical foundation
2009
A modified statistical theory of turbulence based on a scale invariant model of statistical mechanics is described. Hierarchies of statistical fields from cosmic to Planck scale are examined. The predicted velocity profiles for turbulent boundary layer over a flat plate at four consecutive scales of LED, LCD, LMD, and LAD are shown to be in close agreement with the experimental observations. The generalized definitions of "solid", "liquid", and "gas" phases are introduced and a modified quantum mechanical criterion for transition to turbulence based on Poincaré stress is presented.
Quantum theory of fields from Planck to cosmic scales
WSEAS Transactions on Mathematics archive, 2010
A scale invariant model of statistical mechanics is applied to describe a modified statistical theory of turbulence and its quantum mechanical foundations. Hierarchies of statistical fields from cosmic to Planck scales are described. Energy spectrum of equilibrium isotropic turbulence is shown to follow Planck law. Predicted velocity profiles of turbulent boundary layer over a flat plate at four consecutive scales of LED, LCD, LMD, and LAD are shown to be in close agreement with the experimental observations in the literature. The physical and quantum nature of time is described and a scale-invariant definition of time is presented and its relativistic behavior is examined. New paradigms for physical foundations of quantum mechanics as well as derivation of Dirac relativistic wave equation are introduced.
Scale-Invariant Model of Boltzmann Statistical Mechanics and Generalized Thermodynamics
Entropie : thermodynamique – énergie – environnement – économie, 2021
Some implications of a scale-invariant model of Boltzmann statistical mechanics to the laws of generalized thermodynamics are investigated. Through definition of stochastic Planck and Boltzmann universal constants, dimension of Kelvin absolute temperature T (degrees kelvin) is identified as a length (meters) associated with Wien wavelength w T = of particle thermal oscillations. Hence, thermodynamic temperature and atomic mass of the field at scale provide internal measures of (extension, duration) of background space 1 needed to define external space and time coordinates and atomic-mass-unit of 1 . Introduction of invariant internal thermodynamic spacetime and Boltzmann factor are in harmony with modern concepts of quantum gravity as deterministic dissipative dynamic system [73]. The connections between de Pretto number 8338 and Joule-Mayer mechanical equivalent of heat c = 4.169 kJ / kcal J
Continuum versus quantum fields viewed through a scale invariant model of statistical mechanics
2010
The implications of a scale invariant model of statistical mechanics to the physical foundations of quantum mechanics and continuum mechanics are described. The nature of the connections between discrete versus continuum fields will be examined. Also, some of the implications of a scale-invariant model of statistical mechanics to the physical foundation of analysis, the continuum hypothesis, and the prime number theory will be addressed.
Chaos, Solitons & Fractals, 1999
Fractal structures are observed in the universe in two very different ways. Firstly, in the gas forming the cold interstellar medium in scales from 10 −4 pc till 100pc. Secondly, the galaxy distribution has been observed to be fractal in scales up to hundreds of Mpc. We give here a short review of the statistical mechanical (and field theoretical) approach developed by us for the cold interstellar medium (ISM) and large structure of the universe. We consider a non-relativistic self-gravitating gas in thermal equilibrium at temperature T inside a volume V. The statistical mechanics of such system has special features and, as is known, the thermodynamical limit does not exist in its customary form. Moreover, the treatments through microcanonical, canonical and grand canonical ensembles yield different results. We present here for the first time the equation of state for the self-gravitating gas in the canonical ensemble. We find that it has the form p = [N T /V ]f (η), where p is the pressure, N is the number of particles and η ≡ G m 2 N V 1/3 T. The N → ∞ and V → ∞ limit exists keeping η fixed. We compute the function f (η) using Monte Carlo simulations and for small η, analytically. We compute the thermodynamic quantities of the system as free energy, entropy, chemical potential, specific heat, compressibility and speed of sound. We reproduce the well-known gravitational phase transition associated to the Jeans' instability. Namely, a gaseous phase for η < η c and a condensed phase for η > η c. Moreover, we derive the precise behaviour of the physical quantities near the transition. In particular, the pressure vanishes as p ∼ (η c − η) B with B ∼ 0.2 and η c ∼ 1.6 and the energy fluctuations diverge as ∼ (η c − η) B−1. The speed of sound decreases monotonically with η and approaches the value T /6 at the transition.
Turbulence and scale relativity
Physics of Fluids
We develop a new formalism for the study of turbulence using the scale relativity framework (applied in v-space, following de Montera's proposal). We first review some of the various ingredients which are at the heart of the scale relativity approach (scale dependence and fractality, chaotic paths, irreversibility) and recall that they indeed characterize fully developed turbulent flows. Then, we show that, in this framework, the time derivative of the Navier-Stokes equation can be transformed into a macroscopic Schrödinger-like equation. The local velocity Probability Distribution Function (PDF), Pv(v), is given by the squared modulus of a solution of this equation. This implies the presence of null minima Pv(vi) ≈ 0 in this PDF. We also predict a new acceleration component, Aq(v) = ±Dv ∂v ln Pv, which is divergent in these minima. Then, we check these theoretical predictions by data analysis of available turbulence experiments: (1) Empty zones are in effect detected in observed Lagrangian velocity PDFs. (2) A direct proof of the existence of the new acceleration component is obtained by identifying it in the data of a laboratory turbulence experiment. (3) It precisely accounts for the intermittent bursts of the acceleration observed in experiments, separated by calm zones which correspond to Aq ≈ 0 and are shown to remain perfectly Gaussian. (4) Moreover, the shape of the acceleration PDF can be analytically predicted from Aq, and this theoretical PDF precisely fits the experimental data, including the large tails. (5) Finally, numerical simulations of this new process allow us to recover the observed autocorrelation functions of acceleration magnitude and the exponents of structure functions.