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

Sohrab, Siavash H.

doi:10.2514/6.2012-467

Cited by 5 publications

(34 citation statements)

References 53 publications

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“…It has been mentioned that there is a fundamental problem to connect the reversible physical description of a microscopic system with its macroscopic analog usually described by statistical mechanics, because it appears to be difficult or impossible to connect a reversible description with an irreversible one [2,12]. In the approach presented, this problem is still puzzling as demonstrated with the example above on the expansion of the idea gas and as discussed in the following.…”

Section: The Problem Of Time-irreversibility In Time Reversible Descrmentioning

confidence: 90%

“…By using in the definition the word "microscopic", we follow a suggestion by Prigogine [12] because S n is a non ensemble-averaged term. Note, since we are in a time reversible description, the microscopic entropy is in general of reversible character and thus not a monotonously increasing quantity (see below and Section 2.11).…”

Section: The Microscopic Entropy Of a Single Particlementioning

confidence: 99%

“…The latter argument brings the term entropy S back to its roots, where Clausius tried to design and understand "heat engines", which are cyclic machines for the conversion of heat Q into useful work, and found when averaged over many cycles that for an irreversible machine ΔQ T = ΔS > 0 [8]. Although the statistical mechanics argument of entropy increase with time by describing a system tendency towards its most probable state, which is the equilibrium state, is sound, Loschmidt and Zermelo's reversal and recurrence objections remain as powerful as ever [9][10][11][12]. Both objections have been extensively discussed and are just mentioned here in short.…”

Section: Introductionmentioning

confidence: 99%

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A Derivation of a Microscopic Entropy and Time Irreversibility From the Discreteness of Time

Riek

2014

Entropy

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The basic microsopic physical laws are time reversible. In contrast, the second law of thermodynamics, which is a macroscopic physical representation of the world, is able to describe irreversible processes in an isolated system through the change of entropy S > 0. It is the attempt of the present manuscript to bridge the microscopic physical world with its macrosocpic one with an alternative approach than the statistical mechanics theory of Gibbs and Boltzmann. It is proposed that time is discrete with constant step size. Its consequence is the presence of time irreversibility at the microscopic level if the present force is of complex nature (F (r) = const). In order to compare this discrete time irreversible mechamics (for simplicity a "classical", single particle in a one dimensional space is selected) with its classical Newton analog, time reversibility is reintroduced by scaling the time steps for any given time step n by the variable s n leading to the Nosé-Hoover Lagrangian. The corresponding Nosé-Hoover Hamiltonian comprises a term N df k B T ln s n (k B the Boltzmann constant, T the temperature, and N df the number of degrees of freedom) which is defined as the microscopic entropyS n at time point n multiplied by T . Upon ensemble averaging this microscopic entropy S n in equilibrium for a system which does not have fast changing forces approximates its macroscopic counterpart known from thermodynamics. The presented derivation with the resulting analogy between the ensemble averaged microscopic entropy and its thermodynamic analog suggests that the original description of the entropy by Boltzmann and Gibbs is just an ensemble averaging of the time scaling variable s n which is in equilibrium close to 1, but that the entropy term itself has its root not in statistical mechanics but rather in the discreteness of time.

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Section: The Problem Of Time-irreversibility In Time Reversible Descrmentioning

confidence: 90%

Section: The Microscopic Entropy Of a Single Particlementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A Derivation of a Microscopic Entropy and Time Irreversibility From the Discreteness of Time

Riek

2014

Entropy

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“…Each element is composed of an ensemble of small particles called the "atoms" of the field that are governed by distribution function mp, tp) and viewed ' as point-mass. The most probable element (system) velocity of the smaller scale j becomes the velocity of the atom (element) of the larger scale j + 1 [52]. Since invariant Schrodinger equation was recently derived from invariant Bernoulli equation [51], the entire hierarchy of statistical fields shown in Fig.…”

Section: A Scale Invariant Model Of Statistical Mechanicsmentioning

confidence: 99%

“…Similarities between such statistical fields shown in Fig. 1 resulted in recent introduction of a scale-invariant model of statistical mechanics [44] and its application to the fields of thermodynamics [45][46][47], fluid mechanics [48,49], and quantum mechanics [50][51][52].…”

Section: Introductionmentioning

confidence: 99%

Invariant Forms of Conservation Equations for Reactive Fields and Hydro-Thermo-Diffusive Theory of Laminar Flames

Sohrab

2014

Journal of Energy Resources Technology

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A scale-invariant model of statistical mechanics is described leading to invariant Enskog equation o f change that is applied to derive invariant forms of conservation equations for mass, thermal energy, linear momentum, and angular momentum in chemically reactive fields. Modified hydro-thermo-diffusive theories of laminar premixed flames for (1) rigidbody and (2) Brownian-motion flame propagation models are presented and are shown to be mathematically equivalent. The predicted temperature profile, thermal thickness, and propagation speed of laminar methane-air premixed flame are found to be in good agree ment with existing experimental observations.

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Invariant Forms of Conservation Equations and Some Examples of Their Exact Solutions

Sohrab

2014

Journal of Energy Resources Technology

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A scale-invariant model o f statistical mechanics is described leading to invariant Boltz mann equation and the corresponding invariant Enskog equation o f change. A modified form o f Cauchy stress tensor fo r fluid is presented such that in the limit o f vanishing intermolecular spacing, all tangential forces vanish in accordance with perceptions o f Cauchy and Poisson. The invariant form s o f mass, thermal energy, linear momentum, and angu lar momentum conservation equations derived from invariant Enskog equation o f change are described. Also, some exact solutions o f the conservation equations fo r the problems o f normal shock, laminar, and turbulent flow over aflat plate, and flow within a single or multiple concentric spherical liquid droplets made o f immiscible fluids located at the stagnation point o f opposed cylindrically symmetric gaseous finite jets are presented. Jo u rn a l o f E n erg y R eso u rces T e c h n o lo g y C o p y rig h t © 2014 by A S M E SEPTEMBER 2014, Vol. 136 / 032002-1 GALACTIC CLUSTER EGD (J +5) GALAXY EPD (J+4) PLANET EHD (J +3) HYDRO-SYSTEM EFD (J +2) FLUID ELEMENT EED (J+l) EDDY ECD (J) CLUSTER EMD (J -1) MOLECULE EAD (J -2) ATOM ESD (J -3) SUB-PARTICLE EKD (J-4) PHOTON ETD (J -5) TACHYON UNIVERSE LGD (J +11/2) GALACTIC CLUSTER LPD (J + 9/2) GALAXY LHD (J + 7/2) PLANET LFD (J + 5/2) HYDRO-SYSTEM LED (J + 3/2) FLUID ELEMENT LCD (J +1/2) EDDY LMD (J -1/2) CLUSTER LAD (J -3/2) MOLECULE LSD (J -5/2) ATOM LKD (J -7/2) SUB-PARTICLE LTD (J -9/2) PHOTON F ig . 1 A s c a l e -i n v a r i a n t m o d e l o f s t a t i s t i c a l m e c h a n i c s . E q u i l i b r i u m -/ i -d y n a m i c s o n t h e l e f t -h a n d s i d e a n d n o n e q u i i i b r i u m l a m i n a r -/ i -d y n a m i c s o n t h e r i g h t -h a n d s i d e f o rNomenclature cp = specific heat at constant pressure (J/kg K) D = diffusion coefficient (m2/s)

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

Cited by 5 publications

References 53 publications

A Derivation of a Microscopic Entropy and Time Irreversibility From the Discreteness of Time

A Derivation of a Microscopic Entropy and Time Irreversibility From the Discreteness of Time

Invariant Forms of Conservation Equations for Reactive Fields and Hydro-Thermo-Diffusive Theory of Laminar Flames

Invariant Forms of Conservation Equations and Some Examples of Their Exact Solutions

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