Showing papers on "Boltzmann constant published in 2012"

Energy flow in non-equilibrium conformal field theory

Abstract: We study the energy current and its fluctuations in quantum gapless 1d systems far from equilibrium modeled by conformal field theory, where two separated halves are prepared at distinct temperatures and glued together at a point contact. We prove that these systems converge towards steady states, and give a general description of such non-equilibrium steady states in terms of quantum field theory data. We compute the large deviation function, also called the full counting statistics, of energy transfer through the contact. These are universal and satisfy fluctuation relations. We provide a simple representation of these quantum fluctuations in terms of classical Poisson processes whose intensities are proportional to Boltzmann weights.

Semiclassical model of the structure of matter

Abstract: The modern semiclassical method developed over the past few decades and used for describing the properties of the electronic subsystems of matter is reviewed, and its application to quantum physics problems is illustrated. The method involves the Thomas–Fermi statistical model and allows an extension by including additive corrections that take the shell structure of the electronic spectrum and other physical effects into account. Applying the method to the study of matter and finite systems allowed the following, inter alia: (1) an analysis of the total electron energy oscillations as a function of the number of particles in a 1D quantum dot; (2) a description of spatial oscillations of the electron density in atoms and atomic clusters; (3) a description of the stepwise temperature dependence of the ionicity and ionization energy in a Boltzmann plasma; (4) an evaluation of free ion ionization potentials; (5) an interpretation and evaluation of the difference in the patterns of oscillations in the mass spectra of metal clusters.

Energy flow in non-equilibrium conformal field theory

Abstract: We study the energy current and its fluctuations in quantum gapless 1d systems far from equilibrium modeled by conformal field theory, where two separated halves are prepared at distinct temperatures and glued together at a point contact. We prove that these systems converge towards steady states, and give a general description of such non-equilibrium steady states in terms of quantum field theory data. We compute the large deviation function, also called the full counting statistics, of energy transfer through the contact. These are universal and satisfy fluctuation relations. We provide a simple representation of these quantum fluctuations in terms of classical Poisson processes whose intensities are proportional to Boltzmann weights.

Electronic transport and quantum localization effects in organic semiconductors

Abstract: We explore the charge transport mechanism in organic semiconductors based on a model that accounts for the thermal intermolecular disorder at work in pure crystalline compounds, as well as extrinsic sources of disorder that are present in current experimental devices. Starting from the Kubo formula, we describe a theoretical framework that relates the time-dependent quantum dynamics of electrons to the frequency-dependent conductivity. The electron mobility is then calculated through a relaxation time approximation that accounts for quantum localization corrections beyond Boltzmann theory, and allows us to efficiently address the interplay between highly conducting states in the band range and localized states induced by disorder in the band tails. The emergence of a ``transient localization'' phenomenon is shown to be a general feature of organic semiconductors that is compatible with the bandlike temperature dependence of the mobility observed in pure compounds. Carrier trapping by extrinsic disorder causes a crossover to a thermally activated behavior at low temperature, which is progressively suppressed upon increasing the carrier concentration, as is commonly observed in organic field-effect transistors. Our results establish a direct connection between the localization of the electronic states and their conductive properties, formalizing phenomenological considerations that are commonly used in the literature.

Reconciling Planck constant determinations via watt balance and enriched-silicon measurements at NRC Canada

Abstract: The next revision to the International System of Units will emphasize the relationship between the base units (kilogram, metre, second, ampere, kelvin, candela and mole) and fundamental constants of nature (the speed of light, c, the Planck constant, h, the elementary charge, e, the Boltzmann constant, kB, the Avogadro constant, NA, etc). The redefinition cannot proceed without consistency between two complementary metrological approaches to measuring h: a 'physics' approach, using watt balances and the equivalence principle between electrical and mechanical force, and a 'chemistry' approach that can be viewed as determining the mass of a single atom of silicon. We report the first high precision physics and chemistry results that agree within 12 parts per billion: h (watt balance) = 6.626 070 63(43) × 10−34 J s and h(silicon) = 6.626 070 55(21) × 10−34 J s. When combined with values determined by other metrology laboratories, this work helps to constrain our knowledge of h to 20 parts per billion, moving us closer to a redefinition of the metric system used around the world.

The Law of Mass Action

Abstract: 1 Maxwell - Boltzmann Statitics.- 1.1 Thermodynamics and probab ility. The Boltzmann - Planck theorem.- 1.1.1 The Boltzmann - Planck theorem 5.- 1.2 The Maxwell - Boltzmann distribution law.- 1.2.1 Contin uous Maxwell - Boltzmann distribution.- 1.3 Calculation of most probable and mean values.- 1.4 Indistinguishable molecules. The Gibbs' paradox.- 1.5 Phase volume and the number of quantum states.- 1.6 Quantum statistics.- 1.6.1 Bose - Einstein statistics.- 1.6.2 Ferm i - Dirac statistics.- 1.6.3 Comparison of the three types of statis tics.- 1.6.4 Degenera te ideal gas.- 1.6.5 App lications of Bose - Einstein statistics: black-body radiation.- 1.6.6 Applications of Bose - Einstein statistics: heat capacity of solids.- 2 Ensembles, Partition Functions, and Thermodynamic Functions.- 2.1 Gibbs- approach, or how to avoid molecular interactions.- 2.2 The process of equilibration and increasing entropy.- 2.3 Microcanonical distribution.- 2.4. Canonical distribution.- 2.5 The probability of a macrostate.- 2.6 Thermodynamic functions derived from a canonical distribution.- 2.7 Some molecular partition functions.- 2.7.1 Degeneracy.- 2.7.2 Translational motion.- 2.7.3 Free rotation.- 2.7.4 Vibrational motion: linear harmonic oscillator.- 2.7.5 Total parti tion function of an ideal system.- 2.8 Fluctuations.- 2.9 Conclusions.- 3 The Law of Mass Action for Ideal Systems.- 3.1 The law of mass action, its origin and formal thermodynamic derivation.- 3.2 Statistical formulae for free energy.- 3.3 Statis tical formul ae for ideal sys tems.- 3.4 The law of mass action for ideal gases.- 3.4.1 Conversion to molar concent rations.- 3.4.2 Conversion to mole fractions.- 3.4.3 Standard sta tes and standard free energies of reaction.- 3.5 The law of mass act ion for an ideal crys tal. Spin crossover equilibria.- 3.6 Liquids.- 3.6.1 The law of mass action for an 'ideal liquid'.- 3.7 'Breakdown' of the law of mass action.- 3.8 Conclusions.- 4 Reactions in Imperfect Condensed Systems. Free Volume.- 4.1 Additive volume: a semi-empirical model of repulsive interactions.- 4.1.1 Binary equilibrium in a liquid with repul sive interactions.- 4.1.1 Non-isomolar equilibrium in a liquid with repulsive interactions.- 4.2. Lattice theories of the liquid state.- 4.3 The Lennard-Jones and Devonshir e model.- 4.4 Chemica l equi libria in Lennard-Jones and Devon shire liquids.- 4.5 The non-id eal law of mass action, activities, and standard states.- 4.6 Kinetic law of mass action.- 4.7 Conclusions.- 5 Molecular Interactions.- 5.1 Introduction.- 5.2 Empirical binary potentials.- 5.3 Taking into account nearest, next nearest, and longer range interactions in the conde nsed phase.- 5.4 Frequency of vibrations.- 5.5 The shape of the potential wcll in a cell.- 5.6 Free volume of a Lennard-Jones and Devons hire liquid.- 5.7 Experimental determ ination of parameters of the Lennard-Jones potential.- 5.7.1 Compressibility: thc Born' Lande method.- 5.7.2 Acoustical meas urements: the B.B. Kudryavtsev method.- 5.7.3 Viscosity of gases: the Rayleigh' Chapman method.- 5.8 Conclusions.- 6 Imperfect Gases..- 6.1 Introduction. The Virial Theorem.- 6.2 The Rayleigh equation.- 6.2.1 Virial coefficients: the Lennard-Jones method for the determination of the parameters of a binary potential.- 6.2.2 Free energy der ived from the Rayleigh equation of state.- 6.3 A gas with weak binary interactions: a statistical thermodynamics approach.- 6.4 Van der Waals equation of state.- 6.5 Chemical equilibria in imperfect gases.- 6.5.1 Isomolar equilibria in imperfect gases.- 6.5.2 A non-isomolar reaction in an imperfect gas.- 6.5.3 Separate conditions of ideal behaviour for attractive and repul sive molecular interactions.- 6.5.4 Associat ive equilibria in the gaseous phase.- 6.5.5 Mole cular interaction via a chemical reaction.- 6.6 Conclusions.- 7 Reactions in Imperferct Condensed Systems. Lattice Energy.- 7.1 Exchange energy 203.- 7.2 Non-ideality as a result of dependence of the partition function on the nature of the surroundings.- 7.3 Exchange free energy.- 7.4 Phase separations in binary mixtures.- 7.5 The law of mass action for an imperfect mixture in the condensed state.- 7.6 The regular solut ion model of steep spin crossover.- 7.7 Heat capacity changes in spin crossover.- 7.8 Negative exchange energy. Ordering . The Bragg - approach, or how to avoid molecular interactions.- 2.2 The process of equilibration and increasing entropy.- 2.3 Microcanonical distribution.- 2.4. Canonical distribution.- 2.5 The probability of a macrostate.- 2.6 Thermodynamic functions derived from a canonical distribution.- 2.7 Some molecular partition functions.- 2.7.1 Degeneracy.- 2.7.2 Translational motion.- 2.7.3 Free rotation.- 2.7.4 Vibrational motion: linear harmonic oscillator.- 2.7.5 Total parti tion function of an ideal system.- 2.8 Fluctuations.- 2.9 Conclusions.- 3 The Law of Mass Action for Ideal Systems.- 3.1 The law of mass action, its origin and formal thermodynamic derivation.- 3.2 Statistical formulae for free energy.- 3.3 Statis tical formul ae for ideal sys tems.- 3.4 The law of mass action for ideal gases.- 3.4.1 Conversion to molar concent rations.- 3.4.2 Conversion to mole fractions.- 3.4.3 Standard sta tes and standard free energies of reaction.- 3.5 The law of mass act ion for an ideal crys tal. Spin crossover equilibria.- 3.6 Liquids.- 3.6.1 The law of mass action for an 'ideal liquid'.- 3.7 'Breakdown' of the law of mass action.- 3.8 Conclusions.- 4 Reactions in Imperfect Condensed Systems. Free Volume.- 4.1 Additive volume: a semi-empirical model of repulsive interactions.- 4.1.1 Binary equilibrium in a liquid with repul sive interactions.- 4.1.1 Non-isomolar equilibrium in a liquid with repulsive interactions.- 4.2. Lattice theories of the liquid state.- 4.3 The Lennard-Jones and Devonshir e model.- 4.4 Chemica l equi libria in Lennard-Jones and Devon shire liquids.- 4.5 The non-id eal law of mass action, activities, and standard states.- 4.6 Kinetic law of mass action.- 4.7 Conclusions.- 5 Molecular Interactions.- 5.1 Introduction.- 5.2 Empirical binary potentials.- 5.3 Taking into account nearest, next nearest, and longer range interactions in the conde nsed phase.- 5.4 Frequency of vibrations.- 5.5 The shape of the potential wcll in a cell.- 5.6 Free volume of a Lennard-Jones and Devons hire liquid.- 5.7 Experimental determ ination of parameters of the Lennard-Jones potential.- 5.7.1 Compressibility: thc Born' Lande method.- 5.7.2 Acoustical meas urements: the B.B. Kudryavtsev method.- 5.7.3 Viscosity of gases: the Rayleigh' Chapman method.- 5.8 Conclusions.- 6 Imperfect Gases..- 6.1 Introduction. The Virial Theorem.- 6.2 The Rayleigh equation.- 6.2.1 Virial coefficients: the Lennard-Jones method for the determination of the parameters of a binary potential.- 6.2.2 Free energy der ived from the Rayleigh equation of state.- 6.3 A gas with weak binary interactions: a statistical thermodynamics approach.- 6.4 Van der Waals equation of state.- 6.5 Chemical equilibria in imperfect gases.- 6.5.1 Isomolar equilibria in imperfect gases.- 6.5.2 A non-isomolar reaction in an imperfect gas.- 6.5.3 Separate conditions of ideal behaviour for attractive and repul sive molecular interactions.- 6.5.4 Associat ive equilibria in the gaseous phase.- 6.5.5 Mole cular interaction via a chemical reaction.- 6.6 Conclusions.- 7 Reactions in Imperferct Condensed Systems. Lattice Energy.- 7.1 Exchange energy 203.- 7.2 Non-ideality as a result of dependence of the partition function on the nature of the surroundings.- 7.3 Exchange free energy.- 7.4 Phase separations in binary mixtures.- 7.5 The law of mass action for an imperfect mixture in the condensed state.- 7.6 The regular solut ion model of steep spin crossover.- 7.7 Heat capacity changes in spin crossover.- 7.8 Negative exchange energy. Ordering . The Bragg - Williams approximation.- 7.9. Description of order ing taking into account triple interactions.- 7.10 Chemica l equilibrium in ordered systems. Two-step spin crossover.- 7.11 Diluted systems.- 7.12 Conclusions.- 8 Chemical Correlations.- 8.1 Studies of variations of chemical reactivity.- 8.1.1 Molecular parameters governing variations of chemical reactivity.- 8.1.2. Solvent effects.- 8.1.3. Kinetic studies.- 8.1.4. Multidimensionality of var iations. Reference reactions.- 8.2 Linear free energy relationship. Modification of reactants.- 8.3 Linear free energy relationship. Variation of solvent.- 8.4 Isoequilibrium and isokinetic relationships.- 8.4.1 Statistical-mechanical model of the IER in ideal systems.- 8.4.2 The IER in gas-phase reactions.- 8.4.3 lsokinetic relationships.- 8.4.4 Non-ideality as a source of an IER.- 8.4.5 lER and exchange energy.- 8.5 Conclusions.- 9 Concluding Remarks.- 10 Appendices.- 10.1 Lagrange equations and Hamilt on (canonical) equations.- 10.2 Phase space.- 10.2.1 The phase space of a harmonic oscillator.- 10.2.2 The phase space of an ideal gas.- 10.3 Derivation of the canonical distribution.- 10.4 Free volume assoc iated with vibrations.- 10.5 Rotational con tribution to the equilibrium constant of the ionisation of water.- 10.6 Forms of the law of mass action employing the function approximation of the factorial.- 10.7 Derivation of the van der Waals equation of state.- 10.8 Exchange energy.- 10.9 Activity coefficients derived from the non-ideality resulting from triple interactions.- 10.10 The law of mass action for a binary equilibrium in a sys tem with non- additive volume and lattice energy.- 10.11 Physico-chemical constants and units of energy.

Derivation of a BGK model for mixtures

Abstract: The aim of this article is to construct a BGK operator for gas mixtures starting from the true Navier-Stokes equations. That is the ones with transport coefficients given by the hydrodynamic limit of the Boltzmann equation(s). Here the same hydrodynamic limit is obtained by introducing relaxation coefficients on certain moments of the distribution functions. Next the whole model is set by using entropy minimization under moment constraints as in Brull and Schneider (2008. 2009) [23,24]. In our case the BGK operator allows to recover the exact Fick and Newton laws and satisfy the classical properties of the Boltzmann equations for inert gas mixtures. (C) 2012 Elsevier Masson SAS. All rights reserved.

Stability of the nonrelativistic vlasov-maxwell-boltzmann system for angular non-cutoff potentials

Abstract: Although there recently have been extensive studies on the perturbation theory of the angular non-cutoff Boltzmann equation (cf. [4] and [17]), it remains mathematically unknown when there is a self-consistent Lorentz force coupled with the Maxwell equations in the nonrelativistic approximation. In the paper, for perturbative initial data with suitable regularity and integrability, we establish the large time stability of solutions to the Cauchy problem of the Vlasov-Maxwell-Boltzmann system with physical angular non-cutoff intermolecular collisions including the inverse power law potentials, and also obtain as a byproduct the convergence rates of solutions. The proof is based on a new time-velocity weighted energy method with two key technical parts: one is to introduce the exponentially weighted estimates into the non-cutoff Boltzmann operator and the other to design a delicate temporal energy $X(t)$-norm to obtain its uniform bound. The result also extends the case of the hard sphere model considered by Guo [Invent. Math. 153(3): 593--630 (2003)] to the general collision potentials.

Diffusion asymptotics of a kinetic model for gaseous mixtures

Abstract: In this work, we investigate the asymptotic behaviour of the solutions to the non-reactive fully elastic Boltzmann equations for mixtures in the diffusive scaling. We deal with cross sections such as hard spheres or cut-off power law potentials. We use Hilbert expansions near the common thermodynamic equilibrium granted by the H-theorem. The lower-order non trivial equality obtained from the Boltzmann equations leads to a linear functional equation in the velocity variable which is solved thanks to the Fredholm alternative. Since we consider multicomponent mixtures, the classical techniques introduced by Grad cannot be applied, and we propose a new method to treat the terms involving particles with different masses. The next-order equality in the Hilbert expansion then allows to write the macroscopic continuity equations for each component of the mixture.

Rethinking Technology & Creativity in the 21st Century: On Being In-Disciplined

Abstract: udwig Boltzmann (1844–1906) was one of the greatest scientists of his time. His work on statistical mechanics and the kinetic theory of gases helps explain and predict how the properties of atoms (such as charge and mass) determine the physical properties of gases (such as viscosity, diffusion and temperature). Ludwig Boltzmann was also an accomplished musician. Boltzmann, however, did not see these two interests (in science and music) as being independent of each other. In contrast, he often described a synchronicity between mathematics and music, seeing both as being involved in the creative act of identifying and manipulating underlying rhythms and patterns to create new ones. Moreover, Boltzmann perceived this process as being deeply personal, in how an individual’s creative voice was deeply connected to the final product. This of course is in sharp contrast to the prevailing view of science as being a coolly dispassionate methodology, disconnected from the personality of the scientist. Boltzmann’s viewpoint can be seen in how he described the experience of reading physicist James Clerk Maxwell’s work on the dynamical theory of gases. Note the manner in which Boltzmann connects his reading of mathematics to the experience of hearing a musical composition: The variations of the velocities are, at first, developed majestically: then from one side enter the equations of state: and from the other side, the equations of motion in a central field. Ever higher soars the chaos of formulae. Suddenly we hear, as from kettle drums, the four beats “Put N = 5.” The evil spirit V (relative velocity of molecules) vanishes: and, even as in music a hitherto dominating figure in the bass is suddenly silenced, that which had seemed insuperable has been overcome as if by a stroke of magic...One result after another follows in quick succession till at last, as the unexpected climax, we arrive at the conditions for thermal equilibrium together with the expressions for the transport coefficients. The curtain then falls! (Boltzmann quoted in Root-Bernstein, 1989, p. 334)

Path-integral Mayer-sampling calculations of the quantum Boltzmann contribution to virial coefficients of helium-4.

Abstract: We present Mayer-sampling Monte Carlo calculations of the quantum Boltzmann contribution to the virial coefficients Bn, as defined by path integrals, for n = 2 to 4 and for temperatures from 2.6 K to 1000 K, using state-of-the-art ab initio potentials for interactions within pairs and triplets of helium-4 atoms. Effects of exchange are not included. The vapor-liquid critical temperature of the resulting fourth-order virial equation of state is 5.033(16) K, a value only 3% less than the critical temperature of helium-4: 5.19 K. We describe an approach for parsing the Boltzmann contribution into components that reduce the number of Mayer-sampling Monte Carlo steps required for components with large per-step time requirements. We estimate that in this manner the calculation of the Boltzmann contribution to B3 at 2.6 K is completed at least 100 times faster than the previously reported approach.

Erratum: Hadron spectra and elliptic flow for 200 A GeV Au+Au collisions from viscous hydrodynamics coupled to a Boltzmann cascade [Phys. Rev. C 83 , 054910 (2011)]

Abstract: In the version of VISHNU used in the original paper, the baryon-antibaryon annihilation channels in the hadron cascade module UrQMD were accidentally turned off. When redoing the calculations with those channels turned on, we found that baryon-antibaryon annihilation in the late hadronic stage reduces the final proton and antiproton multiplicities by about 30% in central (0–5% centrality) and by about 15% in peripheral (60–70% centrality) collisions while simultaneously slightly increasing the pion and kaon multiplicities. These observations are consistent with recent analyses presented in Refs. [1,2]. To compensate for the resulting slight overall increase in the final total charged hadron multiplicity when using the corrected version of VISHNU, we had to reduce the normalization of initial entropy density by about 4%. Keeping the original parameter sets for η/s and τ0(η/s), we confirmed that (within the statistical uncertainties of the results presented in the original paper) the changes in the hydrodynamic evolution caused by this slight renormalization of the initial density profile are negligible, and the main effects of including B−B̄ annihilation are a small change in the chemical composition of the hadron gas phase, as well as a renormalization and slight hardening of the proton pT spectra. The hardening of the proton spectra arises from preferential annihilation of low-pT baryons and antibaryons.

High accuracy numerical solutions of the Boltzmann Bhatnagar-Gross-Krook equation for steady and oscillatory Couette flows

Abstract: Modeling gas flows generated by micro- and nano-devices often requires the use of kinetic theory. To facilitate implementation, various approximate formulations have been proposed based on the Bhatnagar-Gross-Krook (BGK) kinetic model, including most recently, the lattice Boltzmann (LB) method. While there exists a comprehensive numerical data set for the hard sphere linearized Boltzmann equation for steady Couette flow, no such set of data is available for the Boltzmann-BGK equation. The purpose of this article is to present a high accuracy data set for the linearized Boltzmann-BGK equation over the full range of Knudsen numbers and normalized oscillation frequencies – this encompasses both steady and unsteady Couette flows. This data set is expected to be of particular value in the benchmarking and validation of computational methods such as the LB method and other approaches based on the Boltzmann-BGK equation.

An entropic picture of emergent quantum mechanics

Abstract: Quantum mechanics emerges a la Verlinde from a foliation of ℝ3 by holographic screens, when regarding the latter as entropy reservoirs that a particle can exchange entropy with. This entropy is quantized in units of Boltzmann's constant kB. The holographic screens can be treated thermodynamically as stretched membranes. On that side of a holographic screen where spacetime has already emerged, the energy representation of thermodynamics gives rise to the usual quantum mechanics. A knowledge of the different surface densities of entropy flow across all screens is equivalent to a knowledge of the quantum-mechanical wavefunction on ℝ3. The entropy representation of thermodynamics, as applied to a screen, can be used to describe quantum mechanics in the absence of spacetime, that is, quantum mechanics beyond a holographic screen, where spacetime has not yet emerged. Our approach can be regarded as a formal derivation of Planck's constant ℏ from Boltzmann's constant kB.

Automated Boltzmann collision integrals for moment equations

Abstract: We present a methodology to evaluate the moments of the Boltzmann collision term, in a general automated way, using the computer algebra software Mathematica. Based on Grad's distribution function with 26-moments, we compute the non-linear production terms for a simple gas and a granular gas, and the linear production terms for a binary mixture of gases. The results can be shown for general interaction potential, but, in this paper, they are given only for hard-sphere interaction potential.

A remark on Cannone-Karch solutions to the homogeneous Boltzmann equation for Maxwellian molecules

Abstract: The purpose of this paper is to extend the result concerning the existence and the uniqueness of infinite energy solutions, given by Cannone-Karch, of the Cauchy problem for the spatially homogeneous Boltzmann equation of Maxwellian molecules without Grad's angular cutoff assumption in the mild singularity case, to the strong singularity case. This extension follows from a simple observation of the symmetry on the unit sphere for the Bobylev formula which is the Fourier transform of the Boltzmann collision term.

A fast spectral algorithm for the quantum Boltzmann collision operator

Abstract: J.H. would like to thank KAUST for their generous support. L.Y. was supported in part by the NSF under CAREER grant DMS-0846501.

Recent Results on the Periodic Lorentz Gas

Abstract: The kinetic theory of gases was proposed by J. Clerk Maxwell [34, 35] and L. Boltzmann [5] in the second half of the XIXth century. Because the existence of atoms, on which kinetic theory rested, remained controversial for some time, it was not until many years later, in the XXth century, that the tools of kinetic theory became of common use in various branches of physics such as neutron transport, radiative transfer, plasma and semiconductor physics, etc.

Speed-dependent effects in NH 3 self-broadened spectra: Towards the determination of the Boltzmann constant

Abstract: In this paper we present an accurate analysis of the shape of an isolated rovibrational ammonia line from the strong nu2 band around 10 µm, recorded by laser absorption spectroscopy. Experimental spectra obtained under controlled temperature and pressure, are confronted to various models that take into account Dicke narrowing or speed-dependent effects. Our results show clear evidence for speed-dependent broadening and shifting, which had never been demonstrated so far in NH3. Accurate lineshape parameters of the nu2 saQ(6,3) line are obtained. Our current project aiming at measuring the Boltzmann constant, kB, by laser spectroscopy will straight away benefit from such knowledge. We anticipate that a first optical determination of kB with a competitive uncertainty of a few ppm is now reachable.

Superaging correlation function and ergodicity breaking for Brownian motion in logarithmic potentials

Abstract: We consider an overdamped Brownian particle moving in a confining asymptotically logarithmic potential, which supports a normalized Boltzmann equilibrium density. We derive analytical expressions for the two-time correlation function and the fluctuations of the time-averaged position of the particle for large but finite times. We characterize the occurrence of aging and nonergodic behavior as a function of the depth of the potential, and we support our predictions with extensive Langevin simulations. While the Boltzmann measure is used to obtain stationary correlation functions, we show how the non-normalizable infinite covariant density is related to the superaging behavior.

From Boltzmann’s Equation to the Incompressible Navier–Stokes–Fourier System with Long-Range Interactions

Abstract: We establish a rigorous demonstration of the hydrodynamic convergence of the Boltzmann equation towards a Navier–Stokes–Fourier system under the presence of long-range interactions. This convergence is obtained by letting the Knudsen number tend to zero and has been known to hold, at least formally, for decades. It is only more recently that a fully rigorous mathematical derivation of this hydrodynamic limit was discovered. However, these results failed to encompass almost all physically relevant collision kernels due to a cutoff assumption, which requires that the cross sections be integrable. Indeed, as soon as long-range intermolecular forces are present, non-integrable collision kernels have to be considered because of the enormous number of grazing collisions in the gas. In this long-range setting, the Boltzmann operator becomes a singular integral operator and the known rigorous proofs of hydrodynamic convergence simply do not carry over to that case. In fact, the DiPerna–Lions renormalized solutions do not even make sense in this situation and the relevant global solutions to the Boltzmann equation are the so-called renormalized solutions with a defect measure developed by Alexandre and Villani. Our work overcomes the new mathematical difficulties coming from the consideration of long-range interactions by proving the hydrodynamic convergence of the Alexandre–Villani solutions towards the Leray solutions.

Thermodynamic description of liquid-state limits.

Abstract: A state of random close packing (RCP) of spheres is found to have a thermodynamic status and a fundamental role in the description of liquid-state equilibria. The RCP limiting amorphous ground state, with reproducible density and well-characterized structure, is obtained by well-defined irreversible and reversible processes. The limiting packing fraction yRCP = 0.6366 ± 0.0005 (Buffon’s constant within the uncertainty), and a residual entropy per sphere ΔS(RCP-FCC) ≃ kB (Boltzmann’s constant). Since the Mayer virial expansion does not represent dense fluid equations-of-state for densities exceeding the available-volume percolation transition (ρpa), we infer that a RCP state belongs to the same thermodynamic phase as prepercolation equilibrium dense hard-sphere fluid and likewise for hard-core fluids with attractive forces. Monte Carlo (MC) calculation of the liquid-state coexistence properties of square-well (SW) attractive spheres, together with existing MC results for liquid–vapor coexistence in the SW ...

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

Abstract: 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 JouleMayer 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 mostprobable 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 Schrodinger 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. Key-Words: Kinetic theory of ideal gas; Thermodynamics; Statistical mechanics; Riemann Hypothesis; TOE.

Phase diagram of the modified Lennard-Jones system

Abstract: An investigation of the precise determination of melting temperature in the modified Lennard-Jones system under pressure-free conditions [Y. Asano and K. Fuchizaki, J. Phys. Soc. Jpn.78, 055002 (2009)10.1143/JPSJ.78.055002] was extended under finite-pressure conditions to obtain the phase diagram. The temperature and pressure of the triple point were estimated to be 0.61 e/k B and 0.0018(5) e/σ3, and those of the critical point were 1.0709(19) e/k B and 0.1228(20) e/σ3, where e and σ are the Lennard-Jones parameters for energy and length scales, respectively, and k B is the Boltzmann constant. The potential used here has a finite attractive tail and does not suffer from cutoff problems. The potential can thus be a useful standard in examining statistical–mechanical problems in which different treatments for the tail would lead to different conclusions. The present phase diagram will then be a useful guide not only for equilibrium calculations but also for nonequilibrium problems such as discussions of the limits of phase (in)stability.

Entropy generation across Earth's collisionless bow shock.

Abstract: Earth's bow shock is a collisionless shock wave but entropy has never been directly measured across it. The plasma experiments on Cluster and Double Star measure 3D plasma distributions upstream and downstream of the bow shock allowing calculation of Boltzmann's entropy function H and his famous H theorem, dH/dt≤0. The collisionless Boltzmann (Vlasov) equation predicts that the total entropy does not change if the distribution function across the shock becomes nonthermal, but it allows changes in the entropy density. Here, we present the first direct measurements of entropy density changes across Earth's bow shock and show that the results generally support the model of the Vlasov analysis. These observations are a starting point for a more sophisticated analysis that includes 3D computer modeling of collisionless shocks with input from observed particles, waves, and turbulences.

Modeling of flows in a microchannel based on the boltzmann lattice equation

Abstract: An approach to the analysis of micro- and macroflows with the use of the method of Boltzmann lattices is considered. The velocity profiles of flows in the entry sections of microchannels under no-slip and slip conditions on the wall have been obtained. The influence of the Knudsen number on the hydrodynamic entrance length of a microchannel and its hydrodynamical resistance have been analyzed.