Showing papers on "Boltzmann constant published in 2010"

Decay and Continuity of the Boltzmann Equation in Bounded Domains

Abstract: Boundaries occur naturally in kinetic equations, and boundary effects are crucial for dynamics of dilute gases governed by the Boltzmann equation. We develop a mathematical theory to study the time decay and continuity of Boltzmann solutions for four basic types of boundary conditions: in-flow, bounce-back reflection, specular reflection and diffuse reflection. We establish exponential decay in the L ∞ norm for hard potentials for general classes of smooth domains near an absolute Maxwellian. Moreover, in convex domains, we also establish continuity for these Boltzmann solutions away from the grazing set at the boundary. Our contribution is based on a new L 2 decay theory and its interplay with delicate L ∞ decay analysis for the linearized Boltzmann equation in the presence of many repeated interactions with the boundary.

On Landauer versus Boltzmann and full band versus effective mass evaluation of thermoelectric transport coefficients

Abstract: transport is mathematically related to the solution of the Boltzmann transport equation, and expressions for the thermoelectric parameters in both formalisms are presented. Quantum mechanical and semiclassical techniques to obtain from a full description of the bandstructure, Ek, the density of modes in the Landauer approach or the transport distribution in the Boltzmann solution are compared and thermoelectric transport coefficients are evaluated. Several example calculations for representative bulk materials are presented and the full band results are related to the more common effective mass formalism. Finally, given a full Ek for a crystal, a procedure to extract an accurate, effective mass level description is presented. © 2010 American Institute of Physics. doi:10.1063/1.3291120

Factorization for non-symmetric operators and exponential H-theorem

Abstract: We present an abstract method for deriving decay estimates on the resolvents and semigroups of non-symmetric operators in Banach spaces in terms of estimates in another smaller reference Banach space. This applies to a class of operators writing as a regularizing part, plus a dissipative part. The core of the method is a high-order quantitative factorization argument on the resolvents and semigroups. We then apply this approach to the Fokker-Planck equation, to the kinetic Fokker- Planck equation in the torus, and to the linearized Boltzmann equation in the torus. We finally use this information on the linearized Boltzmann semi- group to study perturbative solutions for the nonlinear Boltzmann equation. We introduce a non-symmetric energy method to prove nonlinear stability in this context in $L^1_v L^\infty _x (1 + |v|^k)$, $k > 2$, with sharp rate of decay in time. As a consequence of these results we obtain the first constructive proof of exponential decay, with sharp rate, towards global equilibrium for the full nonlinear Boltzmann equation for hard spheres, conditionally to some smoothness and (polynomial) moment estimates. This improves the result in [32] where polynomial rates at any order were obtained, and solves the conjecture raised in [91, 29, 86] about the optimal decay rate of the relative entropy in the H-theorem.

Regularizing Effect and Local Existence for the Non-Cutoff Boltzmann Equation

Abstract: The Boltzmann equation without Grad’s angular cutoff assumption is believed to have a regularizing effect on the solutions because of the non-integrable angular singularity of the cross-section. However, even though this has been justified satisfactorily for the spatially homogeneous Boltzmann equation, it is still basically unsolved for the spatially inhomogeneous Boltzmann equation. In this paper, by sharpening the coercivity and upper bound estimates for the collision operator, establishing the hypo-ellipticity of the Boltzmann operator based on a generalized version of the uncertainty principle, and analyzing the commutators between the collision operator and some weighted pseudo-differential operators, we prove the regularizing effect in all (time, space and velocity) variables on the solutions when some mild regularity is imposed on these solutions. For completeness, we also show that when the initial data has this mild regularity and a Maxwellian type decay in the velocity variable, there exists a unique local solution with the same regularity, so that this solution acquires the C∞ regularity for any positive time.

Nonequilibrium kolmogorov type particle distributions and their applications

Abstract: The paper reviews the current state of research on nonequilibrium (Kolmogorov type) stationary and nonstationary distributions of particles with statically screened Coulomb interaction that are exact solutions of the Boltzmann or Landau collision integral with a source and a sink ensuring the energy flow along the spectrum in momentum space. Analysis is made of the advantages of the new process (based on nonequilibrium distributions) of energy conversion and of the time-dependent nonequilibrium kinetics of an electron-phonon system of a crystal in a strong electric field (electroplastic effect).

Fractional Fokker-Planck equations for subdiffusion with space- and time-dependent forces.

Abstract: We derive a fractional Fokker-Planck equation for subdiffusion in a general space- and time-dependent force field from power law waiting time continuous time random walks biased by Boltzmann weights. The governing equation is derived from a generalized master equation and is shown to be equivalent to a subordinated stochastic Langevin equation.

Infinite covariant density for diffusion in logarithmic potentials and optical lattices.

Abstract: We solve the Fokker-Planck equation for Brownian motion in a logarithmic potential. When the diffusion constant is below a critical value the solution approaches an infinite covariant density. With this non-normalizable solution we obtain the phase diagram of anomalous diffusion for this process. We briefly discuss the physical consequences for atoms in optical lattices and charges in the vicinity of long polyelectrolytes. Our work explains in what sense the infinite covariant density and not Boltzmann's equilibrium describes the long time limit of these systems.

Flavored quantum Boltzmann equations

Abstract: We derive from first principles, using nonequilibrium field theory, the quantum Boltzmann equations that describe the dynamics of flavor oscillations, collisions, and a time-dependent mass matrix in the early universe Working to leading nontrivial order in ratios of relevant time scales, we study in detail a toy model for weak-scale baryogenesis: two scalar species that mix through a slowly varying time-dependent and CP-violating mass matrix, and interact with a thermal bath This model clearly illustrates how the CP asymmetry arises through coherent flavor oscillations in a nontrivial background We solve the Boltzmann equations numerically for the density matrices, investigating the impact of collisions in various regimes

Global classical solutions of the Boltzmann equation with long-range interactions.

Abstract: This is a brief announcement of our recent proof of global existence and rapid decay to equilibrium of classical solutions to the Boltzmann equation without any angular cutoff, that is, for long-range interactions. We consider perturbations of the Maxwellian equilibrium states and include the physical cross-sections arising from an inverse-power intermolecular potential r-(p-1) with p > 2, and more generally. We present here a mathematical framework for unique global in time solutions for all of these potentials. We consider it remarkable that this equation, derived by Boltzmann (1) in 1872 and Maxwell (2) in 1867, grants a basic example where a range of geometric fractional derivatives occur in a physical model of the natural world. Our methods provide a new understanding of the effects due to grazing collisions.

Medium corrections to the CP-violating parameter in leptogenesis

Abstract: In two recent papers [4,5], it has been demonstrated that one can obtain quantum-corrected Boltzmann kinetic equations for leptogenesis using a top-down approach based on the Schwinger-Keldysh/Kadanoff-Baym formalism. These ``Boltzmann-like'' equations are similar to the ones obtained in the conventional bottom-up approach but differ in important details. In particular there is a discrepancy between the $CP$-violating parameter obtained in the first-principle derivation and in the framework of thermal field theory. Here we demonstrate that the two approaches can be reconciled if causal $n$-point functions are used in the thermal field theory approach. The new result for the medium correction to the $CP$-violating parameter is qualitatively different from the conventional one. The analogy to a toy model considered earlier enables us to write down consistent quantum-corrected Boltzmann equations, for thermal leptogenesis in the standard model extended by three right-handed neutrinos, which include quantum statistical terms and medium-corrected expressions for the $CP$-violating parameter.

Entropy and chaos in the Kac model

Abstract: We investigate the behavior in $N$ of the $N$--particle entropy functional for Kac's stochastic model of Boltzmann dynamics, and its relation to the entropy function for solutions of Kac's one dimensional nonlinear model Boltzmann equation. We prove results that bring together the notion of propagation of chaos, which Kac introduced in the context of this model, with the problem of estimating the rate of equilibration in the model in entropic terms, showing that the entropic rate of convergence can be arbitrarily slow. Results proved here show that one can in fact use entropy production bounds in Kac's stochastic model to obtain entropic convergence bounds for his non linear model Boltzmann equation, though the problem of obtaining optimal lower bounds of this sort for the original Kac model remains open and the upper bounds obtained here show that this problem is somewhat subtle.

Global Hilbert Expansion for the Vlasov-Poisson-Boltzmann System

Abstract: The dynamics of an electron gas in a constant ion background can be decribed by the Vlasov-Poisson-Boltzmann system at the kinetic level, or by the compressible Euler-Poisson system at the fluid level. We prove that any solution of the Vlasov-Poisson-Boltzmann system near a smooth local Maxwellian with a small irrotational velocity converges global in time to the corresponding solution to the Euler-Poisson system, as the mean free path e goes to zero. We use a recent L2 − L∞ framework in the Boltzmann theory to control the higher order remainder in the Hilbert expansion uniformly in e and globally in time.

Acoustic Resonator Experiments at the Triple Point of Water: First Results for the Boltzmann Constant and Remaining Challenges

Abstract: We report on two sets of isothermal acoustic measurements made with argon close to the triple point of water using a 50 mm radius, thin-walled, diamond-turned quasisphere. Our two isotherms yielded values for the Boltzmann constant, k B, which differ by 0.9 parts in 106, and have an average value of k B = (1.380 649 6 ± 0.000 004 3) × 10−23J · K−1. The relative uncertainty is 3.1 parts in 106, and the average value is 0.58 parts in 106 below the 2006 CODATA value (Mohr et al. Rev Mod Phys 80:633, 2008), and so the values are consistent within their combined (k = 1) uncertainties.

Global Newtonian Limit for the Relativistic Boltzmann Equation near Vacuum

Abstract: We study the Cauchy problem for the relativistic Boltzmann equation with near vacuum initial data. Unique global-in-time mild solutions are obtained uniformly in the speed of light parameter $c\geq1$. We furthermore prove that solutions to the relativistic Boltzmann equation converge to solutions of the Newtonian Boltzmann equation in the limit as $c\to\infty$ on arbitrary time intervals $[0,T]$, with convergence rate $1/c^{2-\epsilon}$ for any $\epsilon\in(0,2)$. This may be the first proof of unique global-in-time validity of the Newtonian limit for a kinetic equation.

A determination of the Boltzmann constant from speed of sound measurements in helium at a single thermodynamic state

Abstract: We report on acoustic and microwave measurements made with a purified helium sample maintained close to a single thermodynamic state (Texp ~ 27316 K, pexp ~ 410 kPa) within a 21 L volume stainless steel spherical cavity From these measurements and ab initio calculations of the non-ideality and the refractive index of helium, we determine a value for the Boltzmann constant kB which is consistent with the recommended 2006 CODATA value: (kB − k2006)/k2006 = (−75 ± 75) × 10−6 We discuss the current limits of the experiment and the prospects of a further reduction in the uncertainty associated with the determination of kB

Progress towards an accurate determination of the Boltzmann constant by Doppler spectroscopy

Abstract: In this paper, we present significant progress performed on an experiment dedicated to the determination of the Boltzmann constant, k, by accurately measuring the Doppler absorption profile of a line in a gas of ammonia at thermal equilibrium. This optical method based on the first principles of statistical mechanics is an alternative to the acoustical method which has led to the unique determination of k published by the CODATA with a relative accuracy of 1.7 ppm. We report on the first measurement of the Boltzmann constant by laser spectroscopy with a statistical uncertainty below 10 ppm, more specifically 6.4 ppm. This progress results from improvements in the detection method and in the statistical treatment of the data. In addition, we have recorded the hyperfine structure of the probed saQ(6,3) rovibrational line of ammonia by saturation spectroscopy and thus determine very precisely the induced 4.36 (2) ppm broadening of the absorption linewidth. We also show that, in our well chosen experimental conditions, saturation effects have a negligible impact on the linewidth. Finally, we draw the route to future developments for an absolute determination of with an accuracy of a few ppm.

Coupling relativistic viscous hydrodynamics to Boltzmann descriptions

Abstract: Models of relativistic heavy-ion collisions typically involve both a hydrodynamic module to describe the high-density liquidlike phase and a Boltzmann module to simulate the low-density breakup phase, which is gaslike. Coupling the prescriptions is more complicated for viscous prescriptions if one wants to maintain continuity of the entire stress-energy tensor and currents. Derivations for the viscosity for a gas are reviewed, which then lead to expressions for changes in the phase-space occupation based on simple relaxation-time pictures of viscosity. These expressions are shown to consistently reproduce the nonequilibrium components of the stress-energy tensor. An algorithm for generating a Monte Carlo sampling of particles with which to initiate the Boltzmann calculations is also presented.

Gevrey smoothing effect of solutions for spatially homogeneous nonlinear Boltzmann equation without angular cutoff

Abstract: There have been extensive studies on the regularizing effect of solutions to the Boltzmann equation without angular cutoff assumption, for both spatially homogeneous and inhomogeneous cases, by noticing the fact that non cutoff Boltzmann collision operator behaves like the fractional power of the Laplace operator. As a further study on the problem in the spatially homogeneous situation, in this paper, we consider the Gevrey regularity of C ∞ solutions with the Maxwellian decay to the Cauchy problem of spatially homogeneous Boltzmann equation for modified hard potentials, by using analytic techniques developed in Alexandre et al. (J Funct Anal 255:2013–2066, 2008; Arch Ration Mech Anal, doi: 10.1007/s00205-010-0290-1 , 2010), Huo et al. (Kinet Relat Models 1:453–489, 2008) and Morimoto et al. (Discrete Contin Dyn Syst Ser A 24:187–212, 2009).

Effect of short- and long-range scattering on the conductivity of graphene: Boltzmann approach vs tight-binding calculations

Abstract: We present a comparative study of the density dependence of the conductivity of graphene sheets calculated in the tight-binding (TB) Landauer approach and on the basis of the Boltzmann theory. The ...

Formation and Propagation of Discontinuity for Boltzmann Equation in Non-Convex Domains

Abstract: The formation and propagation of singularities for Boltzmann equation in bounded domains has been an important question in numerical studies as well as in theoretical studies. Consider the nonlinear Boltzmann solution near Maxwellians under in-flow, diffuse, or bounce-back boundary conditions. We demonstrate that discontinuity is created at the non-convex part of the grazing boundary, then propagates only along the forward characteristics inside the domain before it hits on the boundary again.

Lattice Boltzmann Simulation of Incompressible Transport Phenomena in Macroscopic Solidification Processes

Abstract: A lattice Boltzmann (LB) simulation strategy is proposed for the incompressible transport phenomena occurring during macroscopic solidification of pure substances. The proposed model is derived by coupling a passive scalar-based thermal LB model with the classical enthalpy–porosity technique for solid–liquid phase-transition problems. The underlying hydrodynamics are monitored by a conventional single-particle density distribution function (DF) through a kinetic equation, whereas the thermal field is obtained from another kinetic equation which is governed by a separate temperature DF. The phase-changing aspects are incorporated into the LB model by inserting appropriate source terms in the respective kinetic equations through the most formal technique following the extended Boltzmann equations along with an appropriate enthalpy updating scheme. The proposed model is validated extensively with one- and two-dimensional solidification problems for which analytical and numerical results are available in the ...

On the Boltzmann-Grad Limit for the Two Dimensional Periodic Lorentz Gas

Abstract: The two-dimensional, periodic Lorentz gas, is the dynamical system corresponding with the free motion of a point particle in a planar system of fixed circular obstacles centered at the vertices of a square lattice in the Euclidean plane. Assuming elastic collisions between the particle and the obstacles, this dynamical system is studied in the Boltzmann-Grad limit, assuming that the obstacle radius r and the reciprocal mean free path are asymptotically equivalent small quantities, and that the particle’s distribution function is slowly varying in the space variable. In this limit, the periodic Lorentz gas cannot be described by a linear Boltzmann equation (see Golse in Ann. Fac. Sci. Toulouse 17:735–749, 2008), but involves an integro-differential equation conjectured in Caglioti and Golse (C. R. Acad. Sci. Ser. I Math. 346:477–482, 2008) and proved in Marklof and Strombergsson (preprint arXiv:0801.0612), set on a phase-space larger than the usual single-particle phase-space. The main purpose of the present paper is to study the dynamical properties of this integro-differential equation: identifying its equilibrium states, proving a H Theorem and discussing the speed of approach to equilibrium in the long time limit. In the first part of the paper, we derive the explicit formula for a transition probability appearing in that equation following the method sketched in Caglioti and Golse (C. R. Acad. Sci. Ser. I Math. 346:477–482, 2008).

Effects of non-uniformity of laser induced plasma on plasma temperature and concentrations determined by the Boltzmann plot method: implications from plasma modeling

Abstract: The validity of the popular Boltzmann plot method was theoretically tested for the case of a non-homogeneous non-isothermal laser-induced plasma. A collisional-dominated plasma model was employed to generate synthetic spectra by solving the radiative transfer equation. The spectra were processed with homemade software that calculated values for the plasma temperature and concentrations using the Boltzmann plot approach. Both static and dynamic plasmas were investigated at various temperature and density gradients. The plasma parameters obtained from the Boltzmann plots were subsequently compared with the exact parameters of the model. The results are shown to have a direct implication for calibration-free laser-induced breakdown spectroscopy (CF-LIBS). For nearly all tested situations, the Boltzmann plot method was capable of semi-quantitative analysis providing the accurate determination of concentrations for main plasma components and failing to accurately predict the concentrations of minor components and trace elements.

Derivation of the Boltzmann principle

Abstract: We derive the Boltzmann principle SB=kB ln W based on classical mechanical models of thermodynamics. The argument is based on the heat theorem and can be traced back to the second half of the 19th century in the works of Helmholtz and Boltzmann. Despite its simplicity, this argument has remained almost unknown. We present it in a contemporary, self-contained, and accessible form. The approach constitutes an important link between classical mechanics and statistical mechanics.

Influence of the line-shape model on the spectroscopic determination of the Boltzmann constant

Abstract: Recent high-resolution spectroscopic experiments demonstrated the possibility of measuring the Boltzmann constant directly from the Doppler width of atomic or molecular lines; however, both experimental setups and data analysis methods still need to be improved to compete with the best acoustic methods. In this paper, the influence of choice of spectral line-shape model for data analysis on systematical error of the Doppler-width determination is analyzed on the example of the oxygen line near $\ensuremath{\lambda}=687$ nm. Profile simulations were performed for many different models at pressures ranging from 0.012 Pa to 120 kPa, taking into account such line-shape effects as the speed dependence of collisional broadening and Dicke narrowing. We discuss limitations of an applicability of the Voigt profile for purposes of the Boltzmann constant determination. The influence of the model line-shape profile and the gas pressure range on the Doppler width extrapolated to the zero-pressure value is investigated.

Fokker-Planck asymptotics for traffic flow models

Abstract: Starting from microscopic interaction rules we derive kinetic models of Fokker-Planck type for vehicular traffic flow. The derivation is based on taking a suitable asymptotic limit of the corresponding Boltzmann model. As particular cases, the derived models comprise existing models. New Fokker-Planck models are also given and their differences to existing models are highlighted. Finally, we report on numerical experiments.

The Michaelis-Menten-Stueckelberg Theorem

Abstract: We study chemical reactions with complex mechanisms under two assumptions: (i) intermediates are present in small amounts (this is the quasi-steady-state hypothesis or QSS) and (ii) they are in equilibrium relations with substrates (this is the quasiequilibrium hypothesis or QE). Under these assumptions, we prove the generalized mass action law together with the basic relations between kinetic factors, which are sufficient for the positivity of the entropy production but hold even without microreversibility, when the detailed balance is not applicable. Even though QE and QSS produce useful approximations by themselves, only the combination of these assumptions can render the possibility beyond the "rarefied gas" limit or the "molecular chaos" hypotheses. We do not use any a priori form of the kinetic law for the chemical reactions and describe their equilibria by thermodynamic relations. The transformations of the intermediate compounds can be described by the Markov kinetics because of their low density ({\em low density of elementary events}). This combination of assumptions was introduced by Michaelis and Menten in 1913. In 1952, Stueckelberg used the same assumptions for the gas kinetics and produced the remarkable semi-detailed balance relations between collision rates in the Boltzmann equation that are weaker than the detailed balance conditions but are still sufficient for the Boltzmann $H$-theorem to be valid. Our results are obtained within the Michaelis-Menten-Stueckelbeg conceptual framework.