Showing papers on "Boltzmann constant published in 2007"

Entropy production and time asymmetry in nonequilibrium fluctuations.

Abstract: The time-reversal symmetry of nonequilibrium fluctuations is experimentally investigated in two out-of-equilibrium systems: namely, a Brownian particle in a trap moving at constant speed and an electric circuit with an imposed mean current. The dynamical randomness of their nonequilibrium fluctuations is characterized in terms of the standard and time-reversed entropies per unit time of dynamical systems theory. We present experimental results showing that their difference equals the thermodynamic entropy production in units of Boltzmann's constant.

Coherent matter wave transport in speckle potentials

Abstract: This paper studies multiple scattering of matter waves by a disordered optical potential in two and in three dimensions. We calculate fundamental transport quantities such as the scattering mean free path ls, the Boltzmann transport mean free path lB, and the Boltzmann diffusion constant DB, using a diagrammatic Green functions approach in the weak-scattering regime. Coherent multiple scattering induces interference corrections known as weak localization which entail a reduced diffusion constant. We derive the corresponding expressions for matter wave transport in a correlated speckle potential and provide the relevant parameter values for a possible experimental study of this coherent transport regime, including the critical crossover to the regime of strong or Anderson localization.

Hydrodynamics beyond Navier-Stokes : Exact Solution to the Lattice Boltzmann Hierarchy

Abstract: The exact solution to the hierarchy of nonlinear lattice Boltzmann (LB) kinetic equations in the stationary planar Couette flow is found at nonvanishing Knudsen numbers. A new method of solving LB kinetic equations which combines the method of moments with boundary conditions for populations enables us to derive closed-form solutions for all higher-order moments. A convergence of results suggests that the LB hierarchy with larger velocity sets is the novel way to approximate kinetic theory.

Direct determination of the boltzmann constant by an optical method

Abstract: We have recorded the Doppler profile of a well-isolated rovibrational line in the nu(2) band of (14)NH(3). Ammonia gas was placed in an absorption cell thermalized by a water-ice bath. By extrapolating to zero pressure, we have deduced the Doppler width which gives a first measurement of the Boltzmann constant k(B) by laser spectroscopy. A relative uncertainty of 2 x 10(-4) has been obtained. The present determination should be significantly improved in the near future and contribute to a new definition of the kelvin.

Quantum Boltzmann equations and leptogenesis

Abstract: The closed time-path formalism is a powerful Green's function formulation to describe non-equilibrium phenomena in field theory and it leads to a complete non-equilibrium quantum kinetic theory. We make use of this formalism to write down the set of quantum Boltzmann equations relevant for leptogenesis. They manifest memory effects and off-shell corrections. In particular, memory effects lead to a time-dependent CP asymmetry whose value at a given instant of time depends upon the previous history of the system. This result is particularly relevant when the asymmetry is generated by the decays of nearly mass-degenerate heavy states, as in resonant or soft leptogenesis.

Polarizability of helium and gas metrology.

Abstract: Using a quasispherical, microwave cavity resonator, we measured the refractive index of helium to deduce its molar polarizability in the limit of zero density. We obtained , where the standard uncertainty (9.1 ppm) is a factor of 3.3 smaller than that of the best previous measurement. If the theoretical value of is accepted, these data determine a value for the Boltzmann constant that is only larger than the accepted value. Our techniques will enable a helium-based pressure standard and measurements of thermodynamic temperatures.

Optimal Decay Estimates on the Linearized Boltzmann Equation with Time Dependent Force and their Applications

Abstract: Although the decay in time estimates of the semi-group generated by the linearized Boltzmann operator without forcing have been well established, there is no corresponding result for the case with general external force. This paper is mainly concerned with the optimal decay estimates on the solution operator in some weighted Sobolev spaces for the linearized Boltzmann equation with a time dependent external force. No time decay assumption is made on the force. The proof is based on both the energy method through the macro-micro decomposition and the L p -L q estimates from the spectral analysis. The decay estimates thus obtained are applied to the study on the global existence of the Cauchy problem to the nonlinear Boltzmann equation with time dependent external force and source. Precisely, for space dimension n ≥ 3, the global existence and decay rates of solutions to the Cauchy problem are obtained under the condition that the force and source decay in time with some rates. This time decay restriction can be removed for space dimension n ≥ 5. Moreover, the existence and asymptotic stability of the time periodic solution are given for space dimension n ≥ 5 when the force and source are time periodic with the same period.

A general multiple-relaxation-time boltzmann collision model

Abstract: We formulate a simple extension to the Bhatnagar-Gross-Krook collision model by expanding the distribution function in Hermite polynomials and assigning a relaxation time to each hydrodynamic moment. By discretizing the velocity space, multiple-relaxation-time lattice Boltzmann models can be constructed. The transport coefficients are analytically calculated and numerically verified. At the lowest order, allowing different relaxation rates for the second and third Hermite components results in a variable Prandtl number. Comparing with the previously proposed multiple-relaxation-time lattice Boltzmann models, the present formulation is general in the sense that it is independent of the underlying lattice structure and does not require a procedure for transformation of base vectors.

Spectral - Lagrangian methods for Collisional Models of Non - Equilibrium Statistical States

Abstract: We propose a new spectral Lagrangian based deterministic solver for the non-linear Boltzmann Transport Equation for Variable Hard Potential (VHP) collision kernels with conservative or non-conservative binary interactions. The method is based on symmetries of the Fourier transform of the collision integral, where the complexity in its computing is reduced to a separate integral over the unit sphere $S^2$. In addition, the conservation of moments is enforced by Lagrangian constraints. The resulting scheme, implemented in free space is very versatile and adjusts in a very simple manner, to several cases that involve energy dissipation due to local micro-reversibility (inelastic interactions) or elastic model of slowing down process. Our simulations are benchmarked with the available exact self-similar solutions, exact moment equations and analytical estimates for homogeneous Boltzmann equation for both elastic and inelastic VHP interactions. Benchmarking of the simulations involves the selection of a time self-similar rescaling of the numerical distribution function which is performed using the continuous spectrum of the equation for Maxwell molecules as studied first in \cite{asympGranular} and generalized to a wide range of related models in \cite{BCG}. The method also produces accurate results in the case of inelastic diffusive Boltzmann equations for hard-spheres (inelastic collisions under thermal bath), where overpopulated non-Gaussian exponential tails have been conjectured in computations by stochastic methods in \cite{vanNoi-Ernst, ernst-brito, MSS, GRW04} and rigourously proven in \cite{diffGran} and \cite{highEnergyTails}.

Coherent Matter Wave Transport in Speckle Potentials

Abstract: This article studies multiple scattering of matter waves by a disordered optical potential in two and in three dimensions. We calculate fundamental transport quantities such as the scattering mean free path $\ell_s$, the Boltzmann transport mean free path $\elltrb$, and the Boltzmann diffusion constant $D_B$, using a diagrammatic Green functions approach. Coherent multiple scattering induces interference corrections known as weak localization which entail a reduced diffusion constant. We derive the corresponding expressions for matter wave transport in an correlated speckle potential and provide the relevant parameter values for a possible experimental study of this coherent transport regime, including the critical crossover to the regime of strong or Anderson localization.

Generalized kinetic equations for charge carriers in graphene

Abstract: A system of generalized kinetic equations for the distribution functions of two-dimensional Dirac fermions scattered by impurities is derived in the Born approximation with respect to short-range impurity potential. It is proven that the conductivity following from classical Boltzmann equation picture, where electrons or holes have scattering amplitude reduced due chirality, is justified except for an exponentially narrow range of chemical potential near the conical point. When in this range, creation of infinite number of electron-hole pairs related to quasi-relativistic nature of electrons in graphene results in a renormalization of minimal conductivity as compared to the Boltzmann term and logarithmic corrections in the conductivity similar to the Kondo effect.

Preparative Steps Towards the New Definition of the Kelvin in Terms of the Boltzmann Constant

Abstract: The International Committee for Weights and Measures (CIPM) approved, in its Recommendation 1 of 2005, preparative steps towards new definitions of the kilogram, the ampere, the kelvin, and the mole in terms of fundamental constants. Within the Consultative Committee for Thermometry (CCT), a task group (TG-SI) has been formed to consider the implications of changing the definitions of the above-mentioned base units of the SI, with particular emphasis on the kelvin and the impact of the changes on metrology in thermometry. The TG-SI has presented the results of its deliberations to the CCT and to the Consultative Committee for Units, CCU, and worked with them to prepare a report to the CIPM. This contribution, authored by the members of TG-SI, solicits input from the wider scientific and technical community on this important matter at the TEMPMEKO 2007 conference. For this purpose, the main details of the report to the CIPM are presented. The unit of temperature T, the kelvin, can be defined in terms of the SI unit of energy, the joule, by fixing the value of the Boltzmann constant k, which is simply the proportionality constant between temperature and thermal energy kT. Currently, several experiments are underway to determine k. The TG-SI is monitoring closely the results of all experiments relevant to the possible new definition of the kelvin, and has identified conditions to be met before proceeding with the proposed redefinition. The TG-SI considers that these conditions will be fulfilled before the 24th General Conference on Weights and Measures in October 2011. Therefore, the TG-SI is recommending a redefinition of the kelvin by fixing the value of the Boltzmann constant. A new definition of the kelvin in terms of the Boltzmann constant does not require the replacement of ITS-90 with an improved temperature scale nor does it prevent such a replacement.

Boltzmann–BGK approach to simulating weakly compressible 3D turbulence: comparison between lattice Boltzmann and gas kinetic methods

Abstract: Recently, the so-called gas-kinetic method (GKM) based on the Boltzmann-BGK equation has been successfully used in a variety of high Mach-number shock and laminar flow calculations. In this paper, we study the viability of extending the applicability of GKM to simulate turbulent flows. We evaluate the capability of GKM by making detailed comparisons against the lattice Boltzmann method and a Navier–Stokes solver. We perform three-dimensional direct numerical simulations (DNS) of decaying isotropic turbulence with the three methods and compare the evolution of kinetic energy, dissipation rate, and energy spectrum. Details of various macroscopic flow variables of interest are also examined. Further, the decay exponent calculated from the GKM is compared with the published results. The agreement in all categories considered is quite good. The results clearly demonstrate the potential promise of GKM for turbulence simulations over a broad range of Mach numbers.

Entropic lattice Boltzmann representations required to recover Navier-Stokes flows.

Abstract: There are two disparate formulations of the entropic lattice Boltzmann scheme: one of these theories revolves around the analog of the discrete Boltzmann H function of standard extensive statistical mechanics, while the other revolves around the nonextensive Tsallis entropy. It is shown here that it is the nonenforcement of the pressure tensor moment constraints that lead to extremizations of entropy resulting in Tsallis-like forms. However, with the imposition of the pressure tensor moment constraint, as is fundamentally necessary for the recovery of the Navier-Stokes equations, it is proved that the entropy function must be of the discrete Boltzmann form. Three-dimensional simulations are performed which illustrate some of the differences between standard lattice Boltzmann and entropic lattice Boltzmann schemes, as well as the role played by the number of phase-space velocities used in the discretization.

On the Cauchy problem of the Boltzmann and Landau equations with soft potentials

Abstract: , Global classical solutions near Maxwellians are constructed for the Boltzmann and Landau equations with soft potentials in the whole space. The construction of global solutions is based on refined energy analysis. Our results generalize the classical results in Ukai and Asano (Publ. Res. Inst. Math. Sci. 18 (1982), 477-519) to the very soft potentials for the Boltzmann equation and also extend the classical results in Caflisch (Comm. Math. Phys. 74 (1980), 97-107), Guo (Comm. Math. Phys. 231 (2002), 391-434), and Guo (Arch. Rat. Mech. Anal. 169 (2003), 305-353) in the periodic box to the whole space for the Boltzmann equation and the Landau equation in the Coulomb interaction.

A short review on the derivation of the nonlinear quantum Boltzmann equations

Abstract: In this review paper we describe the problem of deriving a Boltzmann equation for a system of N interacting quantum particles, under the appropriate scaling limits. We mainly follow the approach developed by the authors in previous works. From a rigorous viewpoint, only partial results are available, even for short times, so that the complete problem is still open. 1. Introduction A large quantum system of N identical interacting particles can often be described in terms of a Boltzmann equation. This is an asymptotic model: the equation given from flrst principles is the N body Schrodinger equation. As such, the Boltzmann description only holds in suitable regimes, namely when the number of particles is large, and when the interaction potential between pairs of particles has a small efiect. Concerning this last point, two quite difierent settings are relevant. In the so-called weak-coupling limit, the interaction potential itself is small, while the gas is dense: the typical distance between particles is of order one. In the low-density regime at variance, the elementary interaction potential is of order one, while the gas is rarefled: the typical distance between particles is large, hence the efiect of the pairwise interactions is small.

Molecular gas dynamics : theory, techniques, and applications

Abstract: Preface Boltzmann Equation Highly Rarefied Gas: Free Molecular Gas and Its Correction Slightly Rarefied Gas: Asymptotic Theory of the Boltzmann System for Small Knudsen Numbers Simple Flows Flows Induced by Temperature Fields Flows with Evaporation and Condensation Bifurcation in the Half-Space Problem of Evaporation and Condensation Ghost Effect and Bifurcation I: Benard and Taylor-Couette Problems Ghost Effect and Bifurcation II: Ghost Effect of Infinitesimal Curvature and Bifurcation of the Plane Couette Flow Appendix A. Supplement to the Boltzmann Equation Appendix B. Methods of Solution Appendix C. Some Data Bibliography List of Symbols Index

Tanaka Theorem for Inelastic Maxwell Models

Abstract: We show that the Euclidean Wasserstein distance is contractive for inelastic homogeneous Boltzmann kinetic equations in the Maxwellian approximation and its associated Kac-like caricature. This property is as a generalization of the Tanaka theorem to inelastic interactions. Even in the elastic classical Boltzmann equation, we give a simpler proof of the Tanaka theorem than the ones in [29, 31]. Consequences are drawn on the asymptotic behavior of solutions in terms only of the Euclidean Wasserstein distance.

Density matrix theory of transport and gain in quantum cascade lasers in a magnetic field

Abstract: A density matrix theory of electron transport and optical gain in quantum cascade lasers in an external magnetic field is formulated. Starting from a general quantum kinetic treatment, we describe the intraperiod and interperiod electron dynamics at the non-Markovian, Markovian, and Boltzmann approximation levels. Interactions of electrons with longitudinal optical phonons and classical light fields are included in the present description. The non-Markovian calculation for a prototype structure reveals a significantly different gain spectra in terms of linewidth and additional polaronic features in comparison to the Markovian and Boltzmann ones. Despite strongly controversial interpretations of the origin of the transport processes in the non- Markovian or Markovian and the Boltzmann approaches, they yield comparable values of the current densities.

Thermoelectric power calculation by the Boltzmann equation: Na(x)CoO(2).

Abstract: A full-potential augmented-plane-wave (FLAPW) band-structure calculation in the local density approximation (LDA) was carried out for hexagonal Na(x)CoO(2) (x = 0.45, 0.55, 0.66 and 0.75). The Seebeck tensor was estimated by the Boltzmann theory, assuming that the relaxation time is constant on the Fermi surface. The Seebeck tensor is extremely anisotropic; the c-axis Seebeck coefficient varies dramatically with the Na content. The calculation reproduces the experiment semiquantitatively.

Integral Representation of the Linear Boltzmann Operator for Granular Gas Dynamics with Applications

Abstract: We investigate the properties of the collision operator Q associated to the linear Boltzmann equation for dissipative hard-spheres arising in granular gas dynamics. We establish that, as in the case of non-dissipative interactions, the gain collision operator is an integral operator whose kernel is made explicit. One deduces from this result a complete picture of the spectrum of Q in an Hilbert space setting, generalizing results from T. Carleman (Publications Scientifiques de l’Institut Mittag-Leffler, vol. 2, 1957) to granular gases. In the same way, we obtain from this integral representation of Q + that the semigroup in L 1(ℝ3×ℝ3,dx ⊗ dv) associated to the linear Boltzmann equation for dissipative hard spheres is honest generalizing known results from Arlotti (Acta Appl. Math. 23:129–144, 1991).

Universality in a 2-component fermi system at finite temperature.

Abstract: Thermodynamic properties of a Fermi system close to the unitarity limit, where the 2-body scattering length a approaches {+-}{infinity}, are studied in the high temperature Boltzmann regime. For dilute systems the virial expansion coefficients in the Boltzmann regime are expected, from general arguments, to be universal. A model independent finite temperature T calculation of the third virial coefficient b{sub 3}(T) is presented. At the unitarity limit, b{sub 3}{sup {infinity}}{approx_equal}1.11 is a universal number. The energy density up to the third virial expansion is derived. These calculations are of interest in dilute neutron matter and could be tested in current atomic experiments on dilute Fermi gases near the Feshbach resonance.

Kinetic Approach to Long time Behavior of Linearized Fast Diffusion Equations

Abstract: We show that the rate of convergence towards the self-similar solution of certain linearized versions of the fast diffusion equation can be related to the number of moments of the initial datum that are equal to the moments of the self-similar solution at a fixed time. As a consequence, we find an improved rate of convergence to self-similarity in terms of a Fourier based distance between two solutions. The results are based on the asymptotic equivalence of a collisional kinetic model of Boltzmann type with a linear Fokker-Planck equation with nonconstant coefficients, and make use of methods first applied to the reckoning of the rate of convergence towards equilibrium for the spatially homogeneous Boltzmann equation for Maxwell molecules.

Mass and energy balance laws derived from High-Field limits of thermostatted Boltzmann equations

Abstract: We derive coupled mass and energy balance laws from a High-Field limit of thermostatted Boltzmann equations. The starting point is a Boltzmann equation for elastic collisions subjected to a large force field. By adding a thermostat correction, it is possible to expand the solutions about a High-Field equilibrium obtained when balancing the thermostatted field drift operator with the elastic collision operator. To this aim, a hydrodynamic type scaling of the thermostatted Boltzmann equation is used, considering that the leading 'collision operator' actually consists of the combination of the thermostatted field operator and of the elastic collision operator. At leading order in the Knudsen number, the resulting model consist of coupled nonlinear first order partial differential equations. We investigate two cases. The first one is based on a one-dimensional BGK-type operator. The second one is three dimensional and concerns a Fokker-Planck collision operator. In both cases, we show that the resulting models are hyperbolic, thereby indicating that they might be appropriate for a use in physically realistic situations.

Boltzmann's Legacy

Abstract: Some of the objects introduced by Boltzmann, entropy in the first place, have proven to be of great inspiration in mathematics, and not only in problems related to physics. I will describe, in a slightly informal style, a few striking examples of the beauty and power of the notions cooked up by Boltzmann.

Direct Solution of the Boltzmann Transport Equation and Poisson–SchrÖdinger Equation for Nanoscale MOSFETs

Abstract: We propose an efficient and fast algorithm to solve the coupled Poisson-Schrodinger and Boltzmann transport equations (BTE) in two dimensions. The BTE is solved in the relaxation time approximation within each subband obtained from the direct solution of the Schrodinger equation. The proposed approach, considering a subband-based transport formalism, allows to fully explore the entire range from drift-diffusion to ballistic regime in nanoscale field-effect transistors. Quantum effects are also fully taken into account by the direct solution of the Schrodinger equation. The model is implemented in the NanoTCAD2D device simulator and used to study the device performance of a 25-nm channel-length MOSFET. The influence of scattering on the electron distribution function and on device characteristics is analyzed in detail.

Theoretical Analysis of Microwave Attenuation Constant of Weakly Ionized Dusty Plasma

Abstract: The analytical formulation of complex conductivity and attenuation constant is deduced for weakly ionized dusty plasma by solving the Boltzmann and Shukla equations given by statistical theory. The concept of response factor of charging is proposed. The theoretical result is applied to calculating the microwave attenuation constant of a solid rocket plume, which is in fact a weakly ionized plasma containing dust particles. It is shown that the densities of both electrons and dust particles as well as the radius of dust particles affect the attenuation constant. The calculated results agree with the observations in experiments.

Dynamic Boltzmann free-energy functional theory

Abstract: We present a generalization of the Density Functional Theory to distributions in μ-space rather than in configuration space. This equilibrium theory is the basic ingredient for constructing a dynamic theory with projection operators. The reversible part of the dynamics is computed exactly while the irreversible part is approximated with a fast momentum relaxation assumption. As a result the irreversible operator is given in terms of a viscosity tensor. We show that the kinetic equation has an H-theorem.

Boltzmann distribution and market temperature

Abstract: We show that the minute fluctuations of S&P 500 and NASDAQ 100 indices show Boltzmann statistics over a wide range of positive as well as negative returns, thus allowing us to define a market temperature for either sign. With increasing time the sharp Boltzmann peak broadens into a Gaussian whose volatility σ measured in 1 / min is related to the temperature T by T = σ / 2 . Plots over the years 1990–2006 show that the arrival of the 2000 crash was preceded by an increase in market temperature, suggesting that this increase can be used as a warning signal for crashes.