Showing papers on "Boltzmann constant published in 1997"

Stochastic particle approximations for generalized Boltzmann models and convergence estimates

Abstract: We specify the Markov process corresponding to a generalized mollified Boltzmann equation with general motion between collisions and nonlinear bounded jump (collision) operator, and give the nonlinear martingale problem it solves. We consider various linear interacting particle systems in order to approximate this nonlinear process. We prove propagation of chaos, in variation norm on path space with a precise rate of convergence, using coupling and interaction graph techniques and a representation of the nonlinear process on a Boltzmann tree. No regularity nor uniqueness assumption is needed. We then consider a nonlinear equation with both Vlasov and Boltzmann terms and give a weak pathwise propagation of chaos result using a compactness-uniqueness method which necessitates some regularity. These results imply functional laws of large numbers and extend to multitype models. We give algorithms simulating or approximating the particle systems.

Diffusive limit for finite velocity Boltzmann kinetic models

Abstract: We investigate, in the diffusive scaling, the limit to the macroscopic description of finite-velocity Boltzmann kinetic models, where the rate coefficient in front of the collision operator is assumed to be dependent of the mass density. It is shown that in the limit the flux vanishes, while the evolution of the mass density is governed by a nonlinear parabolic equation of porous medium type. In the last part of the paper we show that our method adapts to prove the so-called Rosseland approximation in radiative transfer theory.We investigate, in the diffusive scaling, the limit to the macroscopic description of finite-velocity Boltzmann kinetic models, where the rate coefficient in front of the collision operator is assumed to be dependent of the mass density. It is shown that in the limit the flux vanishes, while the evolution of the mass density is governed by a nonlinear parabolic equation of porous medium type. In the last part of the paper we show that our method adapts to prove the so-called Rosseland approximation in radiative transfer theory.

Modelling plasma discharges at high electronegativity

Abstract: Macroscopic models for the equilibrium of a three-component electronegative gas discharge are developed. Assuming the electrons and the negative ions to be in Boltzmann equilibrium, a positive ion ambipolar diffusion equation is derived. Such a discharge can consist of an electronegative core and may have electropositive edge regions, but the electropositive regions become small for the highly electronegative plasma considered here. In the parameter range for which the negative ions are Boltzmann, the electron density in the core is nearly uniform, allowing the nonlinear diffusion equation to be solved in terms of elliptic integrals. If the loss of positive ions to the walls dominates the recombination loss, a simpler parabolic solution can be obtained. If recombination loss dominates the loss to the walls, the assumption that the negative ions are in Boltzmann equilibrium is not justified, requiring coupled differential equations for positive and negative ions. Three parameter ranges are distinguished corresponding to a range in which a parabolic approximation is appropriate, a range for which the recombination significantly modifies the ion profiles, but the electron profile is essentially flat, and a range where the electron density variation influences the solution. The more complete solution of the coupled ion equations with the electrons in Boltzmann equilibrium, but not at constant density, is numerically obtained and compared with the more approximate solutions. The theoretical considerations are illustrated using a plane parallel discharge with chlorine feedstock gas of p = 30, 300 and 2000 mTorr and , corresponding to the three parameter regimes. A heuristic model is constructed which gives reasonably accurate values of the plasma parameters in regimes for which the parabolic profile is not adequate.

Moment inequalities for the boltzmann equation and applications to spatially homogeneous problems

Abstract: Some inequalities for the Boltzmann collision integral are proved. These inequalities can be considered as a generalization of the well-known Povzner inequality. The inequalities are used to obtain estimates of moments of the solution to the spatially homogeneous Boltzmann equation for a wide class of intermolecular forces. We obtain simple necessary and sufficient conditions (on the potential) for the uniform boundedness of all moments. For potentials with compact support the following statement is proved: if all moments of the initial distribution function are bounded by the corresponding moments of the MaxwellianA exp(−Bv2), then all moments of the solution are bounded by the corresponding moments of the other MaxwellianA1 exp[−B1(t)v2] for anyt > 0; moreoverB(t) = const for hard spheres. An estimate for a collision frequency is also obtained.

Relaxation Schemes for Nonlinear Kinetic Equations

Abstract: A class of numerical schemes for nonlinear kinetic equations of Boltzmann type is described. Following Wild's approach, the solution is represented as a power series with parameter depending exponentially on the Knudsen number. This permits us to derive accurate and stable time discretizations for all ranges of the mean free path. These schemes preserve the main physical properties: positivity, conservation of mass, momentum, and energy. Moreover, for some particular models, the entropy property is also shown to hold.

On boltzmann equations and fokker—planck asymptotics: Influence of grazing collisions

Abstract: In this paper, we are interested in the influence of grazing collisions, with deflection angle near π/2, in the space-homogeneous Boltzmann equation. We consider collision kernels given by inverse-sth-power force laws, and we deal with general initial data with bounded mass, energy, and entropy. First, once a suitable weak formulation is defined, we prove the existence of solutions of the spatially homogeneous Boltzmann equation, without angular cutoff assumption on the collision kernel, fors ≥ 7/3. Next, the convergence of these solutions to solutions of the Landau-Fokker-Planck equation is studied when the collision kernel concentrates around the value π/2. For very soft interactions, 2

Adiabatically changing the phase-space density of a trapped bose gas

Abstract: We show that the degeneracy parameter of a trapped Bose gas can be changed adiabatically in a reversible way, both in the Boltzmann regime and in the degenerate Bose regime. We have performed measurements on spin-polarized atomic hydrogen in the Boltzmann regime, demonstrating reversible changes of the degeneracy parameter (phase-space density) by more than a factor of 2. This result is in good agreement with theory. By extending our theoretical analysis to the quantum degenerate regime we predict that, starting close enough to the Bose-Einstein phase transition, one can cross the transition by an adiabatic change of the trap shape. [S0031-9007(97)02357-0] PACS numbers: 03.75.Fi, 67.65.+z, 32.80.Pj

Quantum kinetic theory. ii. simulation of the quantum boltzmann master equation

Abstract: We present results of simulations of a quantum Boltzmann master equation (QBME) describing the kinetics of a dilute Bose gas confined in a trapping potential in the regime of Bose condensation. The QBME is the simplest version of a quantum kinetic master equation derived in previous work. We consider two cases of trapping potentials: a three-dimensional square-well potential with periodic boundary conditions and an isotropic harmonic oscillator. We discuss the stationary solutions and relaxation to equilibrium. In particular, we calculate particle distribution functions, fluctuations in the occupation numbers, the time between collisions, and the mean occupation numbers of the one-particle states in the regime of onset of Bose condensation.

2-D MOSFET modeling including surface effects and impact ionization by self-consistent solution of the Boltzmann, Poisson, and hole-continuity equations

Abstract: We present a new two-dimensional (2-D) MOSFET simulation method achieved by directly solving the Boltzmann transport equation for electrons, the hole-current continuity equation, and the Poisson equation self-consistently. The spherical harmonic method is used for the solution of the Boltzmann equation. The solution directly gives the electron distribution function, electrostatic potential, and the hole concentration for the entire 2-D MOSFET. Average quantities such as electron concentration and electron temperature are obtained directly from the integration of the distribution function. The collision integral is formulated to arbitrarily high spherical harmonic order, and new collision terms are included that incorporate effects of surface scattering and electron-hole pair recombination/generation. I-V characteristics, which agree with experiment, are calculated directly from the distribution function for an LDD submicron MOSFET. Electron-hole pair generation due to impact ionization is also included by direct application of the collision integral. The calculations are efficient enough for day-to-day engineering design on workstation-type computers.

Nonequilibrium positive column

Abstract: The dc positive column is modeled with a system of balance equations based on moments of the radially dependent Boltzmann equation taken after the two-term Legendre expansion of the electron energy distribution function is made The importance of the electron energy balance equation, which is frequently ignored in positive column analysis, is emphasized A key assumption is that electron transport coefficients and collision frequencies in the nonequilibrium regime have the same relation to the average energy as in the equilibrium regime, according to a zero-dimensional Boltzmann solution for a particular value of average energy Because of this assumption, the model makes a smooth transition to the traditional equilibrium model with radially constant average energy at sufficiently high pressure Model results in the nonequilibrium regime agree closely with published results of a numerical solution of the one-dimensional Boltzmann equation, including results for radial heat flow in the electron gas with radially varying average energy It is shown that three separate processes account for radial heat flow: convection, conduction, and diffusion In the example chosen for illustration of the method, the convection component is small, while the conduction and diffusion components are large and opposite in direction, nearly canceling each other {copyright}more » {ital 1997} {ital The American Physical Society}« less

Measurement and analysis of elastic and anelastic properties of alumina and silicon carbide

Abstract: Measurements of dynamic Young's modulus, E, and damping as a function of temperature, T, were made for alumina and silicon carbide. The Young's modulus data were compared with some from the literature, and analysed in terms of a theoretical framework relating the Debye temperature, θD, with the elastic constants. For both materials this analysis yielded a ratio T0/θD which was near 0.4, where T0 is an empirical fitting constant for the plot of (E(0)−E)/T versus 1/T (E(0) is the value of E at 0 K). The analysis of the damping data in terms of an Arrhenius type dependence led to effective activation energies near kT, where k is Boltzmann's constant.

Numerical Analysis of Levermore's Moment System

Abstract: The simulation of transitional gas flow near rarefied regimes requires new models which can describe kinetic effects but which are less complex than the original Boltzmann equation. The strategy introduced by Levermore [1] rewrites the Boltzmann equations as a system of moment equations, with a new closure procedure. We recall here the mathematical properties of this Levermore's moment systems. Boundary conditions derived from the kinetic theory are proposed. Based on these properties, we present an original first-order kinetic scheme with explicit flux splitting and implicit source terms. A 14-moments system model is tested to simulate one-dimensional gas flows (Couette flow and normal shock wave) near the transitional regime.

Boltzmann asymptotics with diffuse reflection boundary conditions

Abstract: StrongL 1-convergence towards a stationary solution when time tends to infinity is established for the solutions of the time-dependent nonlinear Boltzmann equation in a bounded domain Ω ⊂ ℝ3 with constant temperature on the boundary. The collisionless case is first investigated in the varying temperature case.

GENERIC Formulation of Boltzmann's Kinetic Equation

Abstract: We express Boltzmann's kinetic equation in the form of the recently proposed General Equation for the Non-Equilibrium Reversible-Irreversible Coupling (GENERIC). This GENERIC formulation demonstrates that no dissipative potential is required for representing Boltzmann's kinetic equation in a general framework for non-equilibrium systems.

The consistent Boltzmann algorithm for the van der Waals equation of state

Abstract: The direct-simulation Monte Carlo method is generalized by introducing an advection displacement that models a hard-core exclusion with a weak and constant interparticle attraction. Simulation results demonstrate that both the van der Waals equation of state and its Maxwell tie-line construction can be obtained.

Convergence of Discrete-Velocity Schemes for the Boltzmann Equation

Abstract: In this paper we prove the convergence of two discrete-velocity deterministic schemes for the Boltzmann equation, namely, Buet's scheme and a new finite-volume scheme that we introduce here. We write the discretized equation in the form of a Boltzmann continuous equation in order to be in the framework of the DiPerna-Lions theory of renormalized solutions. In order to prove convergence we have to overcome two difficulties: the convergence of the discretized collision kernel is very weak and the lemma on the compactness of velocity averages can be recovered only asymptotically when the parameter of discretization tends to zero.

Approximate Solution of the Enskog Equation Far from Equilibrium

Abstract: The hard-sphere potential has been the most intensively studied interaction model in many areas of statistical mechanics for at least two reasons. The first reason is that hard-spheres exhibit many phenomena observed in real systems, such as the existence of liquid, solid, and metastable phases, and indeed provide a good first approximation for real systems of such properties as liquid structure, transport properties, and both liquid- and solid-phase thermodynamic properties. The second reason is that the hard-sphere interaction is the only one for which a tractable kinetic equation applicable at moderate densities, the Enskog equation [1‐ 3], exists. The Enskog equation was originally proposed on physical grounds as a finite-density generalization of the Boltzmann equation which, although applicable to arbitrary two-body interaction models, is restricted to low densities. The generalization of the Enskog equation by van Beijeren and Ernst [4] is capable of giving a unified description of liquid, solid, and metastable states [5]. However, because of the complexity of the Enskog equation, the only analytic solutions available are perturbative and recently developed numerical techniques [6] have yet to be widely applied so that little is known about its description of systems far from equilibrium. The standard method of analyzing either the Boltzmann or the Enskog equation is the Chapman-Enskog expansion [2,3] which is a perturbative expansion of the one-body distribution function, and the kinetic equation describing it, in terms of the uniformity of the system. For example, in a fluid undergoing uniform shear flow (USF), in which the local macroscopic flow velocity along the x axis varies linearly with position along the y axis, $ us$ rd › ay ˆ x, this amounts to an expansion in powers of the shear rate, a. This expansion has only been performed in the general case to third order in the uniformity parameter and the range of validity of such results is presumably limited to near-equilibrium states; the analytic complexity of the Chapman-Enskog procedure has proven prohibitive of the study of higher-order effects. An alternative method of analysis of the Boltzmann equation is the moment method of Grad [2,7,8] according to which the distribution is expressed as an expansion in terms of velocity about local equilibrium. Keeping all terms in the expansion gives an infinite set of coupled equations for the generally spaceand time-dependent coefficients which, assuming the validity of the expansion, is equivalent to the Boltzmann equation. Approximations are then introduced to truncate or decouple the equations allowing for an approximate solution. The method has found particular use in the study of small-wavelength hydrodynamics near equilibrium [9] where the close connection between the moment method and kinetic models has been exploited. The purpose of this Letter is to show that the method may be used to obtain good approximate solutions to the Enskog equation for systems far from equilibrium. In the following, attention is focused on USF since sheared fluids may be reliably simulated and the connection between theory and simulation is well understood. Indeed, for these reasons, USF has been the subject of numerous investigations over the last 20 years (see, e.g., Ref. [10]), and is often viewed as a prototypical nonequilibrium state free from complicating features such as boundary effects. In addition, it has been known for some time that the hard-sphere system is unstable at high shear rates [11] and theories of the instability depend on knowledge of the one-body distribution [12,13]. Since only perturbative results, which lack important physical effects such as shear thinning, have been available, these theories remain tentative. One of the motivations for the present work has been to allow for a more detailed study of this problem. Both the Boltzmann and Enskog equations may be written in the form

Nonlinear transport for a dilute gas in steady Couette flow

Abstract: Transport properties of a dilute gas subjected to arbitrarily large velocity and temperature gradients (steady planar Couette flow) are determined. The results are obtained from the so-called ellipsoidal statistical (ES) kinetic model, which is an extension of the well-known BGK kinetic model to account for the correct Prandtl number. At a hydrodynamic level, the solution is characterized by constant pressure, and linear velocity and parabolic temperature profiles with respect to a scaled variable. The transport coefficients are explicitly evaluated as nonlinear functions of the shear rate. A comparison with previous results derived from a perturbative solution of the Boltzmann equation as well as from other kinetic models is carried out. Such a comparison shows that the ES predictions are in better agreement with the Boltzmann results than those of the other approximations. In addition, the velocity distribution function is also computed. Although the shear rates required for observing non-Newtonian effects are experimentally unrealizable, the conclusions obtained here may be relevant for analyzing computer results.

Boltzmann-grad limit for a particle system in continuum

Abstract: We examine a system of one-dimensional particles in which the particles travel deterministically in between stochastic collisions. The collision rates are chosen so that finitely many collisions occur in a unit interval of time. We prove the kinetic limit and subsequently derive the discrete Boltzmann equation.

Space homogeneous solutions to the Cauchy problem for semiconductor Boltzmann equations

Abstract: The nonlinear Boltzmann equation for an electron gas in a semiconductor is investigated. Some meaningful properties of the collision operator are first presented. A large class of kernels is allowed. Then the global existence and uniqueness of bounded, continuous, space-independent solutions to the related Cauchy problem is performed. Finally, the conservation of mass is examined.

A particle method with adjustable transport properties : the generalized consistent boltzmann algorithm

Abstract: The consistent Boltzmann algorithm (CBA) for dense, hard-sphere gases is generalized to obtain the van der Waals equation of state and the corresponding exact viscosity at all densities except at the highest temperatures. A general scheme for adjusting any transport coefficients to higher values is presented.

Conservative and Entropy Schemes for Boltzmann Collision Operator of Polyatomic Gases

Abstract: We propose two discrete velocity models derived from the Boltzmann equation of Larsen–Borgnakke type for polyatomic gases. These two models are natural extensions of previously discussed discrete velocity models used for monoatomic gases. These two models have the same properties as the continuous one, which are conservation of mass, momentum and energy, discrete Maxwellians as equilibrium states and H-theorems.

Statistical-mechanically exact simulation of polymer conformation in an external field

Abstract: A mathematically exact method is presented for sampling conformations of polymer molecules in an external field with fixed energy or energy range in accord with the formulation of statistical mechanics for a microcanonical ensemble. As a consequence, conformations of negligible Boltzmann weight can be selectively eliminated from simulations for efficient calculation of macroscopic polymer properties. The method is applicable for conformations that are described by a stochastic differential equation along the contour length in the field-free situation. It is based on the concept of a stochastic bridge process for which a new stochastic differential equation is derived that has stipulations at both ends of the process. This idea is exploited on a pair of stochastic differential equations in the conformation vector X and an augmented variable Z which represents the running Boltzmann weight in the given field, transforming to a new pair of equations for which the terminal Boltzmann weight can be arbitrarily s...