Boltzmann System Research Articles

Effect of a temperature difference between the cylinders on the Taylor–Couette bifurcation in a gas

A gas in a time-independent state between rotating coaxial circular cylinders with different temperatures is considered. The effect of the temperature difference on the Taylor–Couette bifurcation in the continuum limit is studied on the basis of the fluid-dynamic-type equations and their boundary conditions derived from the Boltzmann system. The behavior of the bifurcation point for a wide range of temperature differences is presented, and the mechanism of the shift of the bifurcation point owing to a small temperature difference is clarified analytically. The effect of the temperature difference on the bifurcation cannot be explained correctly by the guess by the analogy with the Bénard problem.

Ghost effect and bifurcation in a gas between coaxial circular cylinders with different temperatures

A gas in a time-independent state between rotating coaxial circular cylinders with different temperatures is considered. The bifurcation of the field in the continuum limit is studied on the basis of the system of fluid-dynamic-type equations and their boundary conditions derived from the Boltzmann system [Phys. Fluids 8, 628 (1996)]. When the ratio of the temperatures of the two cylinders is not close to unity, the bifurcation occurs at infinitesimal speeds of rotation of the cylinders of the first order of the Knudsen number. The temperature field in the continuum limit is determined together with the infinitesimal velocity field, and a bifurcated temperature field, as well as an axially uniform and symmetric field, exists in the absence of the flow velocity (ghost effect). Further, thermal stress, as well as viscous stress, produces a finite effect on the bifurcated temperature field (a non-Navier–Stokes effect). For example, when the ratio of the temperatures of the two cylinders is ten or one tenth, the non-Navier–Stokes effect on the bifurcated temperature field amounts to 10%.

SELF-CONSISTENT MODELING OF MOSFET QUANTUM EFFECTS BY SOLVING THE SCHRÖDINGER AND BOLTZMANN SYSTEM OF EQUATIONS

A new method for investigating quantum confinement effects in MOSFETs is presented. The method is based on the numerical solution of the Schrödinger and Boltzmann system of equations. A quantum Boltzmann equation is developed by replacing the classical potential with a quantum potential. The technique naturally accounts for highly nonequilibrium effects including velocity overshoot. The subbands are calculated and then populated using nonequilibrium statistics. Modeling results show the electron concentration and electron current density peaks shifted away from the interface. Calculations show reduced channel concentration which leads to lower drain current for the quantum case. A quantum distribution function is obtained.

Bifurcation of and ghost effect on the temperature field in the Bénard problem of a gas in the continuum limit

A gas in a time-independent state under a uniform weak gravity in a general domain is considered. The asymptotic behavior of the gas in the limit that the Knudsen number of the system tends to zero (or in the continuum limit) is investigated on the basis of the Boltzmann system for the case where the flow velocity vanishes in this limit, and the fluid-dynamic-type equations and their associated boundary conditions describing the behavior of the gas in the continuum limit are derived. The equations, different from the Navier–Stokes ones, contain thermal stress and infinitesimal velocity amplified by the inverse of the Knudsen number. The system is applied to analysis of the behavior of a gas between two parallel plane walls heated from below (Bénard problem), and a bifurcated strongly distorted temperature field is found in infinitesimal velocity and gravity. This is an example showing that the Navier–Stokes system fails to describe the correct behavior of a gas in the continuum limit.

Spectral problem for dilute atomic gases using discrete Uhling-Uhlenbeck operators.

Effects of free orientations (theta which is related to the relative direction of scattering of particles with respect to the normal of the propagating plane-wave front) using four-velocity model for the dispersion relationship of ultrasound propagation in dilute atomic (hard-sphere particles of Bose and Fermi statistics) gases are presented. We address the dispersion relations thus obtained by the relevant parameter rho (a blocking factor) or B which describes the Bose and Fermi particles for the quantum analog of the discrete Boltzmann system when B is positive (and rho=1) and negative (and rho=-1), respectively. The preliminary results show that there is no attenuation when the parameter theta equals pi/4 for all Bs.

The Vlasov-Poisson-Boltzmann System Near Vacuum

Global classical solutions with small amplitude are constructed for the Cauchy problem to the Vlasov–Poisson–Boltzmann system, which describes the dynamics of charged particles interacting with their self-consistent electrostatic potential as well as with themselves through collisions.

Asymptotic theory of the Boltzmann system, for a steady flow of a slightly rarefied gas with a finite Mach number: General theory

A steady rarefied gas flow with Mach number of the order of unity around a body or bodies is considered. The general behaviour of the gas for small Knudsen numbers is studied by asymptotic analysis of the boundary-value problem of the Boltzmann equation for a general domain. The effect of gas rarefaction (or Knudsen number) is expressed as a power series of the square root of the Knudsen number of the system. A series of fluid-dynamic type equations and their associated boundary conditions that determine the component functions of the expansion of the density, flow velocity, and temperature of the gas is obtained by the analysis. The equations up to the order of the square root of the Knudsen number do not contain non-Navier–Stokes stress and heat flow, which differs from the claim by Darrozes (in Rarefied Gas Dynamics, Academic Press, New York, 1969). The contributions up to this order, except in the Knudsen layer, are included in the system of the Navier–Stokes equations and the slip boundary conditions consisting of tangential velocity slip due to the shear of flow and temperature jump due to the temperature gradient normal to the boundary.

On the Initial Boundary Value Problem for the Vlasov-Poisson-Boltzmann System

We prove existence of DiPerna–Lions renormalized solutions to the Boltzmann equation and to the Vlasov–Poisson–Boltzmann system for the initial boundary value problem.

Computer simulation of DNA interacting electrostatically with phosphatidylcholine and trimethylammoniumpropane interfaces

The region of a lipid membrane, with all hydrocarbon chains equal, embedded in an aqueous solution and interacting with a single DNA molecule, has been modelled. The lattice model used assigned an area characteristic of a lipid molecule in a gel phase to each lattice site. In a fluid phase two lipid molecules occupy three sites. We studied a membrane composed of lipids with phosphatidylcholine (PC) and trimethylammoniumpropane (TAP) headgroups. Lipid headgroup states were enumerated as described elsewhere (Pink et al., Biochim. Biophys. Acta, 1988, 1368, 289). Here charged lipid moieties were represented by point charges inside an excluded volume. The aqueous solution was modelled as a linearized Poisson–Boltzmann system characterized by a Debye screening length, κ−1. We employed standard Monte Carlo computer simulation techniques. We came to the following conclusions. (a) In the absence of DNA, PC and TAP headgroup pairs formed dynamic bound states in a gel phase. These did not occur if the PC was represented as an object carrying no charges. Accordingly, although PC carries zero net charge, it is important to represent the charged moieties explicitly. The gel–fluid phase transition in a 1:1 PC–TAP membrane (with equal hydrocarbon chain lengths) might thus involve not only hydrocarbon chain disordering but also the break-up of the dynamical PC–TAP bound pairs. (b) Increasing TAP concentration resulted in changing the orientation of the PC dipole. (c) DNA binding is a complex process and can involve weak binding even to a pure PC interface (which could, however, be disrupted by membrane undulations not modelled here) and tighter binding when TAP is present. The minimum concentration for the latter depends upon κ. DNA binding at low TAP concentrations would be changed if the PC was represented by a chargeless object. (d) A bound DNA changes the headgroup orientation in its proximity and also results in increased headgroup lateral packing in the immediate neighbourhood of the DNA. The latter could result in denser lateral hydrocarbon chain packing in the neighbourhood of the DNA. Both of these phenomena exhibited a dependence upon the TAP concentration. (e) The average lipid–DNA binding energies per DNA PO2− group can be as large as 2kBT or more depending upon the TAP concentration. (f ) DNA can be unbound by changing the ionic concentration. We compare our results with experimental data and other simulations.

The Cauchy Problem for the Einstein–Boltzmann System

We prove an existence and uniqueness of solutions for the Einstein–Boltzmann system locally in time. We restrict our attention to initial data which gives a solution in harmonic coordinates.

Stationary Solutions of Boundary Value Problems for a Maxwell–Boltzmann System Modelling Degenerate Semiconductors

One of the most accurate models for carrier transport in semiconductors is based on the Maxwell-Boltzmann system. Degeneracy effects are taken into account by the nonlinearity of the collisions operator. We use two recent techniques developed for the study of kinetic models, upper solutions and mean compactness results, to prove existence of stationary solutions with arbitrary large boundary data, in any kind of geometries.

Perturbation theory based on the Einstein–Boltzmann system. I. Illustration of the theory for a Robertson–Walker geometry

This article is a discussion of how it is possible to do perturbation theory for the Einstein–Boltzmann system about a dust solution. Explicitly expressed, the background Robertson–Walker universe model and one-parameter families of exact solutions are applied to the Einstein–Boltzmann system to obtain a closed-form solution of the equations governing linearized perturbations at late times, when the nonzero pressure of the gas of massive particles may be regarded as being significantly smaller than the energy density. After splitting the distribution function into two structurally different parts, the analysis given here provides a means of deriving the equations of linear hydrodynamics. In fact, due to the specific properties of the background chosen, one can prove that the evolution of suitably defined hydrodynamic variables is exactly decoupled from the evolution of the phase-space function satisfying the linearized Boltzmann equation. For simplicity, the workings of the method are illustrated by assuming that the perturbed metric is also of the Robertson–Walker form. A detailed treatment of the effect of inhomogeneities in an almost-Robertson–Walker universe model will be the subject of the last article in this series.

Perturbation theory based on the Einstein–Boltzmann system. II. Illustration of the theory for an almost-Robertson–Walker geometry

This is the second in a pair of articles, the overall objective of which is to describe within the framework of the Einstein–Boltzmann system a self-consistent perturbation method which leads to a tractable set of integrodifferential equations for the rate of change of the metric and the distribution function. The main purpose here is to prove that, for cases where the pressure of the gas of massive particles vanishes in the background, the treatment of the Einstein–Boltzmann system by means of a suitable perturbation method automatically produces a complete scheme of hydrodynamics, consisting of a closed set of partial differential equations for the evaluation of the mean velocity, the mass density, the temperature or the pressure, and the metric. The growing hydrodynamic modes are systematically derived for an almost-Robertson–Walker universe model, and the calculations are proposed without making any restrictions on the form of the perturbed metric. To summarize, the present article suggests a scheme of hydrodynamics for the late stages of cosmic expansion and calls attention to the support and interpretation given by the general-relativistic kinetic theory of monatomic gases to this scheme. Comparison with the predictions of the Eckart and/or Landau–Lifshitz theories of dissipative fluids is also briefly presented.

INITIAL-BOUNDARY VALUE PROBLEMS FOR THE BOLTZMANN SYSTEM OF MOMENT EQUATIONS IN AN ARBITRARY APPROXIMATION

Both exterior and interior boundary conditions are formulated for the Boltzmann system of moment equations, and it is proved that these boundary conditions are dissipative. It is proved that the initial-boundary value problem is solvable for the Boltzmann system of moment equations, and that the moment method converges. The boundary condition of general form is approximated. Bibliography: 22 titles.

On the Boltzmann system for a mixture of reacting gases

Exact analytical solutions to the nonlinear spatially homogeneous integro-partiaS differential Boltzmann system, governing the distribution functions of the rarefied gases of a given mixture in the presence of scattering, removal and chemical reaction effects, are derived upon the hypothesis of constant collision frequencies, BGK-type scattering and creation laws, and particles possessing only translational energy. Four classes of second-order chemical reactions are investigated together with the relevant continuity system for the number densities of the participating species.

A system of conservation equations arising in nonlinear dynamics of gas mixtures

On the basis of appropriate hypotheses, concerning the collision frequencies, the scattering probability distributions and the initial conditions, a system ofN conservation equations is derived starting from the Boltzmann system governing a nonlinear evolution problem for theN gases of an assigned mixture. The physical and mathematical constraints connected with the solution of the system so obtained are then discussed for the two cases of only removal and, respectively, of both removal and scattering between the different species of particles, which the mixture considered consists of.

Extended kinetic theory for gas mixtures in the presence of removal and regeneration effects

The continuity system, governing the number densitiesϱ1(t),ϱ2(t),...,ϱN(t) for a mixture ofN gases of interacting particles, is first derived from the corresponding nonlinear Boltzmann system for the relevant distribution functionsf1(¯v, t),f2(¯v, t),...fN(¯v, t), and then solved analytically in several meaningful cases forN=2 on the basis of the theory of nonlinear second-order ordinary differential equations.

On the dynamics of the open bose gas: E. Buffet, Department of Mathematical Physics, University College, Belfield, Dublin, 4, Ireland; Ph. de Smedt, Instituut voor Theoretische Fysica, Universiteit Leuven, B-3030 Leuven, Belgium; and J. V. Pulè, Dublin Institute for Advanced Studies, 10 Burlington Road, Dublin, 4, Ireland

In this paper we study the Einstein–Boltzmann system with Bianchi I symmetry. We show that for small initial data the corresponding solutions of the Einstein–Boltzmann system are future geodesically complete and that they isotropize and have a dust-like behaviour at late times. Detailed information about the metric and the matter terms is obtained, and the results show that the solutions tend asymptotically to the Einstein–de Sitter solution.

Collisional quenching of electronically excited chlorine atoms, Cl[3p 5(2 P 1/2)], by atmospheric gases studied by time-resolved atomic resonance absorption spectroscopy in the vacuum ultraviolet

The collisional quenching of electronically excited chlorine atoms, Cl[3p5(2P1/2)], has been studied by time-resolved atomic resonance absorption spectroscopy in the vacuum ultraviolet at λ= 136.34 nm {Cl[3p4 4s(2P3/2)]â†� Cl[3p5(2P01/2)]}. Cl(3 2P1/2) was generated by the repetitive pulsed irradiation of CCl4 in the presence of excess helium buffer gas. The excited atom was then monitored photoelectrically using signal averaging in the short-time domain before the onset of Boltzmann equilibrium. The following absolute second-order rate constants for the removal of Cl(3 2P1/2), 882 cm–1 above the 3 2P3/2 ground state, are reported (kQ/cm3 molecules–1 s–1, 300 K, errors 1 σ) for a wide range of gases of atmospheric interest: N2, (6.3 ± 1.0)× 10–13; CO2, <5 × 10–13; O2, (2.3 ± 0.3)× 10–11; N2O, (3.7 ± 0.6)× 10–13; H2, <6 × 10–13; H2O, (2.6 ± 0.5)× 10–12; HCl, (1.1 ± 0.1)× 10–12; CH4, (3.9 ± 0.8)× 10–12; CO, ca. 6 × 10–12. These rate data are compared with the collisional-quenching rate constants for the other gases that we have reported previously using this technique which is, at present, the only practical method for monitoring Cl(3 2P1/2) out of Boltzmann equilibrium for fundamental reasons discussed in this paper. General considerations of the chemistry of a Boltzmann system of Cl(3 2PJ) and the roles of the specific spin–orbit states, Cl(3 2P1/2) and Cl(3 2P3/2), are presented with special reference to CH4 and the atmospheric gases in general. Relative rate data for the quenching of the transient precursor, generated from the photolysis of CCl4 and from which Cl(3 2P1/2) is derived, are also presented for most of the gases studied.

The divergence of the partition function of the attractive Frisch-Lloyd model of disordered systems

The path-integral method is used to calculate the partition function of a particle moving in a one-dimensional Frisch-Lloyd disordered system. It is found that the partition function is divergent. The fact that the partition function does not exist for this model disordered system therefore implies that the famous Frisch-Lloyd model for attractive potentials does not represent a stable Boltzmann system. A physical argument for the causes of divergence is outlined.