Modified Boltzmann Equation Research Articles

Self-similar Profiles for Homoenergetic Solutions of the Boltzmann Equation for Non-cutoff Maxwell Molecules

We consider a modified Boltzmann equation which contains, together with the collision operator, an additional drift term which is characterized by a matrix A. Furthermore, we consider a Maxwell gas, where the collision kernel has an angular singularity. Such an equation is used in the study of homoenergetic solutions to the Boltzmann equation. Under smallness assumptions on the drift term, we prove that the longtime asymptotics is given by self-similar solutions. We work in the framework of measure-valued solutions with finite moments of order p>2 and show existence, uniqueness and stability of these self-similar solutions for sufficiently small A. Furthermore, we prove that they have finite moments of arbitrary order if A is small enough. In addition, the singular collision operator allows to prove smoothness of these self-similar solutions. Finally, we study the asymptotics of particular homoenergetic solutions. This extends previous results from the cutoff case to non-cutoff Maxwell gases.

On Solutions of the Modified Boltzmann Equation

On Solutions of the Modified Boltzmann Equation

Optical emission spectroscopy study of plasma parameters in low-pressure hollow cathode plasma jet and planar magnetron powered by DC and pulsed DC supply

Optical emission spectroscopy (OES) has become a non-intrusive and versatile method for plasma parameters determination. Each spectral line in emission spectroscopy corresponds to an optical transition between two quantum levels of atom/molecule. The spectral line intensities are controlled by densities of species in various upper states. We calculated the electron temperature (Te) and electron density (ne) in low-pressure discharge systems based on the modified Boltzmann equation and line intensity ratios. We have evaluated the mentioned plasma parameters in the low-pressure hollow cathode plasma jet (HC) and planar magnetron cathode system (PM) in the continuous DC and the pulsed-DC regime. We present the dependences of the electron temperature (Te) and electron density (ne) on the applied power and the argon flow rate in the mentioned two systems together with a qualitative discussion of the obtained results.

Study of optical emission spectroscopy using modified Boltzmann plot in dual-frequency synchronized pulsed capacitively coupled discharges with DC bias at low-pressure in Ar/O2/C4F8 plasma etching process.

We consider the corona model and local thermal equilibrium approximations of a real plasma to measure the electron temperature (Te) and density (ne), respectively, using the optical emission spectroscopy (OES) method in dual-frequency pulsed capacitively coupled plasmas (CCPs) in a reactive mixture of Ar/O2/C4F8 at a low operating pressure. The operation conditions such as DC continuous and synchronized were used for the study and plasma characterization for the intended plasma application such as high aspect ratio etching (HARE). We show that the present plasma conditions are dominated by a corona balance rather than the supremacy of multi-step excitation. This fact has enabled us to utilize the modified Boltzmann plot technique to evaluate the Te values. In the second method, we simultaneously used the Boltzmann and Saha equations to determine the ne value using the line intensity ratio and the value of Te. Time-resolved measurements of Te and ne were performed for completeness, and the insight of the pulsed discharge was investigated. Time evolution of ne and Te using the OES method revealed a similar trend in the change of plasma parameters, indicating electron impact ionization during the pulse on phase. It was seen that ne in the afterglow speedily decreased within a short time of ∼5 μs. Analysis suggests the formation of afterglow plasmas, which are composed of positive and negative ions with very low electron density. The results revealed that the DC-synchronized operation could be useful for plasma application such as HARE due to different plasma characteristics. It also suggests the production of ion-ion plasmas by the effective utilization of negative ions in the afterglow phase. The corona balance condition was validated in our experiments, and the results were compared with the existing literature.

Modified Boltzmann equation and extended Navier–Stokes equations

We refine the derivation of the Boltzmann equation by considering that the molecules passing through the interfaces of a volume element of physical space and velocity space exhibit different velocity distribution functions and number densities. The resulting equation has a time parameter close to the relaxation time and degenerates into the conventional Boltzmann equation when this parameter takes a value of zero. By considering the macroscopic averaging of mass, momentum, and energy, the corresponding continuity, momentum, and energy equations are obtained. Compared with the extended Navier–Stokes equations, the momentum and energy equations contain additional terms to represent the external forces.

Dark matter freeze-out in modified cosmological scenarios

We study the effects of modifying the expansions history of the Universe on Dark Matter freezeout. We derived a modified Boltzmann equation for freeze-out for an arbitrary energy density in the early Universe and provide an analytic approach using some approximations. We then look at the required thermally averaged cross sections needed to obtain the correct relic density for the specific case where the energy density consists of radiation plus one extra component which cools faster. We compare our analytic approximation to a numerical solutions. We find that it gives reasonable results for most of the parameter space explored, being at most a factor of order one away from the measured value. We find that if the new contribution to the energy density is comparable to the radiation density, then a much smaller cross section for Dark Matter annihilation is required. This would lead to weak scale Dark Matter being much more difficult to detect and opens up the possibility that much heavier Dark Matter could undergo freezeout without violating perturbative unitarity.

Features of the rarefied gas description in terms of a distribution function

The aim of the paper is to determine the relationship of the classical model for describing a rarefied gas using the distribution function and its discrete representation. Here we study the role of discreteness in the description of a medium in the kinetic theory and the interrelation between interaction of the discreteness and the “continuity” of a media. The question of the relationship between the discreteness of a medium and its description with the help of continuum mechanics is important both when processing experimental data and when going from a continuum model to discrete one in mechanics and physics. There are many studies devoted to the influence of the transition from a continuous to discrete medium in computational mathematics, but there is no study of inverse processes. The work related to the formulation of conservation laws as conditions of the equilibrium of forces and moments of forces, as well as the action of additional flows on the sides of an elementary volume, was carried out earlier. After refinement, non -symmetric stress tensor was obtained. The method for calculating this tensor was proposed. The equations for a gas were found from the modified Boltzmann equation and from the phenomenological theory. Inaccuracy leading to the symmetry of the stress tensor arises when calculating the Lagrange function of particles as the sum of pairwise interacting particles and the unchanged position of the inertia system center.

Electrophoresis of hydrophilic/hydrophobic rigid colloid with effects of relaxation and ion size

This article deals with a semi-analytical study on the electrophoresis of charged spherical rigid colloid by considering the effects of relaxation and ion size. The particle surface is taken to be either hydrophilic or hydrophobic in nature. In order to consider the ion size effect we have invoked the Carnahan and Starling model (J. Chem. Phys. 1969, 51, 635-636). The mathematical model is based on Stokes equation for fluid flow, modified Boltzmann equation for spatial distribution of ionic species and Poisson equation for electric potential. We adopt a linear perturbation technique under a weak electric field assumption. An iterative numerical technique in employed to solve the coupled set of perturbed equations. We have validated the numerically obtained electrophoretic mobility with the corresponding analytical solution derived under low potential limit. Going beyond the widely employed Debye-Hückel linearization, we have presented the results for a wide range of surface charge density, electrolyte concentration, and slip length to Debye length ratio. We have also identified several interesting features including occurrence of local maxima and minima in the mobility for critical choice of pertinentparameters.

A Gas-Kinetic BGK Scheme for Natural Convection in a Rotating Annulus

In this paper, a gas-kinetic Bhatnagar–Gross–Krook (BGK) scheme is developed for simulating natural convection in a rotating annulus, which arises in many scientific and engineering fields. Different from most existing methods for the solution of the incompressible Navier–Stokes (N–S) equations with the Boussinesq approximation, compressible full N–S equations with allowable density variation are concerned. An appropriate BGK model is constructed for the macroscopic equations defined in a rotating frame of reference. In particular, in order to account for the source (non-inertial) effects in the BGK model, a microscopic source term is introduced into the modified Boltzmann equation. By using the finite volume method and assuming the flow is smooth, the time-dependent distribution function is simply obtained, from which the fluxes at the cell interface can be evaluated. For validation, a closed rotating annulus with differentially heated cylindrical walls is studied. A conventional N–S solver with the preconditioner is used for comparison. The numerical results show that the present method can accurately predict the variation of the Nusselt number with the Rayleigh number, but no preconditioning is required due to its delicate dissipation property. The computed instantaneous streamlines and temperature contours are also investigated, and it is verified that the Rayleigh–Bénard convection in a rotating annulus is very similar to that in a classical stationary horizontal enclosure.

Development of circular function‐based gas‐kinetic scheme (CGKS) on moving grids for unsteady flows through oscillating cascades

AbstractIn this paper, the circular function‐based gas‐kinetic scheme (CGKS), which was originally developed for simulation of flows on stationary grids, is extended to solve moving boundary problems on moving grids. Particularly, the unsteady flows through oscillating cascades are our major interests. The main idea of the CGKS is to discretize the macroscopic equations by the finite volume method while the fluxes at the cell interface are evaluated by locally reconstructing the solution of the continuous Boltzmann Bhatnagar–Gross–Krook equation. The present solver is based on the fact that the modified Boltzmann equation, which is expressed in a moving frame of reference, can recover the corresponding macroscopic equations with Chapman–Enskog expansion analysis. Different from the original Maxwellian function‐based gas‐kinetic scheme, in improving the computational efficiency, a simple circular function is used to describe the equilibrium state of distribution function. Considering that the concerned cascade oscillating problems belong to cases that the motion of surface boundary is known a priori, the dynamic mesh method is suitable and is adopted in the present work. In achieving the mesh deformation with high quality and efficiency, a hybrid dynamic mesh method named radial basic functions‐transfinite interpolation is presented and applied for cascade geometries. For validation, several numerical test cases involving a wide range are investigated. Numerical results show that the developed CGKS on moving grids is well applied for cascade oscillating flows. And for some cases where nonlinear effects are strong, the solution accuracy could be effectively improved by using the present method. Copyright © 2017 John Wiley & Sons, Ltd.

Power distributions for self-gravitating astrophysical systems based on nonextensive Tsallis kinetics

The long-time development of self-gravitating gaseous astrophysical systems (in particular, the evolution of the protoplanet accretion disk) is mainly determined by relatively fast processes of the collision relaxation of particles. However, slower dynamical processes related to force (Newton or Coulomb) interactions between particles should be included (as q-collisions) in the nonextensive kinetic theory as well. In the present paper, we propose a procedure to include the Newton self-gravity potential and the centrifugal potential in the near-equilibrium power-like q-distribution in the phase space, obtained (in the framework of nonextensive statistics) by means of the modified Boltzmann equation averaged with respect to an unnormalized distribution. We show that if the power distribution satisfies the stationary q-kinetic equation, then the said equation imposes clear restrictions on the character of the long-term force field and on the possible dependence of hydrodynamic parameters of the coordinates: it determines those parameters uniquely. We provide a thermodynamic stability criterion for the equilibrium of the nonextensive system. The results allow us to simulate the evolution of gaseous astrophysical systems (in particular, the gravitational stability of rotating protoplanet accretion disks) more adequately.

Dispersion Effects in the Falkner-Skan Problem and in the Kinetic Theory

The conservation laws of continuum mechanics and of the kinetic theory with the influence of the angular momentum and associated with its rotation of the elementary volume are considered, the variant of accounting lag is investigated for discrete environment. The analysis of the recording of the Lagrangian function for the collective interaction of the particles with the change of the center of inertia of the moving particles and the effect influence of the angular momentum were used. The equations for gas are calculated from the modified Boltzmann equation and the phenomenological theory. For a rigid body the equations were used of the phenomenological theory, but their interpretation was changed. The nonsymmetric stress tensor was obtained. The Boltzmann equation is written with an additional summand. This situation is typical for discrete environment as the transition from discrete to continuous environment is a key to the issue of mechanics. Summary records of all effects lead to a cumbersome system of equations and therefore require the selection of main effects in a particular situation. The Hilbert paradox was being solved. The simplest problem of the boundary layer continuum (the Falkner-Skan task) and the kinetic theory are discussed. A draw attention at the delay process would be suggested for the description of discrete environment. Results are received for some special cases.

Asymptotic behaviour of the Boltzmann equation as a cosmological model

As a Newtonian cosmological model the Vlasov-Poisson-Boltzmann system is considered, and a slightly modified Boltzmann equation, which describes the stability of an expanding universe, is derived. Asymptotic behaviour of solutions turns out to depend on the expansion of the universe, and in this paper we consider the soft potential case and will obtain asymptotic behaviour.

Uniqueness of the self-similar profile for a kinetic annihilation model

We prove the uniqueness of the self-similar profile solution for a modified Boltzmann equation describing probabilistic ballistic annihilation. Such a model describes a system of hard spheres such that, whenever two particles meet, they either annihilate with probability α∈(0,1) or they undergo an elastic collision with probability 1−α. The existence of a self-similar profile for α smaller than an explicit threshold value α_1 has been obtained in our previous contribution [6]. We complement here our analysis of such a model by showing that, for some α♯ explicit, the self-similar profile is unique for α∈(0,α♯).

Multicomponent modified Boltzmann equation and thermalization

The existence of stationary distributions in a multicomponent Boltzmann equation using a non-additive kinetic energy composition rule for binary collisions is discussed. It is found that detailed balance is not achieved when —in contrast to the case of a single rule— several different composition rules are considered. The long-time behaviour of a simple momentum space model is explored numerically: saturating, heating and cooling solutions are presented. These results may be used in modelling the kinetics of multicomponent systems, such as hadronic fireballs or quark-gluon plasma.

Finslerian Perturbation for the λCDM Model

We present Finslerian perturbation for the λCDM model, which breaks the isotropic symmetry of the universe. The analysis on the Killing vectors shows that the Randers—Finsler spacetime breaks the isotropic symmetry even if the scalar perturbations of the FRW metric vanish. In Randers—Finsler spacetime, the modified geodesic equation deduces a modified Boltzmann equation. We propose a perturbational version of the gravitational field equation in Randers—Finsler spacetime, where we have omitted the curvature tensor that does not belong to the base space of the tangent bundle. The gravitational field equations for the gravitational wave are also presented. The primordial power spectrum of the gravitational wave is investigated. We show that the primordial power spectrum for super-horizon perturbations is unchanged. For sub-horizon perturbations, however, the power spectrum is modified.

Existence of self-similar profile for a kinetic annihilation model

We show the existence of a self-similar solution for a modified Boltzmann equation describing probabilistic ballistic annihilation. Such a model describes a system of hard spheres such that, whenever two particles meet, they either annihilate with probability α∈(0,1) or they undergo an elastic collision with probability 1−α. For such a model, the number of particles, the linear momentum and the kinetic energy are not conserved. We show that, for α smaller than some explicit threshold value α⁎, a self-similar solution exists.

An effective potential approach to modeling 25 nm MOSFET devices

We present a thermodynamic approach to introducing quantum corrections to the classical transport picture in semiconductor device simulation. The approach leads to a modified Boltzmann equation with an effective quantum potential. We study the quantum interaction of electrons with a gate oxide barrier potential in a 25 nm MOSFET.

Optical Sensor for Characterizing the Phase Transition in Salted Solutions

We propose a new optical sensor to characterize the solid-liquid phase transition in salted solutions. The probe mainly consists of a Raman spectrometer that extracts the vibrational properties from the light scattered by the salty medium. The spectrum of the O – H stretching band was shown to be strongly affected by the introduction of NaCl and the temperature change as well. A parameter SD defined as the ratio of the integrated intensities of two parts of this band allows to study the temperature and concentration dependences of the phase transition. Then, an easy and efficient signal processing and the exploitation of a modified Boltzmann equation give information on the phase transition. Validations were done on solutions with varying concentration of NaCl.

Non-extensive statistics and its effects on cosmology

We apply non-extensive statistics, namely Tsallis statistics, on a case of supercritical string cosmology (SSC) as the one studied in [1] and obtain interesting cosmological modifications. At the beginning, we seek for the non-extensive corrections to the dilaton energy density and the off-shell (off-critical) terms and by using the set of dynamical equations of [1], we derive the modified evolution equation for the radiation energy density in a r.d.e. (radiation dominated era). This in fact is characterised by fractal (‘exotic’) scaling and this seems to be a generic result of our analysis. The modified Boltzmann equation was also considered, giving us the effects on dark matter candidates relic abundances of non-extensitivity besides the SSC effects. An effort to give a physical interpretation to the essential results of models such as the Tsallis statistical model has been done in collaboration with N. Mavromatos and S. Sarkar by using the D-particles foam model which is presented at the last part of this work.