Iterative solution of the Boltzmann equation

We define an iterative scheme to solve the nonlinear Boltzmann equation. Conservation rules are maintained at each iterative step. We apply this method to a spatially uniform and isotropic velocity distribution function on the Maxwell and very-hard-particle models. A particular example is evaluated and results are compared with the exact solution. It shows to be a very fast convergent approach.     

Wild's solution of the nonlinear Boltzmann equation

The relaxation to equilibrium of a spatially uniform pseudo-Maxwellian gas is considered. A modified Wild expansion is defined for solving the nonlinear Boltzmann equation. The positivity and asymptotic conditions, as well as the conservation rules, are maintained at each truncation order. Some particular examples are evaluated. The comparison with exact solutions illustrates the very fast convergence of this method.     

Stochastic solution of the linearized Boltzmann equation

For the linearized Boltzmann equation with finite cross section, the solution is represented as an integral over the paths of a Markov jump process. The integral is only shown to converge conditionally, where the limiting process is defined by an increasing sequence of stopping times. The notion of local martingale plays an important role. A number of related kinetic models are also mentioned.Supported by NSF Grant GP 28576.     

On the solution to the Boltzmann kinetic equation in calculating the heat flux in polyatomic gases

The solution of the linearized Wang Chang-Uhlenbeck equation using the Hanson-Morse model for calculating the heat flux in a plane molecular gas layer has been considered. General expressions (independent of the form and method for solving the kinetic equation) of the dependence of heat flux on the energy accommodation coefficients have been derived. The values of the temperature jump coefficient for specific gases have been obtained.     

Generalized relativistic Chapman-Enskog solution of the Boltzmann equation

The Chapman-Enskog method of solution of the relativistic Boltzmann equation is generalized in order to admit a time-derivative term associated to a thermodynamic force in its first order solution. Both existence and uniqueness of such a solution are proved based on the standard theory of integral equations. The mathematical implications of the generalization introduced here are thoroughly discussed regarding the nature of heat as chaotic energy transfer in the context of relativity theory.     

Is the solution of the Boltzmann equation positive?

A proof is given that the solution of the general non-linear space- and time-dependent Boltzmann equation, combined with suitable boundary conditions, in non-negative. For the spatially homogeneous situation the proof is extended to show that the solution is always positive.     

Exact solution of the linearized Boltzmann equation in p dimensions

The solution of the transport in p-dimensional space is given for a system of convex boundary surfaces with arbitrary shape and finite diameter. It satisfies homogeneous or non-homogeneous Dirichlet boundary conditions. The equation for the determination of the point spectrum of the Boltzmann operator is also given.     

Exact solution of the non-linear Boltzmann equation for Maxwell models

The exact solution of the Boltzmann equation, obtained recently for Maxwell molecules, holds for various models with arbitrary dimensionality. Furthermore, simple non-linear equations are derived for the ordinary and the Sonine moments, which can be solved subsequentially.     

A perturbation solution to generalized linear Boltzmann equation

A perturbation method to solve the generalized linear Boltzmann equation is presented for the calculation of dynamic conductivity of a normal metal at low temperatures. The method is applied to the ordinary Boltzmann equation. The following three cases are considered: i) when only electron-impurity interactions are dominant, ii) when the metal may be considered as pure plasma, and iii) when both of the interactions are present. The expressions for conductivity are presented in the binary collision approximation. The connected diagram expansion method is extended for the direct calculation of the conductivity. Finally the method is extended for the calculation of conductivity by the correlation function formula.     

Quantum kinetic Boltzmann equation taking into account the resonant exchange of excitations

A derivation of the quantum Boltzmann equation is given for identical particles with internal degrees of freedom. It is shown that the off-diagonal (with respect to the internal degrees of freedom) term of the equation contains an energy pole term, which is not present in the most commonly used kinetic equation, known as the Waldmann-Snider equation. The physical conditions underlying the occurrence of the pole term in the quantum kinetic equation are analyzed. Zh. éksp. Teor. Fiz. 111, 831–837 (March 1997)     

Finite volume solution of the relativistic Boltzmann equation forelectron avalanche studies

The Boltzmann equation for a population of relativistic electrons in a constant external electric field in air is solved numerically in order to determine parameters of the resulting runaway electron beam. The numerical solution takes advantage of algorithms from computational fluid dynamics. The work revises the results from the original work where the numerical solution was found to be unstable     

A steady state, supersonic flow solution of the Boltzmann equation

Nonclassical effects which appear in the supersonic nonequilibrium gas flow of a nonuniform relaxation described by the Boltzmann equation are studied. For such a flow, in particular, the heat flux and the temperature gradient have the same signs. Analytical and numerical results are presented. Possible experimental verification is discussed.     

Exact moment solution of the Boltzmann equation for uniform shear flow

The Ikenberry-Truesdell exact solution to the Boltzmann equation for Maxwell molecules is revisited. This solution refers to a state characterized by a linear profile of the velocity flow and spatially uniform density and temperature. The solution is extended to include explicit expressions for the fourth-degree moments. It is shown that if the shear rate is larger than a certain critical value, the fourth-degree moments do not reach stationary values, even when the temperature is kept constant. The explicit shear-rate dependence of the moments below this critical value are obtained.     

Multiplication of high-energy electrons in irradiated materials studied using the Boltzmann kinetic equation

Processes involved in the formation of electron collision cascades created by nonrelativistic high-energy electrons, which can develop in materials exposed to electron and gamma radiation fluxes, have been considered. The problem is solved using the Boltzmann kinetic equation for high-energy electrons moving in a medium. A model scattering indicatrix is constructed for this equation with an arbitrary potential of interaction between colliding particles. Using this scattering indicatrix, the distribution of the particle energies is obtained. Based on this energy distribution (with an arbitrary interparticle interaction potential), a cascade function is found that describes the multiplication of knock-out electrons (electron cascade) generated when a high-energy electron with a certain energy is scattered on the electron subsystem of the irradiated material. The cascade function has been calculated for the Coulomb potential of the interaction between a high-energy electron and atomic-shell electrons.     

Exact solution of the Boltzmann equation in the homogeneous color conductivity problem

An exact solution of the Boltzmann equation for a binary mixture of colored Maxwell molecules is found. The solution corresponds to a nonequilibrium homogeneous steady state created by a nonconservative external force. Explicit expressions for the moments of the distribution function are obtained. By using information theory, an approximate velocity distribution function is constructed, which is exact in the limits of small and large field strengths. Comparison is made between the exact energy flux and the one obtained from the information theory distribution.     

Conversion between kinetic energy and potential energy in the classical nonlocal Boltzmann equation

An important property of the classical Boltzmann equation is that kinetic energy is conserved. This is closely connected to the fact that the Boltzmann equation describes the nonequilibrium properties of an ideal gas. Generalizations of the Boltzmann equation to higher density involve, among other things, allowing the colliding particles to be at different positions. This spatial nonlocality is known to contribute to the density corrections of gas transport properties. For soft potentials such a spatial separation of the particles also leads to a conversion between kinetic and potential energy. In evaluating these effects the classical dynamics of the whole collision trajectory must be taken into account, involving also the time for the collision process. The resulting time nonlocality has usually been reinterpreted in terms of a spatial nonlocality. However, for a homogeneous system this is not possible and only the time nonlocality remains, this then being responsible for the conversion between kinetic and potential energy. This paper aims to clarify these properties of the nonlocal corrections to the classical mechanical Boltzmann collision term. Comments on the corresponding problem for the quantum Boltzmann equation are also made.     

Monte Carlo solution of the Boltzmann equation via a discrete velocity model

A new discrete velocity scheme for solving the Boltzmann equation is described. Directly solving the Boltzmann equation is computationally expensive because, in addition to working in physical space, the nonlinear collision integral must also be evaluated in a velocity space. Collisions between each point in velocity space with all other points in velocity space must be considered in order to compute the collision integral most accurately, but this is expensive. The computational costs in the present method are reduced by randomly sampling a set of collision partners for each point in velocity space analogous to the Direct Simulation Monte Carlo (DSMC) method. The present method has been applied to a traveling 1D shock wave. The jump conditions across the shock wave match the Rankine–Hugoniot jump conditions. The internal shock wave structure was compared to DSMC solutions, and good agreement was found for Mach numbers ranging from 1.2 to 10. Since a coarse velocity discretization is required for efficient calculation, the effects of different velocity grid resolutions are examined. Additionally, the new scheme’s performance is compared to DSMC and it was found that upstream of the shock wave the new scheme performed nearly an order of magnitude faster than DSMC for the same upstream noise. The noise levels are comparable for the same computational effort downstream of the shock wave.     

Convective scheme solution of the Boltzmann transport equation for nanoscale semiconductor devices

A model for the simulation of the electron energy distribution in nanoscale metal–oxide–semiconductor field-effect transistor (MOSFET) devices, using a kinetic simulation technique, is implemented. The convective scheme (CS), a method of characteristics, is an accurate method of solving the Boltzmann transport equation, a nonlinear integrodifferential equation, for the distribution of electrons in a MOSFET device. The method is used to find probabilities for use in an iterative scheme which iterates to find collision rates in cells. The CS is also a novel approach to 2D semiconductor device simulation. The CS has been extended to handle boundary conditions in 2D as well as to calculation of polygon overlap for polygons of more than three sides. Electron energy distributions in the channel of a MOSFET are presented.