Calculation of total thermal conductivity of ionized gases
Boltzmann equation for calculating transport coefficients and thermal conductivity of partially ionized gases
Ahtye, W. F.↗
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Transport processes in the primordial fireball.
Possible initial perturbations development into galaxies with dissipative effects associated with transport processes treated via Boltzmann equation
BOLTZMANN EQUATION↗
A quick introduction to linear transport phenomena
The goal of this report is to provide an introductory-level overview of the linear Boltzmann equation in the context of neutron transport. After deriving the transport equation, we discuss some of its basic applications in reactor theory. Finally, we review several simplifying approximations of the linearized Boltzmann equation that are essential to solving it in many applications. Although the context of neutron transport is called upon to add concreteness to our discussion, many of the concepts and approximations that we discuss remain relevant in other many other phenomena.
73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
Kinetic theory of dense gases. iii - the generalized enskog equation.
Statistical mechanics theory for gas transport phenomena, obtaining Enskog equation from generalization of Boltzmann equation approximation of Liouville equation
BOLTZMANN EQUATION↗
Energy Models for One-Carrier Transport in Semiconductor Devices
Moment models of carrier transport, derived from the Boltzmann equation, made possible the simulation of certain key effects through such realistic assumptions as energy dependent mobility functions. This type of global dependence permits the observation of velocity overshoot in the vicinity of device junctions, not discerned via classical drift-diffusion models, which are primarily local in nature. It was found that a critical role is played in the hydrodynamic model by the heat conduction term. When ignored, the overshoot is inappropriately damped. When the standard choice of the Wiedemann-Franz law is made for the conductivity, spurious overshoot is observed. Agreement with Monte-Carlo simulation in this regime required empirical modification of this law, or nonstandard choices. Simulations of the hydrodynamic model in one and two dimensions, as well as simulations of a newly developed energy model, the RT model, are presented. The RT model, intermediate between the hydrodynamic and drift-diffusion model, was developed to eliminate the parabolic energy band and Maxwellian distribution assumptions, and to reduce the spurious overshoot with physically consistent assumptions. The algorithms employed for both models are the essentially non-oscillatory shock capturing algorithms. Some mathematical results are presented and contrasted with the highly developed state of the drift-diffusion model.
Jerome, Joseph W.↗
Energetic particle transport in relativistic flows
A discussion is given of pitch angle dependent and diffusive transport equations for cosmic rays applicable for both special relativistic and general relativistic flows, derived from the relativistic Boltzmann equation. As an example of particle transport in a curved spacetime we give a pitch angle dependent transport equation appropriate for radial accretion onto a Schwarzschild black hole. The roles of fluid shear, acceleration and compression on the energy changes of particles in the diffusive transport equations are emphasized. Also discussed are special flows (e.g. rigidly rotating flows) associated with a Killing vector for which a constant of the motion of the particles can be identified, and for which simplified transport equations can be constructed.
Webb, G. M.↗
The effect of convection upon charged particle transport in random magnetic fields
In a coordinate system moving with the plasma and random magnetic fields of a wind that blows with constant velocity in the direction of the guiding field, transport of energetic particles is described by a Boltzmann equation which is similar to the one that describes unconvected transport. Although this formulation is mathematically identical to that developed by Luhmann, which refers to the system where the guiding field is static, there are both practical and fundamental reasons to adopt the new approach. It leads to first-order approximate transport equations which are similar to those that apply in the absence of convection. However, these equations are more general than Parker's description of diffusion and convection, for they describe the coherent modes of transport that appear when the mean free path is large compared to the scale length for spatial variations of the guiding field, and they are valid for arbitrary wind velocity. The latter characteristic opens up new possibilities for analyzing particle transport in relativistic flows seen in some astronomical objects.
Earl, J. A.↗
Transport methods and interactions for space radiations
This report presents a brief history leading to the involvement of the Langley Research Center of the National Aeronautics and Space Administration (NASA) in space-radiation physics and protection. Indeed, a relatively complete summary of technical capability as of the summer of 1990 is given. The Boltzmann equations for coupled ionic and neutronic fields are presented and inversion techniques for the Boltzmann operator are discussed. Errors generated by the straight ahead approximation are derived and are shown to be negligible for most problems of space-radiation protection. A decoupling of projectile propagation from the target fields greatly simplifies the Boltzmann equations and allows an analytic solution of the target fragment transport. Analytic and numerical methods of solving the projectile transport equations are discussed. The nuclear physics underlying the coefficients in the Boltzmann equation is discussed. A coupled-channel optical model is found as a consequence of the loose binding of nuclear matter and closure of the nuclear states in high-energy reactions. Transport solutions with the developed data base are used with laboratory experiments to validate both the transport code and the data base. Numerical benchmarks and comparison with Monte Carlo calculations are also used for code validation.
Wilson, John W.↗
A geometric perspective on kinetic matter–radiation interaction and moment systems
Here, in this paper, we provide a geometric perspective on the kinetic interaction of matter and radiation, based on a pair bracket approach. We discuss the interaction of kinetic theories via dissipative brackets, with our fundamental example being the coupling of matter, described by the Boltzmann equation, and radiation, described by the radiation transport equation. We explore the transition from kinetic systems to their corresponding moment systems, provide a Hamiltonian description of such moment systems, and give a geometric interpretation of the moment closure problem for kinetic theories. As an application, we discuss in detail diffusion radiation hydrodynamics as an example of a pair bracket formulation on a space of moments corresponding to kinetic matter–radiation interaction. Additionally, using the variable moment closure framework of Burby [J. W. Burby, Variable-moment fluid closures with Hamiltonian structure, Sci. Rep. 13 (2023) 18286, doi:10.1038/s41598-023-45416-5], we show how to construct Hamiltonian moment closures for kinetic transport equations with arbitrary Hamiltonian. Using this general construction, we derive novel Hamiltonian moment closures for pure radiation transport.
97 MATHEMATICS AND COMPUTING↗
A Stochastic Calculus Approach to Boltzmann Transport
Traditional Monte Carlo methods for particle transport utilize source iteration to express the solution, the flux density, of the transport equation as a Neumann series. Our contribution is to show that the particle paths simulated within source iteration are associated with the adjoint flux density and the adjoint particle paths are associated with the flux density. Here, we make our assertion rigorous through the use of stochastic calculus by representing the particle path used in source iteration as a solution to a stochastic differential equation (SDE). The solution to the adjoint Boltzmann equation is then expressed in terms of the same SDE, and the solution to the Boltzmann equation is expressed in terms of the SDE associated with the adjoint particle process. An important consequence is that the particle paths used within source iteration simultaneously provide Monte Carlo samples of the flux density and adjoint flux density in the detector and source regions, respectively. The significant practical implication is that particle trajectories can be reused to obtain both forward and adjoint quantities of interest. To the best our knowledge, the reuse of entire particles paths has not appeared in the literature. Monte Carlo simulations are presented to support the reuse of the particle paths.
Boltzmann transport↗
The lower ionosphere of Titan
Ionization of the atmosphere of Titan by galactic cosmic rays is a very significant process throughout the altitude range from 100 to 400 km. An approximate form of the Boltzmann equation for cosmic-ray transport has been used to obtain local ionization rates. Models of both ion and neutral chemistry have been employed to compute electron and ion density profiles for three different values of the H2/CH4 abundance ratio. The peak electron density is of the order of 1000 per cu cm. The most abundant positive ions are C2/H9(+) and C3H9(+), while the predicted densities of the negative ions H(-) and CH3(-) are very small (less than one ten-thousandth that of the positive ions). It is suggested that inclusion of the ion chemistry is important in the computation of the H and CH3 density profiles in the lower ionosphere.
Capone, L. A.↗
A model for the high-temperature transport properties of heavily doped n-type silicon-germanium alloys
A model is presented for the high-temperature transport properties of large-grain-size, heavily doped n-type silicon-germanium alloys. Electron and phonon transport coefficients are calculated using standard Boltzmann equation expressions in the relaxation time approximation. Good agreement with experiment is found by considering acoustic phonon and ionized impurity scattering for electrons, and phonon-phonon, point defect, and electron-phonon scattering for phonons. The parameters describing electron transport in heavily doped and lightly doped materials are significantly different and suggest that most carriers in heavily doped materials are in a band formed largely from impurity states. The maximum dimensionless thermoelectric figure of merit for single-crystal, n-type Si(0.8)Ge(0.2) at 1300 K is estimated at ZT about 1.13 with an optimum carrier concentration of n about 2.9 x 10 to the 20th/cu cm.
Vining, Cronin B.↗
Low-temperature T -linear resistivity due to umklapp scattering from a critical mode
In this study, we consider the transport properties of a model of fermions scattered by a critical bosonic mode. The mode is overdamped and scattering is mainly in the forward direction. Such a mode appears at the quantum critical point for an electronic nematic phase transition and in gauge theories for a U(1) spin liquid. It leads to a short fermion lifetime, violating Landau's criterion for a Fermi liquid. In spite of this, transport can be described by a Boltzmann equation. We include momentum relaxation by umklapp scattering, supplemented by weak impurity scattering. We find that above a very low temperature which scales with Δ q 3 , where Δ q is the minimum umklapp scattering vector, the resistivity is linear in T with a coefficient which is independent of the amount of disorder. This picture holds until the temperature dependent part of the resistivity exceeds that due to impurity scattering. We compare the relaxation time approximation with an exact numerical solution of the Boltzmann equation. Surprisingly we find that, unlike the resistivity, the Hall coefficient strongly deviates from the relaxation time approximation and shows a strong reduction with increasing temperature. We comment on possible comparisons with experiments on high T c cuprates.
36 MATERIALS SCIENCE↗
Numerical modeling of ion transport and adsorption in porous media: A pore-scale study for capacitive deionization desalination
In this report a pore-scale model is presented to simulate the dynamic ion transport and adsorption processes in porous electrodes used for capacitive deionization (CDI). The Stokes equation governing water flow is solved using the lattice Boltzmann method and Nernst-Planck equation describing ion transport is solved using the finite volume method. The ion adsorption process is considered at the surface of carbon electrodes. After validation against analytical solutions and published results, the model is used to study the coupled water flow, ion transport and adsorption in both two-dimensional and three-dimensional porous CDI electrodes at the pore scale. The effect of electrode microstructure, electrical potential and flow velocity on the adsorption processes is quantitatively investigated, and the relative importance of various parameters is determined. The presented model can be a powerful numerical tool to quantitatively analyze ion transport and adsorption in porous electrodes, and may provide useful information for the design and optimization of CDI electrodes and operating conditions for desired desalination efficiency, water throughput and cost.
32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗
Theory of the lattice Boltzmann Method: Dispersion, Dissipation, Isotropy, Galilean Invariance, and Stability
The generalized hydrodynamics (the wave vector dependence of the transport coefficients) of a generalized lattice Boltzmann equation (LBE) is studied in detail. The generalized lattice Boltzmann equation is constructed in moment space rather than in discrete velocity space. The generalized hydrodynamics of the model is obtained by solving the dispersion equation of the linearized LBE either analytically by using perturbation technique or numerically. The proposed LBE model has a maximum number of adjustable parameters for the given set of discrete velocities. Generalized hydrodynamics characterizes dispersion, dissipation (hyper-viscosities), anisotropy, and lack of Galilean invariance of the model, and can be applied to select the values of the adjustable parameters which optimize the properties of the model. The proposed generalized hydrodynamic analysis also provides some insights into stability and proper initial conditions for LBE simulations. The stability properties of some 2D LBE models are analyzed and compared with each other in the parameter space of the mean streaming velocity and the viscous relaxation time. The procedure described in this work can be applied to analyze other LBE models. As examples, LBE models with various interpolation schemes are analyzed. Numerical results on shear flow with an initially discontinuous velocity profile (shock) with or without a constant streaming velocity are shown to demonstrate the dispersion effects in the LBE model; the results compare favorably with our theoretical analysis. We also show that whereas linear analysis of the LBE evolution operator is equivalent to Chapman-Enskog analysis in the long wave-length limit (wave vector k = 0), it can also provide results for large values of k. Such results are important for the stability and other hydrodynamic properties of the LBE method and cannot be obtained through Chapman-Enskog analysis.
Lallemand, Pierre↗
The effect of solar-wind convection on charged particle transport in interplanetary space
In the formulation of focused transport without convection, approximate solutions of the Boltzmann equation are described in terms of small perturbations of its two fundamental steady-state solutions. In the presence of convection, the same approach is applicable, provided that the analysis is carried out in a system moving with the solar wind. The approximate transport equations that apply in this system are very similar to those that apply in the absence of convection. The new equations assume that the wind blows in the direction of the guiding field and is constant in space and time. Nevertheless, these equations are more general than existing formulations, for they retain their validity when the solar wind velocity is large, and they describe the coherent modes of transport that appear when the mean free path is greater than 1 AU. An interesting implication of the new formulation is that coherent disturbances are swept along with the wind. In this respect, they differ from truly scatter-free modes, which are not affected by the wind.
Earl, J. A.↗
Analytical description of charged particle transport along arbitrary guiding-field configurations
Focused particle transport is analyzed in terms of partial differential equations which are applicable to arbitrary spatial variations of the mean free path and the focusing length. A distribution function for transport is written in two forms of the Boltzmann equation and a coefficient of spatial diffusion is defined. Changes from a nearly coherent propagation of flare particles in regions of strong focusing near the sun to ordinary diffusion in the outer solar system are described as supercoherent transitions. A solution to the steady state problem is developed and results are presented for a monopolar guiding field configuration with isotropic diffusion gradually giving way to anisotropy in a region suggestive of a collimated beam of particles streaming away from its source.
Earl, J. A.↗
Path integral molecular dynamics: A high-fidelity approach to quantum dynamics of electrons
We investigate electron transport in the uniform electron gas using ring-polymer molecular dynamics (RPMD). Working in the weakly coupled, non-degenerate regime, we use RPMD to probe how the onset of quantum diffraction effects at high temperature reshapes electron–electron collisions and leads to a classical-to-quantum crossover in macroscopic transport properties. Static thermodynamics obtained with RPMD are consistent with the weak-coupling equation of state, confirming correct quantum Boltzmann sampling. Real-time transport extracted from mean square displacements exhibits the expected ballistic-to-diffusive transition and a systematic reduction of the electronic self-diffusivity as quantum effects strengthen, due to quantum diffraction modifying electron–electron collisions. Direct ring-polymer scattering simulations reveal diffractive “softening” of binary deflections, providing a micro-to-macro link between collision physics and diffusion. The present study establishes RPMD as a quantitative, trajectory-based tool for electron transport across the classical–quantum crossover and furnishes benchmarks for improving Coulomb-log interpolation models. We outline extensions to multi-component plasmas and a path to incorporate Fermi–Dirac statistics within path-integral dynamics.
Electronic transport↗