The Boltzmann Equation and Transport Equations of Charged Particles

2014 ◽  
pp. 147-179
Keyword(s):  
Boltzmann Equation ◽  
Charged Particles ◽  
Transport Equations ◽  
The Boltzmann Equation

scholarly journals A Thermodynamic Treatment of Anisotropic Diffusion in an Electric Field

1972 ◽  
Vol 25 (6) ◽  
pp. 685 ◽  
Author(s):  
RE Robson
Keyword(s):  
Boltzmann Equation ◽  
Electric Field ◽  
Drift Velocity ◽  
Anisotropic Diffusion ◽  
Diffusion Tensor ◽  
Charged Particles ◽  
Equilibrium Thermodynamics ◽  
Neutral Gas ◽  
The Boltzmann Equation ◽  
Diffusive Processes

Non-equilibrium thermodynamics is used to analyse the diffusive processes associated with a swarm of charged particles (ions or electrons) drifting in a neutral gas under the influence of an electric field. A simple approximate phenomenological relationship connecting components of the diffusion tensor with the drift velocity of the swarm is derived and the utility of the formula is illustrated in several cases where previous analyses have been carried out using the Boltzmann equation.

scholarly journals Thermophoresis of a spherical particle: modelling through moment-based, macroscopic transport equations

2019 ◽  
Vol 862 ◽  
pp. 312-347 ◽  
Author(s):  
Juan C. Padrino ◽  
James E. Sprittles ◽  
Duncan A. Lockerby
Keyword(s):  
Boltzmann Equation ◽  
Kinetic Theory ◽  
Gas Dynamics ◽  
Transport Equations ◽  
The Body ◽  
Closed Form Expression ◽  
Moment Equations ◽  
Thermophoretic Force ◽  
Knudsen Numbers ◽  
The Boltzmann Equation

We consider the linearized form of the regularized 13-moment equations (R13) to model the slow, steady gas dynamics surrounding a rigid, heat-conducting sphere when a uniform temperature gradient is imposed far from the sphere and the gas is in a state of rarefaction. Under these conditions, the phenomenon of thermophoresis, characterized by forces on the solid surfaces, occurs. The R13 equations, derived from the Boltzmann equation using the moment method, provide closure to the mass, momentum and energy conservation laws in the form of constitutive, transport equations for the stress and heat flux that extend the Navier–Stokes–Fourier model to include non-equilibrium effects. We obtain analytical solutions for the field variables that characterize the gas dynamics and a closed-form expression for the thermophoretic force on the sphere. We also consider the slow, streaming flow of gas past a sphere using the same model resulting in a drag force on the body. The thermophoretic velocity of the sphere is then determined from the balance between thermophoretic force and drag. The thermophoretic force is compared with predictions from other theories, including Grad’s 13-moment equations (G13), variants of the Boltzmann equation commonly used in kinetic theory, and with recently published experimental data. The new results from R13 agree well with results from kinetic theory up to a Knudsen number (based on the sphere’s radius) of approximately 0.1 for the values of solid-to-gas heat conductivity ratios considered. However, in this range of Knudsen numbers, where for a very high thermal conductivity of the solid the experiments show reversed thermophoretic forces, the R13 solution, which does result in a reversal of the force, as well as the other theories predict significantly smaller forces than the experimental values. For Knudsen numbers between 0.1 and 1 approximately, the R13 model of thermophoretic force qualitatively shows the trend exhibited by the measurements and, among the various models considered, results in the least discrepancy.

scholarly journals On the Complex-Valued Distribution Function of Charged Particles in Magnetic Fields

Mathematics ◽  
2021 ◽  
Vol 9 (19) ◽  
pp. 2382
Author(s):  
Andrey Saveliev
Keyword(s):  
Distribution Function ◽  
Boltzmann Equation ◽  
Magnetic Fields ◽  
Kinetic Theory ◽  
Charged Particles ◽  
Kinetic Theory Of Gases ◽  
The Boltzmann Equation ◽  
Ideal Magnetohydrodynamics ◽  
Complex Valued

In this work, we revisit Boltzmann’s distribution function, which, together with the Boltzmann equation, forms the basis for the kinetic theory of gases and solutions to problems in hydrodynamics. We show that magnetic fields may be included as an intrinsic constituent of the distribution function by theoretically motivating, deriving and analyzing its complex-valued version in its most general form. We then validate these considerations by using it to derive the equations of ideal magnetohydrodynamics, thus showing that our method, based on Boltzmann’s formalism, is suitable to describe the dynamics of charged particles in magnetic fields.

The Boltzmann Equation and the H Theorem (1872–1875)

2018 ◽  
Author(s):  
Olivier Darrigol
Keyword(s):  
Heat Conduction ◽  
Boltzmann Equation ◽  
Transport Phenomena ◽  
Dilute Gas ◽  
The Boltzmann Equation ◽  
H Theorem ◽  
Irreversible Evolution

This chapter covers Boltzmann’s writings about the Boltzmann equation and the H theorem in the period 1872–1875, through which he succeeded in deriving the irreversible evolution of the distribution of molecular velocities in a dilute gas toward Maxwell’s distribution. Boltzmann also used his equation to improve on Maxwell’s theory of transport phenomena (viscosity, diffusion, and heat conduction). The bulky memoir of 1872 and the eponymous equation probably are Boltzmann’s most famous achievements. Despite the now often obsolete ways of demonstration, despite the lengthiness of the arguments, and despite hidden difficulties in the foundations, Boltzmann there displayed his constructive skills at their best.

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