scholarly journals Self-Similar Profiles for Homoenergetic Solutions of the Boltzmann Equation: Particle Velocity Distribution and Entropy

2018 ◽  
Vol 231 (2) ◽  
pp. 787-843 ◽  
Author(s):  
Richard D. James ◽  
Alessia Nota ◽  
Juan J. L. Velázquez
Keyword(s):  
Boltzmann Equation ◽  
Velocity Distribution ◽  
Particle Velocity ◽  
The Boltzmann Equation ◽  
Self Similar

scholarly journals The Boltzman Equation Theory of Charged Particle Transport

1983 ◽  
Vol 36 (2) ◽  
pp. 163 ◽  
Author(s):  
DRA McMahon
Keyword(s):  
Distribution Function ◽  
Charged Particle ◽  
Boltzmann Equation ◽  
Velocity Distribution ◽  
Particle Transport ◽  
Transport Theory ◽  
Numerical Solutions ◽  
Response Theory ◽  
The Boltzmann Equation ◽  
Charged Particle Transport

It is shown how a formally exact Kubo-Iike response theory equivalent to the Boltzmann equation theory of charged particle transport can be constructed. Our response theory gives� the general wavevector and time-dependent velocity distribution at any time in terms of an initial distribution function, to which is added the 'response' induced by a generalized 'perturbation' over the intervening time. The usual Kubo linear response result for the distribution function is recovered by choosing the initial velocity distribution to be Maxwellian. For completeness the response theory introduces an exponential convergence function into the 'response' time integral. This is equivalent to using a modified Boltzmann equation but the general form of the transport theory is not changed. The modified transport theory can be used to advantage where possible convergence difficulties occur in numerical solutions of the Boltzmann equation. This paper gives a systematic development of the modified transport theory and shows how our response theory fits into the broader scheme of solving the Boltzmann equation. Our discussion extends both the work of Kumar et al. (1980), where the distribution function is expanded out in terms of tensor functions pj), and the propagator description where the non-hydrodynamic time development of the distribution function is related to the wavevector dependent Green function of the Boltzmann equation.

Particle dynamics in a turbulent particle–gas suspension at high Stokes number. Part 2. The fluctuating-force model

2010 ◽  
Vol 646 ◽  
pp. 91-125 ◽  
Author(s):  
PARTHA S. GOSWAMI ◽  
V. KUMARAN
Keyword(s):  
Relaxation Time ◽  
Boltzmann Equation ◽  
Velocity Distribution ◽  
Particle Velocity ◽  
Fluid Velocity ◽  
Stokes Number ◽  
Force Distribution ◽  
Force Model ◽  
Gas Suspension ◽  
Velocity Fluctuations

A fluctuating-force model is developed for representing the effect of the turbulent fluid velocity fluctuations on the particle phase in a turbulent gas–solid suspension in the limit of high Stokes number, where the particle relaxation time is large compared with the correlation time for the fluid velocity fluctuations. In the model, a fluctuating force is incorporated in the equation of motion for the particles, and the force distribution is assumed to be an anisotropic Gaussian white noise. It is shown that this is equivalent to incorporating a diffusion term in the Boltzmann equation for the particle velocity distribution functions. The variance of the force distribution, or equivalently the diffusion coefficient in the Boltzmann equation, is related to the time correlation functions for the fluid velocity fluctuations. The fluctuating-force model is applied to the specific case of a Couette flow of a turbulent particle–gas suspension, for which both the fluid and particle velocity distributions were evaluated using direct numerical simulations by Goswami & Kumaran (2010). It is found that the fluctuating-force simulation is able to quantitatively predict the concentration, mean velocity profiles and the mean square velocities, both at relatively low volume fractions, where the viscous relaxation time is small compared with the time between collisions, and at higher volume fractions, where the time between collisions is small compared with the viscous relaxation time. The simulations are also able to predict the velocity distributions in the centre of the Couette, even in cases in which the velocity distribution is very different from a Gaussian distribution.

Analysis of Electron and Heavy Particle Velocity Distribution under Consideration of Non-thermal Equilibrium Arc

2019 ◽  
Vol 139 (9) ◽  
pp. 562-567
Author(s):  
Yusuke Nemoto ◽  
Soshi Iwata ◽  
Yoshifumi Maeda ◽  
Toru Iwao
Keyword(s):  
Velocity Distribution ◽  
Particle Velocity ◽  
Thermal Equilibrium ◽  
Heavy Particle

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.

Approach to Equilibrium, the H-Theorem and Irreversibility

2018 ◽  
Author(s):  
Sauro Succi
Keyword(s):  
Boltzmann Equation ◽  
Detailed Knowledge ◽  
Approach To Equilibrium ◽  
The Boltzmann Equation ◽  
H Theorem ◽  
Irreversible Evolution

Like most of the greatest equations in science, the Boltzmann equation is not only beautiful but also generous. Indeed, it delivers a great deal of information without imposing a detailed knowledge of its solutions. In fact, Boltzmann himself derived most if not all of his main results without ever showing that his equation did admit rigorous solutions. This Chapter illustrates one of the most profound contributions of Boltzmann, namely the famous H-theorem, providing the first quantitative bridge between the irreversible evolution of the macroscopic world and the reversible laws of the underlying microdynamics.

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