scholarly journals A Fokker–Planck Model of the Boltzmann Equation with Correct Prandtl Number

2015 ◽  
Vol 162 (2) ◽  
pp. 397-414 ◽  
Author(s):  
J. Mathiaud ◽  
L. Mieussens
Keyword(s):  
Boltzmann Equation ◽  
Prandtl Number ◽  
The Boltzmann Equation ◽  
Fokker Planck

Stochastic solution of the Boltzmann equation for thermal electrons in an attractive Coulombic potential

1969 ◽  
Vol 3 (3) ◽  
pp. 377-385
Author(s):  
M. J. Bell ◽  
M. D. Kostin
Keyword(s):  
Distribution Function ◽  
Boltzmann Equation ◽  
Diffusion Approximation ◽  
Stochastic Method ◽  
Free Electrons ◽  
The Boltzmann Equation ◽  
Stochastic Solution ◽  
Positive Ion ◽  
Fokker Planck

A stochastic method has been employed to investigate the distribution function of free electrons in gases in the field of a positive ion. Stochastic results have been obtained for several pressures and for both large (Langevin) and small ions, and the results have been compared with the solutions of the diffusion approximation and Pitaevskii's Fokker—Planck approximation to the Boltzmann equation.

A Particle Fokker-Planck Algorithm with Multiscale Temporal Discretization for Rarefied and Continuum Gas Flows

2017 ◽  
Vol 22 (2) ◽  
pp. 338-374 ◽  
Author(s):  
Fei Fei ◽  
Zhaohui Liu ◽  
Jun Zhang ◽  
Chuguang Zheng
Keyword(s):  
Boltzmann Equation ◽  
Planck Equation ◽  
Benchmark Problems ◽  
Integration Scheme ◽  
Gas Flows ◽  
Time Step ◽  
Langevin Model ◽  
Temporal Discretization ◽  
The Boltzmann Equation ◽  
Fokker Planck

AbstractFor gas flows with moderate and low Knudsen numbers, pair-wise collisions in the Boltzmann equation can be approximated by the Langevin model corresponding to the Fokker-Planck equation. Using this simplified collision model, particle numerical schemes, e.g. the Fokker-Planck model (FPM) method, can simulate low Knudsen number gas flows more efficient than those based on the Boltzmann equation, such as the Direct Simulation Monte Carlo (DSMC) method. However, as analyzed in this paper, the transport properties of the FPM method deviate from the physical values as the time step increases, and this problem affects its computational accuracy and efficiency for the simulation of multi-scale flows. Herewe propose a particle Fokker-Planck algorithm with multiscale temporal discretization (MTD-FPM) to overcome the drawbacks of the original FPM method. In the MTD-FPM method, the molecular motion is tracked following the integration scheme of the Langevin model in analogy to the original FPM method. However, to ensure consistent transport coefficients for arbitrary temporal discretization, a time step dependent friction coefficient has been implemented. Several benchmark problems, including Couette, thermal Couette, Poiseuille, and Sod tube flows, are simulated to validate the proposed MTD-FPM method.

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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