scholarly journals Moment inequalities for the boltzmann equation and applications to spatially homogeneous problems

1997 ◽  
Vol 88 (5-6) ◽  
pp. 1183-1214 ◽  
Author(s):  
A. V. Bobylev
Keyword(s):  
Boltzmann Equation ◽  
Moment Inequalities ◽  
The Boltzmann Equation ◽  
Spatially Homogeneous

scholarly journals SECOND-ORDER SCHEME FOR THE SPATIALLY HOMOGENEOUS BOLTZMANN EQUATION WITH MAXWELLIAN MOLECULES

1996 ◽  
Vol 06 (01) ◽  
pp. 137-147 ◽  
Author(s):  
JENS STRUCKMEIER ◽  
KONRAD STEINER
Keyword(s):  
Boltzmann Equation ◽  
Particle Method ◽  
Time Discretization ◽  
Second Order ◽  
Particle Methods ◽  
Homogeneous Boltzmann Equation ◽  
The Boltzmann Equation ◽  
Spatially Homogeneous ◽  
Maxwellian Molecules ◽  
Spatially Homogeneous Boltzmann Equation

In the standard approach particle methods for the Boltzmann equation are obtained using an explicit time discretization of the spatially homogeneous Boltzmann equation. This kind of discretization leads to a restriction on the discretization parameter as well as on the differential cross-section in the case of the general Boltzmann equation. Recently, construction of an implicit particle scheme for the Boltzmann equation with Maxwellian molecules was shown. This paper combines both approaches using a linear combination of explicit and implicit discretizations. It is shown that the new method leads to a second-order particle method when using an equiweighting of explicit and implicit discretization.

THE BOLTZMANN EQUATION AND THE GROUP TRANSFORMATIONS

1993 ◽  
Vol 03 (04) ◽  
pp. 443-476 ◽  
Author(s):  
A.V. BOBYLEV
Keyword(s):  
Boltzmann Equation ◽  
Special Role ◽  
Spatially Inhomogeneous ◽  
The Boltzmann Equation ◽  
Symmetry Properties ◽  
Full Equation ◽  
Spatially Inhomogeneous Boltzmann Equation ◽  
Spatially Homogeneous ◽  
Spatially Homogeneous Boltzmann Equation

This paper is devoted to the investigation of group properties of the nonlinear Boltzmann equation. The complete Lie group of invariant transformations for the spatially inhomogeneous Boltzmann equation is constructed. The generalization to the Lie-Backlund groups is given for the spatially homogeneous case. It is shown that there are only two non-trivial group transformations for the Boltzmann equation in the wide class of Lie and Lie-Backlund transformations. Some consequences of these symmetry properties are discussed. The special role of Galileo group and the analogy between the spatially homogeneous Boltzmann equation and the full equation are also investigated.

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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