Non-existence of a gap in the eigenvalue spectrum of the linearized collision operator of Peierls' phonon-Boltzmann equation

1970 ◽  
Vol 11 (2) ◽  
pp. 139-143 ◽  
Author(s):  
Josef Jäckle
Keyword(s):  
Boltzmann Equation ◽  
Collision Operator ◽  
Eigenvalue Spectrum ◽  
Phonon Boltzmann Equation ◽  
Linearized Collision Operator

Kinetic Theory for a Dilute Gas of Particles with Spin. II. Relaxation Coefficients

1968 ◽  
Vol 23 (12) ◽  
pp. 1893-1902
Author(s):  
S. Hess ◽  
L. Waldmann
Keyword(s):  
Boltzmann Equation ◽  
Kinetic Theory ◽  
Interaction Potential ◽  
Collision Operator ◽  
Dilute Gas ◽  
Order Of Magnitude ◽  
Generalized Boltzmann Equation ◽  
Linearized Collision Operator ◽  
Relaxation Coefficients

The relaxation coefficients to be discussed are given by collision brackets pertaining to the linearized collision operator of the generalized Boltzmann equation for particles with spin. The order of magnitude of various nondiagonal relaxation coefficients which are of interest for the SENFTLEBEN-BEENAKKER effect is investigated. Those nondiagonal relaxation coefficients which are linear in the nonsphericity parameter ε (ε essentially measures the ratio of the nonspherical and the spherical parts of the interaction potential), as well as some diagonal relaxation coefficients are expressed in terms of generalized Omega-integrals.

scholarly journals Higher order slip according to the linearized Boltzmann equation with general boundary conditions

2011 ◽  
Vol 369 (1944) ◽  
pp. 2228-2236 ◽  
Author(s):  
Silvia Lorenzani
Keyword(s):  
Boltzmann Equation ◽  
Collision Operator ◽  
Velocity Slip ◽  
Lattice Boltzmann Methods ◽  
Linearized Boltzmann Equation ◽  
Potential Applications ◽  
Microscale Flows ◽  
Scattering Kernel ◽  
Linearized Collision Operator ◽  
Slip Coefficients

In the present paper, we provide an analytical expression for the first- and second-order velocity slip coefficients by means of a variational technique that applies to the integrodifferential form of the Boltzmann equation based on the true linearized collision operator and the Cercignani–Lampis scattering kernel of the gas–surface interaction. The polynomial form of the Knudsen number obtained for the Poiseuille mass flow rate and the values of the velocity slip coefficients are analysed in the frame of potential applications of the lattice Boltzmann methods in simulations of microscale flows.

UNIFORM L2-STABILITY ESTIMATES FOR THE RELATIVISTIC BOLTZMANN EQUATION

2009 ◽  
Vol 06 (02) ◽  
pp. 295-312 ◽  
Author(s):  
SEUNG-YEAL HA ◽  
HO LEE ◽  
XIONGFENG YANG ◽  
SEOK-BAE YUN
Keyword(s):  
Boltzmann Equation ◽  
Collision Operator ◽  
Stability Estimate ◽  
Classical Solutions ◽  
Stability Estimates ◽  
Type Estimate ◽  
Relativistic Boltzmann Equation ◽  
The Stability ◽  
L2 Stability ◽  
Linearized Collision Operator

In this paper, we derive an a prioriL2-stability estimate for classical solutions to the relativistic Boltzmann equation, when the initial datum is a small perturbation of a global relativistic Maxwellian. For the stability estimate, we use the dissipative property of the linearized collision operator and a Strichartz type estimate for classical solutions. As a direct application of our stability estimates, we establish that classical solutions in Glassey–Strauss and Hsiao–Yu's frameworks satisfy a uniform L2-stability estimate.

scholarly journals Thermal transport at the nanoscale: A Fourier's law vs. phonon Boltzmann equation study

2017 ◽  
Vol 121 (4) ◽  
pp. 044302 ◽  
Author(s):  
J. Kaiser ◽  
T. Feng ◽  
J. Maassen ◽  
X. Wang ◽  
X. Ruan ◽  
...  
Keyword(s):  
Boltzmann Equation ◽  
Thermal Transport ◽  
Fourier’S Law ◽  
Phonon Boltzmann Equation ◽  
Fourier's Law

On the Inversion of the Linearized Collision Operator

1981 ◽  
Vol 36 (2) ◽  
pp. 113-120 ◽  
Author(s):  
Ulrich Weinert
Keyword(s):  
Kernel Function ◽  
Inverse Operator ◽  
Integral Kernel ◽  
Collision Operator ◽  
Boltzmann Collision Operator ◽  
Linearized Collision Operator

Abstract Some features are discussed in connection with the representation of the linearized Boltzmann collision operator and its inversion. It is shown that under certain assumptions the inverse operator can be given explicitly as an integral kernel function.

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