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.

Kinetic Theory for a Dilute Gas of Particles with Spin

1966 ◽  
Vol 21 (10) ◽  
pp. 1529-1546 ◽  
Author(s):  
S. Hess ◽  
L. Waldmann
Keyword(s):  
Kinetic Theory ◽  
Collision Operator ◽  
Momentum Conservation ◽  
Dilute Gas ◽  
Irreducible Tensors ◽  
Reciprocity Relations ◽  
Complete Set ◽  
Expansion Coefficients ◽  
The One ◽  
Linearized Collision Operator

The kinetic theory of particles with spin previously developed for a LORENTzian gas is extended to the case of a pure gas. In part A the transport (BOLTZMANN) equation for the one particle distribution operator is stated and discussed (conservation laws, Η-theorem). A magnetic field acting on the magnetic moment of the particles is incorporated throughout. In part B the pertaining linearized collision operator and certain bracket expressions linked with this operator are considered. Part C deals with the expansion of the distribution operator and of the linearized transport equation with respect to a complete set of composite irreducible tensors built from the components of particle velocity and spin. Thus, the distribution operator is replaced by a set of tensors depending only on time and space-coordinates. The physical meaning of these tensors (expansion coefficients) is invoked. They obey a set of coupled first-order differential equations (transport-relaxation equations) . The reciprocity relations for the relaxation matrices are stated. Finally a detailed discussion of angular momentum conservation is given.

Boltzmann’s Kinetic Theory

2018 ◽  
Author(s):  
Sauro Succi
Keyword(s):  
Boltzmann Equation ◽  
Kinetic Theory ◽  
Statistical Physics ◽  
Neutron Transport ◽  
Condensed Matter ◽  
Relativistic Fluids ◽  
Classical Statistical Mechanics ◽  
Dilute Gas ◽  
Equilibrium Processes ◽  
The Boltzmann Equation

Kinetic theory is the branch of statistical physics dealing with the dynamics of non-equilibrium processes and their relaxation to thermodynamic equilibrium. Established by Ludwig Boltzmann (1844–1906) in 1872, his eponymous equation stands as its mathematical cornerstone. Originally developed in the framework of dilute gas systems, the Boltzmann equation has spread its wings across many areas of modern statistical physics, including electron transport in semiconductors, neutron transport, quantum-relativistic fluids in condensed matter and even subnuclear plasmas. In this Chapter, a basic introduction to the Boltzmann equation in the context of classical statistical mechanics shall be provided.

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.

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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