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.

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.

Modelling Techniques in Applied Glaciology: Numerical Modelling of Avalanches

1983 ◽  
Vol 4 ◽  
pp. 297-297
Author(s):  
G. Brugnot
Keyword(s):  
Finite Difference Method ◽  
Transverse Velocity ◽  
Longitudinal Velocity ◽  
Momentum Conservation ◽  
Dimensional Model ◽  
One Dimensional ◽  
Modelling Techniques ◽  
Reference Grid ◽  
The One ◽  
Near Future

We consider the paper by Brugnot and Pochat (1981), which describes a one-dimensional model applied to a snow avalanche. The main advance made here is the introduction of the second dimension in the runout zone. Indeed, in the channelled course, we still use the one-dimensional model, but, when the avalanche spreads before stopping, we apply a (x, y) grid on the ground and six equations have to be solved: (1) for the avalanche body, one equation for continuity and two equations for momentum conservation, and (2) at the front, one equation for continuity and two equations for momentum conservation. We suppose the front to be a mobile jump, with longitudinal velocity varying more rapidly than transverse velocity.We solve these equations by a finite difference method. This involves many topological problems, due to the actual position of the front, which is defined by its intersection with the reference grid (SI, YJ). In the near future our two directions of research will be testing the code on actual avalanches and improving it by trying to make it cheaper without impairing its accuracy.

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 On the connection between quark propagation and hadronization

2020 ◽  
Vol 80 (9) ◽  
Author(s):  
Alberto Accardi ◽  
Andrea Signori
Keyword(s):  
Sum Rules ◽  
Mass Term ◽  
Final State ◽  
Mass Generation ◽  
Gauge Invariant ◽  
Dynamical Generation ◽  
Dynamical Quark Mass ◽  
Complete Set ◽  
Fragmentation Functions ◽  
The One

AbstractWe investigate the properties and structure of the recently discussed “fully inclusive jet correlator”, namely, the gauge-invariant field correlator characterizing the final state hadrons produced by a free quark as this propagates in the vacuum. Working at the operator level, we connect this object to the single-hadron fragmentation correlator of a quark, and exploit a novel gauge invariant spectral decomposition technique to derive a complete set of momentum sum rules for quark fragmentation functions up to twist-3 level; known results are recovered, and new sum rules proposed. We then show how one can explicitly connect quark hadronization and dynamical quark mass generation by studying the inclusive jet’s gauge-invariant mass term. This mass is, on the one hand, theoretically related to the integrated chiral-odd spectral function of the quark, and, on the other hand, is experimentally accessible through the E and $${\widetilde{E}}$$ E ~ twist-3 fragmentation function sum rules. Thus, measurements of these fragmentation functions in deep inelastic processes provide one with an experimental gateway into the dynamical generation of mass in Quantum Chromodynamics.

scholarly journals XII. The kinetic theory of a gas constituted of spherically symmetrical molecules

1912 ◽  
Vol 211 (471-483) ◽  
pp. 433-483 ◽  
Keyword(s):  
Kinetic Theory ◽  
The Other ◽  
Mathematical Form ◽  
The Real ◽  
Numerical Errors ◽  
Diffusion Of Knowledge ◽  
Descriptive Theory ◽  
The One ◽  
Physical Reasons ◽  
High Degree

The principal kinetic theories of a gas proceed either on the hypothesis that the molecules are rigid elastic spheres, or that they are point centres of forces which vary inversely as the fifth power of the distance. Maxwell has worked out the consequences of the letter hypothesis in his well-known theory, which is unrivalled in its high degree of accuracy and (after some improvements by Boltzmann) in its perfection of mathematical form. All the quantities not taken account of in the theory (such as the time occupied by molecular encounters, and the effect of collisions in which more than two molecules take part) are properly negligible under ordinary conditions. The theory has the disadvantage, however, that the underlying hypothesis is highly artificial (being chosen chiefly on account of mathematical simplifications connected with it, rather than from any physical reasons), and does not represent the real facts at all adequately. The other hypothesis referred to seems to be much more in agreement with fact, but its consequences have been worked out less accurately. The method which has almost always been used is the one originally devised by Clausius and Maxwell; Maxwell abandoned it later, however, as it had “led him at times into grave error.” In spite of its apparent simplicity, numerical errors of large amount may undoubtedly creep in in a very subtle way. Hence the theory of a gas whose molecules are elastic spheres remains in a rather unsatisfactory state. As a “descriptive” theory (to use Meyer’s apt term) it has, however, served a useful purpose; the general laws of gaseous phenomena have been developed by its aid in an elementary way, which has conduced to a wider diffusion of knowledge of the kinetic theory than would have been possible if the sole line of development had been by the more mathematical and accurate methods used by Maxwell and Boltzmann.

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.

scholarly journals Evolution of the microstructure, residual stresses, and mechanical properties of W–Si–N coatings after thermal annealing

2005 ◽  
Vol 20 (5) ◽  
pp. 1356-1368 ◽  
Author(s):  
A. Cavaleiro ◽  
A.P. Marques ◽  
J.V. Fernandes ◽  
N.J.M. Carvalho ◽  
J.Th. De Hosson
Keyword(s):  
Residual Stress ◽  
Thermal Annealing ◽  
Amorphous Film ◽  
Amorphous Coating ◽  
X Ray Diffraction ◽  
Thermal Expansion Coefficients ◽  
Expansion Coefficients ◽  
Composition Structure ◽  
The One ◽  
Deposited Film

W–Si–N films were deposited by reactive sputtering in a Ar + N2 atmosphere from a W target encrusted with different number of Si pieces and followed by a thermal annealing at increasing temperatures up to 900 °C. Three iron-based substrates with different thermal expansion coefficients, in the range of 1.5 × 10−6 to 18 × 10−6 K−1 were used. The chemical composition, structure, residual stress, hardness (H), and Young’s modulus (E) were evaluated after all the annealing steps. The as-deposited film with low N and Si contents was crystalline whereas the one with higher contents was amorphous. After thermal annealing at 900 °C the amorphous film crystallized as body-centered cubic α–W. The crystalline as-deposited film presented the same phase even after annealing. There were no significant changes in the properties of both films up to 800 °C annealing. However, at 900 °C, a strong decrease and increase in the hardness were observed for the crystalline and amorphous films, respectively. It was possible to find a good correlation between the residual stress and the hardness of the films. In several cases, particularly for the amorphous coating, H/E higher than 0.1 was reached, which envisages good tribological behavior. The two methods (curvature and x-ray diffraction) used for calculation of the residual stress of the coatings showed fairly good agreement in the results.

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