General solution of a Boltzmann equation, and the formation of Maxwellian tails

1981 ◽  
Vol 374 (1758) ◽  
pp. 371-400 ◽  
Keyword(s):  
Boltzmann Equation ◽  
General Solution ◽  
Cross Section ◽  
Relative Velocity ◽  
Collision Cross Section ◽  
Two Dimensions ◽  
Time Interval ◽  
Initial Condition ◽  
Finite Time Interval ◽  
Classical Boltzmann Equation

A classical Boltzmann equation is studied. The equation describes the evolution towards the Maxwellian equilibrium state of a homogeneous, isotropic gas where the collision cross section is inversely proportional to the relative velocity of the colliding particles. After Tjon & Wu (1979), the problem is transformed into a mathematically equivalent one, itself a model Boltzmann equation in two dimensions. Working in the context of the latter equation, a formal derivation of the general solution is presented. First a countable ensemble of particular solutions, called pure solutions , is constructed. From these, via a non-linear combination mechanism, the general solution is obtained in a form appropriate for direct numerical computation. The validity of the solution depends upon its containment in a well defined Hilbert space H~ Given that the initial condition lies within H~ it is proved that at least for a small finite time interval it remains in H~.

scholarly journals THE EINSTEIN–BOLTZMANN SYSTEM AND POSITIVITY

2013 ◽  
Vol 10 (01) ◽  
pp. 77-104 ◽  
Author(s):  
HO LEE ◽  
ALAN D. RENDALL
Keyword(s):  
Cauchy Problem ◽  
General Relativity ◽  
Boltzmann Equation ◽  
Cross Section ◽  
Collision Cross Section ◽  
Curved Spacetime ◽  
The Boltzmann Equation ◽  
The Cauchy Problem

The Einstein–Boltzmann (EB) system is studied, with particular attention to the non-negativity of the solution of the Boltzmann equation. A new parametrization of post-collisional momenta in general relativity is introduced and then used to simplify the conditions on the collision cross-section given by Bancel and Choquet-Bruhat. The non-negativity of solutions of the Boltzmann equation on a given curved spacetime has been studied by Bichteler and Tadmon. By examining to what extent the results of these authors apply in the framework of Bancel and Choquet-Bruhat, the non-negativity problem for the EB system is resolved for a certain class of scattering kernels. It is emphasized that it is a challenge to extend the existing theory of the Cauchy problem for the EB system so as to include scattering kernels which are physically well-motivated.

scholarly journals Some Recent Studies of Electron Swarms in Gases

1992 ◽  
Vol 45 (3) ◽  
pp. 365 ◽  
Author(s):  
H Tagashira
Keyword(s):  
Boltzmann Equation ◽  
Cross Section ◽  
Electric Fields ◽  
Cross Sections ◽  
Collision Cross Section ◽  
Vibrational Excitation ◽  
Time Spectrum ◽  
Electron Collision ◽  
Electron Drift ◽  
Radio Frequency Electric Fields

Some recent studies of electron swarms in gases under the action of an electric field are introduced. The studies include a new type of continuity equation for electrons having a form in which the partial derivative of the electron density with respect to position and to time are interchanged, a method to deduce the time-of-flight and arrival-time-spectrum swarm parameters based on a Fourier-transformed Boltzmann equation, an examination of the correspondence between experimental and theoretical electron drift velocities, and an automatic technique to deduce the electron-gas molecule collision cross section from electron drift velocity data. We also briefly introduce a method for the deduction of electron collision cross sections with gas molecules having vibrational excitation cross sections greater than the elastic momentum transfer cross section by using a gas mixture technique, an integral type of method for solution of the Boltzmann equation with salient numerical stability, a quantitative analysis of the effect of Penning ionisation, and the behaviour of electron swarms under radio frequency electric fields.

Charge Changing Collision Cross-Section of Atomic Ions

1981 ◽  
Vol 23 (2) ◽  
pp. 184-187
Author(s):  
S Bliman ◽  
S Dousson ◽  
R Geller ◽  
B Jacquot ◽  
D van Houtte
Keyword(s):  
Cross Section ◽  
Collision Cross Section ◽  
Atomic Ions

scholarly journals Transient and Stationary Losses in a Finite-Buffer Queue with Batch Arrivals

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Andrzej Chydzinski ◽  
Blazej Adamczyk
Keyword(s):  
Batch Size ◽  
Buffer Size ◽  
Arrival Process ◽  
Time Interval ◽  
Finite Buffer ◽  
Finite Time Interval ◽  
Batch Arrivals ◽  
Batch Markovian Arrival Process ◽  
Buffer Overflows ◽  
Finite Buffer Queue

We present an analysis of the number of losses, caused by the buffer overflows, in a finite-buffer queue with batch arrivals and autocorrelated interarrival times. Using the batch Markovian arrival process, the formulas for the average number of losses in a finite time interval and the stationary loss ratio are shown. In addition, several numerical examples are presented, including illustrations of the dependence of the number of losses on the average batch size, buffer size, system load, autocorrelation structure, and time.

Saint–Venant torsion of anisotropic elliptical bar

2017 ◽  
Vol 45 (3) ◽  
pp. 286-294 ◽  
Author(s):  
István Ecsedi ◽  
Attila Baksa
Keyword(s):  
General Solution ◽  
Cross Section ◽  
Elliptical Cross Section ◽  
Elliptical Cross

The object of this article is the Saint–Venant torsion of anisotropic, homogeneous bar with solid elliptical cross section. A general solution of the Saint–Venant torsion for anisotropic elliptical cross section is presented and some known results are reformulated. The case of non-warping cross section is analysed.

Argon Repulsive Potential from Collision Cross‐Section Measurements

1950 ◽  
Vol 18 (4) ◽  
pp. 525-528 ◽  
Author(s):  
I. Amdur ◽  
D. E. Davenport ◽  
M. C. Kells
Keyword(s):  
Cross Section ◽  
Collision Cross Section ◽  
Repulsive Potential
Sign in / Sign up

Export Citation Format

Share Document