scholarly journals Classical Boltzmann equation and high-temperature QED

2015 ◽  
Vol 91 (4) ◽  
Author(s):  
F. T. Brandt ◽  
R. B. Ferreira ◽  
J. F. Thuorst
Keyword(s):  
High Temperature ◽  
Boltzmann Equation ◽  
Classical Boltzmann Equation

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~.

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