Boltzmann equation approximate solution methods

Author: Tato Pilauri
Keywords: Boltzmann, equation, particle
Annotation:

This paper discusses kinetic equation approximate solution by one of the methods, which is suitable for the monoatomic gases; the distribution of molecules ƒ (υ, R, t) in speeds describes non-equilibrium processes in low-density gases. The function ƒ determines the number of particles with speeds(υ,υ+Δυ) in a small interval (V,V+ΔV). If the distribution function depends on x and υ(x) we will have: ძƒ/ძt + υx ძƒ/ძx + F/m ძƒ/ υx =ძƒ/ძt m – is particle Mass. The distribution function of the speed of time is characterized by the private derivatives ძƒ/ძt. The second member determines the outcome of the particle movement in space. The third member determines the distribution function change, which is conditioned by external F power. Standing on the right side member defines distribution function change at the expense of particles collisions. It depends on and the temper of the forces acting between the particles and is equal to: (refer paper for exvation) ƒƒ1ƒ'ƒ'1 - are distribution functions of the molecules before collision and after the collision.  υ υ1 –are speeds of molecules before collision and after the collision. dσ – is scattering differential effective cut dΩ in corporal corner (in the laboratory coordinate system), which is depended on the molecules interaction law. For the model of molecules, which have a spherical form with R radius, Ω= 4R2cosα where α|υ-υ1|is the relative speed and the angle between the line connecting molecules. This equation was brought by Boltzmann in 1872. The different generalizations of the equation describes electrical gases behavior in metals, crystal lattice quantum and etc. Boltzmann equation is often referred as the equation of relocation. Due to complexity of molecules interaction law (especially for polyatomic), Boltzmann equation in its essence cannot be recorded for specific gases in accurate manner, besides the difficulty of molecular-kinetic equation mathematical structure makes it impossible to solve it in accurate analytical form. Also in the kinetic theory of gases, the Boltzmann equation approximate solution, considerable attention is given to the efficient methods of finding solution. One of such approximate solutions method is used for Monohydric gas (Chapman 1916).