The collision integral kernels of the scalar nonlinear Boltzmann equation for pseudopower potentials

An Algorithm for the calculation of non-isotropic collision integral matrix elements of the non-linear Boltzmann equation by the use of recurrence relations

“…We will prove that (17) are satisfied identically, and both procedures (as described in the [11,18] for power potential and proposed in this article) are equivalent. To this end, we show that if the starting isotropic MEs satisfy temperature relations (17), then all MEs satisfy these relations.…”

“…Thus, property (39) is valid for all kernels in the layer λ. From the equivalence of (39) and (17), it follows that the temperature relations hold for all MEs with all indices l, l 1 , l 2 .…”

“…Here G a a,b (v, v 1 , v 2 ) is the kernel of collision integral depending on velocity vectors [17]. This kernel contains delta-function which provides compliance with the energy conservation law.…”

“…Therefore Thus, all MEs with indices corresponding to the layer λ can be found from the recurrence procedure based on velocity relations. Note that at the stage of a non-isotropic MEs calculation temperature relations (17) are not used.…”

“…Laguerre polynomials L α r (x) coincide with Sonine polynomials S r l+1/2 (x) at α = l + 1/2, where l is integer. Since isotropic MEs satisfy (17), then the kernel G 0 0,0 satisfies (39). We show that if (39) holds for kernels in the layer λ− 1, then it is true for kernels in the layer λ, and hence corresponding matrix elements satisfy temperature relations.…”