Maple code for the calculation of the matrix elements of the Boltzmann collision operators for mixtures

Abstract

A Maple code is provided which is used to compute the matrix elements of the collision operators in the Boltzmann equation for arbitrary differential elastic collision cross section. The present paper describes an efficient method for the calculation of the matrix elements of the collision operators in the Sonine basis set. The method employs the generating functions for these polynomials. The transport properties of gaseous mixtures of atoms and/or ions are generally determined from solutions of the Boltzmann equation. The solution of the Boltzmann equation for the velocity distribution functions requires a representation of the integral collision operators defined by the differential cross sections describing collisions between pairs of particles. Many applications have considered either the simple hard sphere cross section or the cross section corresponding to the inverse fourth power of the inter-particle distance (“Maxwell molecules”). There are a few applications where realistic quantum mechanical cross sections have been used. The basis set of Sonine (or Laguerre) polynomials is the basis set of choice used to represent the distribution functions. The Maple code provided is used to express the matrix elements of the collision operators in terms of a finite sum of the omega integrals of transport theory and defined by the differential cross section. Thus the matrix representations of the collision operators are applicable to arbitrary interaction potentials. Program summary Program title: MCBC Catalogue identifier: AEGJ_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEGJ_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 2422 No. of bytes in distributed program, including test data, etc.: 48 653 Distribution format: tar.gz Programming language: Maple Computer: Personal computer or workstation Operating system: Windows or Linux RAM: bytes Classification: 12 Nature of problem: The representation of the collision operators in the Boltzmann equation in the basis set of Sonine (Laguerre) polynomials. Solution method: The matrix element of the generating function for the Sonine polynomials are evaluated analytically. The individual matrix elements are the coefficients in the double power series expansion in two parameters. The Maple code is used to effect this expansion and extract the coefficients of the omega integrals. The omega integrals are defined by the differential cross section. Running time: A few seconds for the examples provided.