Polynomial expansions in kinetic theory of gases

Abstract

Three systems of polynomials, introduced in kinetic theory by Burnett (Chapman and Cowling), by Grad, and by Ikenberry and Truesdell have been studied with a view to shortening the algebraic work involved. Considerations of irreducibility show that the system suggested by Burnett and based on the Sonine polynomials is the most economical. Its effectiveness can be enhanced by a systematic notation based on the algebra of irreducible tensors as developed by Racah and others in connection with atomic and nuclear problems. In particular, certain transformations used by Talmi in connection with the harmonic oscillator shell model may be used to simplify the calculation of the collision integral. Each coefficient in a certain polynomial expansion of this integral is expressed as a product of two Talmi coefficients, an integral involving the dynamics of collision and some moments of the distribution function. The solution of the eigenvalue problem of the linearized collision operator for Maxwellian molecules follows as a simple consequence. An algebraic interpretation and discussion of the properties of Talmi transformation coefficients is given.