Symbolic manipulation techniques for plasma kinetic theory derivations

Abstract

Much of the analysis of instabilities and wave propagation in warm plasmas involves perturbative solutions of the Vlasov or collisionless Boltzmann equation. In general, a dielectric tensor which describes the response of the plasma to perturbations must be derived. A typical derivation involves choosing an equilibrium distribution function which models some plasma of interest, calculating a perturbed distribution function by solving a partial differential equation (usually by the method of characteristics), calculating density and current moments of this perturbed distribution function, and using these quantities in Maxwell's equations. For many equilibrium models of physical interest, this process can be completely carried out analytically. However, even in simple cases, the procedure is tedious-involving hundreds or thousands of separate algebraic or calculus operations and is fraught with opportunity for error. It will be illustrated here how the symbolic manipulation language MACSYMA can be used to automate many of the steps involved in such derivations. By way of example, the classic problem of oscillations in a homogeneous, uniformly magnetized plasma will be considered.