12 - The classical theory of plasmas  pp. 464-506

The classical theory of plasmas

By T. J. M. Boyd and J. J. Sanderson

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Introduction

In this final chapter an attempt is made to sketch the classical mathematical structure underlying the various theoretical models which have been used throughout the book. The knowledge of where a particular model fits within the overall picture helps us both to understand the relationship to other models and to appreciate its limitations. Of course, we have touched upon these relationships and limitations already so the task remaining is to construct the framework of classical plasma theory and show how it all fits together.

Since collisional kinetic theory is the most comprehensive of the models that we have discussed we could begin with it as the foundation of the structure we wish to build. Indeed, we shall demonstrate its pivotal position. This would, however, be less than satisfactory for two reasons. The first and basic objection is that, so far, we have merely assumed a physically appropriate model for collisions. We have not carried out a mathematical derivation of the collision term. In fact, enormous effort has gone into this task though we shall present only a brief resumé. In doing so, we shall show how the separation of the effects of the Coulomb force into a macroscopic component (self-consistent field) and a microscopic component (collisions) appears quite naturally in the mathematical derivation of the collisional kinetic equation. This is the second reason for starting at a more fundamental level than the collisional kinetic equation itself.

To lighten the burden of the mathematical analysis we have, wherever convenient, restricted calculations to a one-component (electron) plasma. The ions, however, are not ignored but treated as a uniform background of positive charge.