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In the previous chapters, the description of the plasma state was refined step by step. In the single-particle model, we were interested in the motion of individual particles in typical magnetic field configurations, but the interaction between the particles and the modification of the fields by the presence and motion of charged particles was neglected. In the fluid model, we had considered the average behavior of particles filling a small volume of space. In this approximation, only moments of a shifted Maxwell distribution, like mean flow velocity or gas temperature, were retained, but, by combining with Maxwell’s equations, the model became self-consistent. The fluid model goes beyond the single-particle model in that pressure effects are now included. This fluid model, and its formulation in terms of MHD-equations, became capable to describe the combined macroscopic motion of plasma and magnetic field lines. A first attempt to deal with non-Maxwellian velocity distributions was the introduction of a beam-plasma system, which generates self-excited electrostatic waves near the electron plasma frequency.

The description of the hot gas of electrons and ions forming a plasma involves a number of non-deterministic or stochastic processes that require a statistical description. The plasma constituents have a wide spread of velocities and perform collisions between the charged particles, or with the gas atoms of the parent gas. The behavior of the plasma as a whole can no longer be reduced to the deterministic motion of individual particles in prescribe felds. Rather, the large number of particles introduces uncertainties that force us to describe the plasma by average quantities. For example, the average motion depends on macroscopic quantities like temperature and density gradients, which generate particle fluxes or electric currents. This Section discusses the stochastic motion of particles and introduces simple statistical concepts to describe typical transport processes in gas discharges. To illustrate the concepts, typical applications are given in gas discharges, in ion thrusters design for spacecrafts, or in the heat balance for nuclear fusion.