The transport theory of the previous chapter dealt mostly with the response to static fields. Here we allow the fields to vary with time. In order to be consistent, we describe the electrons with Maxwell's equations. Since electrons are charged we are taking some of the electron-electron interaction into account. As we will see later, only the long-wavelength properties can be got at this way. First we will deal with the so-called longitudinal response, namely the response to the introduction of an external charge density which may be time and space-dependent. Later we will turn to the transverse response to external fields such as those in a light wave.

Dielectric functions

We start with the simple observation that a fixed positive charge in a system of mobile electrons polarizes the environment. It attracts the electrons so that there is an induced negative charge. The net field produced by the positive charge plus the induced charge is smaller than the positive charge would produce alone: this is called screening. This is familiar in electrostatics; thus we try to think about these effects as a generalization of electrostatics.

We start by being quite general. Suppose we have a medium and we introduce an external charge density, ρe. We will suppose that the external charge is small, and that we can assume the medium is linear. What we are doing here is an example of a general method called linear response theory.