Lorentz model

In a famous sequence of papers of 1906, Hendrik Lorentz introduced a version of the following problem. Large (heavy) particles are distributed about ℝd. A small (light) particle is fired through ℝd, with a trajectory comprising straight-line segments between the points of interaction with the heavy particles. When the small particle hits a heavy particle, the small particle is reflected at its surface, and the large particle remains motionless. See Figure 12.1 for an illustration.

We may think of the heavy particles as objects bounded by reflecting surfaces, and the light particle as a photon. The problem is to say something non-trivial about how the trajectory of the photon depends on the ‘environment’ of heavy particles. Conditional on the environment, the photon pursues a deterministic path about which the natural questions include:

1. Is the path unbounded?

2. How distant is the photon from its starting point after time t?

For simplicity, we assume henceforth that the large particles are identical to one another, and that the small particle has negligible volume.