We discuss a number of mathematical results that are relevant to the statistical
mechanics of a model of radiant cavity in which the electromagnetic field
interacts with a nonlinear charged oscillator. In particular, we show that energy
equipartition in the sense of Jeans would exclude local exponential instability of
orbits; it would also prevent the existence of significant finite invariant measures
on a given energy surface. A phase space of infinite total energy is defined, and
an invariant measure in it is built, for which different modes of the field are
statistically independent.