Infinities in the statistical theory of radiative processes in plasmas

Abstract

The paper deals with two kinds of infinities occurring in Prigogine’s master equation when it is applied to a system containing charged particles (one electron andN ions) and radiation. The first kind is caused by the presence of the radiation field since the usual Hamiltonian for an electron and a radiation field necessarily contains also the self-field of the electron, which is infinite on the electron’s world line. We then analyse how precisely this infinity enters into the formalism. The analysis leads to a prescription for rewriting the Liouville operator which allows relativistic and radiation reaction effects to be approximately included and which yields a theory without divergences due to the presence of a radiation field. Another kind of infinity is shown to be present. It occurs upon integrating over the particle co-ordinates, as is necessary for many applications. The immediate cause of this infinity is shown to be an asymptotic time integration employed to derive the master equation. However, reasons are given to show that the asymptotic time integration cannot simply be replaced by some other formalism leading to finite results; the convergence of the series for the momentum distribution function is indeed poor for small relative velocities between electron and ions, so that only the sum of a large number of terms in the series can provide a correct (and finite) result for small velocities.