In general, transformation of the linear Boltzmann integral operator to a differential operator leads to a differential operator of infinite order. For purposes of mathematical tractability this operator is usually truncated at a finite order and thus questions arise as to the validity of the resulting approximation. In this paper we show that the linear Boltzmann equation can be properly approximated only by the first two terms of the Kramers-Moyal expansion; i.e., the Fokker-Planck equation, with the retention of a finite number of higher-order terms leading to a logical inconsistency.

DOI:https://doi.org/10.1103/PhysRev.162.186