Parameter of Discontinuity and Differential-Operator Expansion of the Linear Boltzmann or Master Operator

Abstract

The ratio of the average microscopic agitation interval to the macroscopic relaxation time is proposed as the expansion parameter of linear Boltzmann or master operators. This parameter is interpretable physically both as a measure of the discontinuity of the random process, and as an inverse measure of the size of the fluctuating system. In the limit when the expansion parameter is zero, the process becomes continuous and is described by the Fokker-Planck equation. When the parameter is nonvanishing, the expansion of the master operator in terms of it is, in three representative cases, a ``CD expansion'' in products of creation and destruction operators for Hermite functions; the dominant term is usually the Fokker-Planck operator. These results are considered in relation to van Kampen's hypothesis for small-parameter expansions of the same operators. It is found that the CD expansion fits the available model processes exactly, and that these processes do not satisfy van Kampen's hypothesis. As a new application, the explicit CD series is given for the density fluctuation model. Special cases of the model include the density fluctuations studied by van Kampen and the Ehrenfest urn model.