The Gordeyev integral for plasma particles colliding with neutrals is obtained using a particle dynamics formalism in which the collisions are modeled as a discrete Poisson process. The result leads to an electron density fluctuation spectrum model for partially ionized plasmas which is identical with the spectral model obtained from BGK plasma kinetic equations. This isomorphism between the Poisson process and the BGK operator is analogous to a similar relation between the Brownian motion process and the Fokker-Planck operator with constant coefficients. We take advantage of this analogy to derive a collisional ISR spectrum model that takes into account collisions with both neutrals and charged species. Copyright 2009 by the American Geophysical Union.