Correlation effects and nonlocal velocity distribution functions

In the present paper we consider the functional equation describing the collisionless particle velocity distribution function f(V) in the framework of probabilistic approach. The key element of the collisionless particles description is using the waiting time distribution ψ(t). The solution of the considered functional is obtained for several model functions ψ(t) and it leads to the power form tails of the velocity distribution f(V). It is possible to adopt considered functional to the Laplace transformation form that allows us to accord “collision” and “collisionless” description. This Laplace form of the functional yields the Levy–Smirnov velocity distribution function with the characteristic exponent aL=12. The possibility to accord the model functional equation and the non-local Einstein–Smoluchowski equation is discussed.