Non-local velocity distribution function and one-flight approximation

The functional equation describing the collisionless particle velocity distribution function f(V) is considered 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=1/2