move and follows the statistics of each particle individually. Initially, the particle sits at a location x’ (x1, x2, x3) at time t’. The motion of the particle is described by its probability of existing in a specific volume element at time t, that is described by Ψ(x1, x2, x3, t) dx1 dx2 dx3 = Ψ(x,t)dx which follows the probability density function (pdf) such that: Where function Q is the probably density for particle transition. The concentration of particles at a location x and time t can then be calculated by summing the probabilities of the number of particles observed as follows: Which is then evaluated by returning to the pdf integral Thus, this approach is used to evaluate the position and velocity of particles relative to their neighbors and environment, and approximates the random concentrations and velocities associated with turbulent diffusion in the statistics of their motion. Solutions The resulting solution for solving the final equations listed above for both the Eulerian and Lagrangian models for analyzing the statistics of species in turbulent flow, both result in very similar expressions for calculating the average concentration at a location from a continuous source. Both solutions develop a Gaussian Plume and are virtually identical under the assumption that the variances in the x,y,z directions are related to the eddy diffusivity: where q= species emission rate, u = wind speed, σi2 = variance in i direction. Under various external conditions such as directional flow speed (wind) and environmental conditions, the variances and diffusivities of turbulent diffusion are measured and used to calculate a good estimate of concentrations at a specific point from a source. This model is very useful in atmospheric sciences, especially when dealing with concentrations of contaminants in air pollution that emanate from sources such as combustion stacks, rivers, or strings of automobiles on a road. Future Research Because applying mathematical equations to turbulent flow and diffusion is so difficult, research in this area has been lacking until recently. In the past, laboratory efforts have used data from steady flow in streams or from fluids, that have a high Reynolds number, flowing through pipes, but it is difficult to obtain accurate data from these methods. This is because these methods involve ideal flow, which cannot simulate the conditions of turbulent flow necessary for developing turbulent diffusion models. With the advancement in computer-aided modeling and programming, scientists have been able to