Probability and repeatibility: One particle diffusion

Many multiphase flow systems involve complex geometry and/or chaotic flows. For this reason, exact solutions to initial value problems (giving, for example, the exact initial positions of all bubbles flowing in a system) contain both too much information and too little information, in a sense that will be made clear. For many purposes, flow details (such as ripples on bubbles) can be neglected, or their overall effect need be considered. Information about such details is extraneous. For some systems, the boundary conditions and initial conditions cannot be controlled sufficiently to allow repeatable experiments. A correct detailed prediction for such a system cannot be made because the initial and boundary conditions cannot be specified. Moreover, predictions for such a system cannot be compared in detail to the experiment. Consequently, predictions for averages have become the standard for complex flow systems. In this paper, the averaging process is elucidated for situations where the flow is simple. The central concept is the evolution of an appropriate probability density function. The evolution (rate of change) of the probability density function is studied for linear, logistic, and Lorenz dynamics. We build the concepts and solutions toward the problem of turbulent dispersion of particles. The effect of uncertainty in the initial conditions, and the dispersion by random flow is described for the evolution of a swarm of particles.