Generalized PDF models for randomly forced point-particles

Langevin (stochastic differential) equations are routinely used to describe particle-laden flows. They predict Gaussian probability density functions (PDFs) of a particle's trajectory and velocity, even though experimentally observed dynamics might be highly non-Gaussian. Our Liouville approach overcomes this dichotomy by replacing the Wiener process in the Langevin models with a (small) set of random variables, whose distributions are tuned to match the observed statistics. This strategy gives rise to an exact (de erministic, first-order, hyperbolic) Liouville equation that describes the evolution of a joint PDF in the augmented phase-space spanned by the random variables and the particle position and velocity. This way, difficulties related to sampling and solving high-dimensional advective-diffusive PDEs are circumvented by using deterministic flow maps. In this talk, I will describe how the Liouville approach can be used to generalize classical solutions of Fokker-Planck and Langevin equations for non-Gaussian systems analytically. The development of analytical PDF models from limited data will be discussed.