Diffusion and Viscoelastic Media
Human lung mucus is a complex polymer fluid which does not obey Newtonian dynamics. In this project, we want to understand the diffusion of small bodies such as bacteria and viruses through this mucus. Diffusion in such complex fluids is frequently modelled using a generalized Langevin equation. These equations are stochastic integro-differential equations governing the velocity of the particle; they include a memory kernel determined by the interaction between the body and the complex fluid. One often selects families of parameterized kernels where the parameters have certain physical interpretations.
Recently, it has become possible to take position data of micron and smaller beads in mucus at millisecond level sampling. The goal is then to fit the paramters given the data. Until now, this type of data has often been analyzed by rudimentary moment matching methods. Our work has been to adapt for these Generalized Langevin models traditional statistical time series methodology such as state space models and ARIMA to allow for maximum likelihood fits of the data. Maximum likelihood methods have a number of advantages which include the capability of error estimates for the parameter fitting and to improve model selection.