UBC Mathematics Colloquium: Khanh Dao Duc

In this talk, I will present two recent studies that respectively revisit the theories of first passage time, and interacting particle process. First, I will analyze a generic planar diffusion process, inspired by neuronal dynamics. The specificity of this process is that the density of escape time from a characteristic boundary is non-Poissonian, and presents an oscillatory decay, which is quantified by using asymptotic methods to derive the spectrum of the Fokker-Planck operator. Secondly, I will present a generalization of the totally asymmetric simple exclusion process (TASEP), which is a widely used model for transport phenomena in non-equilibrium particle systems. To model protein synthesis, the TASEP needs to be generalized, such that particles occupy an extended number of sites and move at arbitrary site-dependent rates. While classical ansatz and mean-field approximations cannot be applied to analyze this generalized model, we developed another method based on the hydrodynamic limit of the process. Deriving and solving this limit yield analytical formulas for the particle flux and density, and a complete description of phase transitions. Surprisingly, the phase diagram and key quantities of the process depend on few parameters, which has important consequences on the control and regulation of traffic in such particle systems.