KINETIC EQUATIONS WITH INTERNAL STATES

Synopsis

Classical kinetic equations govern the evolution of density functions depending on the time, location and velocity of the particles (t, x, v). It is well known that under proper scalings, these equations  can give rise to population-level models in the asymptotic limits. More recently, a new class of kinetic equations containing internal state variables in addition to (t, x, v) have been applied to various scenarios. Examples include modelling of the Lorenz gas, photon transport in clouds, or chemotaxis of bacteria. In the first two examples, the internal state variable is the free transport distance while in the third case, it can be the methylation level in the bacterial cell. In this talk we focus on the applications to biology. In particular, we explain how these extended kinetic equations lead to various population-level models. These include the classical Keller-Segel equation, the flux-limited Keller-Segel and the fractional diffusion equation describing long jumps in the bacterial run-and-tumble motion.