In undergraduate PDE course, one may have learned that the (classical) diffusion equation can be expressed as $u_t=D \Delta u$, where $D$ is a constant diffusivity. This is true for homogeneous environment. However, for (spatially) heterogeneous environment, $D$ is no longer a constant, and diffusion phenomena in those environments such as fractionation, or Soret effect, cannot be explained with the classical diffusion equation. In this talk, I will first discuss how to model and derive some of the diffusion equations in heterogeneous environment by using basic random walk theory. We will see that the heterogeneity of components, such as speed, walk length, sojourn time, etc, can explain the diffusion phenomena. Then, I will give some specific examples how such models can be applied in science, based on my recent works.
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