Particle models for the Stefan problem

I will report on some recent works on stochastic particle systems whose evolution mimics a free boundary problem. Prototype is the d=1 simple symmetric exclusion process (SSEP) restricted to configurations which have a rightmost particle and a leftmost hole.

The process is defined by adding to the SSEP a birth-death generator which at some small rate kills the rightmost particle and, independently, the leftmost hole. The model thus describes a microscopic diffusion (SSEP) in the free boundary interval between the leftmost hole and rightmost particle. Hydrodynamic limit (with proper scaling of space and time) is studied and established.