Microscopic description of moving interfaces – MICMOV

Physicists are able to describe the world at different scales. At the microscopic level, systems are composed of a huge number of atoms
and are studied on time scales which are very large compared to the typical time scales of the atomic degrees of freedom. At the macroscopic level, physical phenomena are often modeled by partial differential equations. Mathematically, providing the link between both descriptions involves a space-time scaling limit procedure which accounts for the difference in scales and explains the transition from an atomic description to a continuous one.

In particular, the rigorous microscopic description of moving interfaces has recently been the subject of an intense research activity, and still necessitates deep studies. To that aim, the class of kinetically constrained lattice gases, which has been introduced in the 1980's in glassy dynamics, is a good candidate to accurately illustrate moving interfaces. If the microscopic mechanisms feature an active-absorbing phase transition, the macroscopic evolution of such particle systems is usually expected to be the solution to a Stefan problem, with a free boundary between the active and frozen regions.

In this proposal we intend to: (i) start with the well-known porous medium equation, and take advantage of its underlying mathematical structure of gradient flow in order to apprehend better the microscopic mechanisms at play ; (ii) derive Stefan problems as hydrodynamic limits of kinetically
constrained lattice gases which contain active-absorbing phase transitions; (iii) derive new hydrodynamic equations, which
combine nonlocal macroscopic effects (such as fractional diffusion) and porous medium type propagation, or involve multiphase flows (with several conservation laws); (iv) improve our understanding of interface fluctuations thanks to the class of exactly solvable particle systems.