Laboratoire de Mathématiques Raphaël Salem

UMR 6085 CNRS - Université de Rouen Normandie

The emergence of mean-curvature flow of an interface between different phases or populations is a phenomenon of long standing interest in statistical physics.  
In this talk, we review recent progress with respect to a class of reaction-diffusion stochastic particle systems on an $n$-dimensional lattice.
In such a process, particles can move across sites as well as be created/annihilated according to diffusion and reaction rates.  
These rates will be chosen so that there are two preferred particle mass density levels $a_1$, $a_2$ in `balance'.

In the evolution, one may understand, when the diffusion and reaction schemes are appropriately scaled, that a rough interface forms between the regions where the mass density is close to $a_1$ or $a_2$.  Via notions in the theory of hydrodynamic limits, we discuss that the scaled limit of the particle mass density field in $n\geq 2$ is a sharp interface flow by mean-curvature. We also discuss the fluctuation field limit of the mass near the forming interface, which informs on the approach to the continuum view in a certain stationary regime in $n=1,2$.