Reaction-diffusion problems : from front dynamics to genealogies (3)

The Ising model, introduced a century ago, is one of the most widely studied models in statistical physics. At equilibrium, it can be used to understand the ferromagnetic-paramagnetic transition, the liquid-gas transition or the order-disorder transition of an alloy. Depending on the case under consideration, the appropriate dynamics must be used (Glauber dynamics for magnetic systems, Kawasaki dynamics for an alloy, molecular dynamics for the liquid-gas transition). It can be shown that, on a large scale, Glauber dynamics leads to the Allen-Cahn equation and Kawasaki dynamics to the Cahn-Hilliard equation. For inhomogeneous initial conditions, in the case of the Allen-Cahn equation, domain walls are bistable fronts whose velocity and shape can be calculated. The lecture continued with a discussion of stable-instable fronts in the pushed case, and ended with a description of a free-boundary model that allows many explicit calculations, including that of the shape of traveling waves in the case of pulled and pushed fronts.