Allen–Cahn equation

The Allen–Cahn equation (after John W. Cahn and Sam Allen) is a reaction-diffusion equation of mathematical physics which describes the process of phase separation in multi-component alloy systems, including order-disorder transitions.

The equation is given by:

∂ η ∂ t = M η [ ϵ η 2 ∇ 2 η − f ′ ( η ) ]

where M η is the mobility, f is the free energy density, and η is the nonconserved order parameter.

It is the L2 gradient flow of the Ginzburg–Landau–Wilson Free Energy Functional. It is closely related to the Cahn–Hilliard equation. In one space-dimension, a very detailed account is given by a paper by Xinfu Chen.