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 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 describes the time evolution of a scalar-valued state variable η {displaystyle eta } on a domain Ω {displaystyle Omega } during a time interval T {displaystyle {mathcal {T}}} , and is given by: where M η {displaystyle M_{eta }} is the mobility, f {displaystyle f} is a double potential well, η ¯ {displaystyle {ar {eta }}} is the control on the state variable at the portion of the boundary ∂ η Ω {displaystyle partial _{eta }Omega } , q {displaystyle q} is the source control at ∂ q Ω {displaystyle partial _{q}Omega } , η o {displaystyle eta _{o}} is the initial condition,and m {displaystyle m} is the outward normal to ∂ Ω {displaystyle partial Omega } . It is the L2 gradient flow of the Ginzburg–Landau 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.