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 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: ∂ η ∂ t = M η [ div ( ε η 2 ∇ η ) − f ′ ( η ) ] on Ω × T , η = η ¯ on ∂ η Ω × T , − ( ε η 2 ∇ η ) ⋅ m = q on ∂ q Ω × T , η = η o on Ω × { 0 } , {\displaystyle {\begin{aligned}{{\partial \eta } \over {\partial t}}={}&M_{\eta }[\operatorname {div} (\varepsilon _{\eta }^{2}\,\nabla \,\eta )-f'(\eta )]\quad {\text{on }}\Omega \times {\mathcal {T}},\quad \eta ={\bar {\eta }}\quad {\text{on }}\partial _{\eta }\Omega \times {\mathcal {T}},\\[5pt]&{-(\varepsilon _{\eta }^{2}\,\nabla \,\eta )}\cdot m=q\quad {\text{on }}\partial _{q}\Omega \times {\mathcal {T}},\quad \eta =\eta _{o}\quad {\text{on }}\Omega \times \{0\},\end{aligned}}} where M η {\displaystyle M_{\eta }} is the mobility, f {\displaystyle f} is a double-well potential, η ¯ {\displaystyle {\bar {\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 } .