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 ${\displaystyle \eta }$ on a domain ${\displaystyle \Omega }$ during a time interval ${\displaystyle {\mathcal {T}}}$, and is given by:^[1][2]

${\displaystyle {\begin{aligned}{{\partial \eta } \over {\partial t}}={}&M_{\eta }[\operatorname {div} (\varepsilon _{\eta }^{2}\,\nabla \,\eta )-f'(\eta )]\quad {\text{on }}\Omega \times {\mathcal {T}},\quad \eta ={\bar {\eta }}\quad {\text{on }}\partial _{\eta }\Omega \times {\mathcal {T}},\\[5pt]&{-(\varepsilon _{\eta }^{2}\,\nabla \,\eta )}\cdot m=q\quad {\text{on }}\partial _{q}\Omega \times {\mathcal {T}},\quad \eta =\eta _{o}\quad {\text{on }}\Omega \times \{0\},\end{aligned}}}$

where ${\displaystyle M_{\eta }}$ is the mobility, ${\displaystyle f}$ is a double-well potential, ${\displaystyle {\bar {\eta }}}$ is the control on the state variable at the portion of the boundary ${\displaystyle \partial _{\eta }\Omega }$, ${\displaystyle q}$ is the source control at ${\displaystyle \partial _{q}\Omega }$, ${\displaystyle \eta _{o}}$ is the initial condition, and ${\displaystyle m}$ is the outward normal to ${\displaystyle \partial \Omega }$.

It is the L² gradient flow of the Ginzburg–Landau free energy functional.^[3] It is closely related to the Cahn–Hilliard equation.