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.

A numerical solution to the one dimensional Allen–Cahn equation

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.

Mathematical description

Let

• ${\displaystyle \Omega \subset \mathbb {R} ^{n}}$ be an open set,
• ${\displaystyle v_{0}(x)\in L^{2}(\Omega )}$ an arbitrary initial function,
• ${\displaystyle \varepsilon >0}$ and ${\displaystyle T>0}$ two constants.

A function ${\displaystyle v(x,t):\Omega \times [0,T]\to \mathbb {R} }$ is a solution to the Allen–Cahn equation if it solves^[4]

${\displaystyle \partial _{t}v-\Delta _{x}v=-{\frac {1}{\varepsilon ^{2}}}f(v),\quad \Omega \times [0,T]}$

where

• ${\displaystyle \Delta _{x}}$ is the Laplacian with respect to the space ${\displaystyle x}$,
• ${\displaystyle f(v)=F'(v)}$ is the derivative of a non-negative ${\displaystyle F\in C^{1}(\mathbb {R} )}$ with two minima ${\displaystyle F(\pm 1)=0}$.

Usually, one has the following initial condition with the Neumann boundary condition

${\displaystyle {\begin{cases}v(x,0)=v_{0}(x),&\Omega \times \{0\}\\\partial _{n}v=0,&\partial \Omega \times [0,T]\\\end{cases}}}$

where ${\displaystyle \partial _{n}v}$ is the outer normal derivative.

For ${\displaystyle F(v)}$ one popular candidate is

${\displaystyle F(v)={\frac {(v^{2}-1)^{2}}{4}},\qquad f(v)=v^{3}-v.}$