Allen–Cahn equation

A numerical solution to the one dimensional 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 $\eta$ on a domain $\mathrm{\Omega }$ during a time interval $𝒯$, and is given by:

$\frac{\partial \eta }{\partial t}=M_{\eta }[\mathrm{div}({\epsilon }_{\eta }^2\mathrm{\nabla }\eta )-f^{\prime }(\eta )]\text{on }\mathrm{\Omega }\times 𝒯,\eta =\overline{\eta }\text{on }{\partial }_{\eta }\mathrm{\Omega }\times 𝒯,$

$-({\epsilon }_{\eta }^2\mathrm{\nabla }\eta )\cdot m=q\text{on }{\partial }_q\mathrm{\Omega }\times 𝒯,\eta ={\eta }_o\text{on }\mathrm{\Omega }\times \{0\},$

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

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

Mathematical description

Let

• $\mathrm{\Omega }\subset {ℝ}^n$ be an open set,
• $v_0(x)\in L^2(\mathrm{\Omega })$ an arbitrary initial function,
• $\epsilon >0$ and $T>0$ two constants.

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

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

where

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

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

$\{\begin{array}{cc}v(x,0)=v_0(x),& \mathrm{\Omega }\times \{0\}\\ {\partial }_{n}v=0,& \partial \mathrm{\Omega }\times [0,T]\end{array}$

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

For $F(v)$ one popular candidate is

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

Further reading

• http://www.ctcms.nist.gov/~wcraig/variational/node10.html
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External links

• Simulation by Nils Berglund of a solution of the Allen–Cahn equation

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