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

{{\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}},

\quad -(\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\},

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

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

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

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

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

where

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

Usually on has the following initial condition with the Neumann boundary condition

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

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

For $F(v)$ one popular candidate is

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

References

Allen, S. M.; Cahn, J. W. (1972). "Ground State Structures in Ordered Binary Alloys with Second Neighbor Interactions". Acta Metall. 20 (3): 423–433. doi:10.1016/0001-6160(72)90037-5.
Allen, S. M.; Cahn, J. W. (1973). "A Correction to the Ground State of FCC Binary Ordered Alloys with First and Second Neighbor Pairwise Interactions". Scripta Metallurgica. 7 (12): 1261–1264. doi:10.1016/0036-9748(73)90073-2.
Veerman, Frits (March 8, 2016). "What is the L² gradient flow?". MathOverflow.
Bartels, Sören (2015). Numerical Methods for Nonlinear Partial Differential Equations. Deutschland: Springer International Publishing. p. 153.

External links

Simulation by Nils Berglund of a solution of the Allen-Cahn equation

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[1] Allen, S. M.; Cahn, J. W. (1972). "Ground State Structures in Ordered Binary Alloys with Second Neighbor Interactions". Acta Metall. 20 (3): 423–433. doi:10.1016/0001-6160(72)90037-5.

[2] Allen, S. M.; Cahn, J. W. (1973). "A Correction to the Ground State of FCC Binary Ordered Alloys with First and Second Neighbor Pairwise Interactions". Scripta Metallurgica. 7 (12): 1261–1264. doi:10.1016/0036-9748(73)90073-2.

[3] Veerman, Frits (March 8, 2016). "What is the L² gradient flow?". MathOverflow.

[Bartels-4] Bartels, Sören (2015). Numerical Methods for Nonlinear Partial Differential Equations. Deutschland: Springer International Publishing. p. 153.

Allen–Cahn equation

Mathematical description

References

Further reading

External links