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 $\Omega$ during a time interval $\mathcal$, and is given by:

:
\begin
= & M_\eta operatorname(\varepsilon^2_\eta \, \nabla\,\eta)-f'(\eta)quad \text \Omega\times\mathcal,
\quad \eta=\bar\eta\quad\text\partial_\eta\Omega\times\mathcal, \\ pt& \cdot m = q\quad\text \partial_q \Omega \times \mathcal,
\quad \eta=\eta_o \quad\text \Omega\times\,
\end

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. It is closely related to the Cahn–Hilliard equation.

Mathematical description

Let
*$\Omega\subset \R^n$ 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 ,Tto \R is a solution to the Allen–Cahn equation if it solves
:
\partial_t v-\Delta_x v = -\fracf(v),\quad \Omega \times ,T
where
*$\Delta_x$ is the Laplacian with respect to the space $x$,
*$f(v)=F'(v)$ is the derivative of a non-negative $F\in C^1(\R)$ with two minima $F(\pm 1)=0$.

Usually, one has the following initial condition with the Neumann boundary condition
:\begin
v(x,0) = v_0(x), & \Omega \times \\\
\partial_n v = 0, & \partial \Omega \times ,T\
\end

where $\partial_n v$ is the outer normal derivative.

For $F(v)$ one popular candidate is
:$F(v)=\frac,\qquad f(v)=v^3-v.$

References

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

Equations of fluid dynamics
Partial differential equations
Equations

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