Allen–Cahn equation explained

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

η

on a domain

\Omega

during a time interval

l{T}

, and is given by:[1] [2]

{{\partialη}\over{\partial

2
t}}=M
η

\nablaη)-f'(η)]on\Omega x l{T},   η=\barη   on\partialη\Omega x l{T},

2
-(\varepsilon
η\nablaη)

m=qon\partialq\Omega x l{T},   η=ηoon\Omega x \{0\},

where

Mη

is the mobility,

f

is a double-well potential,

\barη

is the control on the state variable at the portion of the boundary

\partialη\Omega

,

q

is the source control at

\partialq\Omega

,

ηo

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\Rn

be an open set,

v0(x)\inL2(\Omega)

an arbitrary initial function,

\varepsilon>0

and

T>0

two constants.

A function

v(x,t):\Omega x [0,T]\to\R

is a solution to the Allen–Cahn equation if it solves[4]

\partialtv-\Deltaxv=-

1
\varepsilon2

f(v),\Omega x [0,T]

where

\Deltax

is the Laplacian with respect to the space

x

,

f(v)=F'(v)

is the derivative of a non-negative

F\inC1(\R)

with two minima

F(\pm1)=0

.

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

\begin{cases} v(x,0)=v0(x),&\Omega x \{0\}\\ \partialnv=0,&\partial\Omega x [0,T]\\ \end{cases}

where

\partialnv

is the outer normal derivative.

For

F(v)

one popular candidate is
F(v)=(v2-1)2
4

,    f(v)=v3-v.

Further reading

External links

Notes and References

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