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
η
{\displaystyle \eta }
on a domain
Ω
{\displaystyle \Omega }
during a time interval
T
{\displaystyle {\mathcal {T}}}
, and is given by:[1] [2]
∂
η
∂
t
=
M
η
[
div
(
ε
η
2
∇
η
)
−
f
′
(
η
)
]
on
Ω
×
T
,
η
=
η
¯
on
∂
η
Ω
×
T
,
{\displaystyle {{\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}},}
−
(
ε
η
2
∇
η
)
⋅
m
=
q
on
∂
q
Ω
×
T
,
η
=
η
o
on
Ω
×
{
0
}
,
{\displaystyle \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
η
{\displaystyle M_{\eta }}
is the mobility,
f
{\displaystyle f}
is a double-well potential,
η
¯
{\displaystyle {\bar {\eta }}}
is the control on the state variable at the portion of the boundary
∂
η
Ω
{\displaystyle \partial _{\eta }\Omega }
,
q
{\displaystyle q}
is the source control at
∂
q
Ω
{\displaystyle \partial _{q}\Omega }
,
η
o
{\displaystyle \eta _{o}}
is the initial condition, and
m
{\displaystyle m}
is the outward normal to
∂
Ω
{\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 .