Английская Википедия: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 <math>\eta</math> on a domain <math>\Omega</math> during a time interval <math>\mathcal{T}</math>, and is given by:[1][2]
- <math>{{\partial \eta}\over{\partial t}}=M_\eta[\operatorname{div}(\varepsilon^{2}_{\eta}\nabla\,\eta)-f'(\eta)]\quad \text{on } \Omega\times\mathcal{T},
\quad \eta=\bar\eta\quad\text{on }\partial_\eta\Omega\times\mathcal{T},</math>
- <math>\quad -(\varepsilon^2_\eta\nabla\,\eta)\cdot m = q\quad\text{on } \partial_q \Omega \times \mathcal{T},
\quad \eta=\eta_o \quad\text{on } \Omega\times\{0\},</math> where <math>M_{\eta}</math> is the mobility, <math>f</math> is a double-well potential, <math>\bar\eta</math> is the control on the state variable at the portion of the boundary <math>\partial_\eta\Omega</math>, <math>q</math> is the source control at <math>\partial_q\Omega</math>, <math>\eta_o</math> is the initial condition, and <math>m</math> is the outward normal to <math>\partial\Omega</math>.
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
- <math>\Omega\subset \R^n</math> be an open set,
- <math>v_0(x)\in L^2(\Omega)</math> an arbitrary initial function,
- <math>\varepsilon>0</math> and <math>T>0</math> two constants.
A function <math>v(x,t):\Omega\times [0,T]\to \R</math> is a solution to the Allen–Cahn equation if it solves[4]
- <math>
\partial_t v-\Delta_x v = \frac{1}{\varepsilon^2}f(v),\quad \Omega \times[0,T]
</math> where
- <math>\Delta_x</math> is the Laplacian with respect to the space <math>x</math>,
- <math>f(v)=F'(v)</math> is the derivative of a non-negative <math>F\in C^1(\R)</math> with two minima <math>F(\pm 1)=0</math>.
Usually, one has the following initial condition with the Neumann boundary condition
- <math>\begin{cases}
v(x,0) = v_0(x), & \Omega \times \{0\}\\
\partial_n v = 0, & \partial \Omega \times [0,T]\\
\end{cases} </math> where <math>\partial_n v</math> is the outer normal derivative.
For <math>F(v)</math> one popular candidate is
- <math> F(v)=\frac{(v^2-1)^2}{4},\qquad f(v)=v^3-v.</math>
References
Further reading
- http://www.ctcms.nist.gov/~wcraig/variational/node10.html
- Шаблон:Cite journal
- Шаблон:Cite journal
- Шаблон:Cite journal
- Шаблон:Cite journal
- Шаблон:Cite journal
- Шаблон:Cite journal
External links
- Simulation by Nils Berglund of a solution of the Allen–Cahn equation
