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:

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

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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