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Allen–Cahn equation
Equation in mathematical physics From Wikipedia, the free encyclopedia
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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 on a domain during a time interval , and is given by:[1][2]
where is the mobility, is a double-well potential, is the control on the state variable at the portion of the boundary , is the source control at , is the initial condition, and is the outward normal to .
It is the L2 gradient flow of the Ginzburg–Landau free energy functional.[3] It is closely related to the Cahn–Hilliard equation.
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Mathematical description
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Let
- be an open set,
- an arbitrary initial function,
- and two constants.
A function is a solution to the Allen–Cahn equation if it solves[4]
where
- is the Laplacian with respect to the space ,
- is the derivative of a non-negative with two minima .
Usually, one has the following initial condition with the Neumann boundary condition
where is the outer normal derivative.
For one popular candidate is
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