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Allen–Cahn equation

Equation in mathematical physics From Wikipedia, the free encyclopedia

Allen–Cahn equation
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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.

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A numerical solution to the one dimensional Allen–Cahn equation

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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References

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