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 on a domain during a time interval , and is given by:

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. It is closely related to the Cahn–Hilliard equation.

Mathematical description

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

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

References

Further reading

• http://www.ctcms.nist.gov/~wcraig/variational/node10.html

External links

• Simulation by Nils Berglund of a solution of the Allen–Cahn equation