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