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

during a time interval

is the control on the state variable at the portion of the boundary

is the source control at

is the initial condition, and

It is the L2 gradient flow of the Ginzburg–Landau free energy functional.

[3] It is closely related to the Cahn–Hilliard equation.

is a solution to the Allen–Cahn equation if it solves[4] where Usually, one has the following initial condition with the Neumann boundary condition where

is the outer normal derivative.

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