Allen–Cahn equation
Appearance
The Allen–Cahn equation (after
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
.
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[4]
where
- is the Laplacianwith 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
- .
- .
- ^ Veerman, Frits (March 8, 2016). "What is the L2 gradient flow?". MathOverflow.
- ^ Bartels, Sören (2015). Numerical Methods for Nonlinear Partial Differential Equations. Deutschland: Springer International Publishing. p. 153.
Further reading
- http://www.ctcms.nist.gov/~wcraig/variational/node10.html
- Allen, S. M.; Cahn, J. W. (1975). "Coherent and Incoherent Equilibria in Iron-Rich Iron-Aluminum Alloys". Acta Metall. 23 (9): 1017. .
- Allen, S. M.; Cahn, J. W. (1976). "On Tricritical Points Resulting from the Intersection of Lines of Higher-Order Transitions with Spinodals". Scripta Metallurgica. 10 (5): 451–454. .
- Allen, S. M.; Cahn, J. W. (1976). "Mechanisms of Phase Transformation Within the Miscibility Gap of Fe-Rich Fe-Al Alloys". Acta Metall. 24 (5): 425–437. .
- Cahn, J. W.; Allen, S. M. (1977). "A Microscopic Theory of Domain Wall Motion and Its Experimental Verification in Fe-Al Alloy Domain Growth Kinetics". Journal de Physique. 38: C7–51.
- Allen, S. M.; Cahn, J. W. (1979). "A Microscopic Theory for Antiphase Boundary Motion and Its Application to Antiphase Domain Coarsening". Acta Metall. 27 (6): 1085–1095. .
- S2CID 123291032.
External links
- Simulation by Nils Berglund of a solution of the Allen–Cahn equation