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.

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.

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

 

References

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  1. ^ Allen, S. M.; Cahn, J. W. (1972). "Ground State Structures in Ordered Binary Alloys with Second Neighbor Interactions". Acta Metall. 20 (3): 423–433. doi:10.1016/0001-6160(72)90037-5.
  2. ^ Allen, S. M.; Cahn, J. W. (1973). "A Correction to the Ground State of FCC Binary Ordered Alloys with First and Second Neighbor Pairwise Interactions". Scripta Metallurgica. 7 (12): 1261–1264. doi:10.1016/0036-9748(73)90073-2.
  3. ^ Veerman, Frits (March 8, 2016). "What is the L2 gradient flow?". MathOverflow.
  4. ^ Bartels, Sören (2015). Numerical Methods for Nonlinear Partial Differential Equations. Deutschland: Springer International Publishing. p. 153.

Further reading

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  • 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. doi:10.1016/0001-6160(75)90106-6.
  • 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. doi:10.1016/0036-9748(76)90171-x.
  • 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. doi:10.1016/0001-6160(76)90063-8.
  • 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. doi:10.1016/0001-6160(79)90196-2.
  • Bronsard, L.; Reitich, F. (1993). "On three-phase boundary motion and the singular limit of a vector valued Ginzburg–Landau equation". Arch. Rat. Mech. Anal. 124 (4): 355–379. Bibcode:1993ArRMA.124..355B. doi:10.1007/bf00375607. S2CID 123291032.
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  • Simulation by Nils Berglund of a solution of the Allen–Cahn equation