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 iron alloys, including order-disorder transitions.

The equation is given by:[1][2]

{{\partial \eta}\over{\partial t}}=M_{\eta}[\epsilon^{2}_{\eta}\nabla^{2}\eta-f'(\eta)]

where M_{\eta} is the mobility, f is the free energy density, and \eta is the nonconserved order parameter.

It is the L2 gradient flow of the Ginzburg–Landau–Wilson Free Energy Functional. It is closely related to the Cahn–Hilliard equation. In one space-dimension, a very detailed account is given by a paper by Xinfu Chen.[3]

References

  1. S. M. Allen and J. W. Cahn, "Ground State Structures in Ordered Binary Alloys with Second Neighbor Interactions," Acta Met. 20, 423 (1972).
  2. S. M. Allen and J. W. Cahn, "A Correction to the Ground State of FCC Binary Ordered Alloys with First and Second Neighbor Pairwise Interactions," Scripta Met. 7, 1261 (1973).
  3. X. Chen, "Generation, propagation, and annihilation of metastable patterns", J. Differential Equations 206, 399–437 (2004).
This article is issued from Wikipedia - version of the Tuesday, May 06, 2014. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.