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The Allen–Cahn equation (after John W. Cahn and Sam Allen) is a reaction–diffusion equation of
mathematical physics Mathematical physics refers to the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and t ...
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 \eta on a domain \Omega during a time interval \mathcal, and is given by: :=M_\eta operatorname(\varepsilon^_\nabla\,\eta)-f'(\eta)quad \text \Omega\times\mathcal, \quad \eta=\bar\eta\quad\text\partial_\eta\Omega\times\mathcal, :\quad -(\varepsilon^2_\eta\nabla\,\eta)\cdot m = q\quad\text \partial_q \Omega \times \mathcal, \quad \eta=\eta_o \quad\text \Omega\times\, where M_ is the mobility, f is a double-well potential, \bar\eta is the control on the state variable at the portion of the boundary \partial_\eta\Omega, q is the source control at \partial_q\Omega, \eta_o is the initial condition, and m is the outward normal to \partial\Omega. It is the L2 gradient flow of the Ginzburg–Landau free energy functional. It is closely related to the
Cahn–Hilliard equation The Cahn–Hilliard equation (after John W. Cahn and John E. Hilliard) is an equation of mathematical physics which describes the process of phase separation, by which the two components of a binary fluid spontaneously separate and form domains p ...
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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 Equations of fluid dynamics Partial differential equations Equations {{CMP-stub