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 pure in each component. If ${\displaystyle c}$ is the concentration of the fluid, with ${\displaystyle c=\pm 1}$ indicating domains, then the equation is written as

${\displaystyle {\frac {\partial c}{\partial t}}=D\nabla ^{2}\left(c^{3}-c-\gamma \nabla ^{2}c\right),}$

where ${\displaystyle D}$ is a diffusion coefficient with units of ${\displaystyle {\text{Length}}^{2}/{\text{Time}}}$ and ${\displaystyle {\sqrt {\gamma }}}$ gives the length of the transition regions between the domains. Here ${\displaystyle \partial /{\partial t}}$ is the partial time derivative and ${\displaystyle \nabla ^{2}}$ is the Laplacian in ${\displaystyle n}$ dimensions. Additionally, the quantity ${\displaystyle \mu =c^{3}-c-\gamma \nabla ^{2}c}$ is identified as a chemical potential.

Related to it is the Allen–Cahn equation, as well as the stochastic Cahn–Hilliard Equation and the stochastic Allen–Cahn equation.