Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 344 / 2

TakeIts variational chemical potential isConservation gives $\dot{\varphi }=-\nabla \cdot \mathbf{J}$. Assuming an isotropic constant mobility $M>0$, local linear irreversible thermodynamics gives $\mathbf{J}=-M\nabla \mu$. Hence the noiseless conserved order-parameter dynamics isIt decreases the free energy because $\dot{F}=-\int M|\nabla \mu {|}^{2}d\mathbf{r}\le 0$ under closed or no-flux boundary conditions.

For $a<0$ and $b>0$, planar coexistence occurs at $\varphi =\pm {\varphi }_B$ with ${\varphi }_B=\sqrt{-a/b}$. Put $\alpha =f^{\prime \prime }({\varphi }_B)=a+3b{\varphi }_B^2=-2a>0$. A common small shift $\delta$ gives equal chemical potentials to first order,The thermodynamic pressure is $P=\varphi \mu -f$. Its difference between the positive interior and negative exterior is thereforeFor a sphere the Young–Laplace equation gives $P_{+}-P_{-}=2\sigma /R$. Consequently the Gibbs--Thomson relation is

The curved interface requires exterior composition $-{\varphi }_B+\delta$, but the far field supplies only $-{\varphi }_B$. Material therefore diffuses away from the droplet, which evaporates. Write $\varphi =-{\varphi }_B+g$ outside. Linearization gives $\mu =\alpha g$, and the quasistatic condition $\dot{\varphi }=M{\nabla }^2\mu \simeq 0$ givesSpherical symmetry yields $g(r)=\delta R/r$. Taking the normal outward from the droplet,The interface converts positive-phase material into negative-phase material. Integrating the continuity equation through the moving interface gives the Stefan conditionwhere $[\varphi ]=2{\varphi }_B$, and hence

For $h=\eta \sin (qx)$, the linearized mean curvature isThe boundary value in the upper phase is thereforeThe decaying harmonic extension into $z>0$ isWith the normal directed into the upper half-space,

Using only this upper-phase flux in $\dot{h}=v_n=-J_n/(2{\varphi }_B)$ givesPositive surface tension raises the chemical potential at a crest, drives material away from it, and therefore smooths the interface, which fixes the negative sign of the decay rate.

The dynamics is nonlocal because the conserved composition must diffuse through the bulk. Solving Laplace's equation maps a boundary Fourier amplitude to its normal derivative by multiplication by $|q|$. Combining this Dirichlet-to-Neumann factor with the curvature factor $q^2$ produces the nonanalytic $|q{|}^3$ rate described by diffusion-limited relaxation of an interface.

Both coexisting phases transport the conserved order parameter. The calculation above included only the harmonic field and normal current in the upper half-space. The lower half-space contributes an equal flux into the interface for equal mobilities and symmetric phases. The total interface speed is therefore twice the one-sided result, so the correct coefficient is