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

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.