Spectral Energy Transfers in Domain Growth Problems

In the domain growth process, small structures gradually vanish, leaving behind larger ones. We investigate spectral energy transfers in two standard models for domain growth: (a) the {\it Cahn-Hilliard} (CH) equation with conserved dynamics, and (b) the {\it time-dependent Ginzburg-Landau} (TDGL) equation with non-conserved dynamics. The nonlinear terms in these equations dissipate fluctuations and facilitate energy transfers among Fourier modes. In the TDGL equation, only the $\phi (\mathbf{k}=0,t)$ mode survives, and the order parameter $\phi (\mathbf{r},t)$ approaches a uniform state with $\phi =+1$ or $-1$. On the other hand, there is no dynamics of the $\phi (\mathbf{k}=0,t)$ mode in the CH equation due to the conservation law, highlighting the different dynamics of these equations.