Pattern Size in Gaussian Fields from Spinodal Decomposition

Abstract

We study the two-dimensional snake-like pattern that arises in phase separation of alloys described by spinodal decomposition in the Cahn-Hilliard model. These are somewhat universal patterns due to an overlay of eigenfunctions of the Laplacian with a similar wave-number. Similar structures appear in other models like reaction-diffusion systems describing animal coats' patterns or vegetation patterns in desertification. Our main result studies random functions given by cosine Fourier series with independent Gaussian coefficients, that dominate the dynamics in the Cahn-Hilliard model. This is not a cosine process, as the sum is taken over domains in Fourier space that not only grow and scale with a parameter of order $1/\varepsilon$, but also move to infinity. Moreover, the model under consideration is neither stationary nor isotropic. To study the pattern size of nodal domains we consider the density of zeros on any straight line through the spatial domain. Using a theorem by Edelman and Kostlan and weighted ergodic theorems that ensure the convergence of the moving sums, we show that the average distance of zeros is asymptotically of order $\varepsilon$ with a precisely given constant.