Bethe-Guggenheim Approximation for Non-uniform Systems

Abstract

Building upon the previous chapter, we here apply the BG approximation to non-uniform lattice systems. We consider an Ising model with anisotropic coupling $(J_{x},J_{y})$ in the horizontal and vertical direction and calculate the partition function for a fixed magnetization per spin block. Upon taking the thermodynamic limit, we obtain a Cahn-Hilliard free energy functional with a concentration-dependent gradient energy coefficient. Subsequently, we determine the equilibrium concentration profile and stable wavelength perturbations within the spinodal region. Here, we encounter a remarkable new feature of the BG concentration profile named interface broadening. This phenomenon is not present in the MF approximation, and will be further investigated in Chap. 4. In the final section we provide a comparison between the BG and MF results, and show how their concentration profiles differ qualitatively in the strong coupling limit. Part of this chapter has been published in Physical Review Research under the terms of the Creative Commons Attribution 4.0 International license [1].

Well, I come down in the morning and I take up a pencil and I try to THINK.

Hans Bethe in Hans Bethe: Prophet of Energy