A derivation and comparison of two equations (Landau–Ginzburg and Cahn) for the kinetics of phase transitions

We postulate a master equation and derive by path integral methods the Cahn equation for the most probable evolution of a nonuniform system. To do so we need to impose a constraint of local conservation of the variable of interest. The same derivation without this constraint has been shown previously to lead to a Landau–Ginzburg equation. Thus the two equations have a common origin. The Cahn equation is applicable to conserved variables in systems in which spatial inhomogeneities occur over distances much larger than a characteristic distance λ of the system (like mean free path, or nearest neighbor separation). The Landau–Ginzburg equation is applicable to nonconserved variables, and to conserved variables when spatial inhomogeneities occur over distances much smaller than the characteristic distances for those variables. For conserved variables a precursor of the Cahn equation in the derivation reduces to the Landau–Ginzburg equation as λ is increased.