Emergence of order in textured patterns

A characterization of textured patterns, referred to as the disorder function $\overline{\delta }\left(\beta \right)$, is used to study properties of patterns generated in the Swift-Hohenberg equation (SHE). It is shown to be an intensive, configuration-independent measure. The evolution of random initial states under the SHE exhibits two stages of relaxation. The initial phase, where local striped domains emerge from a noisy background, is quantified by a power-law decay $\overline{\delta }\left(\beta \right)\sim t^{-\mathrm{}(1/2)\mathrm{}\beta }$. Beyond a sharp transition, a slower power-law decay of $\overline{\delta }\left(\beta \right)$, which corresponds to the coarsening of striped domains, is observed. The transition between the phases advances as the system is driven further from the onset of patterns, and suitable scaling of time and $\overline{\delta }\left(\beta \right)$ leads to the collapse of distinct curves. The decay of $\overline{\delta }\left(\beta \right)$ during the initial phase remains unchanged when nonvariational terms are added to the underlying equations, suggesting the possibility of observing it in experimental systems. In contrast, the rate of relaxation during domain coarsening increases with the coefficient of the nonvariational term.