Emergence of order in textured patterns

Area of Science:

• Nonlinear dynamics
• Pattern formation
• Statistical physics

Background:

• The Swift-Hohenberg equation (SHE) models pattern formation in various physical systems.
• Understanding the dynamics of pattern evolution, especially from random initial states, is crucial.
• Characterizing textured patterns requires robust quantitative measures.

Purpose of the Study:

• To characterize textured patterns using a novel disorder function, delta(beta).
• To investigate the relaxation dynamics of patterns generated by the Swift-Hohenberg equation.
• To analyze the influence of nonvariational terms on pattern evolution.

Main Methods:

• Utilized the disorder function delta(beta) as an intensive, configuration-independent measure.
• Studied pattern evolution from random initial states under the Swift-Hohenberg equation.
• Analyzed two distinct stages of relaxation: initial domain emergence and domain coarsening.
• Investigated the effect of adding nonvariational terms to the SHE.

Main Results:

• The disorder function delta(beta) quantifies two relaxation stages: initial power-law decay (delta(beta) ~ t-(1/2)beta) and slower coarsening.
• A sharp transition separates these two relaxation phases.
• The transition point shifts with system drive, and scaling collapses distinct curves.
• Initial phase decay is robust to nonvariational terms, while coarsening rate increases with them.

Conclusions:

• The disorder function delta(beta) effectively characterizes pattern evolution in the SHE.
• The observed two-stage relaxation process is a key feature of SHE dynamics.
• The robustness of initial pattern formation suggests potential experimental observability.
• Nonvariational terms significantly impact domain coarsening dynamics.