Fractal feature analysis based on phase transitions of the Allen-Cahn and Cahn-Hilliard equations

This paper explores the fractal characteristics of phase value time series in phase field models. The phase value time series are obtained from the corresponding phase separation process. Phase field models, such as the conservative Allen-Cahn (CAC) and Cahn-Hilliard (CH) equations, are adopted. The fractal features are calculated by the multifractal detrend fluctuation analysis (MF-DFA) model. We observe that during the phase separation evolution of the CAC and CH models, the Hurst exponent H(2) of the time series of phase values gradually increases from 0.5 of the sequence under random walk, until the evolution tends to stabilize, and H(2) also becomes stable. Additionally, we find that the curve behaviors of 2 - H(2) for both phase field models are consistent with the energy dissipation curves in the phase separation process. The characteristics of H(2) show that the sequence of phase values exhibits fractal characteristics, and the persistence of self-similarity increases with the number of numerical simulation iterations. Our study can provide time series with a specified Hurst exponent for the research field of unstable series. Furthermore, our research finds that the energy dissipation system in the phase field evolution process is closely related to self-similarity.