Open Access

Dirk Blömker, Evelyn Sander, Thomas Wanner

• Phase separation in metal alloys is an important pattern forming physical process with applications in materials science, both for understanding materials structure and for the design of new materials. The Cahn-Hilliard equation is a deterministic model for the dynamics of alloys which has proven to be fundamental for the understanding of several types of phase separation behavior. However, stochastic effects occur in any physical experiment, and thus need to be incorporated into models. While white noise is one standardly chosen option, it is not immediately clear in which way the noise characteristics affect the resulting patterns. In this paper we study the effects of not necessarily small colored noise on pattern formation in a stochastic Cahn-Hilliard model in the nucleation regime. More precisely, we focus on degenerate noise which acts either on isolated eigenmodes, or with a well-defined spatial wavelength. Our studies show that the types of resulting patterns depend criticallyPhase separation in metal alloys is an important pattern forming physical process with applications in materials science, both for understanding materials structure and for the design of new materials. The Cahn-Hilliard equation is a deterministic model for the dynamics of alloys which has proven to be fundamental for the understanding of several types of phase separation behavior. However, stochastic effects occur in any physical experiment, and thus need to be incorporated into models. While white noise is one standardly chosen option, it is not immediately clear in which way the noise characteristics affect the resulting patterns. In this paper we study the effects of not necessarily small colored noise on pattern formation in a stochastic Cahn-Hilliard model in the nucleation regime. More precisely, we focus on degenerate noise which acts either on isolated eigenmodes, or with a well-defined spatial wavelength. Our studies show that the types of resulting patterns depend critically on the spatial noise frequency, and we can explain via numerical continuation methods that if this frequency is too high, then pattern formation is significantly impaired by the underlying structure of the system. In addition, we provide rigorous bounds on the nucleation time frame in the degenerate stochastic setting.…