Understanding the properties of the stochastic phase field models is crucial to model processes in several practical applications, such as soft matters and phase separation in random environments. To describe such random evolution, this work proposes and studies two mathematical models and their numerical approximations for parabolic stochastic partial differential equation (SPDE) with a logarithmic Flory--Huggins energy potential. These multiscale models are built based on a regularized energy technique and thus avoid possible singularities of coefficients. According to the large deviation principle, we show that the limit of the proposed models with small noise naturally recovers the classical dynamics in deterministic case. Moreover, when the driving noise is multiplicative, the Stampacchia maximum principle holds which indicates the robustness of the proposed model. One of the main advantages of the proposed models is that they can admit the energy evolution law and asymptotically preserve the Stampacchia maximum bound of the original problem. To numerically capture these asymptotic behaviors, we investigate the semi-implicit discretizations for regularized logrithmic SPDEs. Several numerical results are presented to verify our theoretical findings.

翻译：理解随机相场模型的特性对于模拟诸多实际应用中的过程至关重要，例如软物质和随机环境中的相分离。为描述此类随机演化，本文提出并研究了两个数学模型及其数值逼近，用于求解具有对数型Flory-Huggins能量势的抛物型随机偏微分方程。这些多尺度模型基于能量正则化技术构建，从而避免了系数可能出现的奇异性。根据大偏差原理，我们证明了所提模型在小噪声条件下的极限能够自然恢复确定性情形下的经典动力学行为。此外，当驱动噪声为乘性时，Stampacchia极大值原理成立，这体现了所提模型的稳健性。所提模型的主要优势之一在于能够满足能量演化律，并渐近保持原问题的Stampacchia极大值界限。为在数值上捕捉这些渐近行为，我们研究了正则化对数型随机偏微分方程的半隐式离散格式。最后给出若干数值结果以验证我们的理论发现。