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软物质及随机环境中的相分离等实际应用过程至关重要。为描述此类随机演化过程，本文提出并研究了具有对数Flory-Huggins能量势的抛物型随机偏微分方程的两个数学模型及其数值逼近方法。这些多尺度模型基于正则化能量技术构建，从而避免了系数可能存在的奇异性。根据大偏差原理，我们证明了所提出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，当驱动噪声为乘性噪声时，Stampacchia最大值原理成立，表明所提出模型具有鲁棒性。该模型的主要优势之一在于能够满足能量演化定律，并渐近保持原始问题的Stampacchia最大值界。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。多个数值结果验证了我们的理论发现。