We introduce a finite-volume numerical scheme for solving stochastic gradient-flow equations. Such equations are of crucial importance within the framework of fluctuating hydrodynamics and dynamic density functional theory. Our proposed scheme deals with general free-energy functionals, including, for instance, external fields or interaction potentials. This allows us to simulate a range of physical phenomena where thermal fluctuations play a crucial role, such as nucleation and other energy-barrier crossing transitions. A positivity-preserving algorithm for the density is derived based on a hybrid space discretization of the deterministic and the stochastic terms and different implicit and explicit time integrators. We show through numerous applications that not only our scheme is able to accurately reproduce the statistical properties (structure factor and correlations) of the physical system, but, because of the multiplicative noise, it allows us to simulate energy barrier crossing dynamics, which cannot be captured by mean-field approaches.
翻译:我们引入了解决随机梯度流方程式的有限量数字计划。这种方程式在波动的流体动力学和动态密度功能理论的框架内至关重要。我们提议的方程式涉及一般的自由能源功能,包括外部领域或互动潜力等。这使我们能够模拟一系列物理现象,热波动在其中起着关键作用,例如核聚变和其他能源屏障交叉转换。密度的假设-保护算法基于确定性和随机性术语以及不同隐含和明确时间集成器的混合空间分解。我们通过许多应用显示,我们的方程式不仅能够准确地复制物理系统的统计特性(结构因素和相关性),而且由于多倍增的噪音,它使我们能够模拟能量屏的交叉动态,而这种动态无法通过中位法来捕捉。