In micro-fluidics not only does capillarity dominate but also thermal fluctuations become important. On the level of the lubrication approximation, this leads to a quasi-linear fourth-order parabolic equation for the film height $h$ driven by space-time white noise. The gradient flow structure of its deterministic counterpart, the thin-film equation, which encodes the balance between driving capillary and limiting viscous forces, provides the guidance for the thermodynamically consistent introduction of fluctuations. We follow this route on the level of a spatial discretization of the gradient flow structure. Starting from an energetically conformal finite-element (FE) discretization, we point out that the numerical mobility function introduced by Gr\"un and Rumpf can be interpreted as a discretization of the metric tensor in the sense of a mixed FE method with lumping. While this discretization was devised in order to preserve the so-called entropy estimate, we use this to show that the resulting high-dimensional stochastic differential equation (SDE) preserves pathwise and pointwise strict positivity, at least in case of the physically relevant mobility function arising from the no-slip boundary condition. As a consequence, this discretization gives rise to a consistent invariant measure, namely a discretization of the Brownian excursion (up to the volume constraint), and thus features an entropic repulsion. The price to pay over more naive discretizations is that when writing the SDE in It\^o's form, which is the basis for the Euler-Mayurama time discretization, a correction term appears. To conclude, we perform various numerical experiments to compare the behavior of our discretization to that of the more naive finite difference discretization of the equation.

中文翻译：

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具有热噪声的薄膜方程的热力学一致和保正离散化

在微流体中，不仅毛细作用占主导地位，而且热波动也变得重要。在润滑近似水平上，这导致时空白噪声驱动的薄膜高度 $h$ 的准线性四阶抛物线方程。其确定性对应物的梯度流动结构，薄膜方程，编码驱动毛细管和限制粘性力之间的平衡，为热力学一致引入波动提供指导。我们在梯度流结构的空间离散化水平上遵循这条路线。从能量保形有限元 (FE) 离散化开始，我们指出 Gr 引入的数值迁移率函数\" 在具有集总的混合有限元方法的意义上，un 和 Rumpf 可以被解释为度量张量的离散化。虽然设计这种离散化是为了保留所谓的熵估计，但我们使用它来表明由此产生的高维随机微分方程 (SDE) 保留了路径和逐点严格的正性，至少在物理相关的迁移率函数的情况下由无滑移边界条件引起。因此，这种离散化产生了一致的不变度量，即布朗偏移的离散化（达到体积约束），因此具有熵斥力。为更幼稚的离散化付出的代价是，当以 It\^o 的形式编写 SDE 时，它是 Euler-Mayurama 时间离散化的基础，会出现一个修正项。