Thin liquid films are ubiquitous in natural phenomena and technological applications. They have been extensively studied via deterministic hydrodynamic equations, but thermal fluctuations often play a crucial role that needs to be understood. An example of this is dewetting, which involves the rupture of a thin liquid film and the formation of droplets. Such a process is thermally activated and requires fluctuations to be taken into account self-consistently. In this work we present an analytical and numerical study of a stochastic thin-film equation derived from first principles. Following a brief review of the derivation, we scrutinise the behaviour of the equation in the limit of perfectly correlated noise along the wall-normal direction, as opposed to the perfectly uncorrelated limit studied by Grün et al. (J Stat Phys 122(6):1261–1291, 2006). We also present a numerical scheme based on a spectral collocation method, which is then utilised to simulate the stochastic thin-film equation. This scheme seems to be very convenient for numerical studies of the stochastic thin-film equation, since it makes it easier to select the frequency modes of the noise (following the spirit of the long-wave approximation). With our numerical scheme we explore the fluctuating dynamics of the thin film and the behaviour of its free energy in the vicinity of rupture. Finally, we study the effect of the noise intensity on the rupture time, using a large number of sample paths as compared to previous studies.

中文翻译：

---

薄液膜的不稳定性、破裂和波动：理论与计算

薄液膜在自然现象和技术应用中无处不在。它们已经通过确定性流体动力学方程进行了广泛的研究，但热波动通常起着需要理解的关键作用。这方面的一个例子是去湿，它涉及薄液膜的破裂和液滴的形成。这种过程是热激活的，需要自洽地考虑波动。在这项工作中，我们对从第一性原理导出的随机薄膜方程进行了分析和数值研究。在对推导进行简要回顾之后，我们仔细研究了方程在沿壁法线方向完全相关的噪声极限下的行为，而不是 Grün 等人研究的完全不相关的极限。(J Stat Phys 122(6):1261–1291, 2006)。我们还提出了一种基于光谱搭配方法的数值方案，然后用于模拟随机薄膜方程。这种方案对于随机薄膜方程的数值研究似乎非常方便，因为它可以更容易地选择噪声的频率模式（遵循长波近似的精神）。通过我们的数值方案，我们探索了薄膜的波动动力学及其在破裂附近的自由能行为。最后，与之前的研究相比，我们使用了大量的样本路径来研究噪声强度对破裂时间的影响。这种方案对于随机薄膜方程的数值研究似乎非常方便，因为它可以更容易地选择噪声的频率模式（遵循长波近似的精神）。通过我们的数值方案，我们探索了薄膜的波动动力学及其在破裂附近的自由能行为。最后，与之前的研究相比，我们使用了大量的样本路径来研究噪声强度对破裂时间的影响。这种方案对于随机薄膜方程的数值研究似乎非常方便，因为它可以更容易地选择噪声的频率模式（遵循长波近似的精神）。通过我们的数值方案，我们探索了薄膜的波动动力学及其在破裂附近的自由能行为。最后，与之前的研究相比，我们使用了大量的样本路径来研究噪声强度对破裂时间的影响。